892019 First Order(Predicate)Logic

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983107983144983137983152983156983141983154 983088983096

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139

Instructor

LE Thanh Sach PhD

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 2

983113983150983155983156983154983157983139983156983151983154991257983155 983113983150983142983151983154983149983137983156983145983151983150

LE Thanh Sach PhDOffice

Department of Computer Science

Faculty of Computer Science and Engineering

HoChiMinh City University of Technology

Office Address

268 LyThuongKiet Str Dist 10 HoChiMinh City

VietnamE-mail LTSACHcsehcmuteduvn

E-home httpcsehcmuteduvn~ltsach

Tel (+84) 83-864-7256 (Ext 5839)

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Artificial Intelligence Predicate Logic Slide 3

983105983139983147983150983151983159983148983141983140983143983149983141983150983156

The slides in this PPT file are composed using thematerials supplied by

Prof Stuart Russell and Peter Norvig They

are currently from University of CaliforniaBerkeley They are also the author of the book ldquoArtificial Intelligence A Modern Approachrdquo whichis used as the textbook for the course

Prof Tom Lenaerts from Universiteacute Libre deBruxelles

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Artificial Intelligence Predicate Logic Slide 4

983119983157983156983148983145983150983141

Why FOLSyntax and semantics of FOL

Using FOL

Wumpus world in FOL

Knowledge engineering in FOL

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Artificial Intelligence Predicate Logic Slide 5

983127983144983161 983110983119983116983103983120983154983151983155 983137983150983140 983139983151983150983155 983151983142 983152983154983151983152983151983155983145983156983145983151983150983137983148 983148983151983143983145983139

Propositional logic is declarative

Propositional logic allows partialdisjunctivenegated information

(unlike most data structures and databases)

Propositional logic is compositional meaning of B 11 and P 12 is derived from meaning of B 11 and of P 12

Meaning in propositional logic is context-

independent

(unlike natural language where meaning depends on context)

Propositional logic has very limited expressive power (unlike natural language) Eg cannot say pits cause breezes in adjacent squaresldquo

983141983160983139983141983152983156 983138983161 983159983154983145983156983145983150983143 983151983150983141 983155983141983150983156983141983150983139983141 983142983151983154 983141983137983139983144 983155983153983157983137983154983141

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Artificial Intelligence Predicate Logic Slide 6

983127983144983161 983110983119983116983103983110983145983154983155983156983085983151983154983140983141983154 983148983151983143983145983139

Whereas propositional logic assumes the worldcontains facts

first-order logic (like natural language) assumes

the world containsObjects people houses numbers colors baseballgames wars hellip

Relations red round prime brother of bigger

than part of comes between hellipFunctions father of best friend one more than

plus hellip

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Artificial Intelligence Predicate Logic Slide 7

983120983154983141983152983151983155983145983156983145983151983150983137983148 983116983151983143983145983139983098 983109983160983137983149983152983148983141983155

MORTALMANAll men are mortal

PLATOMANPlato is a man

SOCRATESMANSocrates is a manSentence in Prepositional LogicSentence in natural language

Disadvantages1 Socrates and Plato are human but the representation is not similar

2 Can not state that each person is mortal Difficult for reasoning

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Artificial Intelligence Predicate Logic Slide 8

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forallX (man(X) mortal(X))All men are mortal

man(plato)Plato is a man

man(socrates)Socrates is a man

Sentence in Predicate LogicSentence in natural language

Using predicate logic

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Artificial Intelligence Predicate Logic Slide 9

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983137983149983152983148983141983155

Ability of reasoning

All men are mortal Socrates is mortal

Socrates is a man

ConclusionSentence in natural language

forallX (man(X) mortal(X))mortal(socrates)

man(socrates)

ConclusionSentence in Predicate Logic

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Artificial Intelligence Predicate Logic Slide 10

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983137983149983152983148983141983155

13 MPmortal(socrates)4

man(socrates)mortal(socrates)

forallX (man(X) mortal(X))

man(socrates)

Representation

2 X= ldquosocratesrdquo UI3

Axiom2

Axiom1

Rule

Socrates is mortal

Backward mapping (to natural language)

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Artificial Intelligence Predicate Logic Slide 11

983123983161983150983156983137983160 983151983142 983110983119983116983098 983106983137983155983145983139 983141983148983141983149983141983150983156983155

Constants KingJohn 2 NUS

Predicates Brother gt

Functions

Sqrt LeftLegOf Variables

x y a b

Connectives not rArr and or hArr

Equality =

Quantifiers Universal Quantifier forall

Existential Quantifier exist

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Artificial Intelligence Predicate Logic Slide 12

983105983156983151983149983145983139 983155983141983150983156983141983150983139983141983155

Term function (term1termn)or constant or variable

Atomic sentence predicate (term1termn)

or term1 = term

2

Eg

Brother(KingJohnRichardTheLionheart) gt (Length(LeftLegOf(Richard)) Length(LeftLegOf(KingJohn)))

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Artificial Intelligence Predicate Logic Slide 13

983107983151983149983152983148983141983160 983155983141983150983156983141983150983139983141983155

Complex sentences are made from atomic sentences usingconnectives examples notS

S1 and S2

S1 or S2

S1rArr S2

S1 hArr S2

Eg Sibling(KingJohnRichard) rArr Sibling(RichardKingJohn)

gt(12) or le (12)

gt(12) and not gt(12)

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Artificial Intelligence Predicate Logic Slide 14

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Sentences are true with respect to a model and an interpretation

Model contains objects (domain elements ) and relations among them

Interpretation specifies referents forconstant symbols rarr objects

predicate symbols rarr relations

function symbols rarr functional relations

An atomic sentence predicate(term1 termn) is true

iff the objects referred to by term1 termn

are in the relation referred to by predicate

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Artificial Intelligence Predicate Logic Slide 15

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forallltvariablesgt ltsentence gt

Everyone at NUS is smart

forallx At(xNUS)rArr Smart(x )

forallx P is true in a model m iff P is true with x being each possible object

in the model Roughly speaking equivalent to the conjunction of instantiations of

P

983105983156983080983115983145983150983143983114983151983144983150983084983118983125983123983081rArr 983123983149983137983154983156983080983115983145983150983143983114983151983144983150983081

and 983105983156983080983122983145983139983144983137983154983140983084983118983125983123983081rArr 983123983149983137983154983156983080983122983145983139983144983137983154983140983081

and 983105983156983080983118983125983123983084983118983125983123983081rArr 983123983149983137983154983156983080983118983125983123983081and 983086983086983086

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Artificial Intelligence Predicate Logic Slide 16

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TypicallyrArr is the main connective with forall

Common mistake using and as the main connective with forall

forallx At(xNUS) and Smart(x)

means ldquoEveryone is at NUS and everyone is smartrdquo

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Artificial Intelligence Predicate Logic Slide 17

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existltvariablesgt ltsentencegt

Someone at NUS is smart

exist x At(xNUS) and Smart(x )$

exist x P is true in a model m iff P is true with x being some possible object

in the model Roughly speaking equivalent to the disjunction of instantiations of P

At(KingJohnNUS) and Smart(KingJohn)or At(RichardNUS) and Smart(Richard)or At(NUSNUS) and Smart(NUS)or

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Artificial Intelligence Predicate Logic Slide 18

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Typically and is the main connective with exist

Common mistake usingrArr as the main connective with

exist

exist x At(xNUS)rArr Smart(x

)is true if there is anyone who is not at NUS

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Artificial Intelligence Predicate Logic Slide 19

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forallx forally is the same as forally forallx

existx existy is the same as existy existx

existx forally is not the same as forally existx

existx forally Loves(xy) ldquoThere is a person who loves everyone in the worldrdquo

forally existx Loves(xy) ldquoEveryone in the world is loved by at least one personrdquo

Quantifier duality each can be expressed using the other

forallx Likes(xIceCream) notexistx notLikes(xIceCream )

existx Likes(xBroccoli) notforallx notLikes(xBroccoli

)

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Artificial Intelligence Predicate Logic Slide 20

983109983153983157983137983148983145983156983161

term1 = term2 is true under a given interpretation if andonly if term1 and term2 refer to the same object

Eg definition of Sibling in terms of Parent

forallxy Sibling(xy) hArr [not(x = y) and existmf not (m = f) and Parent(mx)and Parent(fx) and Parent(my) and Parent(fy)]

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Artificial Intelligence Predicate Logic Slide 21

983125983155983145983150983143 983110983119983116

The kinship domain Brothers are siblings

forallxy Brother(xy) hArr Sibling(xy)

Ones mother is ones female parent

forallmc Mother(c) = m hArr (Female(m) and Parent(mc))

ldquoSiblingrdquo is symmetric

forallxy Sibling(xy) hArr Sibling(yx)

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Artificial Intelligence Predicate Logic Slide 22

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Representing the following sentences in Predicate Logic

1 Marcus was a man

2 Macus was a Pompeian

3 All Pompians were Romans4 Caesar was a ruler

5 All Romans were either loyal to Caesar or hated hime

6 Everyone is loyal to someone

7 People only try to assassinate rulers they are not loyal to

8 Marcus tried to assassinate Caesar

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Artificial Intelligence Predicate Logic Slide 23

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1 Marcus was a man

man(Marcus)

2 Macus was a Pompeian

Pompeian(Marcus)

3 All Pompeians were Romans

forallX Pompeian(X) Roman(X)

4 Caesar was a ruler

ruler(Caesar)

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Artificial Intelligence Predicate Logic Slide 24

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5 All Romans were either loyal to Caesar or hated hime

ldquoorrdquo mean ldquoinclusive orrdquo - OR

forallX Roman(X) loyalto(X Caesar) v hate(X Caesar)

ldquoorrdquo mean ldquoexclusive orrdquo - XOR

forallX Roman(X) [ (loyalto(X Caesar) v hate(X Caesar)) ^

not(loyalto(X Caesar) ^ hate(X Caesar))]

or

forallX Roman(X) [ (loyalto(X Caesar) ^ nothate(X Caesar)) v

(notloyalto(X Caesar) ^ hate(X Caesar))]

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Artificial Intelligence Predicate Logic Slide 25

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

6 Everyone is loyal to someone

forallX exist Y loyalto(XY)

7 People only try to assassinate rulers they are not loyal toforallX forall Y person(X) ^ ruler(Y) ^ tryassassinate(XY)notloyalto(XY)

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Artificial Intelligence Predicate Logic Slide 26

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

8 Marcus tried to assassinate Caesartryassassinate(Marcus Caesar)

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Artificial Intelligence Predicate Logic Slide 27

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

Given the above sentences can we make aconclusion as follows

ldquoMarcus was not loyal to Caesar rdquo

or

notloyalto(MarcusCaesar)

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Artificial Intelligence Predicate Logic Slide 28

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)

(Rule What)

Add an implicit sentence

9 ldquoAll men are peoplerdquo

or

forallforallforallforallX man(X) X man(X) person(X)

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Artificial Intelligence Predicate Logic Slide 29

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)(9 Substitute)

man(Marcus)

(1)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 30

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

How to represent exceptions example

ldquoPaulus was a Pompeian Paulus was neitherloyal nor hated Caesarrdquo

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Artificial Intelligence Predicate Logic Slide 31

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation Semantic Net Simple by adding links

(properties) to subclasses or instances

Predicate Logic Not so simple like

983120983151983149983152983141983145983137983150(983120983137983157983148983157983155) 983134 983212983131983148983151983161983137983148983156983151(983120983137983157983148983157983155 983107983137983141983155983137983154) 983158

983144983137983156983141(983120983157983137983148983157983155 983107983137983141983155983137983154)983133

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 32

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation983118983151983156 983155983151 983155983145983149983152983148983141 983106983141983139983137983157983155983141 983156983144983141983154983141 983145983155 983139983151983150983142983148983145983139983156 983138983141983156983159983141983141983150983098

Pompeian(Paulus) ^ not[loyalto(Paulus Caesar) vhate(Pualus Caesar)]

983137983150983140

forallX Pompeian(X) loyalto(X Caesar) v hate(XCaesar)

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Artificial Intelligence Predicate Logic Slide 33

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation A Better way

forallX Roman(X) ^ noteq(X Paulus) loyalto(X Caesar)v hate(X Caesar

Exception

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Artificial Intelligence Predicate Logic Slide 34

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

In the case of describing relations betweennumbers example 1 983100 2

2 983100983091

983095 983102 983091 +2 983091 983102 1

991270

rArr

983123983144983151983157983148983140 983150983151983156 983141983160983152983148983145983139983145983156983148983161 983140983141983155983139983154983145983138983141983098983148983156(12) 983148983156(2983091) 991270

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 35

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

rArr Need functions and computable predicates

rArr For example 983107983137983148983148 983137 983142983157983150983139983156983145983151983150 983156983151 983139983151983149983152983157983156983141 (983091+2)

983120983137983155983155 983156983144983141 983154983141983155983157983148983156 983156983151 983156983144983141 983152983154983141983140983145983139983137983156983141 991260983143983156991261983098 983143983156(983095 983091+2)

983111983141983156 983156983144983141 983142983145983150983137983148 983154983141983155983157983148983156 (983124983154983157983141)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 36

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

1 Marcus was a man

man(Marcus)

2 Marcus was a PompeianPompeian(Marcus)

3 Marcus was born in 40 AD

born(Marcus40)

4 All men are mortal

forallX man(X) mortal(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 37

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

5 All Pompeian died when the vocano erupted in 79 AD

erupted(vocano 79) ^ forallX [Pompeian(X) died(X 79)]

6 No mortal lives longer then 150 yearsforallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

7 It is now 1991now = 1991

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Artificial Intelligence Predicate Logic Slide 38

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Question

Is Marcus alive

Or

alive(Marcus now)

ornotnotnotnotalive(Marcus now)

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Artificial Intelligence Predicate Logic Slide 39

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Add implicit knowledge

Relation between ldquodeadrdquo and ldquoaliverdquo

8 Alive means not deadforallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

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Artificial Intelligence Predicate Logic Slide 40

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Add implicit knowledge Time properties of an event for example ldquodeadrdquo

9 Is someone dies the he is dead at all later times

forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

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Artificial Intelligence Predicate Logic Slide 41

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Summary of facts1 man(Marcus)

2 Pompeian(Marcus)

3 born(Marcus 40)

4 forallX man(X) mortal(X)

5 forallX Pompeian(X) died(X 79)

6 erupted(volcano 79)

7 forallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

8 now = 19919 forallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

10 forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

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Artificial Intelligence Predicate Logic Slide 42

notalive(Marcus now)

dead(Marcus now)

(9 Substitute)

died(Marcus T1) gt(now T1)

(10 Substitute)

Pompeian(Marcus) gt(now 79)

(5 Substitute)(5 Substitute)

(2)gt(1991 79)

(8 Substitute)

(computation)

ss

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Artificial Intelligence Predicate Logic Slide 43

983125983150983145983158983141983154983155983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983125983113)

Every instantiation of a universally quantified sentence is enta iled by it

forallv α

Subst(vg α)

for any variable v and ground term g

Eg forallx King( x) and Greedy( x)rArr Evil( x

) yieldsKing (John ) and Greedy (John ) rArr Evil (John )

King (Richard ) and Greedy (Richard ) rArr Evil (Richard )

King (Father (John )) and Greedy (Father (John )) rArr Evil (Father (John ))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 44

983109983160983145983155983156983141983150983156983145983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983109983113)

For any sentence α variable v and constant symbol k that

does not

appear elsewhere in the knowledge baseexistv α

Subst(vk α )

Eg exist x Crown( x) and OnHead ( xJohn) yields

Crown(C 1) and OnHead (C 1 John )

provided C 1 is a new constant symbol called a Skolemconstant

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 45

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Suppose the KB contains just the followingforall

x King(x)and

Greedy(x)rArr

Evil(x)King(John)Greedy(John)Brother(RichardJohn)

Instantiating the universal sentence in all possible ways we haveKing(John) and Greedy(John) rArr Evil(John)King(Richard) and Greedy(Richard) rArr Evil(Richard)King(John)Greedy(John)Brother(RichardJohn

)

The new KB is propositionalized proposition symbols are

King(John) Greedy(John) Evil(John) King(Richard

) etc

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 46

983122983141983140983157983139983156983145983151983150 983139983151983150983156983140983086

Every FOL KB can be propositionalized so as to preserve entailment

(A ground sentence is entailed by new KB iff entailed by originalKB)

Idea propositionalize KB and query apply resolution return result

Problem with function symbols there are infinitely many ground

terms

eg Father (Father (Father (John

)))

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Artificial Intelligence Predicate Logic Slide 47

983122983141983155983151983148983157983156983145983151983150

Convert to clause form exampleldquoAll Romans who know Marcus either hate

Caesar or think that anyone who hates anyone is

crazyrdquo

forallX [roman(X) ^ know(X Marcus)]

[hate(X Ceasar) v(forallY existZ hate(YZ)

thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 48

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 49

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

forallX [roman(X) ^ know(X Marcus)][hate(X Ceasar) v

(forallY existZ hate(YZ) thinkcrazy(XY))]

=

forallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 50

983122983141983155983151983148983157983156983145983151983150

2 Move not next to terms

Using the following equivalence

a Double negative

not(notp) = p

b DeMorgan

not(a v b) = nota ^ notb

not(a ^ b) = nota v notb

c Equivalence of qualifiers notforallX P(X) = existX notP(X)

notexistX P(X) = forallX notP(X)

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Artificial Intelligence Predicate Logic Slide 51

983122983141983155983151983148983157983156983145983151983150

2 Move not next to termsforallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

=

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) vthinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 52

983122983141983155983151983148983157983156983145983151983150

3 Change name of variables as follows

forallX P(X) v forallX Q(X)

=forallX P(X) v forallY Q(Y)

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Artificial Intelligence Predicate Logic Slide 53

983122983141983155983151983148983157983156983145983151983150

4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 54

983122983141983155983151983148983157983156983145983151983150

5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

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Artificial Intelligence Predicate Logic Slide 55

983122983141983155983151983148983157983156983145983151983150

6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 56

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

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Artificial Intelligence Predicate Logic Slide 57

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

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Artificial Intelligence Predicate Logic Slide 58

983122983141983155983151983148983157983156983145983151983150

8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

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Artificial Intelligence Predicate Logic Slide 59

983122983141983155983151983148983157983156983145983151983150

9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

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Artificial Intelligence Predicate Logic Slide 60

983122983141983155983151983148983157983156983145983151983150

Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

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Artificial Intelligence Predicate Logic Slide 61

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

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Artificial Intelligence Predicate Logic Slide 62

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

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Artificial Intelligence Predicate Logic Slide 63

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

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Artificial Intelligence Predicate Logic Slide 64

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 65

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

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Artificial Intelligence Predicate Logic Slide 66

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

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Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

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Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

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Artificial Intelligence Predicate Logic Slide 69

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 71

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 72

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 74

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

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Artificial Intelligence Predicate Logic Slide 75

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

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Artificial Intelligence Predicate Logic Slide 76

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

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892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

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Artificial Intelligence Predicate Logic Slide 2

983113983150983155983156983154983157983139983156983151983154991257983155 983113983150983142983151983154983149983137983156983145983151983150

LE Thanh Sach PhDOffice

Department of Computer Science

Faculty of Computer Science and Engineering

HoChiMinh City University of Technology

Office Address

268 LyThuongKiet Str Dist 10 HoChiMinh City

VietnamE-mail LTSACHcsehcmuteduvn

E-home httpcsehcmuteduvn~ltsach

Tel (+84) 83-864-7256 (Ext 5839)

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Artificial Intelligence Predicate Logic Slide 3

983105983139983147983150983151983159983148983141983140983143983149983141983150983156

The slides in this PPT file are composed using thematerials supplied by

Prof Stuart Russell and Peter Norvig They

are currently from University of CaliforniaBerkeley They are also the author of the book ldquoArtificial Intelligence A Modern Approachrdquo whichis used as the textbook for the course

Prof Tom Lenaerts from Universiteacute Libre deBruxelles

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 4

983119983157983156983148983145983150983141

Why FOLSyntax and semantics of FOL

Using FOL

Wumpus world in FOL

Knowledge engineering in FOL

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 5

983127983144983161 983110983119983116983103983120983154983151983155 983137983150983140 983139983151983150983155 983151983142 983152983154983151983152983151983155983145983156983145983151983150983137983148 983148983151983143983145983139

Propositional logic is declarative

Propositional logic allows partialdisjunctivenegated information

(unlike most data structures and databases)

Propositional logic is compositional meaning of B 11 and P 12 is derived from meaning of B 11 and of P 12

Meaning in propositional logic is context-

independent

(unlike natural language where meaning depends on context)

Propositional logic has very limited expressive power (unlike natural language) Eg cannot say pits cause breezes in adjacent squaresldquo

983141983160983139983141983152983156 983138983161 983159983154983145983156983145983150983143 983151983150983141 983155983141983150983156983141983150983139983141 983142983151983154 983141983137983139983144 983155983153983157983137983154983141

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 6

983127983144983161 983110983119983116983103983110983145983154983155983156983085983151983154983140983141983154 983148983151983143983145983139

Whereas propositional logic assumes the worldcontains facts

first-order logic (like natural language) assumes

the world containsObjects people houses numbers colors baseballgames wars hellip

Relations red round prime brother of bigger

than part of comes between hellipFunctions father of best friend one more than

plus hellip

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 7

983120983154983141983152983151983155983145983156983145983151983150983137983148 983116983151983143983145983139983098 983109983160983137983149983152983148983141983155

MORTALMANAll men are mortal

PLATOMANPlato is a man

SOCRATESMANSocrates is a manSentence in Prepositional LogicSentence in natural language

Disadvantages1 Socrates and Plato are human but the representation is not similar

2 Can not state that each person is mortal Difficult for reasoning

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Artificial Intelligence Predicate Logic Slide 8

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983137983149983152983148983141983155

forallX (man(X) mortal(X))All men are mortal

man(plato)Plato is a man

man(socrates)Socrates is a man

Sentence in Predicate LogicSentence in natural language

Using predicate logic

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 9

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983137983149983152983148983141983155

Ability of reasoning

All men are mortal Socrates is mortal

Socrates is a man

ConclusionSentence in natural language

forallX (man(X) mortal(X))mortal(socrates)

man(socrates)

ConclusionSentence in Predicate Logic

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 10

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983137983149983152983148983141983155

13 MPmortal(socrates)4

man(socrates)mortal(socrates)

forallX (man(X) mortal(X))

man(socrates)

Representation

2 X= ldquosocratesrdquo UI3

Axiom2

Axiom1

Rule

Socrates is mortal

Backward mapping (to natural language)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 11

983123983161983150983156983137983160 983151983142 983110983119983116983098 983106983137983155983145983139 983141983148983141983149983141983150983156983155

Constants KingJohn 2 NUS

Predicates Brother gt

Functions

Sqrt LeftLegOf Variables

x y a b

Connectives not rArr and or hArr

Equality =

Quantifiers Universal Quantifier forall

Existential Quantifier exist

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 12

983105983156983151983149983145983139 983155983141983150983156983141983150983139983141983155

Term function (term1termn)or constant or variable

Atomic sentence predicate (term1termn)

or term1 = term

2

Eg

Brother(KingJohnRichardTheLionheart) gt (Length(LeftLegOf(Richard)) Length(LeftLegOf(KingJohn)))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 13

983107983151983149983152983148983141983160 983155983141983150983156983141983150983139983141983155

Complex sentences are made from atomic sentences usingconnectives examples notS

S1 and S2

S1 or S2

S1rArr S2

S1 hArr S2

Eg Sibling(KingJohnRichard) rArr Sibling(RichardKingJohn)

gt(12) or le (12)

gt(12) and not gt(12)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 14

983124983154983157983156983144 983145983150 983142983145983154983155983156983085983151983154983140983141983154 983148983151983143983145983139

Sentences are true with respect to a model and an interpretation

Model contains objects (domain elements ) and relations among them

Interpretation specifies referents forconstant symbols rarr objects

predicate symbols rarr relations

function symbols rarr functional relations

An atomic sentence predicate(term1 termn) is true

iff the objects referred to by term1 termn

are in the relation referred to by predicate

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 15

983125983150983145983158983141983154983155983137983148 983153983157983137983150983156983145983142983145983139983137983156983145983151983150

forallltvariablesgt ltsentence gt

Everyone at NUS is smart

forallx At(xNUS)rArr Smart(x )

forallx P is true in a model m iff P is true with x being each possible object

in the model Roughly speaking equivalent to the conjunction of instantiations of

P

983105983156983080983115983145983150983143983114983151983144983150983084983118983125983123983081rArr 983123983149983137983154983156983080983115983145983150983143983114983151983144983150983081

and 983105983156983080983122983145983139983144983137983154983140983084983118983125983123983081rArr 983123983149983137983154983156983080983122983145983139983144983137983154983140983081

and 983105983156983080983118983125983123983084983118983125983123983081rArr 983123983149983137983154983156983080983118983125983123983081and 983086983086983086

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 16

983105 983139983151983149983149983151983150 983149983145983155983156983137983147983141 983156983151 983137983158983151983145983140

TypicallyrArr is the main connective with forall

Common mistake using and as the main connective with forall

forallx At(xNUS) and Smart(x)

means ldquoEveryone is at NUS and everyone is smartrdquo

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Artificial Intelligence Predicate Logic Slide 17

983109983160983145983155983156983141983150983156983145983137983148 983153983157983137983150983156983145983142983145983139983137983156983145983151983150

existltvariablesgt ltsentencegt

Someone at NUS is smart

exist x At(xNUS) and Smart(x )$

exist x P is true in a model m iff P is true with x being some possible object

in the model Roughly speaking equivalent to the disjunction of instantiations of P

At(KingJohnNUS) and Smart(KingJohn)or At(RichardNUS) and Smart(Richard)or At(NUSNUS) and Smart(NUS)or

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 18

983105983150983151983156983144983141983154 983139983151983149983149983151983150 983149983145983155983156983137983147983141 983156983151 983137983158983151983145983140

Typically and is the main connective with exist

Common mistake usingrArr as the main connective with

exist

exist x At(xNUS)rArr Smart(x

)is true if there is anyone who is not at NUS

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 19

983120983154983151983152983141983154983156983145983141983155 983151983142 983153983157983137983150983156983145983142983145983141983154983155

forallx forally is the same as forally forallx

existx existy is the same as existy existx

existx forally is not the same as forally existx

existx forally Loves(xy) ldquoThere is a person who loves everyone in the worldrdquo

forally existx Loves(xy) ldquoEveryone in the world is loved by at least one personrdquo

Quantifier duality each can be expressed using the other

forallx Likes(xIceCream) notexistx notLikes(xIceCream )

existx Likes(xBroccoli) notforallx notLikes(xBroccoli

)

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Artificial Intelligence Predicate Logic Slide 20

983109983153983157983137983148983145983156983161

term1 = term2 is true under a given interpretation if andonly if term1 and term2 refer to the same object

Eg definition of Sibling in terms of Parent

forallxy Sibling(xy) hArr [not(x = y) and existmf not (m = f) and Parent(mx)and Parent(fx) and Parent(my) and Parent(fy)]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 21

983125983155983145983150983143 983110983119983116

The kinship domain Brothers are siblings

forallxy Brother(xy) hArr Sibling(xy)

Ones mother is ones female parent

forallmc Mother(c) = m hArr (Female(m) and Parent(mc))

ldquoSiblingrdquo is symmetric

forallxy Sibling(xy) hArr Sibling(yx)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 22

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

Representing the following sentences in Predicate Logic

1 Marcus was a man

2 Macus was a Pompeian

3 All Pompians were Romans4 Caesar was a ruler

5 All Romans were either loyal to Caesar or hated hime

6 Everyone is loyal to someone

7 People only try to assassinate rulers they are not loyal to

8 Marcus tried to assassinate Caesar

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Artificial Intelligence Predicate Logic Slide 23

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

1 Marcus was a man

man(Marcus)

2 Macus was a Pompeian

Pompeian(Marcus)

3 All Pompeians were Romans

forallX Pompeian(X) Roman(X)

4 Caesar was a ruler

ruler(Caesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 24

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

5 All Romans were either loyal to Caesar or hated hime

ldquoorrdquo mean ldquoinclusive orrdquo - OR

forallX Roman(X) loyalto(X Caesar) v hate(X Caesar)

ldquoorrdquo mean ldquoexclusive orrdquo - XOR

forallX Roman(X) [ (loyalto(X Caesar) v hate(X Caesar)) ^

not(loyalto(X Caesar) ^ hate(X Caesar))]

or

forallX Roman(X) [ (loyalto(X Caesar) ^ nothate(X Caesar)) v

(notloyalto(X Caesar) ^ hate(X Caesar))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 25

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

6 Everyone is loyal to someone

forallX exist Y loyalto(XY)

7 People only try to assassinate rulers they are not loyal toforallX forall Y person(X) ^ ruler(Y) ^ tryassassinate(XY)notloyalto(XY)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 26

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

8 Marcus tried to assassinate Caesartryassassinate(Marcus Caesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 27

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

Given the above sentences can we make aconclusion as follows

ldquoMarcus was not loyal to Caesar rdquo

or

notloyalto(MarcusCaesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 28

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)

(Rule What)

Add an implicit sentence

9 ldquoAll men are peoplerdquo

or

forallforallforallforallX man(X) X man(X) person(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 29

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)(9 Substitute)

man(Marcus)

(1)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 30

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

How to represent exceptions example

ldquoPaulus was a Pompeian Paulus was neitherloyal nor hated Caesarrdquo

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Artificial Intelligence Predicate Logic Slide 31

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation Semantic Net Simple by adding links

(properties) to subclasses or instances

Predicate Logic Not so simple like

983120983151983149983152983141983145983137983150(983120983137983157983148983157983155) 983134 983212983131983148983151983161983137983148983156983151(983120983137983157983148983157983155 983107983137983141983155983137983154) 983158

983144983137983156983141(983120983157983137983148983157983155 983107983137983141983155983137983154)983133

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 32

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation983118983151983156 983155983151 983155983145983149983152983148983141 983106983141983139983137983157983155983141 983156983144983141983154983141 983145983155 983139983151983150983142983148983145983139983156 983138983141983156983159983141983141983150983098

Pompeian(Paulus) ^ not[loyalto(Paulus Caesar) vhate(Pualus Caesar)]

983137983150983140

forallX Pompeian(X) loyalto(X Caesar) v hate(XCaesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 33

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation A Better way

forallX Roman(X) ^ noteq(X Paulus) loyalto(X Caesar)v hate(X Caesar

Exception

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 34

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

In the case of describing relations betweennumbers example 1 983100 2

2 983100983091

983095 983102 983091 +2 983091 983102 1

991270

rArr

983123983144983151983157983148983140 983150983151983156 983141983160983152983148983145983139983145983156983148983161 983140983141983155983139983154983145983138983141983098983148983156(12) 983148983156(2983091) 991270

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 35

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

rArr Need functions and computable predicates

rArr For example 983107983137983148983148 983137 983142983157983150983139983156983145983151983150 983156983151 983139983151983149983152983157983156983141 (983091+2)

983120983137983155983155 983156983144983141 983154983141983155983157983148983156 983156983151 983156983144983141 983152983154983141983140983145983139983137983156983141 991260983143983156991261983098 983143983156(983095 983091+2)

983111983141983156 983156983144983141 983142983145983150983137983148 983154983141983155983157983148983156 (983124983154983157983141)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 36

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

1 Marcus was a man

man(Marcus)

2 Marcus was a PompeianPompeian(Marcus)

3 Marcus was born in 40 AD

born(Marcus40)

4 All men are mortal

forallX man(X) mortal(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 37

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

5 All Pompeian died when the vocano erupted in 79 AD

erupted(vocano 79) ^ forallX [Pompeian(X) died(X 79)]

6 No mortal lives longer then 150 yearsforallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

7 It is now 1991now = 1991

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Artificial Intelligence Predicate Logic Slide 38

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Question

Is Marcus alive

Or

alive(Marcus now)

ornotnotnotnotalive(Marcus now)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 39

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge

Relation between ldquodeadrdquo and ldquoaliverdquo

8 Alive means not deadforallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 40

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Add implicit knowledge Time properties of an event for example ldquodeadrdquo

9 Is someone dies the he is dead at all later times

forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

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Artificial Intelligence Predicate Logic Slide 41

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Summary of facts1 man(Marcus)

2 Pompeian(Marcus)

3 born(Marcus 40)

4 forallX man(X) mortal(X)

5 forallX Pompeian(X) died(X 79)

6 erupted(volcano 79)

7 forallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

8 now = 19919 forallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

10 forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

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Artificial Intelligence Predicate Logic Slide 42

notalive(Marcus now)

dead(Marcus now)

(9 Substitute)

died(Marcus T1) gt(now T1)

(10 Substitute)

Pompeian(Marcus) gt(now 79)

(5 Substitute)(5 Substitute)

(2)gt(1991 79)

(8 Substitute)

(computation)

ss

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Artificial Intelligence Predicate Logic Slide 43

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Every instantiation of a universally quantified sentence is enta iled by it

forallv α

Subst(vg α)

for any variable v and ground term g

Eg forallx King( x) and Greedy( x)rArr Evil( x

) yieldsKing (John ) and Greedy (John ) rArr Evil (John )

King (Richard ) and Greedy (Richard ) rArr Evil (Richard )

King (Father (John )) and Greedy (Father (John )) rArr Evil (Father (John ))

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Artificial Intelligence Predicate Logic Slide 44

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For any sentence α variable v and constant symbol k that

does not

appear elsewhere in the knowledge baseexistv α

Subst(vk α )

Eg exist x Crown( x) and OnHead ( xJohn) yields

Crown(C 1) and OnHead (C 1 John )

provided C 1 is a new constant symbol called a Skolemconstant

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Artificial Intelligence Predicate Logic Slide 45

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Suppose the KB contains just the followingforall

x King(x)and

Greedy(x)rArr

Evil(x)King(John)Greedy(John)Brother(RichardJohn)

Instantiating the universal sentence in all possible ways we haveKing(John) and Greedy(John) rArr Evil(John)King(Richard) and Greedy(Richard) rArr Evil(Richard)King(John)Greedy(John)Brother(RichardJohn

)

The new KB is propositionalized proposition symbols are

King(John) Greedy(John) Evil(John) King(Richard

) etc

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Artificial Intelligence Predicate Logic Slide 46

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Every FOL KB can be propositionalized so as to preserve entailment

(A ground sentence is entailed by new KB iff entailed by originalKB)

Idea propositionalize KB and query apply resolution return result

Problem with function symbols there are infinitely many ground

terms

eg Father (Father (Father (John

)))

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Artificial Intelligence Predicate Logic Slide 47

983122983141983155983151983148983157983156983145983151983150

Convert to clause form exampleldquoAll Romans who know Marcus either hate

Caesar or think that anyone who hates anyone is

crazyrdquo

forallX [roman(X) ^ know(X Marcus)]

[hate(X Ceasar) v(forallY existZ hate(YZ)

thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 48

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

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Artificial Intelligence Predicate Logic Slide 49

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

forallX [roman(X) ^ know(X Marcus)][hate(X Ceasar) v

(forallY existZ hate(YZ) thinkcrazy(XY))]

=

forallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 50

983122983141983155983151983148983157983156983145983151983150

2 Move not next to terms

Using the following equivalence

a Double negative

not(notp) = p

b DeMorgan

not(a v b) = nota ^ notb

not(a ^ b) = nota v notb

c Equivalence of qualifiers notforallX P(X) = existX notP(X)

notexistX P(X) = forallX notP(X)

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Artificial Intelligence Predicate Logic Slide 51

983122983141983155983151983148983157983156983145983151983150

2 Move not next to termsforallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

=

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) vthinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 52

983122983141983155983151983148983157983156983145983151983150

3 Change name of variables as follows

forallX P(X) v forallX Q(X)

=forallX P(X) v forallY Q(Y)

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Artificial Intelligence Predicate Logic Slide 53

983122983141983155983151983148983157983156983145983151983150

4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 54

983122983141983155983151983148983157983156983145983151983150

5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

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Artificial Intelligence Predicate Logic Slide 55

983122983141983155983151983148983157983156983145983151983150

6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 56

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

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Artificial Intelligence Predicate Logic Slide 57

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

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Artificial Intelligence Predicate Logic Slide 58

983122983141983155983151983148983157983156983145983151983150

8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

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Artificial Intelligence Predicate Logic Slide 59

983122983141983155983151983148983157983156983145983151983150

9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

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Artificial Intelligence Predicate Logic Slide 60

983122983141983155983151983148983157983156983145983151983150

Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

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Artificial Intelligence Predicate Logic Slide 61

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

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Artificial Intelligence Predicate Logic Slide 62

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

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Artificial Intelligence Predicate Logic Slide 63

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

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Artificial Intelligence Predicate Logic Slide 64

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

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Artificial Intelligence Predicate Logic Slide 65

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

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Artificial Intelligence Predicate Logic Slide 66

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

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Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

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Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

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Artificial Intelligence Predicate Logic Slide 69

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

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Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

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Artificial Intelligence Predicate Logic Slide 71

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

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Artificial Intelligence Predicate Logic Slide 72

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

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Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

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Artificial Intelligence Predicate Logic Slide 74

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

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Artificial Intelligence Predicate Logic Slide 75

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

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Artificial Intelligence Predicate Logic Slide 76

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

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Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

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Artificial Intelligence Predicate Logic Slide 3

983105983139983147983150983151983159983148983141983140983143983149983141983150983156

The slides in this PPT file are composed using thematerials supplied by

Prof Stuart Russell and Peter Norvig They

are currently from University of CaliforniaBerkeley They are also the author of the book ldquoArtificial Intelligence A Modern Approachrdquo whichis used as the textbook for the course

Prof Tom Lenaerts from Universiteacute Libre deBruxelles

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Artificial Intelligence Predicate Logic Slide 4

983119983157983156983148983145983150983141

Why FOLSyntax and semantics of FOL

Using FOL

Wumpus world in FOL

Knowledge engineering in FOL

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Artificial Intelligence Predicate Logic Slide 5

983127983144983161 983110983119983116983103983120983154983151983155 983137983150983140 983139983151983150983155 983151983142 983152983154983151983152983151983155983145983156983145983151983150983137983148 983148983151983143983145983139

Propositional logic is declarative

Propositional logic allows partialdisjunctivenegated information

(unlike most data structures and databases)

Propositional logic is compositional meaning of B 11 and P 12 is derived from meaning of B 11 and of P 12

Meaning in propositional logic is context-

independent

(unlike natural language where meaning depends on context)

Propositional logic has very limited expressive power (unlike natural language) Eg cannot say pits cause breezes in adjacent squaresldquo

983141983160983139983141983152983156 983138983161 983159983154983145983156983145983150983143 983151983150983141 983155983141983150983156983141983150983139983141 983142983151983154 983141983137983139983144 983155983153983157983137983154983141

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 680

Artificial Intelligence Predicate Logic Slide 6

983127983144983161 983110983119983116983103983110983145983154983155983156983085983151983154983140983141983154 983148983151983143983145983139

Whereas propositional logic assumes the worldcontains facts

first-order logic (like natural language) assumes

the world containsObjects people houses numbers colors baseballgames wars hellip

Relations red round prime brother of bigger

than part of comes between hellipFunctions father of best friend one more than

plus hellip

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 7

983120983154983141983152983151983155983145983156983145983151983150983137983148 983116983151983143983145983139983098 983109983160983137983149983152983148983141983155

MORTALMANAll men are mortal

PLATOMANPlato is a man

SOCRATESMANSocrates is a manSentence in Prepositional LogicSentence in natural language

Disadvantages1 Socrates and Plato are human but the representation is not similar

2 Can not state that each person is mortal Difficult for reasoning

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 8

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983137983149983152983148983141983155

forallX (man(X) mortal(X))All men are mortal

man(plato)Plato is a man

man(socrates)Socrates is a man

Sentence in Predicate LogicSentence in natural language

Using predicate logic

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 9

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983137983149983152983148983141983155

Ability of reasoning

All men are mortal Socrates is mortal

Socrates is a man

ConclusionSentence in natural language

forallX (man(X) mortal(X))mortal(socrates)

man(socrates)

ConclusionSentence in Predicate Logic

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 10

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983137983149983152983148983141983155

13 MPmortal(socrates)4

man(socrates)mortal(socrates)

forallX (man(X) mortal(X))

man(socrates)

Representation

2 X= ldquosocratesrdquo UI3

Axiom2

Axiom1

Rule

Socrates is mortal

Backward mapping (to natural language)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 11

983123983161983150983156983137983160 983151983142 983110983119983116983098 983106983137983155983145983139 983141983148983141983149983141983150983156983155

Constants KingJohn 2 NUS

Predicates Brother gt

Functions

Sqrt LeftLegOf Variables

x y a b

Connectives not rArr and or hArr

Equality =

Quantifiers Universal Quantifier forall

Existential Quantifier exist

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 12

983105983156983151983149983145983139 983155983141983150983156983141983150983139983141983155

Term function (term1termn)or constant or variable

Atomic sentence predicate (term1termn)

or term1 = term

2

Eg

Brother(KingJohnRichardTheLionheart) gt (Length(LeftLegOf(Richard)) Length(LeftLegOf(KingJohn)))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 13

983107983151983149983152983148983141983160 983155983141983150983156983141983150983139983141983155

Complex sentences are made from atomic sentences usingconnectives examples notS

S1 and S2

S1 or S2

S1rArr S2

S1 hArr S2

Eg Sibling(KingJohnRichard) rArr Sibling(RichardKingJohn)

gt(12) or le (12)

gt(12) and not gt(12)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 14

983124983154983157983156983144 983145983150 983142983145983154983155983156983085983151983154983140983141983154 983148983151983143983145983139

Sentences are true with respect to a model and an interpretation

Model contains objects (domain elements ) and relations among them

Interpretation specifies referents forconstant symbols rarr objects

predicate symbols rarr relations

function symbols rarr functional relations

An atomic sentence predicate(term1 termn) is true

iff the objects referred to by term1 termn

are in the relation referred to by predicate

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 15

983125983150983145983158983141983154983155983137983148 983153983157983137983150983156983145983142983145983139983137983156983145983151983150

forallltvariablesgt ltsentence gt

Everyone at NUS is smart

forallx At(xNUS)rArr Smart(x )

forallx P is true in a model m iff P is true with x being each possible object

in the model Roughly speaking equivalent to the conjunction of instantiations of

P

983105983156983080983115983145983150983143983114983151983144983150983084983118983125983123983081rArr 983123983149983137983154983156983080983115983145983150983143983114983151983144983150983081

and 983105983156983080983122983145983139983144983137983154983140983084983118983125983123983081rArr 983123983149983137983154983156983080983122983145983139983144983137983154983140983081

and 983105983156983080983118983125983123983084983118983125983123983081rArr 983123983149983137983154983156983080983118983125983123983081and 983086983086983086

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 16

983105 983139983151983149983149983151983150 983149983145983155983156983137983147983141 983156983151 983137983158983151983145983140

TypicallyrArr is the main connective with forall

Common mistake using and as the main connective with forall

forallx At(xNUS) and Smart(x)

means ldquoEveryone is at NUS and everyone is smartrdquo

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 17

983109983160983145983155983156983141983150983156983145983137983148 983153983157983137983150983156983145983142983145983139983137983156983145983151983150

existltvariablesgt ltsentencegt

Someone at NUS is smart

exist x At(xNUS) and Smart(x )$

exist x P is true in a model m iff P is true with x being some possible object

in the model Roughly speaking equivalent to the disjunction of instantiations of P

At(KingJohnNUS) and Smart(KingJohn)or At(RichardNUS) and Smart(Richard)or At(NUSNUS) and Smart(NUS)or

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 18

983105983150983151983156983144983141983154 983139983151983149983149983151983150 983149983145983155983156983137983147983141 983156983151 983137983158983151983145983140

Typically and is the main connective with exist

Common mistake usingrArr as the main connective with

exist

exist x At(xNUS)rArr Smart(x

)is true if there is anyone who is not at NUS

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 19

983120983154983151983152983141983154983156983145983141983155 983151983142 983153983157983137983150983156983145983142983145983141983154983155

forallx forally is the same as forally forallx

existx existy is the same as existy existx

existx forally is not the same as forally existx

existx forally Loves(xy) ldquoThere is a person who loves everyone in the worldrdquo

forally existx Loves(xy) ldquoEveryone in the world is loved by at least one personrdquo

Quantifier duality each can be expressed using the other

forallx Likes(xIceCream) notexistx notLikes(xIceCream )

existx Likes(xBroccoli) notforallx notLikes(xBroccoli

)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 20

983109983153983157983137983148983145983156983161

term1 = term2 is true under a given interpretation if andonly if term1 and term2 refer to the same object

Eg definition of Sibling in terms of Parent

forallxy Sibling(xy) hArr [not(x = y) and existmf not (m = f) and Parent(mx)and Parent(fx) and Parent(my) and Parent(fy)]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 21

983125983155983145983150983143 983110983119983116

The kinship domain Brothers are siblings

forallxy Brother(xy) hArr Sibling(xy)

Ones mother is ones female parent

forallmc Mother(c) = m hArr (Female(m) and Parent(mc))

ldquoSiblingrdquo is symmetric

forallxy Sibling(xy) hArr Sibling(yx)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 22

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

Representing the following sentences in Predicate Logic

1 Marcus was a man

2 Macus was a Pompeian

3 All Pompians were Romans4 Caesar was a ruler

5 All Romans were either loyal to Caesar or hated hime

6 Everyone is loyal to someone

7 People only try to assassinate rulers they are not loyal to

8 Marcus tried to assassinate Caesar

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 23

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

1 Marcus was a man

man(Marcus)

2 Macus was a Pompeian

Pompeian(Marcus)

3 All Pompeians were Romans

forallX Pompeian(X) Roman(X)

4 Caesar was a ruler

ruler(Caesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 24

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

5 All Romans were either loyal to Caesar or hated hime

ldquoorrdquo mean ldquoinclusive orrdquo - OR

forallX Roman(X) loyalto(X Caesar) v hate(X Caesar)

ldquoorrdquo mean ldquoexclusive orrdquo - XOR

forallX Roman(X) [ (loyalto(X Caesar) v hate(X Caesar)) ^

not(loyalto(X Caesar) ^ hate(X Caesar))]

or

forallX Roman(X) [ (loyalto(X Caesar) ^ nothate(X Caesar)) v

(notloyalto(X Caesar) ^ hate(X Caesar))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 25

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

6 Everyone is loyal to someone

forallX exist Y loyalto(XY)

7 People only try to assassinate rulers they are not loyal toforallX forall Y person(X) ^ ruler(Y) ^ tryassassinate(XY)notloyalto(XY)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 26

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

8 Marcus tried to assassinate Caesartryassassinate(Marcus Caesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 27

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

Given the above sentences can we make aconclusion as follows

ldquoMarcus was not loyal to Caesar rdquo

or

notloyalto(MarcusCaesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 28

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)

(Rule What)

Add an implicit sentence

9 ldquoAll men are peoplerdquo

or

forallforallforallforallX man(X) X man(X) person(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 29

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)(9 Substitute)

man(Marcus)

(1)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 30

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

How to represent exceptions example

ldquoPaulus was a Pompeian Paulus was neitherloyal nor hated Caesarrdquo

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 31

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation Semantic Net Simple by adding links

(properties) to subclasses or instances

Predicate Logic Not so simple like

983120983151983149983152983141983145983137983150(983120983137983157983148983157983155) 983134 983212983131983148983151983161983137983148983156983151(983120983137983157983148983157983155 983107983137983141983155983137983154) 983158

983144983137983156983141(983120983157983137983148983157983155 983107983137983141983155983137983154)983133

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 32

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation983118983151983156 983155983151 983155983145983149983152983148983141 983106983141983139983137983157983155983141 983156983144983141983154983141 983145983155 983139983151983150983142983148983145983139983156 983138983141983156983159983141983141983150983098

Pompeian(Paulus) ^ not[loyalto(Paulus Caesar) vhate(Pualus Caesar)]

983137983150983140

forallX Pompeian(X) loyalto(X Caesar) v hate(XCaesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 33

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation A Better way

forallX Roman(X) ^ noteq(X Paulus) loyalto(X Caesar)v hate(X Caesar

Exception

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 34

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

In the case of describing relations betweennumbers example 1 983100 2

2 983100983091

983095 983102 983091 +2 983091 983102 1

991270

rArr

983123983144983151983157983148983140 983150983151983156 983141983160983152983148983145983139983145983156983148983161 983140983141983155983139983154983145983138983141983098983148983156(12) 983148983156(2983091) 991270

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 35

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

rArr Need functions and computable predicates

rArr For example 983107983137983148983148 983137 983142983157983150983139983156983145983151983150 983156983151 983139983151983149983152983157983156983141 (983091+2)

983120983137983155983155 983156983144983141 983154983141983155983157983148983156 983156983151 983156983144983141 983152983154983141983140983145983139983137983156983141 991260983143983156991261983098 983143983156(983095 983091+2)

983111983141983156 983156983144983141 983142983145983150983137983148 983154983141983155983157983148983156 (983124983154983157983141)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 36

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

1 Marcus was a man

man(Marcus)

2 Marcus was a PompeianPompeian(Marcus)

3 Marcus was born in 40 AD

born(Marcus40)

4 All men are mortal

forallX man(X) mortal(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 37

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

5 All Pompeian died when the vocano erupted in 79 AD

erupted(vocano 79) ^ forallX [Pompeian(X) died(X 79)]

6 No mortal lives longer then 150 yearsforallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

7 It is now 1991now = 1991

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Artificial Intelligence Predicate Logic Slide 38

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Question

Is Marcus alive

Or

alive(Marcus now)

ornotnotnotnotalive(Marcus now)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 39

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge

Relation between ldquodeadrdquo and ldquoaliverdquo

8 Alive means not deadforallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4080

Artificial Intelligence Predicate Logic Slide 40

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge Time properties of an event for example ldquodeadrdquo

9 Is someone dies the he is dead at all later times

forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4180

Artificial Intelligence Predicate Logic Slide 41

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Summary of facts1 man(Marcus)

2 Pompeian(Marcus)

3 born(Marcus 40)

4 forallX man(X) mortal(X)

5 forallX Pompeian(X) died(X 79)

6 erupted(volcano 79)

7 forallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

8 now = 19919 forallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

10 forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

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Artificial Intelligence Predicate Logic Slide 42

notalive(Marcus now)

dead(Marcus now)

(9 Substitute)

died(Marcus T1) gt(now T1)

(10 Substitute)

Pompeian(Marcus) gt(now 79)

(5 Substitute)(5 Substitute)

(2)gt(1991 79)

(8 Substitute)

(computation)

ss

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Artificial Intelligence Predicate Logic Slide 43

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Every instantiation of a universally quantified sentence is enta iled by it

forallv α

Subst(vg α)

for any variable v and ground term g

Eg forallx King( x) and Greedy( x)rArr Evil( x

) yieldsKing (John ) and Greedy (John ) rArr Evil (John )

King (Richard ) and Greedy (Richard ) rArr Evil (Richard )

King (Father (John )) and Greedy (Father (John )) rArr Evil (Father (John ))

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Artificial Intelligence Predicate Logic Slide 44

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For any sentence α variable v and constant symbol k that

does not

appear elsewhere in the knowledge baseexistv α

Subst(vk α )

Eg exist x Crown( x) and OnHead ( xJohn) yields

Crown(C 1) and OnHead (C 1 John )

provided C 1 is a new constant symbol called a Skolemconstant

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Artificial Intelligence Predicate Logic Slide 45

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Suppose the KB contains just the followingforall

x King(x)and

Greedy(x)rArr

Evil(x)King(John)Greedy(John)Brother(RichardJohn)

Instantiating the universal sentence in all possible ways we haveKing(John) and Greedy(John) rArr Evil(John)King(Richard) and Greedy(Richard) rArr Evil(Richard)King(John)Greedy(John)Brother(RichardJohn

)

The new KB is propositionalized proposition symbols are

King(John) Greedy(John) Evil(John) King(Richard

) etc

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Artificial Intelligence Predicate Logic Slide 46

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Every FOL KB can be propositionalized so as to preserve entailment

(A ground sentence is entailed by new KB iff entailed by originalKB)

Idea propositionalize KB and query apply resolution return result

Problem with function symbols there are infinitely many ground

terms

eg Father (Father (Father (John

)))

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Artificial Intelligence Predicate Logic Slide 47

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Convert to clause form exampleldquoAll Romans who know Marcus either hate

Caesar or think that anyone who hates anyone is

crazyrdquo

forallX [roman(X) ^ know(X Marcus)]

[hate(X Ceasar) v(forallY existZ hate(YZ)

thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 48

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1 Remove

using the equivalence a b = nota v b

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Artificial Intelligence Predicate Logic Slide 49

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

forallX [roman(X) ^ know(X Marcus)][hate(X Ceasar) v

(forallY existZ hate(YZ) thinkcrazy(XY))]

=

forallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 50

983122983141983155983151983148983157983156983145983151983150

2 Move not next to terms

Using the following equivalence

a Double negative

not(notp) = p

b DeMorgan

not(a v b) = nota ^ notb

not(a ^ b) = nota v notb

c Equivalence of qualifiers notforallX P(X) = existX notP(X)

notexistX P(X) = forallX notP(X)

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Artificial Intelligence Predicate Logic Slide 51

983122983141983155983151983148983157983156983145983151983150

2 Move not next to termsforallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

=

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) vthinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 52

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3 Change name of variables as follows

forallX P(X) v forallX Q(X)

=forallX P(X) v forallY Q(Y)

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Artificial Intelligence Predicate Logic Slide 53

983122983141983155983151983148983157983156983145983151983150

4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 54

983122983141983155983151983148983157983156983145983151983150

5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

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Artificial Intelligence Predicate Logic Slide 55

983122983141983155983151983148983157983156983145983151983150

6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 56

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

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Artificial Intelligence Predicate Logic Slide 57

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7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

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Artificial Intelligence Predicate Logic Slide 58

983122983141983155983151983148983157983156983145983151983150

8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

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Artificial Intelligence Predicate Logic Slide 59

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9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

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Artificial Intelligence Predicate Logic Slide 60

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Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

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Artificial Intelligence Predicate Logic Slide 61

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X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

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Artificial Intelligence Predicate Logic Slide 62

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θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

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Artificial Intelligence Predicate Logic Slide 63

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Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

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Artificial Intelligence Predicate Logic Slide 64

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Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

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Artificial Intelligence Predicate Logic Slide 65

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Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

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Artificial Intelligence Predicate Logic Slide 66

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Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

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Artificial Intelligence Predicate Logic Slide 67

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mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

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Artificial Intelligence Predicate Logic Slide 68

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Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

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Artificial Intelligence Predicate Logic Slide 69

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Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

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Artificial Intelligence Predicate Logic Slide 70

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Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

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Artificial Intelligence Predicate Logic Slide 71

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Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

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Artificial Intelligence Predicate Logic Slide 72

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

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Artificial Intelligence Predicate Logic Slide 73

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Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

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Artificial Intelligence Predicate Logic Slide 74

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Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

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Artificial Intelligence Predicate Logic Slide 75

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Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

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Artificial Intelligence Predicate Logic Slide 76

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Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

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Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

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Artificial Intelligence Predicate Logic Slide 4

983119983157983156983148983145983150983141

Why FOLSyntax and semantics of FOL

Using FOL

Wumpus world in FOL

Knowledge engineering in FOL

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Artificial Intelligence Predicate Logic Slide 5

983127983144983161 983110983119983116983103983120983154983151983155 983137983150983140 983139983151983150983155 983151983142 983152983154983151983152983151983155983145983156983145983151983150983137983148 983148983151983143983145983139

Propositional logic is declarative

Propositional logic allows partialdisjunctivenegated information

(unlike most data structures and databases)

Propositional logic is compositional meaning of B 11 and P 12 is derived from meaning of B 11 and of P 12

Meaning in propositional logic is context-

independent

(unlike natural language where meaning depends on context)

Propositional logic has very limited expressive power (unlike natural language) Eg cannot say pits cause breezes in adjacent squaresldquo

983141983160983139983141983152983156 983138983161 983159983154983145983156983145983150983143 983151983150983141 983155983141983150983156983141983150983139983141 983142983151983154 983141983137983139983144 983155983153983157983137983154983141

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Artificial Intelligence Predicate Logic Slide 6

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Whereas propositional logic assumes the worldcontains facts

first-order logic (like natural language) assumes

the world containsObjects people houses numbers colors baseballgames wars hellip

Relations red round prime brother of bigger

than part of comes between hellipFunctions father of best friend one more than

plus hellip

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Artificial Intelligence Predicate Logic Slide 7

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MORTALMANAll men are mortal

PLATOMANPlato is a man

SOCRATESMANSocrates is a manSentence in Prepositional LogicSentence in natural language

Disadvantages1 Socrates and Plato are human but the representation is not similar

2 Can not state that each person is mortal Difficult for reasoning

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Artificial Intelligence Predicate Logic Slide 8

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forallX (man(X) mortal(X))All men are mortal

man(plato)Plato is a man

man(socrates)Socrates is a man

Sentence in Predicate LogicSentence in natural language

Using predicate logic

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httpslidepdfcomreaderfullfirst-orderpredicatelogic 980

Artificial Intelligence Predicate Logic Slide 9

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983137983149983152983148983141983155

Ability of reasoning

All men are mortal Socrates is mortal

Socrates is a man

ConclusionSentence in natural language

forallX (man(X) mortal(X))mortal(socrates)

man(socrates)

ConclusionSentence in Predicate Logic

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 10

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983137983149983152983148983141983155

13 MPmortal(socrates)4

man(socrates)mortal(socrates)

forallX (man(X) mortal(X))

man(socrates)

Representation

2 X= ldquosocratesrdquo UI3

Axiom2

Axiom1

Rule

Socrates is mortal

Backward mapping (to natural language)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 11

983123983161983150983156983137983160 983151983142 983110983119983116983098 983106983137983155983145983139 983141983148983141983149983141983150983156983155

Constants KingJohn 2 NUS

Predicates Brother gt

Functions

Sqrt LeftLegOf Variables

x y a b

Connectives not rArr and or hArr

Equality =

Quantifiers Universal Quantifier forall

Existential Quantifier exist

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 12

983105983156983151983149983145983139 983155983141983150983156983141983150983139983141983155

Term function (term1termn)or constant or variable

Atomic sentence predicate (term1termn)

or term1 = term

2

Eg

Brother(KingJohnRichardTheLionheart) gt (Length(LeftLegOf(Richard)) Length(LeftLegOf(KingJohn)))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 13

983107983151983149983152983148983141983160 983155983141983150983156983141983150983139983141983155

Complex sentences are made from atomic sentences usingconnectives examples notS

S1 and S2

S1 or S2

S1rArr S2

S1 hArr S2

Eg Sibling(KingJohnRichard) rArr Sibling(RichardKingJohn)

gt(12) or le (12)

gt(12) and not gt(12)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 14

983124983154983157983156983144 983145983150 983142983145983154983155983156983085983151983154983140983141983154 983148983151983143983145983139

Sentences are true with respect to a model and an interpretation

Model contains objects (domain elements ) and relations among them

Interpretation specifies referents forconstant symbols rarr objects

predicate symbols rarr relations

function symbols rarr functional relations

An atomic sentence predicate(term1 termn) is true

iff the objects referred to by term1 termn

are in the relation referred to by predicate

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 15

983125983150983145983158983141983154983155983137983148 983153983157983137983150983156983145983142983145983139983137983156983145983151983150

forallltvariablesgt ltsentence gt

Everyone at NUS is smart

forallx At(xNUS)rArr Smart(x )

forallx P is true in a model m iff P is true with x being each possible object

in the model Roughly speaking equivalent to the conjunction of instantiations of

P

983105983156983080983115983145983150983143983114983151983144983150983084983118983125983123983081rArr 983123983149983137983154983156983080983115983145983150983143983114983151983144983150983081

and 983105983156983080983122983145983139983144983137983154983140983084983118983125983123983081rArr 983123983149983137983154983156983080983122983145983139983144983137983154983140983081

and 983105983156983080983118983125983123983084983118983125983123983081rArr 983123983149983137983154983156983080983118983125983123983081and 983086983086983086

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 16

983105 983139983151983149983149983151983150 983149983145983155983156983137983147983141 983156983151 983137983158983151983145983140

TypicallyrArr is the main connective with forall

Common mistake using and as the main connective with forall

forallx At(xNUS) and Smart(x)

means ldquoEveryone is at NUS and everyone is smartrdquo

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 17

983109983160983145983155983156983141983150983156983145983137983148 983153983157983137983150983156983145983142983145983139983137983156983145983151983150

existltvariablesgt ltsentencegt

Someone at NUS is smart

exist x At(xNUS) and Smart(x )$

exist x P is true in a model m iff P is true with x being some possible object

in the model Roughly speaking equivalent to the disjunction of instantiations of P

At(KingJohnNUS) and Smart(KingJohn)or At(RichardNUS) and Smart(Richard)or At(NUSNUS) and Smart(NUS)or

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 18

983105983150983151983156983144983141983154 983139983151983149983149983151983150 983149983145983155983156983137983147983141 983156983151 983137983158983151983145983140

Typically and is the main connective with exist

Common mistake usingrArr as the main connective with

exist

exist x At(xNUS)rArr Smart(x

)is true if there is anyone who is not at NUS

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 19

983120983154983151983152983141983154983156983145983141983155 983151983142 983153983157983137983150983156983145983142983145983141983154983155

forallx forally is the same as forally forallx

existx existy is the same as existy existx

existx forally is not the same as forally existx

existx forally Loves(xy) ldquoThere is a person who loves everyone in the worldrdquo

forally existx Loves(xy) ldquoEveryone in the world is loved by at least one personrdquo

Quantifier duality each can be expressed using the other

forallx Likes(xIceCream) notexistx notLikes(xIceCream )

existx Likes(xBroccoli) notforallx notLikes(xBroccoli

)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 20

983109983153983157983137983148983145983156983161

term1 = term2 is true under a given interpretation if andonly if term1 and term2 refer to the same object

Eg definition of Sibling in terms of Parent

forallxy Sibling(xy) hArr [not(x = y) and existmf not (m = f) and Parent(mx)and Parent(fx) and Parent(my) and Parent(fy)]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 21

983125983155983145983150983143 983110983119983116

The kinship domain Brothers are siblings

forallxy Brother(xy) hArr Sibling(xy)

Ones mother is ones female parent

forallmc Mother(c) = m hArr (Female(m) and Parent(mc))

ldquoSiblingrdquo is symmetric

forallxy Sibling(xy) hArr Sibling(yx)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 22

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

Representing the following sentences in Predicate Logic

1 Marcus was a man

2 Macus was a Pompeian

3 All Pompians were Romans4 Caesar was a ruler

5 All Romans were either loyal to Caesar or hated hime

6 Everyone is loyal to someone

7 People only try to assassinate rulers they are not loyal to

8 Marcus tried to assassinate Caesar

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 23

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

1 Marcus was a man

man(Marcus)

2 Macus was a Pompeian

Pompeian(Marcus)

3 All Pompeians were Romans

forallX Pompeian(X) Roman(X)

4 Caesar was a ruler

ruler(Caesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 24

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

5 All Romans were either loyal to Caesar or hated hime

ldquoorrdquo mean ldquoinclusive orrdquo - OR

forallX Roman(X) loyalto(X Caesar) v hate(X Caesar)

ldquoorrdquo mean ldquoexclusive orrdquo - XOR

forallX Roman(X) [ (loyalto(X Caesar) v hate(X Caesar)) ^

not(loyalto(X Caesar) ^ hate(X Caesar))]

or

forallX Roman(X) [ (loyalto(X Caesar) ^ nothate(X Caesar)) v

(notloyalto(X Caesar) ^ hate(X Caesar))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 25

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

6 Everyone is loyal to someone

forallX exist Y loyalto(XY)

7 People only try to assassinate rulers they are not loyal toforallX forall Y person(X) ^ ruler(Y) ^ tryassassinate(XY)notloyalto(XY)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 26

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

8 Marcus tried to assassinate Caesartryassassinate(Marcus Caesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 27

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

Given the above sentences can we make aconclusion as follows

ldquoMarcus was not loyal to Caesar rdquo

or

notloyalto(MarcusCaesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 28

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)

(Rule What)

Add an implicit sentence

9 ldquoAll men are peoplerdquo

or

forallforallforallforallX man(X) X man(X) person(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 29

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)(9 Substitute)

man(Marcus)

(1)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 30

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

How to represent exceptions example

ldquoPaulus was a Pompeian Paulus was neitherloyal nor hated Caesarrdquo

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 31

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation Semantic Net Simple by adding links

(properties) to subclasses or instances

Predicate Logic Not so simple like

983120983151983149983152983141983145983137983150(983120983137983157983148983157983155) 983134 983212983131983148983151983161983137983148983156983151(983120983137983157983148983157983155 983107983137983141983155983137983154) 983158

983144983137983156983141(983120983157983137983148983157983155 983107983137983141983155983137983154)983133

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 32

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation983118983151983156 983155983151 983155983145983149983152983148983141 983106983141983139983137983157983155983141 983156983144983141983154983141 983145983155 983139983151983150983142983148983145983139983156 983138983141983156983159983141983141983150983098

Pompeian(Paulus) ^ not[loyalto(Paulus Caesar) vhate(Pualus Caesar)]

983137983150983140

forallX Pompeian(X) loyalto(X Caesar) v hate(XCaesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 33

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation A Better way

forallX Roman(X) ^ noteq(X Paulus) loyalto(X Caesar)v hate(X Caesar

Exception

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Artificial Intelligence Predicate Logic Slide 34

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

In the case of describing relations betweennumbers example 1 983100 2

2 983100983091

983095 983102 983091 +2 983091 983102 1

991270

rArr

983123983144983151983157983148983140 983150983151983156 983141983160983152983148983145983139983145983156983148983161 983140983141983155983139983154983145983138983141983098983148983156(12) 983148983156(2983091) 991270

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 35

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

rArr Need functions and computable predicates

rArr For example 983107983137983148983148 983137 983142983157983150983139983156983145983151983150 983156983151 983139983151983149983152983157983156983141 (983091+2)

983120983137983155983155 983156983144983141 983154983141983155983157983148983156 983156983151 983156983144983141 983152983154983141983140983145983139983137983156983141 991260983143983156991261983098 983143983156(983095 983091+2)

983111983141983156 983156983144983141 983142983145983150983137983148 983154983141983155983157983148983156 (983124983154983157983141)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 36

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

1 Marcus was a man

man(Marcus)

2 Marcus was a PompeianPompeian(Marcus)

3 Marcus was born in 40 AD

born(Marcus40)

4 All men are mortal

forallX man(X) mortal(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 37

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

5 All Pompeian died when the vocano erupted in 79 AD

erupted(vocano 79) ^ forallX [Pompeian(X) died(X 79)]

6 No mortal lives longer then 150 yearsforallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

7 It is now 1991now = 1991

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 38

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Question

Is Marcus alive

Or

alive(Marcus now)

ornotnotnotnotalive(Marcus now)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 39

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge

Relation between ldquodeadrdquo and ldquoaliverdquo

8 Alive means not deadforallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 40

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge Time properties of an event for example ldquodeadrdquo

9 Is someone dies the he is dead at all later times

forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4180

Artificial Intelligence Predicate Logic Slide 41

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Summary of facts1 man(Marcus)

2 Pompeian(Marcus)

3 born(Marcus 40)

4 forallX man(X) mortal(X)

5 forallX Pompeian(X) died(X 79)

6 erupted(volcano 79)

7 forallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

8 now = 19919 forallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

10 forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4280

Artificial Intelligence Predicate Logic Slide 42

notalive(Marcus now)

dead(Marcus now)

(9 Substitute)

died(Marcus T1) gt(now T1)

(10 Substitute)

Pompeian(Marcus) gt(now 79)

(5 Substitute)(5 Substitute)

(2)gt(1991 79)

(8 Substitute)

(computation)

ss

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4380

Artificial Intelligence Predicate Logic Slide 43

983125983150983145983158983141983154983155983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983125983113)

Every instantiation of a universally quantified sentence is enta iled by it

forallv α

Subst(vg α)

for any variable v and ground term g

Eg forallx King( x) and Greedy( x)rArr Evil( x

) yieldsKing (John ) and Greedy (John ) rArr Evil (John )

King (Richard ) and Greedy (Richard ) rArr Evil (Richard )

King (Father (John )) and Greedy (Father (John )) rArr Evil (Father (John ))

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4480

Artificial Intelligence Predicate Logic Slide 44

983109983160983145983155983156983141983150983156983145983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983109983113)

For any sentence α variable v and constant symbol k that

does not

appear elsewhere in the knowledge baseexistv α

Subst(vk α )

Eg exist x Crown( x) and OnHead ( xJohn) yields

Crown(C 1) and OnHead (C 1 John )

provided C 1 is a new constant symbol called a Skolemconstant

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4580

Artificial Intelligence Predicate Logic Slide 45

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Suppose the KB contains just the followingforall

x King(x)and

Greedy(x)rArr

Evil(x)King(John)Greedy(John)Brother(RichardJohn)

Instantiating the universal sentence in all possible ways we haveKing(John) and Greedy(John) rArr Evil(John)King(Richard) and Greedy(Richard) rArr Evil(Richard)King(John)Greedy(John)Brother(RichardJohn

)

The new KB is propositionalized proposition symbols are

King(John) Greedy(John) Evil(John) King(Richard

) etc

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Artificial Intelligence Predicate Logic Slide 46

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Every FOL KB can be propositionalized so as to preserve entailment

(A ground sentence is entailed by new KB iff entailed by originalKB)

Idea propositionalize KB and query apply resolution return result

Problem with function symbols there are infinitely many ground

terms

eg Father (Father (Father (John

)))

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Artificial Intelligence Predicate Logic Slide 47

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Convert to clause form exampleldquoAll Romans who know Marcus either hate

Caesar or think that anyone who hates anyone is

crazyrdquo

forallX [roman(X) ^ know(X Marcus)]

[hate(X Ceasar) v(forallY existZ hate(YZ)

thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 48

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1 Remove

using the equivalence a b = nota v b

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Artificial Intelligence Predicate Logic Slide 49

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1 Remove

using the equivalence a b = nota v b

forallX [roman(X) ^ know(X Marcus)][hate(X Ceasar) v

(forallY existZ hate(YZ) thinkcrazy(XY))]

=

forallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 50

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2 Move not next to terms

Using the following equivalence

a Double negative

not(notp) = p

b DeMorgan

not(a v b) = nota ^ notb

not(a ^ b) = nota v notb

c Equivalence of qualifiers notforallX P(X) = existX notP(X)

notexistX P(X) = forallX notP(X)

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Artificial Intelligence Predicate Logic Slide 51

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2 Move not next to termsforallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

=

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) vthinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 52

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3 Change name of variables as follows

forallX P(X) v forallX Q(X)

=forallX P(X) v forallY Q(Y)

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Artificial Intelligence Predicate Logic Slide 53

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4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 54

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5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

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Artificial Intelligence Predicate Logic Slide 55

983122983141983155983151983148983157983156983145983151983150

6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 56

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7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

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Artificial Intelligence Predicate Logic Slide 57

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7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

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Artificial Intelligence Predicate Logic Slide 58

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8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

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Artificial Intelligence Predicate Logic Slide 59

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9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

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Artificial Intelligence Predicate Logic Slide 60

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Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

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Artificial Intelligence Predicate Logic Slide 61

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X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

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Artificial Intelligence Predicate Logic Slide 62

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θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

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Artificial Intelligence Predicate Logic Slide 63

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Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

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Artificial Intelligence Predicate Logic Slide 64

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Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

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Artificial Intelligence Predicate Logic Slide 65

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Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

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Artificial Intelligence Predicate Logic Slide 66

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Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

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Artificial Intelligence Predicate Logic Slide 67

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mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

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Artificial Intelligence Predicate Logic Slide 68

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Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

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Artificial Intelligence Predicate Logic Slide 69

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Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

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Artificial Intelligence Predicate Logic Slide 70

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Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

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Artificial Intelligence Predicate Logic Slide 71

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Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

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Artificial Intelligence Predicate Logic Slide 72

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Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

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Artificial Intelligence Predicate Logic Slide 73

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Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

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Artificial Intelligence Predicate Logic Slide 74

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Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

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Artificial Intelligence Predicate Logic Slide 75

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Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

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Artificial Intelligence Predicate Logic Slide 76

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Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

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892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

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Artificial Intelligence Predicate Logic Slide 5

983127983144983161 983110983119983116983103983120983154983151983155 983137983150983140 983139983151983150983155 983151983142 983152983154983151983152983151983155983145983156983145983151983150983137983148 983148983151983143983145983139

Propositional logic is declarative

Propositional logic allows partialdisjunctivenegated information

(unlike most data structures and databases)

Propositional logic is compositional meaning of B 11 and P 12 is derived from meaning of B 11 and of P 12

Meaning in propositional logic is context-

independent

(unlike natural language where meaning depends on context)

Propositional logic has very limited expressive power (unlike natural language) Eg cannot say pits cause breezes in adjacent squaresldquo

983141983160983139983141983152983156 983138983161 983159983154983145983156983145983150983143 983151983150983141 983155983141983150983156983141983150983139983141 983142983151983154 983141983137983139983144 983155983153983157983137983154983141

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Artificial Intelligence Predicate Logic Slide 6

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Whereas propositional logic assumes the worldcontains facts

first-order logic (like natural language) assumes

the world containsObjects people houses numbers colors baseballgames wars hellip

Relations red round prime brother of bigger

than part of comes between hellipFunctions father of best friend one more than

plus hellip

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Artificial Intelligence Predicate Logic Slide 7

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MORTALMANAll men are mortal

PLATOMANPlato is a man

SOCRATESMANSocrates is a manSentence in Prepositional LogicSentence in natural language

Disadvantages1 Socrates and Plato are human but the representation is not similar

2 Can not state that each person is mortal Difficult for reasoning

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Artificial Intelligence Predicate Logic Slide 8

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forallX (man(X) mortal(X))All men are mortal

man(plato)Plato is a man

man(socrates)Socrates is a man

Sentence in Predicate LogicSentence in natural language

Using predicate logic

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Artificial Intelligence Predicate Logic Slide 9

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Ability of reasoning

All men are mortal Socrates is mortal

Socrates is a man

ConclusionSentence in natural language

forallX (man(X) mortal(X))mortal(socrates)

man(socrates)

ConclusionSentence in Predicate Logic

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Artificial Intelligence Predicate Logic Slide 10

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13 MPmortal(socrates)4

man(socrates)mortal(socrates)

forallX (man(X) mortal(X))

man(socrates)

Representation

2 X= ldquosocratesrdquo UI3

Axiom2

Axiom1

Rule

Socrates is mortal

Backward mapping (to natural language)

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Artificial Intelligence Predicate Logic Slide 11

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Constants KingJohn 2 NUS

Predicates Brother gt

Functions

Sqrt LeftLegOf Variables

x y a b

Connectives not rArr and or hArr

Equality =

Quantifiers Universal Quantifier forall

Existential Quantifier exist

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Artificial Intelligence Predicate Logic Slide 12

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Term function (term1termn)or constant or variable

Atomic sentence predicate (term1termn)

or term1 = term

2

Eg

Brother(KingJohnRichardTheLionheart) gt (Length(LeftLegOf(Richard)) Length(LeftLegOf(KingJohn)))

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Artificial Intelligence Predicate Logic Slide 13

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Complex sentences are made from atomic sentences usingconnectives examples notS

S1 and S2

S1 or S2

S1rArr S2

S1 hArr S2

Eg Sibling(KingJohnRichard) rArr Sibling(RichardKingJohn)

gt(12) or le (12)

gt(12) and not gt(12)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 14

983124983154983157983156983144 983145983150 983142983145983154983155983156983085983151983154983140983141983154 983148983151983143983145983139

Sentences are true with respect to a model and an interpretation

Model contains objects (domain elements ) and relations among them

Interpretation specifies referents forconstant symbols rarr objects

predicate symbols rarr relations

function symbols rarr functional relations

An atomic sentence predicate(term1 termn) is true

iff the objects referred to by term1 termn

are in the relation referred to by predicate

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 15

983125983150983145983158983141983154983155983137983148 983153983157983137983150983156983145983142983145983139983137983156983145983151983150

forallltvariablesgt ltsentence gt

Everyone at NUS is smart

forallx At(xNUS)rArr Smart(x )

forallx P is true in a model m iff P is true with x being each possible object

in the model Roughly speaking equivalent to the conjunction of instantiations of

P

983105983156983080983115983145983150983143983114983151983144983150983084983118983125983123983081rArr 983123983149983137983154983156983080983115983145983150983143983114983151983144983150983081

and 983105983156983080983122983145983139983144983137983154983140983084983118983125983123983081rArr 983123983149983137983154983156983080983122983145983139983144983137983154983140983081

and 983105983156983080983118983125983123983084983118983125983123983081rArr 983123983149983137983154983156983080983118983125983123983081and 983086983086983086

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 16

983105 983139983151983149983149983151983150 983149983145983155983156983137983147983141 983156983151 983137983158983151983145983140

TypicallyrArr is the main connective with forall

Common mistake using and as the main connective with forall

forallx At(xNUS) and Smart(x)

means ldquoEveryone is at NUS and everyone is smartrdquo

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 17

983109983160983145983155983156983141983150983156983145983137983148 983153983157983137983150983156983145983142983145983139983137983156983145983151983150

existltvariablesgt ltsentencegt

Someone at NUS is smart

exist x At(xNUS) and Smart(x )$

exist x P is true in a model m iff P is true with x being some possible object

in the model Roughly speaking equivalent to the disjunction of instantiations of P

At(KingJohnNUS) and Smart(KingJohn)or At(RichardNUS) and Smart(Richard)or At(NUSNUS) and Smart(NUS)or

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 18

983105983150983151983156983144983141983154 983139983151983149983149983151983150 983149983145983155983156983137983147983141 983156983151 983137983158983151983145983140

Typically and is the main connective with exist

Common mistake usingrArr as the main connective with

exist

exist x At(xNUS)rArr Smart(x

)is true if there is anyone who is not at NUS

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 19

983120983154983151983152983141983154983156983145983141983155 983151983142 983153983157983137983150983156983145983142983145983141983154983155

forallx forally is the same as forally forallx

existx existy is the same as existy existx

existx forally is not the same as forally existx

existx forally Loves(xy) ldquoThere is a person who loves everyone in the worldrdquo

forally existx Loves(xy) ldquoEveryone in the world is loved by at least one personrdquo

Quantifier duality each can be expressed using the other

forallx Likes(xIceCream) notexistx notLikes(xIceCream )

existx Likes(xBroccoli) notforallx notLikes(xBroccoli

)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 20

983109983153983157983137983148983145983156983161

term1 = term2 is true under a given interpretation if andonly if term1 and term2 refer to the same object

Eg definition of Sibling in terms of Parent

forallxy Sibling(xy) hArr [not(x = y) and existmf not (m = f) and Parent(mx)and Parent(fx) and Parent(my) and Parent(fy)]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 21

983125983155983145983150983143 983110983119983116

The kinship domain Brothers are siblings

forallxy Brother(xy) hArr Sibling(xy)

Ones mother is ones female parent

forallmc Mother(c) = m hArr (Female(m) and Parent(mc))

ldquoSiblingrdquo is symmetric

forallxy Sibling(xy) hArr Sibling(yx)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 22

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

Representing the following sentences in Predicate Logic

1 Marcus was a man

2 Macus was a Pompeian

3 All Pompians were Romans4 Caesar was a ruler

5 All Romans were either loyal to Caesar or hated hime

6 Everyone is loyal to someone

7 People only try to assassinate rulers they are not loyal to

8 Marcus tried to assassinate Caesar

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 23

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

1 Marcus was a man

man(Marcus)

2 Macus was a Pompeian

Pompeian(Marcus)

3 All Pompeians were Romans

forallX Pompeian(X) Roman(X)

4 Caesar was a ruler

ruler(Caesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 24

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

5 All Romans were either loyal to Caesar or hated hime

ldquoorrdquo mean ldquoinclusive orrdquo - OR

forallX Roman(X) loyalto(X Caesar) v hate(X Caesar)

ldquoorrdquo mean ldquoexclusive orrdquo - XOR

forallX Roman(X) [ (loyalto(X Caesar) v hate(X Caesar)) ^

not(loyalto(X Caesar) ^ hate(X Caesar))]

or

forallX Roman(X) [ (loyalto(X Caesar) ^ nothate(X Caesar)) v

(notloyalto(X Caesar) ^ hate(X Caesar))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 25

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

6 Everyone is loyal to someone

forallX exist Y loyalto(XY)

7 People only try to assassinate rulers they are not loyal toforallX forall Y person(X) ^ ruler(Y) ^ tryassassinate(XY)notloyalto(XY)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 26

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

8 Marcus tried to assassinate Caesartryassassinate(Marcus Caesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 27

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

Given the above sentences can we make aconclusion as follows

ldquoMarcus was not loyal to Caesar rdquo

or

notloyalto(MarcusCaesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 28

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)

(Rule What)

Add an implicit sentence

9 ldquoAll men are peoplerdquo

or

forallforallforallforallX man(X) X man(X) person(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 29

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)(9 Substitute)

man(Marcus)

(1)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 30

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

How to represent exceptions example

ldquoPaulus was a Pompeian Paulus was neitherloyal nor hated Caesarrdquo

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 31

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation Semantic Net Simple by adding links

(properties) to subclasses or instances

Predicate Logic Not so simple like

983120983151983149983152983141983145983137983150(983120983137983157983148983157983155) 983134 983212983131983148983151983161983137983148983156983151(983120983137983157983148983157983155 983107983137983141983155983137983154) 983158

983144983137983156983141(983120983157983137983148983157983155 983107983137983141983155983137983154)983133

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 32

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation983118983151983156 983155983151 983155983145983149983152983148983141 983106983141983139983137983157983155983141 983156983144983141983154983141 983145983155 983139983151983150983142983148983145983139983156 983138983141983156983159983141983141983150983098

Pompeian(Paulus) ^ not[loyalto(Paulus Caesar) vhate(Pualus Caesar)]

983137983150983140

forallX Pompeian(X) loyalto(X Caesar) v hate(XCaesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 33

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation A Better way

forallX Roman(X) ^ noteq(X Paulus) loyalto(X Caesar)v hate(X Caesar

Exception

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 34

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

In the case of describing relations betweennumbers example 1 983100 2

2 983100983091

983095 983102 983091 +2 983091 983102 1

991270

rArr

983123983144983151983157983148983140 983150983151983156 983141983160983152983148983145983139983145983156983148983161 983140983141983155983139983154983145983138983141983098983148983156(12) 983148983156(2983091) 991270

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 35

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

rArr Need functions and computable predicates

rArr For example 983107983137983148983148 983137 983142983157983150983139983156983145983151983150 983156983151 983139983151983149983152983157983156983141 (983091+2)

983120983137983155983155 983156983144983141 983154983141983155983157983148983156 983156983151 983156983144983141 983152983154983141983140983145983139983137983156983141 991260983143983156991261983098 983143983156(983095 983091+2)

983111983141983156 983156983144983141 983142983145983150983137983148 983154983141983155983157983148983156 (983124983154983157983141)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 36

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

1 Marcus was a man

man(Marcus)

2 Marcus was a PompeianPompeian(Marcus)

3 Marcus was born in 40 AD

born(Marcus40)

4 All men are mortal

forallX man(X) mortal(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 37

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

5 All Pompeian died when the vocano erupted in 79 AD

erupted(vocano 79) ^ forallX [Pompeian(X) died(X 79)]

6 No mortal lives longer then 150 yearsforallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

7 It is now 1991now = 1991

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 38

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Question

Is Marcus alive

Or

alive(Marcus now)

ornotnotnotnotalive(Marcus now)

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 3980

Artificial Intelligence Predicate Logic Slide 39

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge

Relation between ldquodeadrdquo and ldquoaliverdquo

8 Alive means not deadforallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4080

Artificial Intelligence Predicate Logic Slide 40

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge Time properties of an event for example ldquodeadrdquo

9 Is someone dies the he is dead at all later times

forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4180

Artificial Intelligence Predicate Logic Slide 41

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Summary of facts1 man(Marcus)

2 Pompeian(Marcus)

3 born(Marcus 40)

4 forallX man(X) mortal(X)

5 forallX Pompeian(X) died(X 79)

6 erupted(volcano 79)

7 forallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

8 now = 19919 forallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

10 forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4280

Artificial Intelligence Predicate Logic Slide 42

notalive(Marcus now)

dead(Marcus now)

(9 Substitute)

died(Marcus T1) gt(now T1)

(10 Substitute)

Pompeian(Marcus) gt(now 79)

(5 Substitute)(5 Substitute)

(2)gt(1991 79)

(8 Substitute)

(computation)

ss

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4380

Artificial Intelligence Predicate Logic Slide 43

983125983150983145983158983141983154983155983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983125983113)

Every instantiation of a universally quantified sentence is enta iled by it

forallv α

Subst(vg α)

for any variable v and ground term g

Eg forallx King( x) and Greedy( x)rArr Evil( x

) yieldsKing (John ) and Greedy (John ) rArr Evil (John )

King (Richard ) and Greedy (Richard ) rArr Evil (Richard )

King (Father (John )) and Greedy (Father (John )) rArr Evil (Father (John ))

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4480

Artificial Intelligence Predicate Logic Slide 44

983109983160983145983155983156983141983150983156983145983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983109983113)

For any sentence α variable v and constant symbol k that

does not

appear elsewhere in the knowledge baseexistv α

Subst(vk α )

Eg exist x Crown( x) and OnHead ( xJohn) yields

Crown(C 1) and OnHead (C 1 John )

provided C 1 is a new constant symbol called a Skolemconstant

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4580

Artificial Intelligence Predicate Logic Slide 45

983122983141983140983157983139983156983145983151983150 983156983151 983152983154983151983152983151983155983145983156983145983151983150983137983148 983145983150983142983141983154983141983150983139983141

Suppose the KB contains just the followingforall

x King(x)and

Greedy(x)rArr

Evil(x)King(John)Greedy(John)Brother(RichardJohn)

Instantiating the universal sentence in all possible ways we haveKing(John) and Greedy(John) rArr Evil(John)King(Richard) and Greedy(Richard) rArr Evil(Richard)King(John)Greedy(John)Brother(RichardJohn

)

The new KB is propositionalized proposition symbols are

King(John) Greedy(John) Evil(John) King(Richard

) etc

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4680

Artificial Intelligence Predicate Logic Slide 46

983122983141983140983157983139983156983145983151983150 983139983151983150983156983140983086

Every FOL KB can be propositionalized so as to preserve entailment

(A ground sentence is entailed by new KB iff entailed by originalKB)

Idea propositionalize KB and query apply resolution return result

Problem with function symbols there are infinitely many ground

terms

eg Father (Father (Father (John

)))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 47

983122983141983155983151983148983157983156983145983151983150

Convert to clause form exampleldquoAll Romans who know Marcus either hate

Caesar or think that anyone who hates anyone is

crazyrdquo

forallX [roman(X) ^ know(X Marcus)]

[hate(X Ceasar) v(forallY existZ hate(YZ)

thinkcrazy(XY))]

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4880

Artificial Intelligence Predicate Logic Slide 48

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 49

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

forallX [roman(X) ^ know(X Marcus)][hate(X Ceasar) v

(forallY existZ hate(YZ) thinkcrazy(XY))]

=

forallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 50

983122983141983155983151983148983157983156983145983151983150

2 Move not next to terms

Using the following equivalence

a Double negative

not(notp) = p

b DeMorgan

not(a v b) = nota ^ notb

not(a ^ b) = nota v notb

c Equivalence of qualifiers notforallX P(X) = existX notP(X)

notexistX P(X) = forallX notP(X)

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Artificial Intelligence Predicate Logic Slide 51

983122983141983155983151983148983157983156983145983151983150

2 Move not next to termsforallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

=

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) vthinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 52

983122983141983155983151983148983157983156983145983151983150

3 Change name of variables as follows

forallX P(X) v forallX Q(X)

=forallX P(X) v forallY Q(Y)

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Artificial Intelligence Predicate Logic Slide 53

983122983141983155983151983148983157983156983145983151983150

4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 54

983122983141983155983151983148983157983156983145983151983150

5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

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Artificial Intelligence Predicate Logic Slide 55

983122983141983155983151983148983157983156983145983151983150

6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 56

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

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Artificial Intelligence Predicate Logic Slide 57

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

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Artificial Intelligence Predicate Logic Slide 58

983122983141983155983151983148983157983156983145983151983150

8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

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Artificial Intelligence Predicate Logic Slide 59

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9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

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Artificial Intelligence Predicate Logic Slide 60

983122983141983155983151983148983157983156983145983151983150

Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

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Artificial Intelligence Predicate Logic Slide 61

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

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Artificial Intelligence Predicate Logic Slide 62

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θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

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Artificial Intelligence Predicate Logic Slide 63

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

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Artificial Intelligence Predicate Logic Slide 64

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 65

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Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

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Artificial Intelligence Predicate Logic Slide 66

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

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Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

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Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

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Artificial Intelligence Predicate Logic Slide 69

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

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Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

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Artificial Intelligence Predicate Logic Slide 71

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Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

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Artificial Intelligence Predicate Logic Slide 72

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Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

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Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

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Artificial Intelligence Predicate Logic Slide 74

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Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

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Artificial Intelligence Predicate Logic Slide 75

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Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

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Artificial Intelligence Predicate Logic Slide 76

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Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

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Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

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Artificial Intelligence Predicate Logic Slide 6

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Whereas propositional logic assumes the worldcontains facts

first-order logic (like natural language) assumes

the world containsObjects people houses numbers colors baseballgames wars hellip

Relations red round prime brother of bigger

than part of comes between hellipFunctions father of best friend one more than

plus hellip

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Artificial Intelligence Predicate Logic Slide 7

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MORTALMANAll men are mortal

PLATOMANPlato is a man

SOCRATESMANSocrates is a manSentence in Prepositional LogicSentence in natural language

Disadvantages1 Socrates and Plato are human but the representation is not similar

2 Can not state that each person is mortal Difficult for reasoning

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Artificial Intelligence Predicate Logic Slide 8

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forallX (man(X) mortal(X))All men are mortal

man(plato)Plato is a man

man(socrates)Socrates is a man

Sentence in Predicate LogicSentence in natural language

Using predicate logic

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Artificial Intelligence Predicate Logic Slide 9

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Ability of reasoning

All men are mortal Socrates is mortal

Socrates is a man

ConclusionSentence in natural language

forallX (man(X) mortal(X))mortal(socrates)

man(socrates)

ConclusionSentence in Predicate Logic

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Artificial Intelligence Predicate Logic Slide 10

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13 MPmortal(socrates)4

man(socrates)mortal(socrates)

forallX (man(X) mortal(X))

man(socrates)

Representation

2 X= ldquosocratesrdquo UI3

Axiom2

Axiom1

Rule

Socrates is mortal

Backward mapping (to natural language)

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Artificial Intelligence Predicate Logic Slide 11

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Constants KingJohn 2 NUS

Predicates Brother gt

Functions

Sqrt LeftLegOf Variables

x y a b

Connectives not rArr and or hArr

Equality =

Quantifiers Universal Quantifier forall

Existential Quantifier exist

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Artificial Intelligence Predicate Logic Slide 12

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Term function (term1termn)or constant or variable

Atomic sentence predicate (term1termn)

or term1 = term

2

Eg

Brother(KingJohnRichardTheLionheart) gt (Length(LeftLegOf(Richard)) Length(LeftLegOf(KingJohn)))

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Artificial Intelligence Predicate Logic Slide 13

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Complex sentences are made from atomic sentences usingconnectives examples notS

S1 and S2

S1 or S2

S1rArr S2

S1 hArr S2

Eg Sibling(KingJohnRichard) rArr Sibling(RichardKingJohn)

gt(12) or le (12)

gt(12) and not gt(12)

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Artificial Intelligence Predicate Logic Slide 14

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Sentences are true with respect to a model and an interpretation

Model contains objects (domain elements ) and relations among them

Interpretation specifies referents forconstant symbols rarr objects

predicate symbols rarr relations

function symbols rarr functional relations

An atomic sentence predicate(term1 termn) is true

iff the objects referred to by term1 termn

are in the relation referred to by predicate

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Artificial Intelligence Predicate Logic Slide 15

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forallltvariablesgt ltsentence gt

Everyone at NUS is smart

forallx At(xNUS)rArr Smart(x )

forallx P is true in a model m iff P is true with x being each possible object

in the model Roughly speaking equivalent to the conjunction of instantiations of

P

983105983156983080983115983145983150983143983114983151983144983150983084983118983125983123983081rArr 983123983149983137983154983156983080983115983145983150983143983114983151983144983150983081

and 983105983156983080983122983145983139983144983137983154983140983084983118983125983123983081rArr 983123983149983137983154983156983080983122983145983139983144983137983154983140983081

and 983105983156983080983118983125983123983084983118983125983123983081rArr 983123983149983137983154983156983080983118983125983123983081and 983086983086983086

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Artificial Intelligence Predicate Logic Slide 16

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TypicallyrArr is the main connective with forall

Common mistake using and as the main connective with forall

forallx At(xNUS) and Smart(x)

means ldquoEveryone is at NUS and everyone is smartrdquo

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Artificial Intelligence Predicate Logic Slide 17

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existltvariablesgt ltsentencegt

Someone at NUS is smart

exist x At(xNUS) and Smart(x )$

exist x P is true in a model m iff P is true with x being some possible object

in the model Roughly speaking equivalent to the disjunction of instantiations of P

At(KingJohnNUS) and Smart(KingJohn)or At(RichardNUS) and Smart(Richard)or At(NUSNUS) and Smart(NUS)or

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Artificial Intelligence Predicate Logic Slide 18

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Typically and is the main connective with exist

Common mistake usingrArr as the main connective with

exist

exist x At(xNUS)rArr Smart(x

)is true if there is anyone who is not at NUS

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 19

983120983154983151983152983141983154983156983145983141983155 983151983142 983153983157983137983150983156983145983142983145983141983154983155

forallx forally is the same as forally forallx

existx existy is the same as existy existx

existx forally is not the same as forally existx

existx forally Loves(xy) ldquoThere is a person who loves everyone in the worldrdquo

forally existx Loves(xy) ldquoEveryone in the world is loved by at least one personrdquo

Quantifier duality each can be expressed using the other

forallx Likes(xIceCream) notexistx notLikes(xIceCream )

existx Likes(xBroccoli) notforallx notLikes(xBroccoli

)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 20

983109983153983157983137983148983145983156983161

term1 = term2 is true under a given interpretation if andonly if term1 and term2 refer to the same object

Eg definition of Sibling in terms of Parent

forallxy Sibling(xy) hArr [not(x = y) and existmf not (m = f) and Parent(mx)and Parent(fx) and Parent(my) and Parent(fy)]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 21

983125983155983145983150983143 983110983119983116

The kinship domain Brothers are siblings

forallxy Brother(xy) hArr Sibling(xy)

Ones mother is ones female parent

forallmc Mother(c) = m hArr (Female(m) and Parent(mc))

ldquoSiblingrdquo is symmetric

forallxy Sibling(xy) hArr Sibling(yx)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 22

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

Representing the following sentences in Predicate Logic

1 Marcus was a man

2 Macus was a Pompeian

3 All Pompians were Romans4 Caesar was a ruler

5 All Romans were either loyal to Caesar or hated hime

6 Everyone is loyal to someone

7 People only try to assassinate rulers they are not loyal to

8 Marcus tried to assassinate Caesar

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 23

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

1 Marcus was a man

man(Marcus)

2 Macus was a Pompeian

Pompeian(Marcus)

3 All Pompeians were Romans

forallX Pompeian(X) Roman(X)

4 Caesar was a ruler

ruler(Caesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 24

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

5 All Romans were either loyal to Caesar or hated hime

ldquoorrdquo mean ldquoinclusive orrdquo - OR

forallX Roman(X) loyalto(X Caesar) v hate(X Caesar)

ldquoorrdquo mean ldquoexclusive orrdquo - XOR

forallX Roman(X) [ (loyalto(X Caesar) v hate(X Caesar)) ^

not(loyalto(X Caesar) ^ hate(X Caesar))]

or

forallX Roman(X) [ (loyalto(X Caesar) ^ nothate(X Caesar)) v

(notloyalto(X Caesar) ^ hate(X Caesar))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 25

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

6 Everyone is loyal to someone

forallX exist Y loyalto(XY)

7 People only try to assassinate rulers they are not loyal toforallX forall Y person(X) ^ ruler(Y) ^ tryassassinate(XY)notloyalto(XY)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 26

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

8 Marcus tried to assassinate Caesartryassassinate(Marcus Caesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 27

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

Given the above sentences can we make aconclusion as follows

ldquoMarcus was not loyal to Caesar rdquo

or

notloyalto(MarcusCaesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 28

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)

(Rule What)

Add an implicit sentence

9 ldquoAll men are peoplerdquo

or

forallforallforallforallX man(X) X man(X) person(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 29

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)(9 Substitute)

man(Marcus)

(1)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 30

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

How to represent exceptions example

ldquoPaulus was a Pompeian Paulus was neitherloyal nor hated Caesarrdquo

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 31

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation Semantic Net Simple by adding links

(properties) to subclasses or instances

Predicate Logic Not so simple like

983120983151983149983152983141983145983137983150(983120983137983157983148983157983155) 983134 983212983131983148983151983161983137983148983156983151(983120983137983157983148983157983155 983107983137983141983155983137983154) 983158

983144983137983156983141(983120983157983137983148983157983155 983107983137983141983155983137983154)983133

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 32

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation983118983151983156 983155983151 983155983145983149983152983148983141 983106983141983139983137983157983155983141 983156983144983141983154983141 983145983155 983139983151983150983142983148983145983139983156 983138983141983156983159983141983141983150983098

Pompeian(Paulus) ^ not[loyalto(Paulus Caesar) vhate(Pualus Caesar)]

983137983150983140

forallX Pompeian(X) loyalto(X Caesar) v hate(XCaesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 33

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation A Better way

forallX Roman(X) ^ noteq(X Paulus) loyalto(X Caesar)v hate(X Caesar

Exception

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Artificial Intelligence Predicate Logic Slide 34

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

In the case of describing relations betweennumbers example 1 983100 2

2 983100983091

983095 983102 983091 +2 983091 983102 1

991270

rArr

983123983144983151983157983148983140 983150983151983156 983141983160983152983148983145983139983145983156983148983161 983140983141983155983139983154983145983138983141983098983148983156(12) 983148983156(2983091) 991270

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 35

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

rArr Need functions and computable predicates

rArr For example 983107983137983148983148 983137 983142983157983150983139983156983145983151983150 983156983151 983139983151983149983152983157983156983141 (983091+2)

983120983137983155983155 983156983144983141 983154983141983155983157983148983156 983156983151 983156983144983141 983152983154983141983140983145983139983137983156983141 991260983143983156991261983098 983143983156(983095 983091+2)

983111983141983156 983156983144983141 983142983145983150983137983148 983154983141983155983157983148983156 (983124983154983157983141)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 36

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

1 Marcus was a man

man(Marcus)

2 Marcus was a PompeianPompeian(Marcus)

3 Marcus was born in 40 AD

born(Marcus40)

4 All men are mortal

forallX man(X) mortal(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 37

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

5 All Pompeian died when the vocano erupted in 79 AD

erupted(vocano 79) ^ forallX [Pompeian(X) died(X 79)]

6 No mortal lives longer then 150 yearsforallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

7 It is now 1991now = 1991

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 38

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Question

Is Marcus alive

Or

alive(Marcus now)

ornotnotnotnotalive(Marcus now)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 39

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge

Relation between ldquodeadrdquo and ldquoaliverdquo

8 Alive means not deadforallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 40

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge Time properties of an event for example ldquodeadrdquo

9 Is someone dies the he is dead at all later times

forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 41

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Summary of facts1 man(Marcus)

2 Pompeian(Marcus)

3 born(Marcus 40)

4 forallX man(X) mortal(X)

5 forallX Pompeian(X) died(X 79)

6 erupted(volcano 79)

7 forallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

8 now = 19919 forallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

10 forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4280

Artificial Intelligence Predicate Logic Slide 42

notalive(Marcus now)

dead(Marcus now)

(9 Substitute)

died(Marcus T1) gt(now T1)

(10 Substitute)

Pompeian(Marcus) gt(now 79)

(5 Substitute)(5 Substitute)

(2)gt(1991 79)

(8 Substitute)

(computation)

ss

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4380

Artificial Intelligence Predicate Logic Slide 43

983125983150983145983158983141983154983155983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983125983113)

Every instantiation of a universally quantified sentence is enta iled by it

forallv α

Subst(vg α)

for any variable v and ground term g

Eg forallx King( x) and Greedy( x)rArr Evil( x

) yieldsKing (John ) and Greedy (John ) rArr Evil (John )

King (Richard ) and Greedy (Richard ) rArr Evil (Richard )

King (Father (John )) and Greedy (Father (John )) rArr Evil (Father (John ))

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4480

Artificial Intelligence Predicate Logic Slide 44

983109983160983145983155983156983141983150983156983145983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983109983113)

For any sentence α variable v and constant symbol k that

does not

appear elsewhere in the knowledge baseexistv α

Subst(vk α )

Eg exist x Crown( x) and OnHead ( xJohn) yields

Crown(C 1) and OnHead (C 1 John )

provided C 1 is a new constant symbol called a Skolemconstant

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4580

Artificial Intelligence Predicate Logic Slide 45

983122983141983140983157983139983156983145983151983150 983156983151 983152983154983151983152983151983155983145983156983145983151983150983137983148 983145983150983142983141983154983141983150983139983141

Suppose the KB contains just the followingforall

x King(x)and

Greedy(x)rArr

Evil(x)King(John)Greedy(John)Brother(RichardJohn)

Instantiating the universal sentence in all possible ways we haveKing(John) and Greedy(John) rArr Evil(John)King(Richard) and Greedy(Richard) rArr Evil(Richard)King(John)Greedy(John)Brother(RichardJohn

)

The new KB is propositionalized proposition symbols are

King(John) Greedy(John) Evil(John) King(Richard

) etc

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4680

Artificial Intelligence Predicate Logic Slide 46

983122983141983140983157983139983156983145983151983150 983139983151983150983156983140983086

Every FOL KB can be propositionalized so as to preserve entailment

(A ground sentence is entailed by new KB iff entailed by originalKB)

Idea propositionalize KB and query apply resolution return result

Problem with function symbols there are infinitely many ground

terms

eg Father (Father (Father (John

)))

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4780

Artificial Intelligence Predicate Logic Slide 47

983122983141983155983151983148983157983156983145983151983150

Convert to clause form exampleldquoAll Romans who know Marcus either hate

Caesar or think that anyone who hates anyone is

crazyrdquo

forallX [roman(X) ^ know(X Marcus)]

[hate(X Ceasar) v(forallY existZ hate(YZ)

thinkcrazy(XY))]

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4880

Artificial Intelligence Predicate Logic Slide 48

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 49

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

forallX [roman(X) ^ know(X Marcus)][hate(X Ceasar) v

(forallY existZ hate(YZ) thinkcrazy(XY))]

=

forallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 50

983122983141983155983151983148983157983156983145983151983150

2 Move not next to terms

Using the following equivalence

a Double negative

not(notp) = p

b DeMorgan

not(a v b) = nota ^ notb

not(a ^ b) = nota v notb

c Equivalence of qualifiers notforallX P(X) = existX notP(X)

notexistX P(X) = forallX notP(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 51

983122983141983155983151983148983157983156983145983151983150

2 Move not next to termsforallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

=

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) vthinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 52

983122983141983155983151983148983157983156983145983151983150

3 Change name of variables as follows

forallX P(X) v forallX Q(X)

=forallX P(X) v forallY Q(Y)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 53

983122983141983155983151983148983157983156983145983151983150

4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 54

983122983141983155983151983148983157983156983145983151983150

5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 55

983122983141983155983151983148983157983156983145983151983150

6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 56

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 57

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 58

983122983141983155983151983148983157983156983145983151983150

8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 59

983122983141983155983151983148983157983156983145983151983150

9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

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Artificial Intelligence Predicate Logic Slide 60

983122983141983155983151983148983157983156983145983151983150

Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 61

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X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

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Artificial Intelligence Predicate Logic Slide 62

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θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 63

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 64

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 65

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 66

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 69

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 71

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 72

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

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Artificial Intelligence Predicate Logic Slide 74

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Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

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Artificial Intelligence Predicate Logic Slide 75

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

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Artificial Intelligence Predicate Logic Slide 76

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Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

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892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

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Artificial Intelligence Predicate Logic Slide 7

983120983154983141983152983151983155983145983156983145983151983150983137983148 983116983151983143983145983139983098 983109983160983137983149983152983148983141983155

MORTALMANAll men are mortal

PLATOMANPlato is a man

SOCRATESMANSocrates is a manSentence in Prepositional LogicSentence in natural language

Disadvantages1 Socrates and Plato are human but the representation is not similar

2 Can not state that each person is mortal Difficult for reasoning

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Artificial Intelligence Predicate Logic Slide 8

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forallX (man(X) mortal(X))All men are mortal

man(plato)Plato is a man

man(socrates)Socrates is a man

Sentence in Predicate LogicSentence in natural language

Using predicate logic

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 9

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Ability of reasoning

All men are mortal Socrates is mortal

Socrates is a man

ConclusionSentence in natural language

forallX (man(X) mortal(X))mortal(socrates)

man(socrates)

ConclusionSentence in Predicate Logic

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Artificial Intelligence Predicate Logic Slide 10

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13 MPmortal(socrates)4

man(socrates)mortal(socrates)

forallX (man(X) mortal(X))

man(socrates)

Representation

2 X= ldquosocratesrdquo UI3

Axiom2

Axiom1

Rule

Socrates is mortal

Backward mapping (to natural language)

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Artificial Intelligence Predicate Logic Slide 11

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Constants KingJohn 2 NUS

Predicates Brother gt

Functions

Sqrt LeftLegOf Variables

x y a b

Connectives not rArr and or hArr

Equality =

Quantifiers Universal Quantifier forall

Existential Quantifier exist

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Artificial Intelligence Predicate Logic Slide 12

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Term function (term1termn)or constant or variable

Atomic sentence predicate (term1termn)

or term1 = term

2

Eg

Brother(KingJohnRichardTheLionheart) gt (Length(LeftLegOf(Richard)) Length(LeftLegOf(KingJohn)))

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Artificial Intelligence Predicate Logic Slide 13

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Complex sentences are made from atomic sentences usingconnectives examples notS

S1 and S2

S1 or S2

S1rArr S2

S1 hArr S2

Eg Sibling(KingJohnRichard) rArr Sibling(RichardKingJohn)

gt(12) or le (12)

gt(12) and not gt(12)

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Artificial Intelligence Predicate Logic Slide 14

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Sentences are true with respect to a model and an interpretation

Model contains objects (domain elements ) and relations among them

Interpretation specifies referents forconstant symbols rarr objects

predicate symbols rarr relations

function symbols rarr functional relations

An atomic sentence predicate(term1 termn) is true

iff the objects referred to by term1 termn

are in the relation referred to by predicate

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Artificial Intelligence Predicate Logic Slide 15

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forallltvariablesgt ltsentence gt

Everyone at NUS is smart

forallx At(xNUS)rArr Smart(x )

forallx P is true in a model m iff P is true with x being each possible object

in the model Roughly speaking equivalent to the conjunction of instantiations of

P

983105983156983080983115983145983150983143983114983151983144983150983084983118983125983123983081rArr 983123983149983137983154983156983080983115983145983150983143983114983151983144983150983081

and 983105983156983080983122983145983139983144983137983154983140983084983118983125983123983081rArr 983123983149983137983154983156983080983122983145983139983144983137983154983140983081

and 983105983156983080983118983125983123983084983118983125983123983081rArr 983123983149983137983154983156983080983118983125983123983081and 983086983086983086

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Artificial Intelligence Predicate Logic Slide 16

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TypicallyrArr is the main connective with forall

Common mistake using and as the main connective with forall

forallx At(xNUS) and Smart(x)

means ldquoEveryone is at NUS and everyone is smartrdquo

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Artificial Intelligence Predicate Logic Slide 17

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existltvariablesgt ltsentencegt

Someone at NUS is smart

exist x At(xNUS) and Smart(x )$

exist x P is true in a model m iff P is true with x being some possible object

in the model Roughly speaking equivalent to the disjunction of instantiations of P

At(KingJohnNUS) and Smart(KingJohn)or At(RichardNUS) and Smart(Richard)or At(NUSNUS) and Smart(NUS)or

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Artificial Intelligence Predicate Logic Slide 18

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Typically and is the main connective with exist

Common mistake usingrArr as the main connective with

exist

exist x At(xNUS)rArr Smart(x

)is true if there is anyone who is not at NUS

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Artificial Intelligence Predicate Logic Slide 19

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forallx forally is the same as forally forallx

existx existy is the same as existy existx

existx forally is not the same as forally existx

existx forally Loves(xy) ldquoThere is a person who loves everyone in the worldrdquo

forally existx Loves(xy) ldquoEveryone in the world is loved by at least one personrdquo

Quantifier duality each can be expressed using the other

forallx Likes(xIceCream) notexistx notLikes(xIceCream )

existx Likes(xBroccoli) notforallx notLikes(xBroccoli

)

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Artificial Intelligence Predicate Logic Slide 20

983109983153983157983137983148983145983156983161

term1 = term2 is true under a given interpretation if andonly if term1 and term2 refer to the same object

Eg definition of Sibling in terms of Parent

forallxy Sibling(xy) hArr [not(x = y) and existmf not (m = f) and Parent(mx)and Parent(fx) and Parent(my) and Parent(fy)]

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Artificial Intelligence Predicate Logic Slide 21

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The kinship domain Brothers are siblings

forallxy Brother(xy) hArr Sibling(xy)

Ones mother is ones female parent

forallmc Mother(c) = m hArr (Female(m) and Parent(mc))

ldquoSiblingrdquo is symmetric

forallxy Sibling(xy) hArr Sibling(yx)

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Artificial Intelligence Predicate Logic Slide 22

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Representing the following sentences in Predicate Logic

1 Marcus was a man

2 Macus was a Pompeian

3 All Pompians were Romans4 Caesar was a ruler

5 All Romans were either loyal to Caesar or hated hime

6 Everyone is loyal to someone

7 People only try to assassinate rulers they are not loyal to

8 Marcus tried to assassinate Caesar

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Artificial Intelligence Predicate Logic Slide 23

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1 Marcus was a man

man(Marcus)

2 Macus was a Pompeian

Pompeian(Marcus)

3 All Pompeians were Romans

forallX Pompeian(X) Roman(X)

4 Caesar was a ruler

ruler(Caesar)

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Artificial Intelligence Predicate Logic Slide 24

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5 All Romans were either loyal to Caesar or hated hime

ldquoorrdquo mean ldquoinclusive orrdquo - OR

forallX Roman(X) loyalto(X Caesar) v hate(X Caesar)

ldquoorrdquo mean ldquoexclusive orrdquo - XOR

forallX Roman(X) [ (loyalto(X Caesar) v hate(X Caesar)) ^

not(loyalto(X Caesar) ^ hate(X Caesar))]

or

forallX Roman(X) [ (loyalto(X Caesar) ^ nothate(X Caesar)) v

(notloyalto(X Caesar) ^ hate(X Caesar))]

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httpslidepdfcomreaderfullfirst-orderpredicatelogic 2580

Artificial Intelligence Predicate Logic Slide 25

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

6 Everyone is loyal to someone

forallX exist Y loyalto(XY)

7 People only try to assassinate rulers they are not loyal toforallX forall Y person(X) ^ ruler(Y) ^ tryassassinate(XY)notloyalto(XY)

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Artificial Intelligence Predicate Logic Slide 26

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

8 Marcus tried to assassinate Caesartryassassinate(Marcus Caesar)

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Artificial Intelligence Predicate Logic Slide 27

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

Given the above sentences can we make aconclusion as follows

ldquoMarcus was not loyal to Caesar rdquo

or

notloyalto(MarcusCaesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 28

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)

(Rule What)

Add an implicit sentence

9 ldquoAll men are peoplerdquo

or

forallforallforallforallX man(X) X man(X) person(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 29

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)(9 Substitute)

man(Marcus)

(1)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 30

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

How to represent exceptions example

ldquoPaulus was a Pompeian Paulus was neitherloyal nor hated Caesarrdquo

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Artificial Intelligence Predicate Logic Slide 31

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation Semantic Net Simple by adding links

(properties) to subclasses or instances

Predicate Logic Not so simple like

983120983151983149983152983141983145983137983150(983120983137983157983148983157983155) 983134 983212983131983148983151983161983137983148983156983151(983120983137983157983148983157983155 983107983137983141983155983137983154) 983158

983144983137983156983141(983120983157983137983148983157983155 983107983137983141983155983137983154)983133

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 32

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation983118983151983156 983155983151 983155983145983149983152983148983141 983106983141983139983137983157983155983141 983156983144983141983154983141 983145983155 983139983151983150983142983148983145983139983156 983138983141983156983159983141983141983150983098

Pompeian(Paulus) ^ not[loyalto(Paulus Caesar) vhate(Pualus Caesar)]

983137983150983140

forallX Pompeian(X) loyalto(X Caesar) v hate(XCaesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 33

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation A Better way

forallX Roman(X) ^ noteq(X Paulus) loyalto(X Caesar)v hate(X Caesar

Exception

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 34

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

In the case of describing relations betweennumbers example 1 983100 2

2 983100983091

983095 983102 983091 +2 983091 983102 1

991270

rArr

983123983144983151983157983148983140 983150983151983156 983141983160983152983148983145983139983145983156983148983161 983140983141983155983139983154983145983138983141983098983148983156(12) 983148983156(2983091) 991270

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 35

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

rArr Need functions and computable predicates

rArr For example 983107983137983148983148 983137 983142983157983150983139983156983145983151983150 983156983151 983139983151983149983152983157983156983141 (983091+2)

983120983137983155983155 983156983144983141 983154983141983155983157983148983156 983156983151 983156983144983141 983152983154983141983140983145983139983137983156983141 991260983143983156991261983098 983143983156(983095 983091+2)

983111983141983156 983156983144983141 983142983145983150983137983148 983154983141983155983157983148983156 (983124983154983157983141)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 36

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

1 Marcus was a man

man(Marcus)

2 Marcus was a PompeianPompeian(Marcus)

3 Marcus was born in 40 AD

born(Marcus40)

4 All men are mortal

forallX man(X) mortal(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 37

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

5 All Pompeian died when the vocano erupted in 79 AD

erupted(vocano 79) ^ forallX [Pompeian(X) died(X 79)]

6 No mortal lives longer then 150 yearsforallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

7 It is now 1991now = 1991

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Artificial Intelligence Predicate Logic Slide 38

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Question

Is Marcus alive

Or

alive(Marcus now)

ornotnotnotnotalive(Marcus now)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 39

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge

Relation between ldquodeadrdquo and ldquoaliverdquo

8 Alive means not deadforallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

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Artificial Intelligence Predicate Logic Slide 40

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge Time properties of an event for example ldquodeadrdquo

9 Is someone dies the he is dead at all later times

forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

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Artificial Intelligence Predicate Logic Slide 41

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Summary of facts1 man(Marcus)

2 Pompeian(Marcus)

3 born(Marcus 40)

4 forallX man(X) mortal(X)

5 forallX Pompeian(X) died(X 79)

6 erupted(volcano 79)

7 forallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

8 now = 19919 forallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

10 forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 42

notalive(Marcus now)

dead(Marcus now)

(9 Substitute)

died(Marcus T1) gt(now T1)

(10 Substitute)

Pompeian(Marcus) gt(now 79)

(5 Substitute)(5 Substitute)

(2)gt(1991 79)

(8 Substitute)

(computation)

ss

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 43

983125983150983145983158983141983154983155983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983125983113)

Every instantiation of a universally quantified sentence is enta iled by it

forallv α

Subst(vg α)

for any variable v and ground term g

Eg forallx King( x) and Greedy( x)rArr Evil( x

) yieldsKing (John ) and Greedy (John ) rArr Evil (John )

King (Richard ) and Greedy (Richard ) rArr Evil (Richard )

King (Father (John )) and Greedy (Father (John )) rArr Evil (Father (John ))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 44

983109983160983145983155983156983141983150983156983145983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983109983113)

For any sentence α variable v and constant symbol k that

does not

appear elsewhere in the knowledge baseexistv α

Subst(vk α )

Eg exist x Crown( x) and OnHead ( xJohn) yields

Crown(C 1) and OnHead (C 1 John )

provided C 1 is a new constant symbol called a Skolemconstant

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 45

983122983141983140983157983139983156983145983151983150 983156983151 983152983154983151983152983151983155983145983156983145983151983150983137983148 983145983150983142983141983154983141983150983139983141

Suppose the KB contains just the followingforall

x King(x)and

Greedy(x)rArr

Evil(x)King(John)Greedy(John)Brother(RichardJohn)

Instantiating the universal sentence in all possible ways we haveKing(John) and Greedy(John) rArr Evil(John)King(Richard) and Greedy(Richard) rArr Evil(Richard)King(John)Greedy(John)Brother(RichardJohn

)

The new KB is propositionalized proposition symbols are

King(John) Greedy(John) Evil(John) King(Richard

) etc

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4680

Artificial Intelligence Predicate Logic Slide 46

983122983141983140983157983139983156983145983151983150 983139983151983150983156983140983086

Every FOL KB can be propositionalized so as to preserve entailment

(A ground sentence is entailed by new KB iff entailed by originalKB)

Idea propositionalize KB and query apply resolution return result

Problem with function symbols there are infinitely many ground

terms

eg Father (Father (Father (John

)))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 47

983122983141983155983151983148983157983156983145983151983150

Convert to clause form exampleldquoAll Romans who know Marcus either hate

Caesar or think that anyone who hates anyone is

crazyrdquo

forallX [roman(X) ^ know(X Marcus)]

[hate(X Ceasar) v(forallY existZ hate(YZ)

thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 48

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 49

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

forallX [roman(X) ^ know(X Marcus)][hate(X Ceasar) v

(forallY existZ hate(YZ) thinkcrazy(XY))]

=

forallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 50

983122983141983155983151983148983157983156983145983151983150

2 Move not next to terms

Using the following equivalence

a Double negative

not(notp) = p

b DeMorgan

not(a v b) = nota ^ notb

not(a ^ b) = nota v notb

c Equivalence of qualifiers notforallX P(X) = existX notP(X)

notexistX P(X) = forallX notP(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 51

983122983141983155983151983148983157983156983145983151983150

2 Move not next to termsforallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

=

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) vthinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 52

983122983141983155983151983148983157983156983145983151983150

3 Change name of variables as follows

forallX P(X) v forallX Q(X)

=forallX P(X) v forallY Q(Y)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 53

983122983141983155983151983148983157983156983145983151983150

4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 54

983122983141983155983151983148983157983156983145983151983150

5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 55

983122983141983155983151983148983157983156983145983151983150

6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 56

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 57

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 58

983122983141983155983151983148983157983156983145983151983150

8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 59

983122983141983155983151983148983157983156983145983151983150

9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 60

983122983141983155983151983148983157983156983145983151983150

Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 61

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 62

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 63

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 64

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 65

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 66

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 69

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 71

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 72

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 74

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 75

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 76

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

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892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 8

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983137983149983152983148983141983155

forallX (man(X) mortal(X))All men are mortal

man(plato)Plato is a man

man(socrates)Socrates is a man

Sentence in Predicate LogicSentence in natural language

Using predicate logic

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Artificial Intelligence Predicate Logic Slide 9

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983137983149983152983148983141983155

Ability of reasoning

All men are mortal Socrates is mortal

Socrates is a man

ConclusionSentence in natural language

forallX (man(X) mortal(X))mortal(socrates)

man(socrates)

ConclusionSentence in Predicate Logic

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 10

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983137983149983152983148983141983155

13 MPmortal(socrates)4

man(socrates)mortal(socrates)

forallX (man(X) mortal(X))

man(socrates)

Representation

2 X= ldquosocratesrdquo UI3

Axiom2

Axiom1

Rule

Socrates is mortal

Backward mapping (to natural language)

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Artificial Intelligence Predicate Logic Slide 11

983123983161983150983156983137983160 983151983142 983110983119983116983098 983106983137983155983145983139 983141983148983141983149983141983150983156983155

Constants KingJohn 2 NUS

Predicates Brother gt

Functions

Sqrt LeftLegOf Variables

x y a b

Connectives not rArr and or hArr

Equality =

Quantifiers Universal Quantifier forall

Existential Quantifier exist

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 12

983105983156983151983149983145983139 983155983141983150983156983141983150983139983141983155

Term function (term1termn)or constant or variable

Atomic sentence predicate (term1termn)

or term1 = term

2

Eg

Brother(KingJohnRichardTheLionheart) gt (Length(LeftLegOf(Richard)) Length(LeftLegOf(KingJohn)))

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Artificial Intelligence Predicate Logic Slide 13

983107983151983149983152983148983141983160 983155983141983150983156983141983150983139983141983155

Complex sentences are made from atomic sentences usingconnectives examples notS

S1 and S2

S1 or S2

S1rArr S2

S1 hArr S2

Eg Sibling(KingJohnRichard) rArr Sibling(RichardKingJohn)

gt(12) or le (12)

gt(12) and not gt(12)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 14

983124983154983157983156983144 983145983150 983142983145983154983155983156983085983151983154983140983141983154 983148983151983143983145983139

Sentences are true with respect to a model and an interpretation

Model contains objects (domain elements ) and relations among them

Interpretation specifies referents forconstant symbols rarr objects

predicate symbols rarr relations

function symbols rarr functional relations

An atomic sentence predicate(term1 termn) is true

iff the objects referred to by term1 termn

are in the relation referred to by predicate

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 15

983125983150983145983158983141983154983155983137983148 983153983157983137983150983156983145983142983145983139983137983156983145983151983150

forallltvariablesgt ltsentence gt

Everyone at NUS is smart

forallx At(xNUS)rArr Smart(x )

forallx P is true in a model m iff P is true with x being each possible object

in the model Roughly speaking equivalent to the conjunction of instantiations of

P

983105983156983080983115983145983150983143983114983151983144983150983084983118983125983123983081rArr 983123983149983137983154983156983080983115983145983150983143983114983151983144983150983081

and 983105983156983080983122983145983139983144983137983154983140983084983118983125983123983081rArr 983123983149983137983154983156983080983122983145983139983144983137983154983140983081

and 983105983156983080983118983125983123983084983118983125983123983081rArr 983123983149983137983154983156983080983118983125983123983081and 983086983086983086

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Artificial Intelligence Predicate Logic Slide 16

983105 983139983151983149983149983151983150 983149983145983155983156983137983147983141 983156983151 983137983158983151983145983140

TypicallyrArr is the main connective with forall

Common mistake using and as the main connective with forall

forallx At(xNUS) and Smart(x)

means ldquoEveryone is at NUS and everyone is smartrdquo

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 17

983109983160983145983155983156983141983150983156983145983137983148 983153983157983137983150983156983145983142983145983139983137983156983145983151983150

existltvariablesgt ltsentencegt

Someone at NUS is smart

exist x At(xNUS) and Smart(x )$

exist x P is true in a model m iff P is true with x being some possible object

in the model Roughly speaking equivalent to the disjunction of instantiations of P

At(KingJohnNUS) and Smart(KingJohn)or At(RichardNUS) and Smart(Richard)or At(NUSNUS) and Smart(NUS)or

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 18

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Typically and is the main connective with exist

Common mistake usingrArr as the main connective with

exist

exist x At(xNUS)rArr Smart(x

)is true if there is anyone who is not at NUS

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 19

983120983154983151983152983141983154983156983145983141983155 983151983142 983153983157983137983150983156983145983142983145983141983154983155

forallx forally is the same as forally forallx

existx existy is the same as existy existx

existx forally is not the same as forally existx

existx forally Loves(xy) ldquoThere is a person who loves everyone in the worldrdquo

forally existx Loves(xy) ldquoEveryone in the world is loved by at least one personrdquo

Quantifier duality each can be expressed using the other

forallx Likes(xIceCream) notexistx notLikes(xIceCream )

existx Likes(xBroccoli) notforallx notLikes(xBroccoli

)

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Artificial Intelligence Predicate Logic Slide 20

983109983153983157983137983148983145983156983161

term1 = term2 is true under a given interpretation if andonly if term1 and term2 refer to the same object

Eg definition of Sibling in terms of Parent

forallxy Sibling(xy) hArr [not(x = y) and existmf not (m = f) and Parent(mx)and Parent(fx) and Parent(my) and Parent(fy)]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 21

983125983155983145983150983143 983110983119983116

The kinship domain Brothers are siblings

forallxy Brother(xy) hArr Sibling(xy)

Ones mother is ones female parent

forallmc Mother(c) = m hArr (Female(m) and Parent(mc))

ldquoSiblingrdquo is symmetric

forallxy Sibling(xy) hArr Sibling(yx)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 22

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

Representing the following sentences in Predicate Logic

1 Marcus was a man

2 Macus was a Pompeian

3 All Pompians were Romans4 Caesar was a ruler

5 All Romans were either loyal to Caesar or hated hime

6 Everyone is loyal to someone

7 People only try to assassinate rulers they are not loyal to

8 Marcus tried to assassinate Caesar

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 23

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

1 Marcus was a man

man(Marcus)

2 Macus was a Pompeian

Pompeian(Marcus)

3 All Pompeians were Romans

forallX Pompeian(X) Roman(X)

4 Caesar was a ruler

ruler(Caesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 24

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

5 All Romans were either loyal to Caesar or hated hime

ldquoorrdquo mean ldquoinclusive orrdquo - OR

forallX Roman(X) loyalto(X Caesar) v hate(X Caesar)

ldquoorrdquo mean ldquoexclusive orrdquo - XOR

forallX Roman(X) [ (loyalto(X Caesar) v hate(X Caesar)) ^

not(loyalto(X Caesar) ^ hate(X Caesar))]

or

forallX Roman(X) [ (loyalto(X Caesar) ^ nothate(X Caesar)) v

(notloyalto(X Caesar) ^ hate(X Caesar))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 25

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

6 Everyone is loyal to someone

forallX exist Y loyalto(XY)

7 People only try to assassinate rulers they are not loyal toforallX forall Y person(X) ^ ruler(Y) ^ tryassassinate(XY)notloyalto(XY)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 26

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

8 Marcus tried to assassinate Caesartryassassinate(Marcus Caesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 27

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

Given the above sentences can we make aconclusion as follows

ldquoMarcus was not loyal to Caesar rdquo

or

notloyalto(MarcusCaesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 28

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)

(Rule What)

Add an implicit sentence

9 ldquoAll men are peoplerdquo

or

forallforallforallforallX man(X) X man(X) person(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 29

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)(9 Substitute)

man(Marcus)

(1)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 30

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

How to represent exceptions example

ldquoPaulus was a Pompeian Paulus was neitherloyal nor hated Caesarrdquo

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Artificial Intelligence Predicate Logic Slide 31

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation Semantic Net Simple by adding links

(properties) to subclasses or instances

Predicate Logic Not so simple like

983120983151983149983152983141983145983137983150(983120983137983157983148983157983155) 983134 983212983131983148983151983161983137983148983156983151(983120983137983157983148983157983155 983107983137983141983155983137983154) 983158

983144983137983156983141(983120983157983137983148983157983155 983107983137983141983155983137983154)983133

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 32

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation983118983151983156 983155983151 983155983145983149983152983148983141 983106983141983139983137983157983155983141 983156983144983141983154983141 983145983155 983139983151983150983142983148983145983139983156 983138983141983156983159983141983141983150983098

Pompeian(Paulus) ^ not[loyalto(Paulus Caesar) vhate(Pualus Caesar)]

983137983150983140

forallX Pompeian(X) loyalto(X Caesar) v hate(XCaesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 33

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation A Better way

forallX Roman(X) ^ noteq(X Paulus) loyalto(X Caesar)v hate(X Caesar

Exception

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 34

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

In the case of describing relations betweennumbers example 1 983100 2

2 983100983091

983095 983102 983091 +2 983091 983102 1

991270

rArr

983123983144983151983157983148983140 983150983151983156 983141983160983152983148983145983139983145983156983148983161 983140983141983155983139983154983145983138983141983098983148983156(12) 983148983156(2983091) 991270

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 35

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

rArr Need functions and computable predicates

rArr For example 983107983137983148983148 983137 983142983157983150983139983156983145983151983150 983156983151 983139983151983149983152983157983156983141 (983091+2)

983120983137983155983155 983156983144983141 983154983141983155983157983148983156 983156983151 983156983144983141 983152983154983141983140983145983139983137983156983141 991260983143983156991261983098 983143983156(983095 983091+2)

983111983141983156 983156983144983141 983142983145983150983137983148 983154983141983155983157983148983156 (983124983154983157983141)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 36

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

1 Marcus was a man

man(Marcus)

2 Marcus was a PompeianPompeian(Marcus)

3 Marcus was born in 40 AD

born(Marcus40)

4 All men are mortal

forallX man(X) mortal(X)

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Artificial Intelligence Predicate Logic Slide 37

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

5 All Pompeian died when the vocano erupted in 79 AD

erupted(vocano 79) ^ forallX [Pompeian(X) died(X 79)]

6 No mortal lives longer then 150 yearsforallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

7 It is now 1991now = 1991

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Artificial Intelligence Predicate Logic Slide 38

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Question

Is Marcus alive

Or

alive(Marcus now)

ornotnotnotnotalive(Marcus now)

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Artificial Intelligence Predicate Logic Slide 39

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge

Relation between ldquodeadrdquo and ldquoaliverdquo

8 Alive means not deadforallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

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Artificial Intelligence Predicate Logic Slide 40

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge Time properties of an event for example ldquodeadrdquo

9 Is someone dies the he is dead at all later times

forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

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Artificial Intelligence Predicate Logic Slide 41

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Summary of facts1 man(Marcus)

2 Pompeian(Marcus)

3 born(Marcus 40)

4 forallX man(X) mortal(X)

5 forallX Pompeian(X) died(X 79)

6 erupted(volcano 79)

7 forallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

8 now = 19919 forallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

10 forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

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Artificial Intelligence Predicate Logic Slide 42

notalive(Marcus now)

dead(Marcus now)

(9 Substitute)

died(Marcus T1) gt(now T1)

(10 Substitute)

Pompeian(Marcus) gt(now 79)

(5 Substitute)(5 Substitute)

(2)gt(1991 79)

(8 Substitute)

(computation)

ss

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 43

983125983150983145983158983141983154983155983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983125983113)

Every instantiation of a universally quantified sentence is enta iled by it

forallv α

Subst(vg α)

for any variable v and ground term g

Eg forallx King( x) and Greedy( x)rArr Evil( x

) yieldsKing (John ) and Greedy (John ) rArr Evil (John )

King (Richard ) and Greedy (Richard ) rArr Evil (Richard )

King (Father (John )) and Greedy (Father (John )) rArr Evil (Father (John ))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 44

983109983160983145983155983156983141983150983156983145983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983109983113)

For any sentence α variable v and constant symbol k that

does not

appear elsewhere in the knowledge baseexistv α

Subst(vk α )

Eg exist x Crown( x) and OnHead ( xJohn) yields

Crown(C 1) and OnHead (C 1 John )

provided C 1 is a new constant symbol called a Skolemconstant

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 45

983122983141983140983157983139983156983145983151983150 983156983151 983152983154983151983152983151983155983145983156983145983151983150983137983148 983145983150983142983141983154983141983150983139983141

Suppose the KB contains just the followingforall

x King(x)and

Greedy(x)rArr

Evil(x)King(John)Greedy(John)Brother(RichardJohn)

Instantiating the universal sentence in all possible ways we haveKing(John) and Greedy(John) rArr Evil(John)King(Richard) and Greedy(Richard) rArr Evil(Richard)King(John)Greedy(John)Brother(RichardJohn

)

The new KB is propositionalized proposition symbols are

King(John) Greedy(John) Evil(John) King(Richard

) etc

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 46

983122983141983140983157983139983156983145983151983150 983139983151983150983156983140983086

Every FOL KB can be propositionalized so as to preserve entailment

(A ground sentence is entailed by new KB iff entailed by originalKB)

Idea propositionalize KB and query apply resolution return result

Problem with function symbols there are infinitely many ground

terms

eg Father (Father (Father (John

)))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 47

983122983141983155983151983148983157983156983145983151983150

Convert to clause form exampleldquoAll Romans who know Marcus either hate

Caesar or think that anyone who hates anyone is

crazyrdquo

forallX [roman(X) ^ know(X Marcus)]

[hate(X Ceasar) v(forallY existZ hate(YZ)

thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 48

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 49

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

forallX [roman(X) ^ know(X Marcus)][hate(X Ceasar) v

(forallY existZ hate(YZ) thinkcrazy(XY))]

=

forallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 50

983122983141983155983151983148983157983156983145983151983150

2 Move not next to terms

Using the following equivalence

a Double negative

not(notp) = p

b DeMorgan

not(a v b) = nota ^ notb

not(a ^ b) = nota v notb

c Equivalence of qualifiers notforallX P(X) = existX notP(X)

notexistX P(X) = forallX notP(X)

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Artificial Intelligence Predicate Logic Slide 51

983122983141983155983151983148983157983156983145983151983150

2 Move not next to termsforallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

=

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) vthinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 52

983122983141983155983151983148983157983156983145983151983150

3 Change name of variables as follows

forallX P(X) v forallX Q(X)

=forallX P(X) v forallY Q(Y)

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Artificial Intelligence Predicate Logic Slide 53

983122983141983155983151983148983157983156983145983151983150

4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 54

983122983141983155983151983148983157983156983145983151983150

5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

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Artificial Intelligence Predicate Logic Slide 55

983122983141983155983151983148983157983156983145983151983150

6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 56

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

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Artificial Intelligence Predicate Logic Slide 57

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

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Artificial Intelligence Predicate Logic Slide 58

983122983141983155983151983148983157983156983145983151983150

8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

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Artificial Intelligence Predicate Logic Slide 59

983122983141983155983151983148983157983156983145983151983150

9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

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Artificial Intelligence Predicate Logic Slide 60

983122983141983155983151983148983157983156983145983151983150

Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

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Artificial Intelligence Predicate Logic Slide 61

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

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Artificial Intelligence Predicate Logic Slide 62

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

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Artificial Intelligence Predicate Logic Slide 63

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

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Artificial Intelligence Predicate Logic Slide 64

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 65

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 66

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

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Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

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Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

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Artificial Intelligence Predicate Logic Slide 69

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

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Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

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Artificial Intelligence Predicate Logic Slide 71

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

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Artificial Intelligence Predicate Logic Slide 72

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 74

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

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Artificial Intelligence Predicate Logic Slide 75

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 76

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

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892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

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Artificial Intelligence Predicate Logic Slide 9

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983137983149983152983148983141983155

Ability of reasoning

All men are mortal Socrates is mortal

Socrates is a man

ConclusionSentence in natural language

forallX (man(X) mortal(X))mortal(socrates)

man(socrates)

ConclusionSentence in Predicate Logic

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Artificial Intelligence Predicate Logic Slide 10

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983137983149983152983148983141983155

13 MPmortal(socrates)4

man(socrates)mortal(socrates)

forallX (man(X) mortal(X))

man(socrates)

Representation

2 X= ldquosocratesrdquo UI3

Axiom2

Axiom1

Rule

Socrates is mortal

Backward mapping (to natural language)

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Artificial Intelligence Predicate Logic Slide 11

983123983161983150983156983137983160 983151983142 983110983119983116983098 983106983137983155983145983139 983141983148983141983149983141983150983156983155

Constants KingJohn 2 NUS

Predicates Brother gt

Functions

Sqrt LeftLegOf Variables

x y a b

Connectives not rArr and or hArr

Equality =

Quantifiers Universal Quantifier forall

Existential Quantifier exist

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Artificial Intelligence Predicate Logic Slide 12

983105983156983151983149983145983139 983155983141983150983156983141983150983139983141983155

Term function (term1termn)or constant or variable

Atomic sentence predicate (term1termn)

or term1 = term

2

Eg

Brother(KingJohnRichardTheLionheart) gt (Length(LeftLegOf(Richard)) Length(LeftLegOf(KingJohn)))

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Artificial Intelligence Predicate Logic Slide 13

983107983151983149983152983148983141983160 983155983141983150983156983141983150983139983141983155

Complex sentences are made from atomic sentences usingconnectives examples notS

S1 and S2

S1 or S2

S1rArr S2

S1 hArr S2

Eg Sibling(KingJohnRichard) rArr Sibling(RichardKingJohn)

gt(12) or le (12)

gt(12) and not gt(12)

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Artificial Intelligence Predicate Logic Slide 14

983124983154983157983156983144 983145983150 983142983145983154983155983156983085983151983154983140983141983154 983148983151983143983145983139

Sentences are true with respect to a model and an interpretation

Model contains objects (domain elements ) and relations among them

Interpretation specifies referents forconstant symbols rarr objects

predicate symbols rarr relations

function symbols rarr functional relations

An atomic sentence predicate(term1 termn) is true

iff the objects referred to by term1 termn

are in the relation referred to by predicate

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Artificial Intelligence Predicate Logic Slide 15

983125983150983145983158983141983154983155983137983148 983153983157983137983150983156983145983142983145983139983137983156983145983151983150

forallltvariablesgt ltsentence gt

Everyone at NUS is smart

forallx At(xNUS)rArr Smart(x )

forallx P is true in a model m iff P is true with x being each possible object

in the model Roughly speaking equivalent to the conjunction of instantiations of

P

983105983156983080983115983145983150983143983114983151983144983150983084983118983125983123983081rArr 983123983149983137983154983156983080983115983145983150983143983114983151983144983150983081

and 983105983156983080983122983145983139983144983137983154983140983084983118983125983123983081rArr 983123983149983137983154983156983080983122983145983139983144983137983154983140983081

and 983105983156983080983118983125983123983084983118983125983123983081rArr 983123983149983137983154983156983080983118983125983123983081and 983086983086983086

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Artificial Intelligence Predicate Logic Slide 16

983105 983139983151983149983149983151983150 983149983145983155983156983137983147983141 983156983151 983137983158983151983145983140

TypicallyrArr is the main connective with forall

Common mistake using and as the main connective with forall

forallx At(xNUS) and Smart(x)

means ldquoEveryone is at NUS and everyone is smartrdquo

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Artificial Intelligence Predicate Logic Slide 17

983109983160983145983155983156983141983150983156983145983137983148 983153983157983137983150983156983145983142983145983139983137983156983145983151983150

existltvariablesgt ltsentencegt

Someone at NUS is smart

exist x At(xNUS) and Smart(x )$

exist x P is true in a model m iff P is true with x being some possible object

in the model Roughly speaking equivalent to the disjunction of instantiations of P

At(KingJohnNUS) and Smart(KingJohn)or At(RichardNUS) and Smart(Richard)or At(NUSNUS) and Smart(NUS)or

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Artificial Intelligence Predicate Logic Slide 18

983105983150983151983156983144983141983154 983139983151983149983149983151983150 983149983145983155983156983137983147983141 983156983151 983137983158983151983145983140

Typically and is the main connective with exist

Common mistake usingrArr as the main connective with

exist

exist x At(xNUS)rArr Smart(x

)is true if there is anyone who is not at NUS

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Artificial Intelligence Predicate Logic Slide 19

983120983154983151983152983141983154983156983145983141983155 983151983142 983153983157983137983150983156983145983142983145983141983154983155

forallx forally is the same as forally forallx

existx existy is the same as existy existx

existx forally is not the same as forally existx

existx forally Loves(xy) ldquoThere is a person who loves everyone in the worldrdquo

forally existx Loves(xy) ldquoEveryone in the world is loved by at least one personrdquo

Quantifier duality each can be expressed using the other

forallx Likes(xIceCream) notexistx notLikes(xIceCream )

existx Likes(xBroccoli) notforallx notLikes(xBroccoli

)

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Artificial Intelligence Predicate Logic Slide 20

983109983153983157983137983148983145983156983161

term1 = term2 is true under a given interpretation if andonly if term1 and term2 refer to the same object

Eg definition of Sibling in terms of Parent

forallxy Sibling(xy) hArr [not(x = y) and existmf not (m = f) and Parent(mx)and Parent(fx) and Parent(my) and Parent(fy)]

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Artificial Intelligence Predicate Logic Slide 21

983125983155983145983150983143 983110983119983116

The kinship domain Brothers are siblings

forallxy Brother(xy) hArr Sibling(xy)

Ones mother is ones female parent

forallmc Mother(c) = m hArr (Female(m) and Parent(mc))

ldquoSiblingrdquo is symmetric

forallxy Sibling(xy) hArr Sibling(yx)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 22

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

Representing the following sentences in Predicate Logic

1 Marcus was a man

2 Macus was a Pompeian

3 All Pompians were Romans4 Caesar was a ruler

5 All Romans were either loyal to Caesar or hated hime

6 Everyone is loyal to someone

7 People only try to assassinate rulers they are not loyal to

8 Marcus tried to assassinate Caesar

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Artificial Intelligence Predicate Logic Slide 23

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

1 Marcus was a man

man(Marcus)

2 Macus was a Pompeian

Pompeian(Marcus)

3 All Pompeians were Romans

forallX Pompeian(X) Roman(X)

4 Caesar was a ruler

ruler(Caesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 24

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

5 All Romans were either loyal to Caesar or hated hime

ldquoorrdquo mean ldquoinclusive orrdquo - OR

forallX Roman(X) loyalto(X Caesar) v hate(X Caesar)

ldquoorrdquo mean ldquoexclusive orrdquo - XOR

forallX Roman(X) [ (loyalto(X Caesar) v hate(X Caesar)) ^

not(loyalto(X Caesar) ^ hate(X Caesar))]

or

forallX Roman(X) [ (loyalto(X Caesar) ^ nothate(X Caesar)) v

(notloyalto(X Caesar) ^ hate(X Caesar))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 25

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

6 Everyone is loyal to someone

forallX exist Y loyalto(XY)

7 People only try to assassinate rulers they are not loyal toforallX forall Y person(X) ^ ruler(Y) ^ tryassassinate(XY)notloyalto(XY)

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Artificial Intelligence Predicate Logic Slide 26

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

8 Marcus tried to assassinate Caesartryassassinate(Marcus Caesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 27

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

Given the above sentences can we make aconclusion as follows

ldquoMarcus was not loyal to Caesar rdquo

or

notloyalto(MarcusCaesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 28

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)

(Rule What)

Add an implicit sentence

9 ldquoAll men are peoplerdquo

or

forallforallforallforallX man(X) X man(X) person(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 29

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)(9 Substitute)

man(Marcus)

(1)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 30

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

How to represent exceptions example

ldquoPaulus was a Pompeian Paulus was neitherloyal nor hated Caesarrdquo

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Artificial Intelligence Predicate Logic Slide 31

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation Semantic Net Simple by adding links

(properties) to subclasses or instances

Predicate Logic Not so simple like

983120983151983149983152983141983145983137983150(983120983137983157983148983157983155) 983134 983212983131983148983151983161983137983148983156983151(983120983137983157983148983157983155 983107983137983141983155983137983154) 983158

983144983137983156983141(983120983157983137983148983157983155 983107983137983141983155983137983154)983133

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 32

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation983118983151983156 983155983151 983155983145983149983152983148983141 983106983141983139983137983157983155983141 983156983144983141983154983141 983145983155 983139983151983150983142983148983145983139983156 983138983141983156983159983141983141983150983098

Pompeian(Paulus) ^ not[loyalto(Paulus Caesar) vhate(Pualus Caesar)]

983137983150983140

forallX Pompeian(X) loyalto(X Caesar) v hate(XCaesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 33

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation A Better way

forallX Roman(X) ^ noteq(X Paulus) loyalto(X Caesar)v hate(X Caesar

Exception

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 34

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

In the case of describing relations betweennumbers example 1 983100 2

2 983100983091

983095 983102 983091 +2 983091 983102 1

991270

rArr

983123983144983151983157983148983140 983150983151983156 983141983160983152983148983145983139983145983156983148983161 983140983141983155983139983154983145983138983141983098983148983156(12) 983148983156(2983091) 991270

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 35

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

rArr Need functions and computable predicates

rArr For example 983107983137983148983148 983137 983142983157983150983139983156983145983151983150 983156983151 983139983151983149983152983157983156983141 (983091+2)

983120983137983155983155 983156983144983141 983154983141983155983157983148983156 983156983151 983156983144983141 983152983154983141983140983145983139983137983156983141 991260983143983156991261983098 983143983156(983095 983091+2)

983111983141983156 983156983144983141 983142983145983150983137983148 983154983141983155983157983148983156 (983124983154983157983141)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 36

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

1 Marcus was a man

man(Marcus)

2 Marcus was a PompeianPompeian(Marcus)

3 Marcus was born in 40 AD

born(Marcus40)

4 All men are mortal

forallX man(X) mortal(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 37

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

5 All Pompeian died when the vocano erupted in 79 AD

erupted(vocano 79) ^ forallX [Pompeian(X) died(X 79)]

6 No mortal lives longer then 150 yearsforallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

7 It is now 1991now = 1991

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Artificial Intelligence Predicate Logic Slide 38

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Question

Is Marcus alive

Or

alive(Marcus now)

ornotnotnotnotalive(Marcus now)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 39

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Add implicit knowledge

Relation between ldquodeadrdquo and ldquoaliverdquo

8 Alive means not deadforallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

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Artificial Intelligence Predicate Logic Slide 40

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Add implicit knowledge Time properties of an event for example ldquodeadrdquo

9 Is someone dies the he is dead at all later times

forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

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Artificial Intelligence Predicate Logic Slide 41

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Summary of facts1 man(Marcus)

2 Pompeian(Marcus)

3 born(Marcus 40)

4 forallX man(X) mortal(X)

5 forallX Pompeian(X) died(X 79)

6 erupted(volcano 79)

7 forallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

8 now = 19919 forallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

10 forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

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Artificial Intelligence Predicate Logic Slide 42

notalive(Marcus now)

dead(Marcus now)

(9 Substitute)

died(Marcus T1) gt(now T1)

(10 Substitute)

Pompeian(Marcus) gt(now 79)

(5 Substitute)(5 Substitute)

(2)gt(1991 79)

(8 Substitute)

(computation)

ss

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Artificial Intelligence Predicate Logic Slide 43

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Every instantiation of a universally quantified sentence is enta iled by it

forallv α

Subst(vg α)

for any variable v and ground term g

Eg forallx King( x) and Greedy( x)rArr Evil( x

) yieldsKing (John ) and Greedy (John ) rArr Evil (John )

King (Richard ) and Greedy (Richard ) rArr Evil (Richard )

King (Father (John )) and Greedy (Father (John )) rArr Evil (Father (John ))

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Artificial Intelligence Predicate Logic Slide 44

983109983160983145983155983156983141983150983156983145983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983109983113)

For any sentence α variable v and constant symbol k that

does not

appear elsewhere in the knowledge baseexistv α

Subst(vk α )

Eg exist x Crown( x) and OnHead ( xJohn) yields

Crown(C 1) and OnHead (C 1 John )

provided C 1 is a new constant symbol called a Skolemconstant

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 45

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Suppose the KB contains just the followingforall

x King(x)and

Greedy(x)rArr

Evil(x)King(John)Greedy(John)Brother(RichardJohn)

Instantiating the universal sentence in all possible ways we haveKing(John) and Greedy(John) rArr Evil(John)King(Richard) and Greedy(Richard) rArr Evil(Richard)King(John)Greedy(John)Brother(RichardJohn

)

The new KB is propositionalized proposition symbols are

King(John) Greedy(John) Evil(John) King(Richard

) etc

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 46

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Every FOL KB can be propositionalized so as to preserve entailment

(A ground sentence is entailed by new KB iff entailed by originalKB)

Idea propositionalize KB and query apply resolution return result

Problem with function symbols there are infinitely many ground

terms

eg Father (Father (Father (John

)))

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Artificial Intelligence Predicate Logic Slide 47

983122983141983155983151983148983157983156983145983151983150

Convert to clause form exampleldquoAll Romans who know Marcus either hate

Caesar or think that anyone who hates anyone is

crazyrdquo

forallX [roman(X) ^ know(X Marcus)]

[hate(X Ceasar) v(forallY existZ hate(YZ)

thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 48

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 49

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

forallX [roman(X) ^ know(X Marcus)][hate(X Ceasar) v

(forallY existZ hate(YZ) thinkcrazy(XY))]

=

forallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 50

983122983141983155983151983148983157983156983145983151983150

2 Move not next to terms

Using the following equivalence

a Double negative

not(notp) = p

b DeMorgan

not(a v b) = nota ^ notb

not(a ^ b) = nota v notb

c Equivalence of qualifiers notforallX P(X) = existX notP(X)

notexistX P(X) = forallX notP(X)

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Artificial Intelligence Predicate Logic Slide 51

983122983141983155983151983148983157983156983145983151983150

2 Move not next to termsforallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

=

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) vthinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 52

983122983141983155983151983148983157983156983145983151983150

3 Change name of variables as follows

forallX P(X) v forallX Q(X)

=forallX P(X) v forallY Q(Y)

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Artificial Intelligence Predicate Logic Slide 53

983122983141983155983151983148983157983156983145983151983150

4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 54

983122983141983155983151983148983157983156983145983151983150

5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

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Artificial Intelligence Predicate Logic Slide 55

983122983141983155983151983148983157983156983145983151983150

6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 56

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

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Artificial Intelligence Predicate Logic Slide 57

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

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Artificial Intelligence Predicate Logic Slide 58

983122983141983155983151983148983157983156983145983151983150

8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

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Artificial Intelligence Predicate Logic Slide 59

983122983141983155983151983148983157983156983145983151983150

9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

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Artificial Intelligence Predicate Logic Slide 60

983122983141983155983151983148983157983156983145983151983150

Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

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Artificial Intelligence Predicate Logic Slide 61

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

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Artificial Intelligence Predicate Logic Slide 62

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

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Artificial Intelligence Predicate Logic Slide 63

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

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Artificial Intelligence Predicate Logic Slide 64

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

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Artificial Intelligence Predicate Logic Slide 65

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

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Artificial Intelligence Predicate Logic Slide 66

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

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Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

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Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

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Artificial Intelligence Predicate Logic Slide 69

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

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Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

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Artificial Intelligence Predicate Logic Slide 71

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

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Artificial Intelligence Predicate Logic Slide 72

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

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Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

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Artificial Intelligence Predicate Logic Slide 74

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

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Artificial Intelligence Predicate Logic Slide 75

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

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Artificial Intelligence Predicate Logic Slide 76

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

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892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

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Artificial Intelligence Predicate Logic Slide 10

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983137983149983152983148983141983155

13 MPmortal(socrates)4

man(socrates)mortal(socrates)

forallX (man(X) mortal(X))

man(socrates)

Representation

2 X= ldquosocratesrdquo UI3

Axiom2

Axiom1

Rule

Socrates is mortal

Backward mapping (to natural language)

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Artificial Intelligence Predicate Logic Slide 11

983123983161983150983156983137983160 983151983142 983110983119983116983098 983106983137983155983145983139 983141983148983141983149983141983150983156983155

Constants KingJohn 2 NUS

Predicates Brother gt

Functions

Sqrt LeftLegOf Variables

x y a b

Connectives not rArr and or hArr

Equality =

Quantifiers Universal Quantifier forall

Existential Quantifier exist

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 12

983105983156983151983149983145983139 983155983141983150983156983141983150983139983141983155

Term function (term1termn)or constant or variable

Atomic sentence predicate (term1termn)

or term1 = term

2

Eg

Brother(KingJohnRichardTheLionheart) gt (Length(LeftLegOf(Richard)) Length(LeftLegOf(KingJohn)))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 13

983107983151983149983152983148983141983160 983155983141983150983156983141983150983139983141983155

Complex sentences are made from atomic sentences usingconnectives examples notS

S1 and S2

S1 or S2

S1rArr S2

S1 hArr S2

Eg Sibling(KingJohnRichard) rArr Sibling(RichardKingJohn)

gt(12) or le (12)

gt(12) and not gt(12)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 14

983124983154983157983156983144 983145983150 983142983145983154983155983156983085983151983154983140983141983154 983148983151983143983145983139

Sentences are true with respect to a model and an interpretation

Model contains objects (domain elements ) and relations among them

Interpretation specifies referents forconstant symbols rarr objects

predicate symbols rarr relations

function symbols rarr functional relations

An atomic sentence predicate(term1 termn) is true

iff the objects referred to by term1 termn

are in the relation referred to by predicate

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 15

983125983150983145983158983141983154983155983137983148 983153983157983137983150983156983145983142983145983139983137983156983145983151983150

forallltvariablesgt ltsentence gt

Everyone at NUS is smart

forallx At(xNUS)rArr Smart(x )

forallx P is true in a model m iff P is true with x being each possible object

in the model Roughly speaking equivalent to the conjunction of instantiations of

P

983105983156983080983115983145983150983143983114983151983144983150983084983118983125983123983081rArr 983123983149983137983154983156983080983115983145983150983143983114983151983144983150983081

and 983105983156983080983122983145983139983144983137983154983140983084983118983125983123983081rArr 983123983149983137983154983156983080983122983145983139983144983137983154983140983081

and 983105983156983080983118983125983123983084983118983125983123983081rArr 983123983149983137983154983156983080983118983125983123983081and 983086983086983086

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 16

983105 983139983151983149983149983151983150 983149983145983155983156983137983147983141 983156983151 983137983158983151983145983140

TypicallyrArr is the main connective with forall

Common mistake using and as the main connective with forall

forallx At(xNUS) and Smart(x)

means ldquoEveryone is at NUS and everyone is smartrdquo

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 17

983109983160983145983155983156983141983150983156983145983137983148 983153983157983137983150983156983145983142983145983139983137983156983145983151983150

existltvariablesgt ltsentencegt

Someone at NUS is smart

exist x At(xNUS) and Smart(x )$

exist x P is true in a model m iff P is true with x being some possible object

in the model Roughly speaking equivalent to the disjunction of instantiations of P

At(KingJohnNUS) and Smart(KingJohn)or At(RichardNUS) and Smart(Richard)or At(NUSNUS) and Smart(NUS)or

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 18

983105983150983151983156983144983141983154 983139983151983149983149983151983150 983149983145983155983156983137983147983141 983156983151 983137983158983151983145983140

Typically and is the main connective with exist

Common mistake usingrArr as the main connective with

exist

exist x At(xNUS)rArr Smart(x

)is true if there is anyone who is not at NUS

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 19

983120983154983151983152983141983154983156983145983141983155 983151983142 983153983157983137983150983156983145983142983145983141983154983155

forallx forally is the same as forally forallx

existx existy is the same as existy existx

existx forally is not the same as forally existx

existx forally Loves(xy) ldquoThere is a person who loves everyone in the worldrdquo

forally existx Loves(xy) ldquoEveryone in the world is loved by at least one personrdquo

Quantifier duality each can be expressed using the other

forallx Likes(xIceCream) notexistx notLikes(xIceCream )

existx Likes(xBroccoli) notforallx notLikes(xBroccoli

)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 20

983109983153983157983137983148983145983156983161

term1 = term2 is true under a given interpretation if andonly if term1 and term2 refer to the same object

Eg definition of Sibling in terms of Parent

forallxy Sibling(xy) hArr [not(x = y) and existmf not (m = f) and Parent(mx)and Parent(fx) and Parent(my) and Parent(fy)]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 21

983125983155983145983150983143 983110983119983116

The kinship domain Brothers are siblings

forallxy Brother(xy) hArr Sibling(xy)

Ones mother is ones female parent

forallmc Mother(c) = m hArr (Female(m) and Parent(mc))

ldquoSiblingrdquo is symmetric

forallxy Sibling(xy) hArr Sibling(yx)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 22

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

Representing the following sentences in Predicate Logic

1 Marcus was a man

2 Macus was a Pompeian

3 All Pompians were Romans4 Caesar was a ruler

5 All Romans were either loyal to Caesar or hated hime

6 Everyone is loyal to someone

7 People only try to assassinate rulers they are not loyal to

8 Marcus tried to assassinate Caesar

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 23

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

1 Marcus was a man

man(Marcus)

2 Macus was a Pompeian

Pompeian(Marcus)

3 All Pompeians were Romans

forallX Pompeian(X) Roman(X)

4 Caesar was a ruler

ruler(Caesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 24

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

5 All Romans were either loyal to Caesar or hated hime

ldquoorrdquo mean ldquoinclusive orrdquo - OR

forallX Roman(X) loyalto(X Caesar) v hate(X Caesar)

ldquoorrdquo mean ldquoexclusive orrdquo - XOR

forallX Roman(X) [ (loyalto(X Caesar) v hate(X Caesar)) ^

not(loyalto(X Caesar) ^ hate(X Caesar))]

or

forallX Roman(X) [ (loyalto(X Caesar) ^ nothate(X Caesar)) v

(notloyalto(X Caesar) ^ hate(X Caesar))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 25

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

6 Everyone is loyal to someone

forallX exist Y loyalto(XY)

7 People only try to assassinate rulers they are not loyal toforallX forall Y person(X) ^ ruler(Y) ^ tryassassinate(XY)notloyalto(XY)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 26

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

8 Marcus tried to assassinate Caesartryassassinate(Marcus Caesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 27

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

Given the above sentences can we make aconclusion as follows

ldquoMarcus was not loyal to Caesar rdquo

or

notloyalto(MarcusCaesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 28

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)

(Rule What)

Add an implicit sentence

9 ldquoAll men are peoplerdquo

or

forallforallforallforallX man(X) X man(X) person(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 29

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)(9 Substitute)

man(Marcus)

(1)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 30

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

How to represent exceptions example

ldquoPaulus was a Pompeian Paulus was neitherloyal nor hated Caesarrdquo

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 31

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation Semantic Net Simple by adding links

(properties) to subclasses or instances

Predicate Logic Not so simple like

983120983151983149983152983141983145983137983150(983120983137983157983148983157983155) 983134 983212983131983148983151983161983137983148983156983151(983120983137983157983148983157983155 983107983137983141983155983137983154) 983158

983144983137983156983141(983120983157983137983148983157983155 983107983137983141983155983137983154)983133

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 32

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation983118983151983156 983155983151 983155983145983149983152983148983141 983106983141983139983137983157983155983141 983156983144983141983154983141 983145983155 983139983151983150983142983148983145983139983156 983138983141983156983159983141983141983150983098

Pompeian(Paulus) ^ not[loyalto(Paulus Caesar) vhate(Pualus Caesar)]

983137983150983140

forallX Pompeian(X) loyalto(X Caesar) v hate(XCaesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 33

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation A Better way

forallX Roman(X) ^ noteq(X Paulus) loyalto(X Caesar)v hate(X Caesar

Exception

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Artificial Intelligence Predicate Logic Slide 34

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

In the case of describing relations betweennumbers example 1 983100 2

2 983100983091

983095 983102 983091 +2 983091 983102 1

991270

rArr

983123983144983151983157983148983140 983150983151983156 983141983160983152983148983145983139983145983156983148983161 983140983141983155983139983154983145983138983141983098983148983156(12) 983148983156(2983091) 991270

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 35

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

rArr Need functions and computable predicates

rArr For example 983107983137983148983148 983137 983142983157983150983139983156983145983151983150 983156983151 983139983151983149983152983157983156983141 (983091+2)

983120983137983155983155 983156983144983141 983154983141983155983157983148983156 983156983151 983156983144983141 983152983154983141983140983145983139983137983156983141 991260983143983156991261983098 983143983156(983095 983091+2)

983111983141983156 983156983144983141 983142983145983150983137983148 983154983141983155983157983148983156 (983124983154983157983141)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 36

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

1 Marcus was a man

man(Marcus)

2 Marcus was a PompeianPompeian(Marcus)

3 Marcus was born in 40 AD

born(Marcus40)

4 All men are mortal

forallX man(X) mortal(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 37

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

5 All Pompeian died when the vocano erupted in 79 AD

erupted(vocano 79) ^ forallX [Pompeian(X) died(X 79)]

6 No mortal lives longer then 150 yearsforallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

7 It is now 1991now = 1991

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 38

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Question

Is Marcus alive

Or

alive(Marcus now)

ornotnotnotnotalive(Marcus now)

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 3980

Artificial Intelligence Predicate Logic Slide 39

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge

Relation between ldquodeadrdquo and ldquoaliverdquo

8 Alive means not deadforallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4080

Artificial Intelligence Predicate Logic Slide 40

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge Time properties of an event for example ldquodeadrdquo

9 Is someone dies the he is dead at all later times

forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4180

Artificial Intelligence Predicate Logic Slide 41

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Summary of facts1 man(Marcus)

2 Pompeian(Marcus)

3 born(Marcus 40)

4 forallX man(X) mortal(X)

5 forallX Pompeian(X) died(X 79)

6 erupted(volcano 79)

7 forallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

8 now = 19919 forallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

10 forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4280

Artificial Intelligence Predicate Logic Slide 42

notalive(Marcus now)

dead(Marcus now)

(9 Substitute)

died(Marcus T1) gt(now T1)

(10 Substitute)

Pompeian(Marcus) gt(now 79)

(5 Substitute)(5 Substitute)

(2)gt(1991 79)

(8 Substitute)

(computation)

ss

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4380

Artificial Intelligence Predicate Logic Slide 43

983125983150983145983158983141983154983155983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983125983113)

Every instantiation of a universally quantified sentence is enta iled by it

forallv α

Subst(vg α)

for any variable v and ground term g

Eg forallx King( x) and Greedy( x)rArr Evil( x

) yieldsKing (John ) and Greedy (John ) rArr Evil (John )

King (Richard ) and Greedy (Richard ) rArr Evil (Richard )

King (Father (John )) and Greedy (Father (John )) rArr Evil (Father (John ))

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4480

Artificial Intelligence Predicate Logic Slide 44

983109983160983145983155983156983141983150983156983145983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983109983113)

For any sentence α variable v and constant symbol k that

does not

appear elsewhere in the knowledge baseexistv α

Subst(vk α )

Eg exist x Crown( x) and OnHead ( xJohn) yields

Crown(C 1) and OnHead (C 1 John )

provided C 1 is a new constant symbol called a Skolemconstant

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4580

Artificial Intelligence Predicate Logic Slide 45

983122983141983140983157983139983156983145983151983150 983156983151 983152983154983151983152983151983155983145983156983145983151983150983137983148 983145983150983142983141983154983141983150983139983141

Suppose the KB contains just the followingforall

x King(x)and

Greedy(x)rArr

Evil(x)King(John)Greedy(John)Brother(RichardJohn)

Instantiating the universal sentence in all possible ways we haveKing(John) and Greedy(John) rArr Evil(John)King(Richard) and Greedy(Richard) rArr Evil(Richard)King(John)Greedy(John)Brother(RichardJohn

)

The new KB is propositionalized proposition symbols are

King(John) Greedy(John) Evil(John) King(Richard

) etc

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 46

983122983141983140983157983139983156983145983151983150 983139983151983150983156983140983086

Every FOL KB can be propositionalized so as to preserve entailment

(A ground sentence is entailed by new KB iff entailed by originalKB)

Idea propositionalize KB and query apply resolution return result

Problem with function symbols there are infinitely many ground

terms

eg Father (Father (Father (John

)))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 47

983122983141983155983151983148983157983156983145983151983150

Convert to clause form exampleldquoAll Romans who know Marcus either hate

Caesar or think that anyone who hates anyone is

crazyrdquo

forallX [roman(X) ^ know(X Marcus)]

[hate(X Ceasar) v(forallY existZ hate(YZ)

thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 48

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

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Artificial Intelligence Predicate Logic Slide 49

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

forallX [roman(X) ^ know(X Marcus)][hate(X Ceasar) v

(forallY existZ hate(YZ) thinkcrazy(XY))]

=

forallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 50

983122983141983155983151983148983157983156983145983151983150

2 Move not next to terms

Using the following equivalence

a Double negative

not(notp) = p

b DeMorgan

not(a v b) = nota ^ notb

not(a ^ b) = nota v notb

c Equivalence of qualifiers notforallX P(X) = existX notP(X)

notexistX P(X) = forallX notP(X)

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Artificial Intelligence Predicate Logic Slide 51

983122983141983155983151983148983157983156983145983151983150

2 Move not next to termsforallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

=

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) vthinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 52

983122983141983155983151983148983157983156983145983151983150

3 Change name of variables as follows

forallX P(X) v forallX Q(X)

=forallX P(X) v forallY Q(Y)

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Artificial Intelligence Predicate Logic Slide 53

983122983141983155983151983148983157983156983145983151983150

4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 54

983122983141983155983151983148983157983156983145983151983150

5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

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Artificial Intelligence Predicate Logic Slide 55

983122983141983155983151983148983157983156983145983151983150

6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 56

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

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Artificial Intelligence Predicate Logic Slide 57

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

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Artificial Intelligence Predicate Logic Slide 58

983122983141983155983151983148983157983156983145983151983150

8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

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Artificial Intelligence Predicate Logic Slide 59

983122983141983155983151983148983157983156983145983151983150

9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

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Artificial Intelligence Predicate Logic Slide 60

983122983141983155983151983148983157983156983145983151983150

Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

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Artificial Intelligence Predicate Logic Slide 61

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

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Artificial Intelligence Predicate Logic Slide 62

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

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Artificial Intelligence Predicate Logic Slide 63

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

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Artificial Intelligence Predicate Logic Slide 64

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

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Artificial Intelligence Predicate Logic Slide 65

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

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Artificial Intelligence Predicate Logic Slide 66

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

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Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

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Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

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Artificial Intelligence Predicate Logic Slide 69

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

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Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

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Artificial Intelligence Predicate Logic Slide 71

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

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Artificial Intelligence Predicate Logic Slide 72

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

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Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

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Artificial Intelligence Predicate Logic Slide 74

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

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Artificial Intelligence Predicate Logic Slide 75

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

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Artificial Intelligence Predicate Logic Slide 76

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

892019 First Order(Predicate)Logic

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892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

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Artificial Intelligence Predicate Logic Slide 11

983123983161983150983156983137983160 983151983142 983110983119983116983098 983106983137983155983145983139 983141983148983141983149983141983150983156983155

Constants KingJohn 2 NUS

Predicates Brother gt

Functions

Sqrt LeftLegOf Variables

x y a b

Connectives not rArr and or hArr

Equality =

Quantifiers Universal Quantifier forall

Existential Quantifier exist

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Artificial Intelligence Predicate Logic Slide 12

983105983156983151983149983145983139 983155983141983150983156983141983150983139983141983155

Term function (term1termn)or constant or variable

Atomic sentence predicate (term1termn)

or term1 = term

2

Eg

Brother(KingJohnRichardTheLionheart) gt (Length(LeftLegOf(Richard)) Length(LeftLegOf(KingJohn)))

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Artificial Intelligence Predicate Logic Slide 13

983107983151983149983152983148983141983160 983155983141983150983156983141983150983139983141983155

Complex sentences are made from atomic sentences usingconnectives examples notS

S1 and S2

S1 or S2

S1rArr S2

S1 hArr S2

Eg Sibling(KingJohnRichard) rArr Sibling(RichardKingJohn)

gt(12) or le (12)

gt(12) and not gt(12)

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Artificial Intelligence Predicate Logic Slide 14

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Sentences are true with respect to a model and an interpretation

Model contains objects (domain elements ) and relations among them

Interpretation specifies referents forconstant symbols rarr objects

predicate symbols rarr relations

function symbols rarr functional relations

An atomic sentence predicate(term1 termn) is true

iff the objects referred to by term1 termn

are in the relation referred to by predicate

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Artificial Intelligence Predicate Logic Slide 15

983125983150983145983158983141983154983155983137983148 983153983157983137983150983156983145983142983145983139983137983156983145983151983150

forallltvariablesgt ltsentence gt

Everyone at NUS is smart

forallx At(xNUS)rArr Smart(x )

forallx P is true in a model m iff P is true with x being each possible object

in the model Roughly speaking equivalent to the conjunction of instantiations of

P

983105983156983080983115983145983150983143983114983151983144983150983084983118983125983123983081rArr 983123983149983137983154983156983080983115983145983150983143983114983151983144983150983081

and 983105983156983080983122983145983139983144983137983154983140983084983118983125983123983081rArr 983123983149983137983154983156983080983122983145983139983144983137983154983140983081

and 983105983156983080983118983125983123983084983118983125983123983081rArr 983123983149983137983154983156983080983118983125983123983081and 983086983086983086

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Artificial Intelligence Predicate Logic Slide 16

983105 983139983151983149983149983151983150 983149983145983155983156983137983147983141 983156983151 983137983158983151983145983140

TypicallyrArr is the main connective with forall

Common mistake using and as the main connective with forall

forallx At(xNUS) and Smart(x)

means ldquoEveryone is at NUS and everyone is smartrdquo

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Artificial Intelligence Predicate Logic Slide 17

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existltvariablesgt ltsentencegt

Someone at NUS is smart

exist x At(xNUS) and Smart(x )$

exist x P is true in a model m iff P is true with x being some possible object

in the model Roughly speaking equivalent to the disjunction of instantiations of P

At(KingJohnNUS) and Smart(KingJohn)or At(RichardNUS) and Smart(Richard)or At(NUSNUS) and Smart(NUS)or

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Artificial Intelligence Predicate Logic Slide 18

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Typically and is the main connective with exist

Common mistake usingrArr as the main connective with

exist

exist x At(xNUS)rArr Smart(x

)is true if there is anyone who is not at NUS

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Artificial Intelligence Predicate Logic Slide 19

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forallx forally is the same as forally forallx

existx existy is the same as existy existx

existx forally is not the same as forally existx

existx forally Loves(xy) ldquoThere is a person who loves everyone in the worldrdquo

forally existx Loves(xy) ldquoEveryone in the world is loved by at least one personrdquo

Quantifier duality each can be expressed using the other

forallx Likes(xIceCream) notexistx notLikes(xIceCream )

existx Likes(xBroccoli) notforallx notLikes(xBroccoli

)

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Artificial Intelligence Predicate Logic Slide 20

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term1 = term2 is true under a given interpretation if andonly if term1 and term2 refer to the same object

Eg definition of Sibling in terms of Parent

forallxy Sibling(xy) hArr [not(x = y) and existmf not (m = f) and Parent(mx)and Parent(fx) and Parent(my) and Parent(fy)]

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Artificial Intelligence Predicate Logic Slide 21

983125983155983145983150983143 983110983119983116

The kinship domain Brothers are siblings

forallxy Brother(xy) hArr Sibling(xy)

Ones mother is ones female parent

forallmc Mother(c) = m hArr (Female(m) and Parent(mc))

ldquoSiblingrdquo is symmetric

forallxy Sibling(xy) hArr Sibling(yx)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 22

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

Representing the following sentences in Predicate Logic

1 Marcus was a man

2 Macus was a Pompeian

3 All Pompians were Romans4 Caesar was a ruler

5 All Romans were either loyal to Caesar or hated hime

6 Everyone is loyal to someone

7 People only try to assassinate rulers they are not loyal to

8 Marcus tried to assassinate Caesar

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 23

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

1 Marcus was a man

man(Marcus)

2 Macus was a Pompeian

Pompeian(Marcus)

3 All Pompeians were Romans

forallX Pompeian(X) Roman(X)

4 Caesar was a ruler

ruler(Caesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 24

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

5 All Romans were either loyal to Caesar or hated hime

ldquoorrdquo mean ldquoinclusive orrdquo - OR

forallX Roman(X) loyalto(X Caesar) v hate(X Caesar)

ldquoorrdquo mean ldquoexclusive orrdquo - XOR

forallX Roman(X) [ (loyalto(X Caesar) v hate(X Caesar)) ^

not(loyalto(X Caesar) ^ hate(X Caesar))]

or

forallX Roman(X) [ (loyalto(X Caesar) ^ nothate(X Caesar)) v

(notloyalto(X Caesar) ^ hate(X Caesar))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 25

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

6 Everyone is loyal to someone

forallX exist Y loyalto(XY)

7 People only try to assassinate rulers they are not loyal toforallX forall Y person(X) ^ ruler(Y) ^ tryassassinate(XY)notloyalto(XY)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 26

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

8 Marcus tried to assassinate Caesartryassassinate(Marcus Caesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 27

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

Given the above sentences can we make aconclusion as follows

ldquoMarcus was not loyal to Caesar rdquo

or

notloyalto(MarcusCaesar)

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 2880

Artificial Intelligence Predicate Logic Slide 28

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)

(Rule What)

Add an implicit sentence

9 ldquoAll men are peoplerdquo

or

forallforallforallforallX man(X) X man(X) person(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 29

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)(9 Substitute)

man(Marcus)

(1)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 30

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

How to represent exceptions example

ldquoPaulus was a Pompeian Paulus was neitherloyal nor hated Caesarrdquo

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 31

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation Semantic Net Simple by adding links

(properties) to subclasses or instances

Predicate Logic Not so simple like

983120983151983149983152983141983145983137983150(983120983137983157983148983157983155) 983134 983212983131983148983151983161983137983148983156983151(983120983137983157983148983157983155 983107983137983141983155983137983154) 983158

983144983137983156983141(983120983157983137983148983157983155 983107983137983141983155983137983154)983133

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 32

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation983118983151983156 983155983151 983155983145983149983152983148983141 983106983141983139983137983157983155983141 983156983144983141983154983141 983145983155 983139983151983150983142983148983145983139983156 983138983141983156983159983141983141983150983098

Pompeian(Paulus) ^ not[loyalto(Paulus Caesar) vhate(Pualus Caesar)]

983137983150983140

forallX Pompeian(X) loyalto(X Caesar) v hate(XCaesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 33

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation A Better way

forallX Roman(X) ^ noteq(X Paulus) loyalto(X Caesar)v hate(X Caesar

Exception

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 34

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

In the case of describing relations betweennumbers example 1 983100 2

2 983100983091

983095 983102 983091 +2 983091 983102 1

991270

rArr

983123983144983151983157983148983140 983150983151983156 983141983160983152983148983145983139983145983156983148983161 983140983141983155983139983154983145983138983141983098983148983156(12) 983148983156(2983091) 991270

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 35

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

rArr Need functions and computable predicates

rArr For example 983107983137983148983148 983137 983142983157983150983139983156983145983151983150 983156983151 983139983151983149983152983157983156983141 (983091+2)

983120983137983155983155 983156983144983141 983154983141983155983157983148983156 983156983151 983156983144983141 983152983154983141983140983145983139983137983156983141 991260983143983156991261983098 983143983156(983095 983091+2)

983111983141983156 983156983144983141 983142983145983150983137983148 983154983141983155983157983148983156 (983124983154983157983141)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 36

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

1 Marcus was a man

man(Marcus)

2 Marcus was a PompeianPompeian(Marcus)

3 Marcus was born in 40 AD

born(Marcus40)

4 All men are mortal

forallX man(X) mortal(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 37

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

5 All Pompeian died when the vocano erupted in 79 AD

erupted(vocano 79) ^ forallX [Pompeian(X) died(X 79)]

6 No mortal lives longer then 150 yearsforallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

7 It is now 1991now = 1991

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Artificial Intelligence Predicate Logic Slide 38

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Question

Is Marcus alive

Or

alive(Marcus now)

ornotnotnotnotalive(Marcus now)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 39

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge

Relation between ldquodeadrdquo and ldquoaliverdquo

8 Alive means not deadforallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 40

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge Time properties of an event for example ldquodeadrdquo

9 Is someone dies the he is dead at all later times

forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 41

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Summary of facts1 man(Marcus)

2 Pompeian(Marcus)

3 born(Marcus 40)

4 forallX man(X) mortal(X)

5 forallX Pompeian(X) died(X 79)

6 erupted(volcano 79)

7 forallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

8 now = 19919 forallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

10 forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4280

Artificial Intelligence Predicate Logic Slide 42

notalive(Marcus now)

dead(Marcus now)

(9 Substitute)

died(Marcus T1) gt(now T1)

(10 Substitute)

Pompeian(Marcus) gt(now 79)

(5 Substitute)(5 Substitute)

(2)gt(1991 79)

(8 Substitute)

(computation)

ss

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4380

Artificial Intelligence Predicate Logic Slide 43

983125983150983145983158983141983154983155983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983125983113)

Every instantiation of a universally quantified sentence is enta iled by it

forallv α

Subst(vg α)

for any variable v and ground term g

Eg forallx King( x) and Greedy( x)rArr Evil( x

) yieldsKing (John ) and Greedy (John ) rArr Evil (John )

King (Richard ) and Greedy (Richard ) rArr Evil (Richard )

King (Father (John )) and Greedy (Father (John )) rArr Evil (Father (John ))

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4480

Artificial Intelligence Predicate Logic Slide 44

983109983160983145983155983156983141983150983156983145983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983109983113)

For any sentence α variable v and constant symbol k that

does not

appear elsewhere in the knowledge baseexistv α

Subst(vk α )

Eg exist x Crown( x) and OnHead ( xJohn) yields

Crown(C 1) and OnHead (C 1 John )

provided C 1 is a new constant symbol called a Skolemconstant

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4580

Artificial Intelligence Predicate Logic Slide 45

983122983141983140983157983139983156983145983151983150 983156983151 983152983154983151983152983151983155983145983156983145983151983150983137983148 983145983150983142983141983154983141983150983139983141

Suppose the KB contains just the followingforall

x King(x)and

Greedy(x)rArr

Evil(x)King(John)Greedy(John)Brother(RichardJohn)

Instantiating the universal sentence in all possible ways we haveKing(John) and Greedy(John) rArr Evil(John)King(Richard) and Greedy(Richard) rArr Evil(Richard)King(John)Greedy(John)Brother(RichardJohn

)

The new KB is propositionalized proposition symbols are

King(John) Greedy(John) Evil(John) King(Richard

) etc

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4680

Artificial Intelligence Predicate Logic Slide 46

983122983141983140983157983139983156983145983151983150 983139983151983150983156983140983086

Every FOL KB can be propositionalized so as to preserve entailment

(A ground sentence is entailed by new KB iff entailed by originalKB)

Idea propositionalize KB and query apply resolution return result

Problem with function symbols there are infinitely many ground

terms

eg Father (Father (Father (John

)))

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4780

Artificial Intelligence Predicate Logic Slide 47

983122983141983155983151983148983157983156983145983151983150

Convert to clause form exampleldquoAll Romans who know Marcus either hate

Caesar or think that anyone who hates anyone is

crazyrdquo

forallX [roman(X) ^ know(X Marcus)]

[hate(X Ceasar) v(forallY existZ hate(YZ)

thinkcrazy(XY))]

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4880

Artificial Intelligence Predicate Logic Slide 48

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 49

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

forallX [roman(X) ^ know(X Marcus)][hate(X Ceasar) v

(forallY existZ hate(YZ) thinkcrazy(XY))]

=

forallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 5080

Artificial Intelligence Predicate Logic Slide 50

983122983141983155983151983148983157983156983145983151983150

2 Move not next to terms

Using the following equivalence

a Double negative

not(notp) = p

b DeMorgan

not(a v b) = nota ^ notb

not(a ^ b) = nota v notb

c Equivalence of qualifiers notforallX P(X) = existX notP(X)

notexistX P(X) = forallX notP(X)

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 5180

Artificial Intelligence Predicate Logic Slide 51

983122983141983155983151983148983157983156983145983151983150

2 Move not next to termsforallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

=

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) vthinkcrazy(XY))]

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 5280

Artificial Intelligence Predicate Logic Slide 52

983122983141983155983151983148983157983156983145983151983150

3 Change name of variables as follows

forallX P(X) v forallX Q(X)

=forallX P(X) v forallY Q(Y)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 53

983122983141983155983151983148983157983156983145983151983150

4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 5480

Artificial Intelligence Predicate Logic Slide 54

983122983141983155983151983148983157983156983145983151983150

5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 5580

Artificial Intelligence Predicate Logic Slide 55

983122983141983155983151983148983157983156983145983151983150

6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 5680

Artificial Intelligence Predicate Logic Slide 56

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 57

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 5880

Artificial Intelligence Predicate Logic Slide 58

983122983141983155983151983148983157983156983145983151983150

8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 59

983122983141983155983151983148983157983156983145983151983150

9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 60

983122983141983155983151983148983157983156983145983151983150

Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 61

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 62

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 63

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 64

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 65

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 66

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 69

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 71

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 72

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 74

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

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Artificial Intelligence Predicate Logic Slide 75

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 76

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

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892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 12

983105983156983151983149983145983139 983155983141983150983156983141983150983139983141983155

Term function (term1termn)or constant or variable

Atomic sentence predicate (term1termn)

or term1 = term

2

Eg

Brother(KingJohnRichardTheLionheart) gt (Length(LeftLegOf(Richard)) Length(LeftLegOf(KingJohn)))

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Artificial Intelligence Predicate Logic Slide 13

983107983151983149983152983148983141983160 983155983141983150983156983141983150983139983141983155

Complex sentences are made from atomic sentences usingconnectives examples notS

S1 and S2

S1 or S2

S1rArr S2

S1 hArr S2

Eg Sibling(KingJohnRichard) rArr Sibling(RichardKingJohn)

gt(12) or le (12)

gt(12) and not gt(12)

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Artificial Intelligence Predicate Logic Slide 14

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Sentences are true with respect to a model and an interpretation

Model contains objects (domain elements ) and relations among them

Interpretation specifies referents forconstant symbols rarr objects

predicate symbols rarr relations

function symbols rarr functional relations

An atomic sentence predicate(term1 termn) is true

iff the objects referred to by term1 termn

are in the relation referred to by predicate

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Artificial Intelligence Predicate Logic Slide 15

983125983150983145983158983141983154983155983137983148 983153983157983137983150983156983145983142983145983139983137983156983145983151983150

forallltvariablesgt ltsentence gt

Everyone at NUS is smart

forallx At(xNUS)rArr Smart(x )

forallx P is true in a model m iff P is true with x being each possible object

in the model Roughly speaking equivalent to the conjunction of instantiations of

P

983105983156983080983115983145983150983143983114983151983144983150983084983118983125983123983081rArr 983123983149983137983154983156983080983115983145983150983143983114983151983144983150983081

and 983105983156983080983122983145983139983144983137983154983140983084983118983125983123983081rArr 983123983149983137983154983156983080983122983145983139983144983137983154983140983081

and 983105983156983080983118983125983123983084983118983125983123983081rArr 983123983149983137983154983156983080983118983125983123983081and 983086983086983086

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Artificial Intelligence Predicate Logic Slide 16

983105 983139983151983149983149983151983150 983149983145983155983156983137983147983141 983156983151 983137983158983151983145983140

TypicallyrArr is the main connective with forall

Common mistake using and as the main connective with forall

forallx At(xNUS) and Smart(x)

means ldquoEveryone is at NUS and everyone is smartrdquo

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Artificial Intelligence Predicate Logic Slide 17

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existltvariablesgt ltsentencegt

Someone at NUS is smart

exist x At(xNUS) and Smart(x )$

exist x P is true in a model m iff P is true with x being some possible object

in the model Roughly speaking equivalent to the disjunction of instantiations of P

At(KingJohnNUS) and Smart(KingJohn)or At(RichardNUS) and Smart(Richard)or At(NUSNUS) and Smart(NUS)or

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 18

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Typically and is the main connective with exist

Common mistake usingrArr as the main connective with

exist

exist x At(xNUS)rArr Smart(x

)is true if there is anyone who is not at NUS

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 19

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forallx forally is the same as forally forallx

existx existy is the same as existy existx

existx forally is not the same as forally existx

existx forally Loves(xy) ldquoThere is a person who loves everyone in the worldrdquo

forally existx Loves(xy) ldquoEveryone in the world is loved by at least one personrdquo

Quantifier duality each can be expressed using the other

forallx Likes(xIceCream) notexistx notLikes(xIceCream )

existx Likes(xBroccoli) notforallx notLikes(xBroccoli

)

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Artificial Intelligence Predicate Logic Slide 20

983109983153983157983137983148983145983156983161

term1 = term2 is true under a given interpretation if andonly if term1 and term2 refer to the same object

Eg definition of Sibling in terms of Parent

forallxy Sibling(xy) hArr [not(x = y) and existmf not (m = f) and Parent(mx)and Parent(fx) and Parent(my) and Parent(fy)]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 21

983125983155983145983150983143 983110983119983116

The kinship domain Brothers are siblings

forallxy Brother(xy) hArr Sibling(xy)

Ones mother is ones female parent

forallmc Mother(c) = m hArr (Female(m) and Parent(mc))

ldquoSiblingrdquo is symmetric

forallxy Sibling(xy) hArr Sibling(yx)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 22

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Representing the following sentences in Predicate Logic

1 Marcus was a man

2 Macus was a Pompeian

3 All Pompians were Romans4 Caesar was a ruler

5 All Romans were either loyal to Caesar or hated hime

6 Everyone is loyal to someone

7 People only try to assassinate rulers they are not loyal to

8 Marcus tried to assassinate Caesar

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Artificial Intelligence Predicate Logic Slide 23

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

1 Marcus was a man

man(Marcus)

2 Macus was a Pompeian

Pompeian(Marcus)

3 All Pompeians were Romans

forallX Pompeian(X) Roman(X)

4 Caesar was a ruler

ruler(Caesar)

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Artificial Intelligence Predicate Logic Slide 24

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

5 All Romans were either loyal to Caesar or hated hime

ldquoorrdquo mean ldquoinclusive orrdquo - OR

forallX Roman(X) loyalto(X Caesar) v hate(X Caesar)

ldquoorrdquo mean ldquoexclusive orrdquo - XOR

forallX Roman(X) [ (loyalto(X Caesar) v hate(X Caesar)) ^

not(loyalto(X Caesar) ^ hate(X Caesar))]

or

forallX Roman(X) [ (loyalto(X Caesar) ^ nothate(X Caesar)) v

(notloyalto(X Caesar) ^ hate(X Caesar))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 25

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

6 Everyone is loyal to someone

forallX exist Y loyalto(XY)

7 People only try to assassinate rulers they are not loyal toforallX forall Y person(X) ^ ruler(Y) ^ tryassassinate(XY)notloyalto(XY)

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Artificial Intelligence Predicate Logic Slide 26

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

8 Marcus tried to assassinate Caesartryassassinate(Marcus Caesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 27

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Given the above sentences can we make aconclusion as follows

ldquoMarcus was not loyal to Caesar rdquo

or

notloyalto(MarcusCaesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 28

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)

(Rule What)

Add an implicit sentence

9 ldquoAll men are peoplerdquo

or

forallforallforallforallX man(X) X man(X) person(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 29

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)(9 Substitute)

man(Marcus)

(1)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 30

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

How to represent exceptions example

ldquoPaulus was a Pompeian Paulus was neitherloyal nor hated Caesarrdquo

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Artificial Intelligence Predicate Logic Slide 31

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation Semantic Net Simple by adding links

(properties) to subclasses or instances

Predicate Logic Not so simple like

983120983151983149983152983141983145983137983150(983120983137983157983148983157983155) 983134 983212983131983148983151983161983137983148983156983151(983120983137983157983148983157983155 983107983137983141983155983137983154) 983158

983144983137983156983141(983120983157983137983148983157983155 983107983137983141983155983137983154)983133

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 32

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation983118983151983156 983155983151 983155983145983149983152983148983141 983106983141983139983137983157983155983141 983156983144983141983154983141 983145983155 983139983151983150983142983148983145983139983156 983138983141983156983159983141983141983150983098

Pompeian(Paulus) ^ not[loyalto(Paulus Caesar) vhate(Pualus Caesar)]

983137983150983140

forallX Pompeian(X) loyalto(X Caesar) v hate(XCaesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 33

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation A Better way

forallX Roman(X) ^ noteq(X Paulus) loyalto(X Caesar)v hate(X Caesar

Exception

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Artificial Intelligence Predicate Logic Slide 34

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

In the case of describing relations betweennumbers example 1 983100 2

2 983100983091

983095 983102 983091 +2 983091 983102 1

991270

rArr

983123983144983151983157983148983140 983150983151983156 983141983160983152983148983145983139983145983156983148983161 983140983141983155983139983154983145983138983141983098983148983156(12) 983148983156(2983091) 991270

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 35

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

rArr Need functions and computable predicates

rArr For example 983107983137983148983148 983137 983142983157983150983139983156983145983151983150 983156983151 983139983151983149983152983157983156983141 (983091+2)

983120983137983155983155 983156983144983141 983154983141983155983157983148983156 983156983151 983156983144983141 983152983154983141983140983145983139983137983156983141 991260983143983156991261983098 983143983156(983095 983091+2)

983111983141983156 983156983144983141 983142983145983150983137983148 983154983141983155983157983148983156 (983124983154983157983141)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 36

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

1 Marcus was a man

man(Marcus)

2 Marcus was a PompeianPompeian(Marcus)

3 Marcus was born in 40 AD

born(Marcus40)

4 All men are mortal

forallX man(X) mortal(X)

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Artificial Intelligence Predicate Logic Slide 37

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Facts

5 All Pompeian died when the vocano erupted in 79 AD

erupted(vocano 79) ^ forallX [Pompeian(X) died(X 79)]

6 No mortal lives longer then 150 yearsforallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

7 It is now 1991now = 1991

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Artificial Intelligence Predicate Logic Slide 38

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Question

Is Marcus alive

Or

alive(Marcus now)

ornotnotnotnotalive(Marcus now)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 39

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge

Relation between ldquodeadrdquo and ldquoaliverdquo

8 Alive means not deadforallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

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Artificial Intelligence Predicate Logic Slide 40

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge Time properties of an event for example ldquodeadrdquo

9 Is someone dies the he is dead at all later times

forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

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Artificial Intelligence Predicate Logic Slide 41

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Summary of facts1 man(Marcus)

2 Pompeian(Marcus)

3 born(Marcus 40)

4 forallX man(X) mortal(X)

5 forallX Pompeian(X) died(X 79)

6 erupted(volcano 79)

7 forallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

8 now = 19919 forallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

10 forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

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Artificial Intelligence Predicate Logic Slide 42

notalive(Marcus now)

dead(Marcus now)

(9 Substitute)

died(Marcus T1) gt(now T1)

(10 Substitute)

Pompeian(Marcus) gt(now 79)

(5 Substitute)(5 Substitute)

(2)gt(1991 79)

(8 Substitute)

(computation)

ss

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Artificial Intelligence Predicate Logic Slide 43

983125983150983145983158983141983154983155983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983125983113)

Every instantiation of a universally quantified sentence is enta iled by it

forallv α

Subst(vg α)

for any variable v and ground term g

Eg forallx King( x) and Greedy( x)rArr Evil( x

) yieldsKing (John ) and Greedy (John ) rArr Evil (John )

King (Richard ) and Greedy (Richard ) rArr Evil (Richard )

King (Father (John )) and Greedy (Father (John )) rArr Evil (Father (John ))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 44

983109983160983145983155983156983141983150983156983145983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983109983113)

For any sentence α variable v and constant symbol k that

does not

appear elsewhere in the knowledge baseexistv α

Subst(vk α )

Eg exist x Crown( x) and OnHead ( xJohn) yields

Crown(C 1) and OnHead (C 1 John )

provided C 1 is a new constant symbol called a Skolemconstant

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 45

983122983141983140983157983139983156983145983151983150 983156983151 983152983154983151983152983151983155983145983156983145983151983150983137983148 983145983150983142983141983154983141983150983139983141

Suppose the KB contains just the followingforall

x King(x)and

Greedy(x)rArr

Evil(x)King(John)Greedy(John)Brother(RichardJohn)

Instantiating the universal sentence in all possible ways we haveKing(John) and Greedy(John) rArr Evil(John)King(Richard) and Greedy(Richard) rArr Evil(Richard)King(John)Greedy(John)Brother(RichardJohn

)

The new KB is propositionalized proposition symbols are

King(John) Greedy(John) Evil(John) King(Richard

) etc

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 46

983122983141983140983157983139983156983145983151983150 983139983151983150983156983140983086

Every FOL KB can be propositionalized so as to preserve entailment

(A ground sentence is entailed by new KB iff entailed by originalKB)

Idea propositionalize KB and query apply resolution return result

Problem with function symbols there are infinitely many ground

terms

eg Father (Father (Father (John

)))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 47

983122983141983155983151983148983157983156983145983151983150

Convert to clause form exampleldquoAll Romans who know Marcus either hate

Caesar or think that anyone who hates anyone is

crazyrdquo

forallX [roman(X) ^ know(X Marcus)]

[hate(X Ceasar) v(forallY existZ hate(YZ)

thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 48

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 49

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

forallX [roman(X) ^ know(X Marcus)][hate(X Ceasar) v

(forallY existZ hate(YZ) thinkcrazy(XY))]

=

forallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 50

983122983141983155983151983148983157983156983145983151983150

2 Move not next to terms

Using the following equivalence

a Double negative

not(notp) = p

b DeMorgan

not(a v b) = nota ^ notb

not(a ^ b) = nota v notb

c Equivalence of qualifiers notforallX P(X) = existX notP(X)

notexistX P(X) = forallX notP(X)

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Artificial Intelligence Predicate Logic Slide 51

983122983141983155983151983148983157983156983145983151983150

2 Move not next to termsforallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

=

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) vthinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 52

983122983141983155983151983148983157983156983145983151983150

3 Change name of variables as follows

forallX P(X) v forallX Q(X)

=forallX P(X) v forallY Q(Y)

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Artificial Intelligence Predicate Logic Slide 53

983122983141983155983151983148983157983156983145983151983150

4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 54

983122983141983155983151983148983157983156983145983151983150

5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

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Artificial Intelligence Predicate Logic Slide 55

983122983141983155983151983148983157983156983145983151983150

6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 56

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

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Artificial Intelligence Predicate Logic Slide 57

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

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Artificial Intelligence Predicate Logic Slide 58

983122983141983155983151983148983157983156983145983151983150

8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

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Artificial Intelligence Predicate Logic Slide 59

983122983141983155983151983148983157983156983145983151983150

9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

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Artificial Intelligence Predicate Logic Slide 60

983122983141983155983151983148983157983156983145983151983150

Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

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Artificial Intelligence Predicate Logic Slide 61

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

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Artificial Intelligence Predicate Logic Slide 62

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

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Artificial Intelligence Predicate Logic Slide 63

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

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Artificial Intelligence Predicate Logic Slide 64

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

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Artificial Intelligence Predicate Logic Slide 65

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

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Artificial Intelligence Predicate Logic Slide 66

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

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Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

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Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

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Artificial Intelligence Predicate Logic Slide 69

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

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Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

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Artificial Intelligence Predicate Logic Slide 71

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 72

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 74

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

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Artificial Intelligence Predicate Logic Slide 75

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 76

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

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892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 13

983107983151983149983152983148983141983160 983155983141983150983156983141983150983139983141983155

Complex sentences are made from atomic sentences usingconnectives examples notS

S1 and S2

S1 or S2

S1rArr S2

S1 hArr S2

Eg Sibling(KingJohnRichard) rArr Sibling(RichardKingJohn)

gt(12) or le (12)

gt(12) and not gt(12)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 14

983124983154983157983156983144 983145983150 983142983145983154983155983156983085983151983154983140983141983154 983148983151983143983145983139

Sentences are true with respect to a model and an interpretation

Model contains objects (domain elements ) and relations among them

Interpretation specifies referents forconstant symbols rarr objects

predicate symbols rarr relations

function symbols rarr functional relations

An atomic sentence predicate(term1 termn) is true

iff the objects referred to by term1 termn

are in the relation referred to by predicate

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 15

983125983150983145983158983141983154983155983137983148 983153983157983137983150983156983145983142983145983139983137983156983145983151983150

forallltvariablesgt ltsentence gt

Everyone at NUS is smart

forallx At(xNUS)rArr Smart(x )

forallx P is true in a model m iff P is true with x being each possible object

in the model Roughly speaking equivalent to the conjunction of instantiations of

P

983105983156983080983115983145983150983143983114983151983144983150983084983118983125983123983081rArr 983123983149983137983154983156983080983115983145983150983143983114983151983144983150983081

and 983105983156983080983122983145983139983144983137983154983140983084983118983125983123983081rArr 983123983149983137983154983156983080983122983145983139983144983137983154983140983081

and 983105983156983080983118983125983123983084983118983125983123983081rArr 983123983149983137983154983156983080983118983125983123983081and 983086983086983086

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 16

983105 983139983151983149983149983151983150 983149983145983155983156983137983147983141 983156983151 983137983158983151983145983140

TypicallyrArr is the main connective with forall

Common mistake using and as the main connective with forall

forallx At(xNUS) and Smart(x)

means ldquoEveryone is at NUS and everyone is smartrdquo

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 17

983109983160983145983155983156983141983150983156983145983137983148 983153983157983137983150983156983145983142983145983139983137983156983145983151983150

existltvariablesgt ltsentencegt

Someone at NUS is smart

exist x At(xNUS) and Smart(x )$

exist x P is true in a model m iff P is true with x being some possible object

in the model Roughly speaking equivalent to the disjunction of instantiations of P

At(KingJohnNUS) and Smart(KingJohn)or At(RichardNUS) and Smart(Richard)or At(NUSNUS) and Smart(NUS)or

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 18

983105983150983151983156983144983141983154 983139983151983149983149983151983150 983149983145983155983156983137983147983141 983156983151 983137983158983151983145983140

Typically and is the main connective with exist

Common mistake usingrArr as the main connective with

exist

exist x At(xNUS)rArr Smart(x

)is true if there is anyone who is not at NUS

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 19

983120983154983151983152983141983154983156983145983141983155 983151983142 983153983157983137983150983156983145983142983145983141983154983155

forallx forally is the same as forally forallx

existx existy is the same as existy existx

existx forally is not the same as forally existx

existx forally Loves(xy) ldquoThere is a person who loves everyone in the worldrdquo

forally existx Loves(xy) ldquoEveryone in the world is loved by at least one personrdquo

Quantifier duality each can be expressed using the other

forallx Likes(xIceCream) notexistx notLikes(xIceCream )

existx Likes(xBroccoli) notforallx notLikes(xBroccoli

)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 20

983109983153983157983137983148983145983156983161

term1 = term2 is true under a given interpretation if andonly if term1 and term2 refer to the same object

Eg definition of Sibling in terms of Parent

forallxy Sibling(xy) hArr [not(x = y) and existmf not (m = f) and Parent(mx)and Parent(fx) and Parent(my) and Parent(fy)]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 21

983125983155983145983150983143 983110983119983116

The kinship domain Brothers are siblings

forallxy Brother(xy) hArr Sibling(xy)

Ones mother is ones female parent

forallmc Mother(c) = m hArr (Female(m) and Parent(mc))

ldquoSiblingrdquo is symmetric

forallxy Sibling(xy) hArr Sibling(yx)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 22

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

Representing the following sentences in Predicate Logic

1 Marcus was a man

2 Macus was a Pompeian

3 All Pompians were Romans4 Caesar was a ruler

5 All Romans were either loyal to Caesar or hated hime

6 Everyone is loyal to someone

7 People only try to assassinate rulers they are not loyal to

8 Marcus tried to assassinate Caesar

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 23

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

1 Marcus was a man

man(Marcus)

2 Macus was a Pompeian

Pompeian(Marcus)

3 All Pompeians were Romans

forallX Pompeian(X) Roman(X)

4 Caesar was a ruler

ruler(Caesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 24

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

5 All Romans were either loyal to Caesar or hated hime

ldquoorrdquo mean ldquoinclusive orrdquo - OR

forallX Roman(X) loyalto(X Caesar) v hate(X Caesar)

ldquoorrdquo mean ldquoexclusive orrdquo - XOR

forallX Roman(X) [ (loyalto(X Caesar) v hate(X Caesar)) ^

not(loyalto(X Caesar) ^ hate(X Caesar))]

or

forallX Roman(X) [ (loyalto(X Caesar) ^ nothate(X Caesar)) v

(notloyalto(X Caesar) ^ hate(X Caesar))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 25

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

6 Everyone is loyal to someone

forallX exist Y loyalto(XY)

7 People only try to assassinate rulers they are not loyal toforallX forall Y person(X) ^ ruler(Y) ^ tryassassinate(XY)notloyalto(XY)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 26

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

8 Marcus tried to assassinate Caesartryassassinate(Marcus Caesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 27

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

Given the above sentences can we make aconclusion as follows

ldquoMarcus was not loyal to Caesar rdquo

or

notloyalto(MarcusCaesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 28

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)

(Rule What)

Add an implicit sentence

9 ldquoAll men are peoplerdquo

or

forallforallforallforallX man(X) X man(X) person(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 29

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)(9 Substitute)

man(Marcus)

(1)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 30

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

How to represent exceptions example

ldquoPaulus was a Pompeian Paulus was neitherloyal nor hated Caesarrdquo

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 31

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation Semantic Net Simple by adding links

(properties) to subclasses or instances

Predicate Logic Not so simple like

983120983151983149983152983141983145983137983150(983120983137983157983148983157983155) 983134 983212983131983148983151983161983137983148983156983151(983120983137983157983148983157983155 983107983137983141983155983137983154) 983158

983144983137983156983141(983120983157983137983148983157983155 983107983137983141983155983137983154)983133

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 32

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation983118983151983156 983155983151 983155983145983149983152983148983141 983106983141983139983137983157983155983141 983156983144983141983154983141 983145983155 983139983151983150983142983148983145983139983156 983138983141983156983159983141983141983150983098

Pompeian(Paulus) ^ not[loyalto(Paulus Caesar) vhate(Pualus Caesar)]

983137983150983140

forallX Pompeian(X) loyalto(X Caesar) v hate(XCaesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 33

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation A Better way

forallX Roman(X) ^ noteq(X Paulus) loyalto(X Caesar)v hate(X Caesar

Exception

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 34

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

In the case of describing relations betweennumbers example 1 983100 2

2 983100983091

983095 983102 983091 +2 983091 983102 1

991270

rArr

983123983144983151983157983148983140 983150983151983156 983141983160983152983148983145983139983145983156983148983161 983140983141983155983139983154983145983138983141983098983148983156(12) 983148983156(2983091) 991270

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 35

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

rArr Need functions and computable predicates

rArr For example 983107983137983148983148 983137 983142983157983150983139983156983145983151983150 983156983151 983139983151983149983152983157983156983141 (983091+2)

983120983137983155983155 983156983144983141 983154983141983155983157983148983156 983156983151 983156983144983141 983152983154983141983140983145983139983137983156983141 991260983143983156991261983098 983143983156(983095 983091+2)

983111983141983156 983156983144983141 983142983145983150983137983148 983154983141983155983157983148983156 (983124983154983157983141)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 36

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

1 Marcus was a man

man(Marcus)

2 Marcus was a PompeianPompeian(Marcus)

3 Marcus was born in 40 AD

born(Marcus40)

4 All men are mortal

forallX man(X) mortal(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 37

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

5 All Pompeian died when the vocano erupted in 79 AD

erupted(vocano 79) ^ forallX [Pompeian(X) died(X 79)]

6 No mortal lives longer then 150 yearsforallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

7 It is now 1991now = 1991

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 38

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Question

Is Marcus alive

Or

alive(Marcus now)

ornotnotnotnotalive(Marcus now)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 39

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge

Relation between ldquodeadrdquo and ldquoaliverdquo

8 Alive means not deadforallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4080

Artificial Intelligence Predicate Logic Slide 40

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge Time properties of an event for example ldquodeadrdquo

9 Is someone dies the he is dead at all later times

forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 41

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Summary of facts1 man(Marcus)

2 Pompeian(Marcus)

3 born(Marcus 40)

4 forallX man(X) mortal(X)

5 forallX Pompeian(X) died(X 79)

6 erupted(volcano 79)

7 forallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

8 now = 19919 forallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

10 forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

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Artificial Intelligence Predicate Logic Slide 42

notalive(Marcus now)

dead(Marcus now)

(9 Substitute)

died(Marcus T1) gt(now T1)

(10 Substitute)

Pompeian(Marcus) gt(now 79)

(5 Substitute)(5 Substitute)

(2)gt(1991 79)

(8 Substitute)

(computation)

ss

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Artificial Intelligence Predicate Logic Slide 43

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Every instantiation of a universally quantified sentence is enta iled by it

forallv α

Subst(vg α)

for any variable v and ground term g

Eg forallx King( x) and Greedy( x)rArr Evil( x

) yieldsKing (John ) and Greedy (John ) rArr Evil (John )

King (Richard ) and Greedy (Richard ) rArr Evil (Richard )

King (Father (John )) and Greedy (Father (John )) rArr Evil (Father (John ))

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Artificial Intelligence Predicate Logic Slide 44

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For any sentence α variable v and constant symbol k that

does not

appear elsewhere in the knowledge baseexistv α

Subst(vk α )

Eg exist x Crown( x) and OnHead ( xJohn) yields

Crown(C 1) and OnHead (C 1 John )

provided C 1 is a new constant symbol called a Skolemconstant

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Artificial Intelligence Predicate Logic Slide 45

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Suppose the KB contains just the followingforall

x King(x)and

Greedy(x)rArr

Evil(x)King(John)Greedy(John)Brother(RichardJohn)

Instantiating the universal sentence in all possible ways we haveKing(John) and Greedy(John) rArr Evil(John)King(Richard) and Greedy(Richard) rArr Evil(Richard)King(John)Greedy(John)Brother(RichardJohn

)

The new KB is propositionalized proposition symbols are

King(John) Greedy(John) Evil(John) King(Richard

) etc

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Artificial Intelligence Predicate Logic Slide 46

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Every FOL KB can be propositionalized so as to preserve entailment

(A ground sentence is entailed by new KB iff entailed by originalKB)

Idea propositionalize KB and query apply resolution return result

Problem with function symbols there are infinitely many ground

terms

eg Father (Father (Father (John

)))

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Artificial Intelligence Predicate Logic Slide 47

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Convert to clause form exampleldquoAll Romans who know Marcus either hate

Caesar or think that anyone who hates anyone is

crazyrdquo

forallX [roman(X) ^ know(X Marcus)]

[hate(X Ceasar) v(forallY existZ hate(YZ)

thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 48

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

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Artificial Intelligence Predicate Logic Slide 49

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

forallX [roman(X) ^ know(X Marcus)][hate(X Ceasar) v

(forallY existZ hate(YZ) thinkcrazy(XY))]

=

forallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 50

983122983141983155983151983148983157983156983145983151983150

2 Move not next to terms

Using the following equivalence

a Double negative

not(notp) = p

b DeMorgan

not(a v b) = nota ^ notb

not(a ^ b) = nota v notb

c Equivalence of qualifiers notforallX P(X) = existX notP(X)

notexistX P(X) = forallX notP(X)

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Artificial Intelligence Predicate Logic Slide 51

983122983141983155983151983148983157983156983145983151983150

2 Move not next to termsforallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

=

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) vthinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 52

983122983141983155983151983148983157983156983145983151983150

3 Change name of variables as follows

forallX P(X) v forallX Q(X)

=forallX P(X) v forallY Q(Y)

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Artificial Intelligence Predicate Logic Slide 53

983122983141983155983151983148983157983156983145983151983150

4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 54

983122983141983155983151983148983157983156983145983151983150

5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

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Artificial Intelligence Predicate Logic Slide 55

983122983141983155983151983148983157983156983145983151983150

6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 56

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

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Artificial Intelligence Predicate Logic Slide 57

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

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Artificial Intelligence Predicate Logic Slide 58

983122983141983155983151983148983157983156983145983151983150

8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

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Artificial Intelligence Predicate Logic Slide 59

983122983141983155983151983148983157983156983145983151983150

9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

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Artificial Intelligence Predicate Logic Slide 60

983122983141983155983151983148983157983156983145983151983150

Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

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Artificial Intelligence Predicate Logic Slide 61

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X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

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Artificial Intelligence Predicate Logic Slide 62

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θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

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Artificial Intelligence Predicate Logic Slide 63

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Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

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Artificial Intelligence Predicate Logic Slide 64

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Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

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Artificial Intelligence Predicate Logic Slide 65

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

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Artificial Intelligence Predicate Logic Slide 66

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Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

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Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

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Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

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Artificial Intelligence Predicate Logic Slide 69

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Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

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Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

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Artificial Intelligence Predicate Logic Slide 71

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

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Artificial Intelligence Predicate Logic Slide 72

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

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Artificial Intelligence Predicate Logic Slide 73

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Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

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Artificial Intelligence Predicate Logic Slide 74

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

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Artificial Intelligence Predicate Logic Slide 75

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Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

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Artificial Intelligence Predicate Logic Slide 76

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Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

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Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

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Artificial Intelligence Predicate Logic Slide 14

983124983154983157983156983144 983145983150 983142983145983154983155983156983085983151983154983140983141983154 983148983151983143983145983139

Sentences are true with respect to a model and an interpretation

Model contains objects (domain elements ) and relations among them

Interpretation specifies referents forconstant symbols rarr objects

predicate symbols rarr relations

function symbols rarr functional relations

An atomic sentence predicate(term1 termn) is true

iff the objects referred to by term1 termn

are in the relation referred to by predicate

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Artificial Intelligence Predicate Logic Slide 15

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forallltvariablesgt ltsentence gt

Everyone at NUS is smart

forallx At(xNUS)rArr Smart(x )

forallx P is true in a model m iff P is true with x being each possible object

in the model Roughly speaking equivalent to the conjunction of instantiations of

P

983105983156983080983115983145983150983143983114983151983144983150983084983118983125983123983081rArr 983123983149983137983154983156983080983115983145983150983143983114983151983144983150983081

and 983105983156983080983122983145983139983144983137983154983140983084983118983125983123983081rArr 983123983149983137983154983156983080983122983145983139983144983137983154983140983081

and 983105983156983080983118983125983123983084983118983125983123983081rArr 983123983149983137983154983156983080983118983125983123983081and 983086983086983086

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Artificial Intelligence Predicate Logic Slide 16

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TypicallyrArr is the main connective with forall

Common mistake using and as the main connective with forall

forallx At(xNUS) and Smart(x)

means ldquoEveryone is at NUS and everyone is smartrdquo

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 17

983109983160983145983155983156983141983150983156983145983137983148 983153983157983137983150983156983145983142983145983139983137983156983145983151983150

existltvariablesgt ltsentencegt

Someone at NUS is smart

exist x At(xNUS) and Smart(x )$

exist x P is true in a model m iff P is true with x being some possible object

in the model Roughly speaking equivalent to the disjunction of instantiations of P

At(KingJohnNUS) and Smart(KingJohn)or At(RichardNUS) and Smart(Richard)or At(NUSNUS) and Smart(NUS)or

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 18

983105983150983151983156983144983141983154 983139983151983149983149983151983150 983149983145983155983156983137983147983141 983156983151 983137983158983151983145983140

Typically and is the main connective with exist

Common mistake usingrArr as the main connective with

exist

exist x At(xNUS)rArr Smart(x

)is true if there is anyone who is not at NUS

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 19

983120983154983151983152983141983154983156983145983141983155 983151983142 983153983157983137983150983156983145983142983145983141983154983155

forallx forally is the same as forally forallx

existx existy is the same as existy existx

existx forally is not the same as forally existx

existx forally Loves(xy) ldquoThere is a person who loves everyone in the worldrdquo

forally existx Loves(xy) ldquoEveryone in the world is loved by at least one personrdquo

Quantifier duality each can be expressed using the other

forallx Likes(xIceCream) notexistx notLikes(xIceCream )

existx Likes(xBroccoli) notforallx notLikes(xBroccoli

)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 20

983109983153983157983137983148983145983156983161

term1 = term2 is true under a given interpretation if andonly if term1 and term2 refer to the same object

Eg definition of Sibling in terms of Parent

forallxy Sibling(xy) hArr [not(x = y) and existmf not (m = f) and Parent(mx)and Parent(fx) and Parent(my) and Parent(fy)]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 21

983125983155983145983150983143 983110983119983116

The kinship domain Brothers are siblings

forallxy Brother(xy) hArr Sibling(xy)

Ones mother is ones female parent

forallmc Mother(c) = m hArr (Female(m) and Parent(mc))

ldquoSiblingrdquo is symmetric

forallxy Sibling(xy) hArr Sibling(yx)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 22

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

Representing the following sentences in Predicate Logic

1 Marcus was a man

2 Macus was a Pompeian

3 All Pompians were Romans4 Caesar was a ruler

5 All Romans were either loyal to Caesar or hated hime

6 Everyone is loyal to someone

7 People only try to assassinate rulers they are not loyal to

8 Marcus tried to assassinate Caesar

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 23

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

1 Marcus was a man

man(Marcus)

2 Macus was a Pompeian

Pompeian(Marcus)

3 All Pompeians were Romans

forallX Pompeian(X) Roman(X)

4 Caesar was a ruler

ruler(Caesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 24

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

5 All Romans were either loyal to Caesar or hated hime

ldquoorrdquo mean ldquoinclusive orrdquo - OR

forallX Roman(X) loyalto(X Caesar) v hate(X Caesar)

ldquoorrdquo mean ldquoexclusive orrdquo - XOR

forallX Roman(X) [ (loyalto(X Caesar) v hate(X Caesar)) ^

not(loyalto(X Caesar) ^ hate(X Caesar))]

or

forallX Roman(X) [ (loyalto(X Caesar) ^ nothate(X Caesar)) v

(notloyalto(X Caesar) ^ hate(X Caesar))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 25

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

6 Everyone is loyal to someone

forallX exist Y loyalto(XY)

7 People only try to assassinate rulers they are not loyal toforallX forall Y person(X) ^ ruler(Y) ^ tryassassinate(XY)notloyalto(XY)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 26

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

8 Marcus tried to assassinate Caesartryassassinate(Marcus Caesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 27

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

Given the above sentences can we make aconclusion as follows

ldquoMarcus was not loyal to Caesar rdquo

or

notloyalto(MarcusCaesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 28

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)

(Rule What)

Add an implicit sentence

9 ldquoAll men are peoplerdquo

or

forallforallforallforallX man(X) X man(X) person(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 29

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)(9 Substitute)

man(Marcus)

(1)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 30

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

How to represent exceptions example

ldquoPaulus was a Pompeian Paulus was neitherloyal nor hated Caesarrdquo

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 31

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation Semantic Net Simple by adding links

(properties) to subclasses or instances

Predicate Logic Not so simple like

983120983151983149983152983141983145983137983150(983120983137983157983148983157983155) 983134 983212983131983148983151983161983137983148983156983151(983120983137983157983148983157983155 983107983137983141983155983137983154) 983158

983144983137983156983141(983120983157983137983148983157983155 983107983137983141983155983137983154)983133

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 32

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation983118983151983156 983155983151 983155983145983149983152983148983141 983106983141983139983137983157983155983141 983156983144983141983154983141 983145983155 983139983151983150983142983148983145983139983156 983138983141983156983159983141983141983150983098

Pompeian(Paulus) ^ not[loyalto(Paulus Caesar) vhate(Pualus Caesar)]

983137983150983140

forallX Pompeian(X) loyalto(X Caesar) v hate(XCaesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 33

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation A Better way

forallX Roman(X) ^ noteq(X Paulus) loyalto(X Caesar)v hate(X Caesar

Exception

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Artificial Intelligence Predicate Logic Slide 34

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

In the case of describing relations betweennumbers example 1 983100 2

2 983100983091

983095 983102 983091 +2 983091 983102 1

991270

rArr

983123983144983151983157983148983140 983150983151983156 983141983160983152983148983145983139983145983156983148983161 983140983141983155983139983154983145983138983141983098983148983156(12) 983148983156(2983091) 991270

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 35

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

rArr Need functions and computable predicates

rArr For example 983107983137983148983148 983137 983142983157983150983139983156983145983151983150 983156983151 983139983151983149983152983157983156983141 (983091+2)

983120983137983155983155 983156983144983141 983154983141983155983157983148983156 983156983151 983156983144983141 983152983154983141983140983145983139983137983156983141 991260983143983156991261983098 983143983156(983095 983091+2)

983111983141983156 983156983144983141 983142983145983150983137983148 983154983141983155983157983148983156 (983124983154983157983141)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 36

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

1 Marcus was a man

man(Marcus)

2 Marcus was a PompeianPompeian(Marcus)

3 Marcus was born in 40 AD

born(Marcus40)

4 All men are mortal

forallX man(X) mortal(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 37

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

5 All Pompeian died when the vocano erupted in 79 AD

erupted(vocano 79) ^ forallX [Pompeian(X) died(X 79)]

6 No mortal lives longer then 150 yearsforallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

7 It is now 1991now = 1991

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 38

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Question

Is Marcus alive

Or

alive(Marcus now)

ornotnotnotnotalive(Marcus now)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 39

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge

Relation between ldquodeadrdquo and ldquoaliverdquo

8 Alive means not deadforallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 40

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge Time properties of an event for example ldquodeadrdquo

9 Is someone dies the he is dead at all later times

forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4180

Artificial Intelligence Predicate Logic Slide 41

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Summary of facts1 man(Marcus)

2 Pompeian(Marcus)

3 born(Marcus 40)

4 forallX man(X) mortal(X)

5 forallX Pompeian(X) died(X 79)

6 erupted(volcano 79)

7 forallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

8 now = 19919 forallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

10 forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4280

Artificial Intelligence Predicate Logic Slide 42

notalive(Marcus now)

dead(Marcus now)

(9 Substitute)

died(Marcus T1) gt(now T1)

(10 Substitute)

Pompeian(Marcus) gt(now 79)

(5 Substitute)(5 Substitute)

(2)gt(1991 79)

(8 Substitute)

(computation)

ss

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4380

Artificial Intelligence Predicate Logic Slide 43

983125983150983145983158983141983154983155983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983125983113)

Every instantiation of a universally quantified sentence is enta iled by it

forallv α

Subst(vg α)

for any variable v and ground term g

Eg forallx King( x) and Greedy( x)rArr Evil( x

) yieldsKing (John ) and Greedy (John ) rArr Evil (John )

King (Richard ) and Greedy (Richard ) rArr Evil (Richard )

King (Father (John )) and Greedy (Father (John )) rArr Evil (Father (John ))

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4480

Artificial Intelligence Predicate Logic Slide 44

983109983160983145983155983156983141983150983156983145983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983109983113)

For any sentence α variable v and constant symbol k that

does not

appear elsewhere in the knowledge baseexistv α

Subst(vk α )

Eg exist x Crown( x) and OnHead ( xJohn) yields

Crown(C 1) and OnHead (C 1 John )

provided C 1 is a new constant symbol called a Skolemconstant

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4580

Artificial Intelligence Predicate Logic Slide 45

983122983141983140983157983139983156983145983151983150 983156983151 983152983154983151983152983151983155983145983156983145983151983150983137983148 983145983150983142983141983154983141983150983139983141

Suppose the KB contains just the followingforall

x King(x)and

Greedy(x)rArr

Evil(x)King(John)Greedy(John)Brother(RichardJohn)

Instantiating the universal sentence in all possible ways we haveKing(John) and Greedy(John) rArr Evil(John)King(Richard) and Greedy(Richard) rArr Evil(Richard)King(John)Greedy(John)Brother(RichardJohn

)

The new KB is propositionalized proposition symbols are

King(John) Greedy(John) Evil(John) King(Richard

) etc

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4680

Artificial Intelligence Predicate Logic Slide 46

983122983141983140983157983139983156983145983151983150 983139983151983150983156983140983086

Every FOL KB can be propositionalized so as to preserve entailment

(A ground sentence is entailed by new KB iff entailed by originalKB)

Idea propositionalize KB and query apply resolution return result

Problem with function symbols there are infinitely many ground

terms

eg Father (Father (Father (John

)))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 47

983122983141983155983151983148983157983156983145983151983150

Convert to clause form exampleldquoAll Romans who know Marcus either hate

Caesar or think that anyone who hates anyone is

crazyrdquo

forallX [roman(X) ^ know(X Marcus)]

[hate(X Ceasar) v(forallY existZ hate(YZ)

thinkcrazy(XY))]

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4880

Artificial Intelligence Predicate Logic Slide 48

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 49

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

forallX [roman(X) ^ know(X Marcus)][hate(X Ceasar) v

(forallY existZ hate(YZ) thinkcrazy(XY))]

=

forallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 50

983122983141983155983151983148983157983156983145983151983150

2 Move not next to terms

Using the following equivalence

a Double negative

not(notp) = p

b DeMorgan

not(a v b) = nota ^ notb

not(a ^ b) = nota v notb

c Equivalence of qualifiers notforallX P(X) = existX notP(X)

notexistX P(X) = forallX notP(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 51

983122983141983155983151983148983157983156983145983151983150

2 Move not next to termsforallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

=

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) vthinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 52

983122983141983155983151983148983157983156983145983151983150

3 Change name of variables as follows

forallX P(X) v forallX Q(X)

=forallX P(X) v forallY Q(Y)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 53

983122983141983155983151983148983157983156983145983151983150

4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 54

983122983141983155983151983148983157983156983145983151983150

5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 55

983122983141983155983151983148983157983156983145983151983150

6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 56

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

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Artificial Intelligence Predicate Logic Slide 57

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

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Artificial Intelligence Predicate Logic Slide 58

983122983141983155983151983148983157983156983145983151983150

8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 59

983122983141983155983151983148983157983156983145983151983150

9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

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Artificial Intelligence Predicate Logic Slide 60

983122983141983155983151983148983157983156983145983151983150

Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 61

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

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Artificial Intelligence Predicate Logic Slide 62

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

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Artificial Intelligence Predicate Logic Slide 63

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 64

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 65

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

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Artificial Intelligence Predicate Logic Slide 66

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

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Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

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Artificial Intelligence Predicate Logic Slide 69

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

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Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

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Artificial Intelligence Predicate Logic Slide 71

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

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Artificial Intelligence Predicate Logic Slide 72

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

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Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

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Artificial Intelligence Predicate Logic Slide 74

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

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Artificial Intelligence Predicate Logic Slide 75

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

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Artificial Intelligence Predicate Logic Slide 76

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

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892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

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Artificial Intelligence Predicate Logic Slide 15

983125983150983145983158983141983154983155983137983148 983153983157983137983150983156983145983142983145983139983137983156983145983151983150

forallltvariablesgt ltsentence gt

Everyone at NUS is smart

forallx At(xNUS)rArr Smart(x )

forallx P is true in a model m iff P is true with x being each possible object

in the model Roughly speaking equivalent to the conjunction of instantiations of

P

983105983156983080983115983145983150983143983114983151983144983150983084983118983125983123983081rArr 983123983149983137983154983156983080983115983145983150983143983114983151983144983150983081

and 983105983156983080983122983145983139983144983137983154983140983084983118983125983123983081rArr 983123983149983137983154983156983080983122983145983139983144983137983154983140983081

and 983105983156983080983118983125983123983084983118983125983123983081rArr 983123983149983137983154983156983080983118983125983123983081and 983086983086983086

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Artificial Intelligence Predicate Logic Slide 16

983105 983139983151983149983149983151983150 983149983145983155983156983137983147983141 983156983151 983137983158983151983145983140

TypicallyrArr is the main connective with forall

Common mistake using and as the main connective with forall

forallx At(xNUS) and Smart(x)

means ldquoEveryone is at NUS and everyone is smartrdquo

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Artificial Intelligence Predicate Logic Slide 17

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existltvariablesgt ltsentencegt

Someone at NUS is smart

exist x At(xNUS) and Smart(x )$

exist x P is true in a model m iff P is true with x being some possible object

in the model Roughly speaking equivalent to the disjunction of instantiations of P

At(KingJohnNUS) and Smart(KingJohn)or At(RichardNUS) and Smart(Richard)or At(NUSNUS) and Smart(NUS)or

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Artificial Intelligence Predicate Logic Slide 18

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Typically and is the main connective with exist

Common mistake usingrArr as the main connective with

exist

exist x At(xNUS)rArr Smart(x

)is true if there is anyone who is not at NUS

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Artificial Intelligence Predicate Logic Slide 19

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forallx forally is the same as forally forallx

existx existy is the same as existy existx

existx forally is not the same as forally existx

existx forally Loves(xy) ldquoThere is a person who loves everyone in the worldrdquo

forally existx Loves(xy) ldquoEveryone in the world is loved by at least one personrdquo

Quantifier duality each can be expressed using the other

forallx Likes(xIceCream) notexistx notLikes(xIceCream )

existx Likes(xBroccoli) notforallx notLikes(xBroccoli

)

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Artificial Intelligence Predicate Logic Slide 20

983109983153983157983137983148983145983156983161

term1 = term2 is true under a given interpretation if andonly if term1 and term2 refer to the same object

Eg definition of Sibling in terms of Parent

forallxy Sibling(xy) hArr [not(x = y) and existmf not (m = f) and Parent(mx)and Parent(fx) and Parent(my) and Parent(fy)]

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Artificial Intelligence Predicate Logic Slide 21

983125983155983145983150983143 983110983119983116

The kinship domain Brothers are siblings

forallxy Brother(xy) hArr Sibling(xy)

Ones mother is ones female parent

forallmc Mother(c) = m hArr (Female(m) and Parent(mc))

ldquoSiblingrdquo is symmetric

forallxy Sibling(xy) hArr Sibling(yx)

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Artificial Intelligence Predicate Logic Slide 22

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Representing the following sentences in Predicate Logic

1 Marcus was a man

2 Macus was a Pompeian

3 All Pompians were Romans4 Caesar was a ruler

5 All Romans were either loyal to Caesar or hated hime

6 Everyone is loyal to someone

7 People only try to assassinate rulers they are not loyal to

8 Marcus tried to assassinate Caesar

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Artificial Intelligence Predicate Logic Slide 23

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1 Marcus was a man

man(Marcus)

2 Macus was a Pompeian

Pompeian(Marcus)

3 All Pompeians were Romans

forallX Pompeian(X) Roman(X)

4 Caesar was a ruler

ruler(Caesar)

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Artificial Intelligence Predicate Logic Slide 24

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5 All Romans were either loyal to Caesar or hated hime

ldquoorrdquo mean ldquoinclusive orrdquo - OR

forallX Roman(X) loyalto(X Caesar) v hate(X Caesar)

ldquoorrdquo mean ldquoexclusive orrdquo - XOR

forallX Roman(X) [ (loyalto(X Caesar) v hate(X Caesar)) ^

not(loyalto(X Caesar) ^ hate(X Caesar))]

or

forallX Roman(X) [ (loyalto(X Caesar) ^ nothate(X Caesar)) v

(notloyalto(X Caesar) ^ hate(X Caesar))]

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Artificial Intelligence Predicate Logic Slide 25

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

6 Everyone is loyal to someone

forallX exist Y loyalto(XY)

7 People only try to assassinate rulers they are not loyal toforallX forall Y person(X) ^ ruler(Y) ^ tryassassinate(XY)notloyalto(XY)

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Artificial Intelligence Predicate Logic Slide 26

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8 Marcus tried to assassinate Caesartryassassinate(Marcus Caesar)

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Artificial Intelligence Predicate Logic Slide 27

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Given the above sentences can we make aconclusion as follows

ldquoMarcus was not loyal to Caesar rdquo

or

notloyalto(MarcusCaesar)

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Artificial Intelligence Predicate Logic Slide 28

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)

(Rule What)

Add an implicit sentence

9 ldquoAll men are peoplerdquo

or

forallforallforallforallX man(X) X man(X) person(X)

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Artificial Intelligence Predicate Logic Slide 29

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)(9 Substitute)

man(Marcus)

(1)

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Artificial Intelligence Predicate Logic Slide 30

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

How to represent exceptions example

ldquoPaulus was a Pompeian Paulus was neitherloyal nor hated Caesarrdquo

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Artificial Intelligence Predicate Logic Slide 31

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation Semantic Net Simple by adding links

(properties) to subclasses or instances

Predicate Logic Not so simple like

983120983151983149983152983141983145983137983150(983120983137983157983148983157983155) 983134 983212983131983148983151983161983137983148983156983151(983120983137983157983148983157983155 983107983137983141983155983137983154) 983158

983144983137983156983141(983120983157983137983148983157983155 983107983137983141983155983137983154)983133

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 3280

Artificial Intelligence Predicate Logic Slide 32

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation983118983151983156 983155983151 983155983145983149983152983148983141 983106983141983139983137983157983155983141 983156983144983141983154983141 983145983155 983139983151983150983142983148983145983139983156 983138983141983156983159983141983141983150983098

Pompeian(Paulus) ^ not[loyalto(Paulus Caesar) vhate(Pualus Caesar)]

983137983150983140

forallX Pompeian(X) loyalto(X Caesar) v hate(XCaesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 33

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation A Better way

forallX Roman(X) ^ noteq(X Paulus) loyalto(X Caesar)v hate(X Caesar

Exception

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 34

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

In the case of describing relations betweennumbers example 1 983100 2

2 983100983091

983095 983102 983091 +2 983091 983102 1

991270

rArr

983123983144983151983157983148983140 983150983151983156 983141983160983152983148983145983139983145983156983148983161 983140983141983155983139983154983145983138983141983098983148983156(12) 983148983156(2983091) 991270

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 35

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

rArr Need functions and computable predicates

rArr For example 983107983137983148983148 983137 983142983157983150983139983156983145983151983150 983156983151 983139983151983149983152983157983156983141 (983091+2)

983120983137983155983155 983156983144983141 983154983141983155983157983148983156 983156983151 983156983144983141 983152983154983141983140983145983139983137983156983141 991260983143983156991261983098 983143983156(983095 983091+2)

983111983141983156 983156983144983141 983142983145983150983137983148 983154983141983155983157983148983156 (983124983154983157983141)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 36

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

1 Marcus was a man

man(Marcus)

2 Marcus was a PompeianPompeian(Marcus)

3 Marcus was born in 40 AD

born(Marcus40)

4 All men are mortal

forallX man(X) mortal(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 37

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

5 All Pompeian died when the vocano erupted in 79 AD

erupted(vocano 79) ^ forallX [Pompeian(X) died(X 79)]

6 No mortal lives longer then 150 yearsforallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

7 It is now 1991now = 1991

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Artificial Intelligence Predicate Logic Slide 38

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Question

Is Marcus alive

Or

alive(Marcus now)

ornotnotnotnotalive(Marcus now)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 39

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge

Relation between ldquodeadrdquo and ldquoaliverdquo

8 Alive means not deadforallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

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Artificial Intelligence Predicate Logic Slide 40

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge Time properties of an event for example ldquodeadrdquo

9 Is someone dies the he is dead at all later times

forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

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Artificial Intelligence Predicate Logic Slide 41

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Summary of facts1 man(Marcus)

2 Pompeian(Marcus)

3 born(Marcus 40)

4 forallX man(X) mortal(X)

5 forallX Pompeian(X) died(X 79)

6 erupted(volcano 79)

7 forallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

8 now = 19919 forallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

10 forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 42

notalive(Marcus now)

dead(Marcus now)

(9 Substitute)

died(Marcus T1) gt(now T1)

(10 Substitute)

Pompeian(Marcus) gt(now 79)

(5 Substitute)(5 Substitute)

(2)gt(1991 79)

(8 Substitute)

(computation)

ss

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 43

983125983150983145983158983141983154983155983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983125983113)

Every instantiation of a universally quantified sentence is enta iled by it

forallv α

Subst(vg α)

for any variable v and ground term g

Eg forallx King( x) and Greedy( x)rArr Evil( x

) yieldsKing (John ) and Greedy (John ) rArr Evil (John )

King (Richard ) and Greedy (Richard ) rArr Evil (Richard )

King (Father (John )) and Greedy (Father (John )) rArr Evil (Father (John ))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 44

983109983160983145983155983156983141983150983156983145983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983109983113)

For any sentence α variable v and constant symbol k that

does not

appear elsewhere in the knowledge baseexistv α

Subst(vk α )

Eg exist x Crown( x) and OnHead ( xJohn) yields

Crown(C 1) and OnHead (C 1 John )

provided C 1 is a new constant symbol called a Skolemconstant

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 45

983122983141983140983157983139983156983145983151983150 983156983151 983152983154983151983152983151983155983145983156983145983151983150983137983148 983145983150983142983141983154983141983150983139983141

Suppose the KB contains just the followingforall

x King(x)and

Greedy(x)rArr

Evil(x)King(John)Greedy(John)Brother(RichardJohn)

Instantiating the universal sentence in all possible ways we haveKing(John) and Greedy(John) rArr Evil(John)King(Richard) and Greedy(Richard) rArr Evil(Richard)King(John)Greedy(John)Brother(RichardJohn

)

The new KB is propositionalized proposition symbols are

King(John) Greedy(John) Evil(John) King(Richard

) etc

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 46

983122983141983140983157983139983156983145983151983150 983139983151983150983156983140983086

Every FOL KB can be propositionalized so as to preserve entailment

(A ground sentence is entailed by new KB iff entailed by originalKB)

Idea propositionalize KB and query apply resolution return result

Problem with function symbols there are infinitely many ground

terms

eg Father (Father (Father (John

)))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 47

983122983141983155983151983148983157983156983145983151983150

Convert to clause form exampleldquoAll Romans who know Marcus either hate

Caesar or think that anyone who hates anyone is

crazyrdquo

forallX [roman(X) ^ know(X Marcus)]

[hate(X Ceasar) v(forallY existZ hate(YZ)

thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 48

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 49

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

forallX [roman(X) ^ know(X Marcus)][hate(X Ceasar) v

(forallY existZ hate(YZ) thinkcrazy(XY))]

=

forallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 50

983122983141983155983151983148983157983156983145983151983150

2 Move not next to terms

Using the following equivalence

a Double negative

not(notp) = p

b DeMorgan

not(a v b) = nota ^ notb

not(a ^ b) = nota v notb

c Equivalence of qualifiers notforallX P(X) = existX notP(X)

notexistX P(X) = forallX notP(X)

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Artificial Intelligence Predicate Logic Slide 51

983122983141983155983151983148983157983156983145983151983150

2 Move not next to termsforallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

=

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) vthinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 52

983122983141983155983151983148983157983156983145983151983150

3 Change name of variables as follows

forallX P(X) v forallX Q(X)

=forallX P(X) v forallY Q(Y)

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Artificial Intelligence Predicate Logic Slide 53

983122983141983155983151983148983157983156983145983151983150

4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 54

983122983141983155983151983148983157983156983145983151983150

5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

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Artificial Intelligence Predicate Logic Slide 55

983122983141983155983151983148983157983156983145983151983150

6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 56

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 57

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 58

983122983141983155983151983148983157983156983145983151983150

8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 59

983122983141983155983151983148983157983156983145983151983150

9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

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Artificial Intelligence Predicate Logic Slide 60

983122983141983155983151983148983157983156983145983151983150

Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

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Artificial Intelligence Predicate Logic Slide 61

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

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Artificial Intelligence Predicate Logic Slide 62

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

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Artificial Intelligence Predicate Logic Slide 63

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

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Artificial Intelligence Predicate Logic Slide 64

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 65

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 66

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

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Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

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Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

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Artificial Intelligence Predicate Logic Slide 69

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

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Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 71

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 72

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 74

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 75

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 76

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

892019 First Order(Predicate)Logic

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892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 16

983105 983139983151983149983149983151983150 983149983145983155983156983137983147983141 983156983151 983137983158983151983145983140

TypicallyrArr is the main connective with forall

Common mistake using and as the main connective with forall

forallx At(xNUS) and Smart(x)

means ldquoEveryone is at NUS and everyone is smartrdquo

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 17

983109983160983145983155983156983141983150983156983145983137983148 983153983157983137983150983156983145983142983145983139983137983156983145983151983150

existltvariablesgt ltsentencegt

Someone at NUS is smart

exist x At(xNUS) and Smart(x )$

exist x P is true in a model m iff P is true with x being some possible object

in the model Roughly speaking equivalent to the disjunction of instantiations of P

At(KingJohnNUS) and Smart(KingJohn)or At(RichardNUS) and Smart(Richard)or At(NUSNUS) and Smart(NUS)or

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 18

983105983150983151983156983144983141983154 983139983151983149983149983151983150 983149983145983155983156983137983147983141 983156983151 983137983158983151983145983140

Typically and is the main connective with exist

Common mistake usingrArr as the main connective with

exist

exist x At(xNUS)rArr Smart(x

)is true if there is anyone who is not at NUS

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 19

983120983154983151983152983141983154983156983145983141983155 983151983142 983153983157983137983150983156983145983142983145983141983154983155

forallx forally is the same as forally forallx

existx existy is the same as existy existx

existx forally is not the same as forally existx

existx forally Loves(xy) ldquoThere is a person who loves everyone in the worldrdquo

forally existx Loves(xy) ldquoEveryone in the world is loved by at least one personrdquo

Quantifier duality each can be expressed using the other

forallx Likes(xIceCream) notexistx notLikes(xIceCream )

existx Likes(xBroccoli) notforallx notLikes(xBroccoli

)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 20

983109983153983157983137983148983145983156983161

term1 = term2 is true under a given interpretation if andonly if term1 and term2 refer to the same object

Eg definition of Sibling in terms of Parent

forallxy Sibling(xy) hArr [not(x = y) and existmf not (m = f) and Parent(mx)and Parent(fx) and Parent(my) and Parent(fy)]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 21

983125983155983145983150983143 983110983119983116

The kinship domain Brothers are siblings

forallxy Brother(xy) hArr Sibling(xy)

Ones mother is ones female parent

forallmc Mother(c) = m hArr (Female(m) and Parent(mc))

ldquoSiblingrdquo is symmetric

forallxy Sibling(xy) hArr Sibling(yx)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 22

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

Representing the following sentences in Predicate Logic

1 Marcus was a man

2 Macus was a Pompeian

3 All Pompians were Romans4 Caesar was a ruler

5 All Romans were either loyal to Caesar or hated hime

6 Everyone is loyal to someone

7 People only try to assassinate rulers they are not loyal to

8 Marcus tried to assassinate Caesar

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 23

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

1 Marcus was a man

man(Marcus)

2 Macus was a Pompeian

Pompeian(Marcus)

3 All Pompeians were Romans

forallX Pompeian(X) Roman(X)

4 Caesar was a ruler

ruler(Caesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 24

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

5 All Romans were either loyal to Caesar or hated hime

ldquoorrdquo mean ldquoinclusive orrdquo - OR

forallX Roman(X) loyalto(X Caesar) v hate(X Caesar)

ldquoorrdquo mean ldquoexclusive orrdquo - XOR

forallX Roman(X) [ (loyalto(X Caesar) v hate(X Caesar)) ^

not(loyalto(X Caesar) ^ hate(X Caesar))]

or

forallX Roman(X) [ (loyalto(X Caesar) ^ nothate(X Caesar)) v

(notloyalto(X Caesar) ^ hate(X Caesar))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 25

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

6 Everyone is loyal to someone

forallX exist Y loyalto(XY)

7 People only try to assassinate rulers they are not loyal toforallX forall Y person(X) ^ ruler(Y) ^ tryassassinate(XY)notloyalto(XY)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 26

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

8 Marcus tried to assassinate Caesartryassassinate(Marcus Caesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 27

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

Given the above sentences can we make aconclusion as follows

ldquoMarcus was not loyal to Caesar rdquo

or

notloyalto(MarcusCaesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 28

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)

(Rule What)

Add an implicit sentence

9 ldquoAll men are peoplerdquo

or

forallforallforallforallX man(X) X man(X) person(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 29

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)(9 Substitute)

man(Marcus)

(1)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 30

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

How to represent exceptions example

ldquoPaulus was a Pompeian Paulus was neitherloyal nor hated Caesarrdquo

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 31

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation Semantic Net Simple by adding links

(properties) to subclasses or instances

Predicate Logic Not so simple like

983120983151983149983152983141983145983137983150(983120983137983157983148983157983155) 983134 983212983131983148983151983161983137983148983156983151(983120983137983157983148983157983155 983107983137983141983155983137983154) 983158

983144983137983156983141(983120983157983137983148983157983155 983107983137983141983155983137983154)983133

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 32

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation983118983151983156 983155983151 983155983145983149983152983148983141 983106983141983139983137983157983155983141 983156983144983141983154983141 983145983155 983139983151983150983142983148983145983139983156 983138983141983156983159983141983141983150983098

Pompeian(Paulus) ^ not[loyalto(Paulus Caesar) vhate(Pualus Caesar)]

983137983150983140

forallX Pompeian(X) loyalto(X Caesar) v hate(XCaesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 33

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation A Better way

forallX Roman(X) ^ noteq(X Paulus) loyalto(X Caesar)v hate(X Caesar

Exception

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 34

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

In the case of describing relations betweennumbers example 1 983100 2

2 983100983091

983095 983102 983091 +2 983091 983102 1

991270

rArr

983123983144983151983157983148983140 983150983151983156 983141983160983152983148983145983139983145983156983148983161 983140983141983155983139983154983145983138983141983098983148983156(12) 983148983156(2983091) 991270

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 35

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

rArr Need functions and computable predicates

rArr For example 983107983137983148983148 983137 983142983157983150983139983156983145983151983150 983156983151 983139983151983149983152983157983156983141 (983091+2)

983120983137983155983155 983156983144983141 983154983141983155983157983148983156 983156983151 983156983144983141 983152983154983141983140983145983139983137983156983141 991260983143983156991261983098 983143983156(983095 983091+2)

983111983141983156 983156983144983141 983142983145983150983137983148 983154983141983155983157983148983156 (983124983154983157983141)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 36

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

1 Marcus was a man

man(Marcus)

2 Marcus was a PompeianPompeian(Marcus)

3 Marcus was born in 40 AD

born(Marcus40)

4 All men are mortal

forallX man(X) mortal(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 37

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

5 All Pompeian died when the vocano erupted in 79 AD

erupted(vocano 79) ^ forallX [Pompeian(X) died(X 79)]

6 No mortal lives longer then 150 yearsforallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

7 It is now 1991now = 1991

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 38

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Question

Is Marcus alive

Or

alive(Marcus now)

ornotnotnotnotalive(Marcus now)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 39

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge

Relation between ldquodeadrdquo and ldquoaliverdquo

8 Alive means not deadforallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4080

Artificial Intelligence Predicate Logic Slide 40

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge Time properties of an event for example ldquodeadrdquo

9 Is someone dies the he is dead at all later times

forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4180

Artificial Intelligence Predicate Logic Slide 41

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Summary of facts1 man(Marcus)

2 Pompeian(Marcus)

3 born(Marcus 40)

4 forallX man(X) mortal(X)

5 forallX Pompeian(X) died(X 79)

6 erupted(volcano 79)

7 forallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

8 now = 19919 forallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

10 forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4280

Artificial Intelligence Predicate Logic Slide 42

notalive(Marcus now)

dead(Marcus now)

(9 Substitute)

died(Marcus T1) gt(now T1)

(10 Substitute)

Pompeian(Marcus) gt(now 79)

(5 Substitute)(5 Substitute)

(2)gt(1991 79)

(8 Substitute)

(computation)

ss

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4380

Artificial Intelligence Predicate Logic Slide 43

983125983150983145983158983141983154983155983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983125983113)

Every instantiation of a universally quantified sentence is enta iled by it

forallv α

Subst(vg α)

for any variable v and ground term g

Eg forallx King( x) and Greedy( x)rArr Evil( x

) yieldsKing (John ) and Greedy (John ) rArr Evil (John )

King (Richard ) and Greedy (Richard ) rArr Evil (Richard )

King (Father (John )) and Greedy (Father (John )) rArr Evil (Father (John ))

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Artificial Intelligence Predicate Logic Slide 44

983109983160983145983155983156983141983150983156983145983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983109983113)

For any sentence α variable v and constant symbol k that

does not

appear elsewhere in the knowledge baseexistv α

Subst(vk α )

Eg exist x Crown( x) and OnHead ( xJohn) yields

Crown(C 1) and OnHead (C 1 John )

provided C 1 is a new constant symbol called a Skolemconstant

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Artificial Intelligence Predicate Logic Slide 45

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Suppose the KB contains just the followingforall

x King(x)and

Greedy(x)rArr

Evil(x)King(John)Greedy(John)Brother(RichardJohn)

Instantiating the universal sentence in all possible ways we haveKing(John) and Greedy(John) rArr Evil(John)King(Richard) and Greedy(Richard) rArr Evil(Richard)King(John)Greedy(John)Brother(RichardJohn

)

The new KB is propositionalized proposition symbols are

King(John) Greedy(John) Evil(John) King(Richard

) etc

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Artificial Intelligence Predicate Logic Slide 46

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Every FOL KB can be propositionalized so as to preserve entailment

(A ground sentence is entailed by new KB iff entailed by originalKB)

Idea propositionalize KB and query apply resolution return result

Problem with function symbols there are infinitely many ground

terms

eg Father (Father (Father (John

)))

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Artificial Intelligence Predicate Logic Slide 47

983122983141983155983151983148983157983156983145983151983150

Convert to clause form exampleldquoAll Romans who know Marcus either hate

Caesar or think that anyone who hates anyone is

crazyrdquo

forallX [roman(X) ^ know(X Marcus)]

[hate(X Ceasar) v(forallY existZ hate(YZ)

thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 48

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

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Artificial Intelligence Predicate Logic Slide 49

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

forallX [roman(X) ^ know(X Marcus)][hate(X Ceasar) v

(forallY existZ hate(YZ) thinkcrazy(XY))]

=

forallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 50

983122983141983155983151983148983157983156983145983151983150

2 Move not next to terms

Using the following equivalence

a Double negative

not(notp) = p

b DeMorgan

not(a v b) = nota ^ notb

not(a ^ b) = nota v notb

c Equivalence of qualifiers notforallX P(X) = existX notP(X)

notexistX P(X) = forallX notP(X)

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Artificial Intelligence Predicate Logic Slide 51

983122983141983155983151983148983157983156983145983151983150

2 Move not next to termsforallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

=

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) vthinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 52

983122983141983155983151983148983157983156983145983151983150

3 Change name of variables as follows

forallX P(X) v forallX Q(X)

=forallX P(X) v forallY Q(Y)

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Artificial Intelligence Predicate Logic Slide 53

983122983141983155983151983148983157983156983145983151983150

4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 54

983122983141983155983151983148983157983156983145983151983150

5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

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Artificial Intelligence Predicate Logic Slide 55

983122983141983155983151983148983157983156983145983151983150

6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 56

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

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Artificial Intelligence Predicate Logic Slide 57

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 58

983122983141983155983151983148983157983156983145983151983150

8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

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Artificial Intelligence Predicate Logic Slide 59

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9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

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Artificial Intelligence Predicate Logic Slide 60

983122983141983155983151983148983157983156983145983151983150

Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

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Artificial Intelligence Predicate Logic Slide 61

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

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Artificial Intelligence Predicate Logic Slide 62

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θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

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Artificial Intelligence Predicate Logic Slide 63

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

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Artificial Intelligence Predicate Logic Slide 64

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

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Artificial Intelligence Predicate Logic Slide 65

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

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Artificial Intelligence Predicate Logic Slide 66

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

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Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

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Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

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Artificial Intelligence Predicate Logic Slide 69

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

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Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

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Artificial Intelligence Predicate Logic Slide 71

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Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

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Artificial Intelligence Predicate Logic Slide 72

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

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Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

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Artificial Intelligence Predicate Logic Slide 74

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

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Artificial Intelligence Predicate Logic Slide 75

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Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

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Artificial Intelligence Predicate Logic Slide 76

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Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

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892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

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Artificial Intelligence Predicate Logic Slide 17

983109983160983145983155983156983141983150983156983145983137983148 983153983157983137983150983156983145983142983145983139983137983156983145983151983150

existltvariablesgt ltsentencegt

Someone at NUS is smart

exist x At(xNUS) and Smart(x )$

exist x P is true in a model m iff P is true with x being some possible object

in the model Roughly speaking equivalent to the disjunction of instantiations of P

At(KingJohnNUS) and Smart(KingJohn)or At(RichardNUS) and Smart(Richard)or At(NUSNUS) and Smart(NUS)or

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Artificial Intelligence Predicate Logic Slide 18

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Typically and is the main connective with exist

Common mistake usingrArr as the main connective with

exist

exist x At(xNUS)rArr Smart(x

)is true if there is anyone who is not at NUS

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Artificial Intelligence Predicate Logic Slide 19

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forallx forally is the same as forally forallx

existx existy is the same as existy existx

existx forally is not the same as forally existx

existx forally Loves(xy) ldquoThere is a person who loves everyone in the worldrdquo

forally existx Loves(xy) ldquoEveryone in the world is loved by at least one personrdquo

Quantifier duality each can be expressed using the other

forallx Likes(xIceCream) notexistx notLikes(xIceCream )

existx Likes(xBroccoli) notforallx notLikes(xBroccoli

)

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Artificial Intelligence Predicate Logic Slide 20

983109983153983157983137983148983145983156983161

term1 = term2 is true under a given interpretation if andonly if term1 and term2 refer to the same object

Eg definition of Sibling in terms of Parent

forallxy Sibling(xy) hArr [not(x = y) and existmf not (m = f) and Parent(mx)and Parent(fx) and Parent(my) and Parent(fy)]

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Artificial Intelligence Predicate Logic Slide 21

983125983155983145983150983143 983110983119983116

The kinship domain Brothers are siblings

forallxy Brother(xy) hArr Sibling(xy)

Ones mother is ones female parent

forallmc Mother(c) = m hArr (Female(m) and Parent(mc))

ldquoSiblingrdquo is symmetric

forallxy Sibling(xy) hArr Sibling(yx)

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Artificial Intelligence Predicate Logic Slide 22

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Representing the following sentences in Predicate Logic

1 Marcus was a man

2 Macus was a Pompeian

3 All Pompians were Romans4 Caesar was a ruler

5 All Romans were either loyal to Caesar or hated hime

6 Everyone is loyal to someone

7 People only try to assassinate rulers they are not loyal to

8 Marcus tried to assassinate Caesar

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Artificial Intelligence Predicate Logic Slide 23

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1 Marcus was a man

man(Marcus)

2 Macus was a Pompeian

Pompeian(Marcus)

3 All Pompeians were Romans

forallX Pompeian(X) Roman(X)

4 Caesar was a ruler

ruler(Caesar)

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Artificial Intelligence Predicate Logic Slide 24

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

5 All Romans were either loyal to Caesar or hated hime

ldquoorrdquo mean ldquoinclusive orrdquo - OR

forallX Roman(X) loyalto(X Caesar) v hate(X Caesar)

ldquoorrdquo mean ldquoexclusive orrdquo - XOR

forallX Roman(X) [ (loyalto(X Caesar) v hate(X Caesar)) ^

not(loyalto(X Caesar) ^ hate(X Caesar))]

or

forallX Roman(X) [ (loyalto(X Caesar) ^ nothate(X Caesar)) v

(notloyalto(X Caesar) ^ hate(X Caesar))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 25

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

6 Everyone is loyal to someone

forallX exist Y loyalto(XY)

7 People only try to assassinate rulers they are not loyal toforallX forall Y person(X) ^ ruler(Y) ^ tryassassinate(XY)notloyalto(XY)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 26

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

8 Marcus tried to assassinate Caesartryassassinate(Marcus Caesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 27

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

Given the above sentences can we make aconclusion as follows

ldquoMarcus was not loyal to Caesar rdquo

or

notloyalto(MarcusCaesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 28

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)

(Rule What)

Add an implicit sentence

9 ldquoAll men are peoplerdquo

or

forallforallforallforallX man(X) X man(X) person(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 29

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)(9 Substitute)

man(Marcus)

(1)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 30

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

How to represent exceptions example

ldquoPaulus was a Pompeian Paulus was neitherloyal nor hated Caesarrdquo

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 31

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation Semantic Net Simple by adding links

(properties) to subclasses or instances

Predicate Logic Not so simple like

983120983151983149983152983141983145983137983150(983120983137983157983148983157983155) 983134 983212983131983148983151983161983137983148983156983151(983120983137983157983148983157983155 983107983137983141983155983137983154) 983158

983144983137983156983141(983120983157983137983148983157983155 983107983137983141983155983137983154)983133

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 32

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation983118983151983156 983155983151 983155983145983149983152983148983141 983106983141983139983137983157983155983141 983156983144983141983154983141 983145983155 983139983151983150983142983148983145983139983156 983138983141983156983159983141983141983150983098

Pompeian(Paulus) ^ not[loyalto(Paulus Caesar) vhate(Pualus Caesar)]

983137983150983140

forallX Pompeian(X) loyalto(X Caesar) v hate(XCaesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 33

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation A Better way

forallX Roman(X) ^ noteq(X Paulus) loyalto(X Caesar)v hate(X Caesar

Exception

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 34

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

In the case of describing relations betweennumbers example 1 983100 2

2 983100983091

983095 983102 983091 +2 983091 983102 1

991270

rArr

983123983144983151983157983148983140 983150983151983156 983141983160983152983148983145983139983145983156983148983161 983140983141983155983139983154983145983138983141983098983148983156(12) 983148983156(2983091) 991270

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 35

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

rArr Need functions and computable predicates

rArr For example 983107983137983148983148 983137 983142983157983150983139983156983145983151983150 983156983151 983139983151983149983152983157983156983141 (983091+2)

983120983137983155983155 983156983144983141 983154983141983155983157983148983156 983156983151 983156983144983141 983152983154983141983140983145983139983137983156983141 991260983143983156991261983098 983143983156(983095 983091+2)

983111983141983156 983156983144983141 983142983145983150983137983148 983154983141983155983157983148983156 (983124983154983157983141)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 36

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

1 Marcus was a man

man(Marcus)

2 Marcus was a PompeianPompeian(Marcus)

3 Marcus was born in 40 AD

born(Marcus40)

4 All men are mortal

forallX man(X) mortal(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 37

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

5 All Pompeian died when the vocano erupted in 79 AD

erupted(vocano 79) ^ forallX [Pompeian(X) died(X 79)]

6 No mortal lives longer then 150 yearsforallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

7 It is now 1991now = 1991

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Artificial Intelligence Predicate Logic Slide 38

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Question

Is Marcus alive

Or

alive(Marcus now)

ornotnotnotnotalive(Marcus now)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 39

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge

Relation between ldquodeadrdquo and ldquoaliverdquo

8 Alive means not deadforallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 40

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge Time properties of an event for example ldquodeadrdquo

9 Is someone dies the he is dead at all later times

forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 41

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Summary of facts1 man(Marcus)

2 Pompeian(Marcus)

3 born(Marcus 40)

4 forallX man(X) mortal(X)

5 forallX Pompeian(X) died(X 79)

6 erupted(volcano 79)

7 forallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

8 now = 19919 forallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

10 forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 42

notalive(Marcus now)

dead(Marcus now)

(9 Substitute)

died(Marcus T1) gt(now T1)

(10 Substitute)

Pompeian(Marcus) gt(now 79)

(5 Substitute)(5 Substitute)

(2)gt(1991 79)

(8 Substitute)

(computation)

ss

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 43

983125983150983145983158983141983154983155983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983125983113)

Every instantiation of a universally quantified sentence is enta iled by it

forallv α

Subst(vg α)

for any variable v and ground term g

Eg forallx King( x) and Greedy( x)rArr Evil( x

) yieldsKing (John ) and Greedy (John ) rArr Evil (John )

King (Richard ) and Greedy (Richard ) rArr Evil (Richard )

King (Father (John )) and Greedy (Father (John )) rArr Evil (Father (John ))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 44

983109983160983145983155983156983141983150983156983145983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983109983113)

For any sentence α variable v and constant symbol k that

does not

appear elsewhere in the knowledge baseexistv α

Subst(vk α )

Eg exist x Crown( x) and OnHead ( xJohn) yields

Crown(C 1) and OnHead (C 1 John )

provided C 1 is a new constant symbol called a Skolemconstant

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 45

983122983141983140983157983139983156983145983151983150 983156983151 983152983154983151983152983151983155983145983156983145983151983150983137983148 983145983150983142983141983154983141983150983139983141

Suppose the KB contains just the followingforall

x King(x)and

Greedy(x)rArr

Evil(x)King(John)Greedy(John)Brother(RichardJohn)

Instantiating the universal sentence in all possible ways we haveKing(John) and Greedy(John) rArr Evil(John)King(Richard) and Greedy(Richard) rArr Evil(Richard)King(John)Greedy(John)Brother(RichardJohn

)

The new KB is propositionalized proposition symbols are

King(John) Greedy(John) Evil(John) King(Richard

) etc

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4680

Artificial Intelligence Predicate Logic Slide 46

983122983141983140983157983139983156983145983151983150 983139983151983150983156983140983086

Every FOL KB can be propositionalized so as to preserve entailment

(A ground sentence is entailed by new KB iff entailed by originalKB)

Idea propositionalize KB and query apply resolution return result

Problem with function symbols there are infinitely many ground

terms

eg Father (Father (Father (John

)))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 47

983122983141983155983151983148983157983156983145983151983150

Convert to clause form exampleldquoAll Romans who know Marcus either hate

Caesar or think that anyone who hates anyone is

crazyrdquo

forallX [roman(X) ^ know(X Marcus)]

[hate(X Ceasar) v(forallY existZ hate(YZ)

thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 48

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 49

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

forallX [roman(X) ^ know(X Marcus)][hate(X Ceasar) v

(forallY existZ hate(YZ) thinkcrazy(XY))]

=

forallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 50

983122983141983155983151983148983157983156983145983151983150

2 Move not next to terms

Using the following equivalence

a Double negative

not(notp) = p

b DeMorgan

not(a v b) = nota ^ notb

not(a ^ b) = nota v notb

c Equivalence of qualifiers notforallX P(X) = existX notP(X)

notexistX P(X) = forallX notP(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 51

983122983141983155983151983148983157983156983145983151983150

2 Move not next to termsforallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

=

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) vthinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 52

983122983141983155983151983148983157983156983145983151983150

3 Change name of variables as follows

forallX P(X) v forallX Q(X)

=forallX P(X) v forallY Q(Y)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 53

983122983141983155983151983148983157983156983145983151983150

4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 54

983122983141983155983151983148983157983156983145983151983150

5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 55

983122983141983155983151983148983157983156983145983151983150

6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 5680

Artificial Intelligence Predicate Logic Slide 56

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 57

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 58

983122983141983155983151983148983157983156983145983151983150

8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 59

983122983141983155983151983148983157983156983145983151983150

9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 60

983122983141983155983151983148983157983156983145983151983150

Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 61

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 62

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6380

Artificial Intelligence Predicate Logic Slide 63

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 64

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 65

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

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Artificial Intelligence Predicate Logic Slide 66

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

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Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

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Artificial Intelligence Predicate Logic Slide 69

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

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Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

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Artificial Intelligence Predicate Logic Slide 71

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

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Artificial Intelligence Predicate Logic Slide 72

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

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Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

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Artificial Intelligence Predicate Logic Slide 74

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

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Artificial Intelligence Predicate Logic Slide 75

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

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Artificial Intelligence Predicate Logic Slide 76

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Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

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892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

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Artificial Intelligence Predicate Logic Slide 18

983105983150983151983156983144983141983154 983139983151983149983149983151983150 983149983145983155983156983137983147983141 983156983151 983137983158983151983145983140

Typically and is the main connective with exist

Common mistake usingrArr as the main connective with

exist

exist x At(xNUS)rArr Smart(x

)is true if there is anyone who is not at NUS

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Artificial Intelligence Predicate Logic Slide 19

983120983154983151983152983141983154983156983145983141983155 983151983142 983153983157983137983150983156983145983142983145983141983154983155

forallx forally is the same as forally forallx

existx existy is the same as existy existx

existx forally is not the same as forally existx

existx forally Loves(xy) ldquoThere is a person who loves everyone in the worldrdquo

forally existx Loves(xy) ldquoEveryone in the world is loved by at least one personrdquo

Quantifier duality each can be expressed using the other

forallx Likes(xIceCream) notexistx notLikes(xIceCream )

existx Likes(xBroccoli) notforallx notLikes(xBroccoli

)

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Artificial Intelligence Predicate Logic Slide 20

983109983153983157983137983148983145983156983161

term1 = term2 is true under a given interpretation if andonly if term1 and term2 refer to the same object

Eg definition of Sibling in terms of Parent

forallxy Sibling(xy) hArr [not(x = y) and existmf not (m = f) and Parent(mx)and Parent(fx) and Parent(my) and Parent(fy)]

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Artificial Intelligence Predicate Logic Slide 21

983125983155983145983150983143 983110983119983116

The kinship domain Brothers are siblings

forallxy Brother(xy) hArr Sibling(xy)

Ones mother is ones female parent

forallmc Mother(c) = m hArr (Female(m) and Parent(mc))

ldquoSiblingrdquo is symmetric

forallxy Sibling(xy) hArr Sibling(yx)

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Artificial Intelligence Predicate Logic Slide 22

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

Representing the following sentences in Predicate Logic

1 Marcus was a man

2 Macus was a Pompeian

3 All Pompians were Romans4 Caesar was a ruler

5 All Romans were either loyal to Caesar or hated hime

6 Everyone is loyal to someone

7 People only try to assassinate rulers they are not loyal to

8 Marcus tried to assassinate Caesar

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Artificial Intelligence Predicate Logic Slide 23

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

1 Marcus was a man

man(Marcus)

2 Macus was a Pompeian

Pompeian(Marcus)

3 All Pompeians were Romans

forallX Pompeian(X) Roman(X)

4 Caesar was a ruler

ruler(Caesar)

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Artificial Intelligence Predicate Logic Slide 24

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5 All Romans were either loyal to Caesar or hated hime

ldquoorrdquo mean ldquoinclusive orrdquo - OR

forallX Roman(X) loyalto(X Caesar) v hate(X Caesar)

ldquoorrdquo mean ldquoexclusive orrdquo - XOR

forallX Roman(X) [ (loyalto(X Caesar) v hate(X Caesar)) ^

not(loyalto(X Caesar) ^ hate(X Caesar))]

or

forallX Roman(X) [ (loyalto(X Caesar) ^ nothate(X Caesar)) v

(notloyalto(X Caesar) ^ hate(X Caesar))]

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Artificial Intelligence Predicate Logic Slide 25

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

6 Everyone is loyal to someone

forallX exist Y loyalto(XY)

7 People only try to assassinate rulers they are not loyal toforallX forall Y person(X) ^ ruler(Y) ^ tryassassinate(XY)notloyalto(XY)

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Artificial Intelligence Predicate Logic Slide 26

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

8 Marcus tried to assassinate Caesartryassassinate(Marcus Caesar)

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Artificial Intelligence Predicate Logic Slide 27

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

Given the above sentences can we make aconclusion as follows

ldquoMarcus was not loyal to Caesar rdquo

or

notloyalto(MarcusCaesar)

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Artificial Intelligence Predicate Logic Slide 28

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)

(Rule What)

Add an implicit sentence

9 ldquoAll men are peoplerdquo

or

forallforallforallforallX man(X) X man(X) person(X)

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Artificial Intelligence Predicate Logic Slide 29

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)(9 Substitute)

man(Marcus)

(1)

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Artificial Intelligence Predicate Logic Slide 30

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

How to represent exceptions example

ldquoPaulus was a Pompeian Paulus was neitherloyal nor hated Caesarrdquo

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Artificial Intelligence Predicate Logic Slide 31

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation Semantic Net Simple by adding links

(properties) to subclasses or instances

Predicate Logic Not so simple like

983120983151983149983152983141983145983137983150(983120983137983157983148983157983155) 983134 983212983131983148983151983161983137983148983156983151(983120983137983157983148983157983155 983107983137983141983155983137983154) 983158

983144983137983156983141(983120983157983137983148983157983155 983107983137983141983155983137983154)983133

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 32

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation983118983151983156 983155983151 983155983145983149983152983148983141 983106983141983139983137983157983155983141 983156983144983141983154983141 983145983155 983139983151983150983142983148983145983139983156 983138983141983156983159983141983141983150983098

Pompeian(Paulus) ^ not[loyalto(Paulus Caesar) vhate(Pualus Caesar)]

983137983150983140

forallX Pompeian(X) loyalto(X Caesar) v hate(XCaesar)

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Artificial Intelligence Predicate Logic Slide 33

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation A Better way

forallX Roman(X) ^ noteq(X Paulus) loyalto(X Caesar)v hate(X Caesar

Exception

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Artificial Intelligence Predicate Logic Slide 34

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

In the case of describing relations betweennumbers example 1 983100 2

2 983100983091

983095 983102 983091 +2 983091 983102 1

991270

rArr

983123983144983151983157983148983140 983150983151983156 983141983160983152983148983145983139983145983156983148983161 983140983141983155983139983154983145983138983141983098983148983156(12) 983148983156(2983091) 991270

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 35

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

rArr Need functions and computable predicates

rArr For example 983107983137983148983148 983137 983142983157983150983139983156983145983151983150 983156983151 983139983151983149983152983157983156983141 (983091+2)

983120983137983155983155 983156983144983141 983154983141983155983157983148983156 983156983151 983156983144983141 983152983154983141983140983145983139983137983156983141 991260983143983156991261983098 983143983156(983095 983091+2)

983111983141983156 983156983144983141 983142983145983150983137983148 983154983141983155983157983148983156 (983124983154983157983141)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 36

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

1 Marcus was a man

man(Marcus)

2 Marcus was a PompeianPompeian(Marcus)

3 Marcus was born in 40 AD

born(Marcus40)

4 All men are mortal

forallX man(X) mortal(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 37

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

5 All Pompeian died when the vocano erupted in 79 AD

erupted(vocano 79) ^ forallX [Pompeian(X) died(X 79)]

6 No mortal lives longer then 150 yearsforallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

7 It is now 1991now = 1991

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Artificial Intelligence Predicate Logic Slide 38

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Question

Is Marcus alive

Or

alive(Marcus now)

ornotnotnotnotalive(Marcus now)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 39

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Add implicit knowledge

Relation between ldquodeadrdquo and ldquoaliverdquo

8 Alive means not deadforallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

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Artificial Intelligence Predicate Logic Slide 40

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Add implicit knowledge Time properties of an event for example ldquodeadrdquo

9 Is someone dies the he is dead at all later times

forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

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Artificial Intelligence Predicate Logic Slide 41

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Summary of facts1 man(Marcus)

2 Pompeian(Marcus)

3 born(Marcus 40)

4 forallX man(X) mortal(X)

5 forallX Pompeian(X) died(X 79)

6 erupted(volcano 79)

7 forallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

8 now = 19919 forallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

10 forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

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Artificial Intelligence Predicate Logic Slide 42

notalive(Marcus now)

dead(Marcus now)

(9 Substitute)

died(Marcus T1) gt(now T1)

(10 Substitute)

Pompeian(Marcus) gt(now 79)

(5 Substitute)(5 Substitute)

(2)gt(1991 79)

(8 Substitute)

(computation)

ss

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Artificial Intelligence Predicate Logic Slide 43

983125983150983145983158983141983154983155983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983125983113)

Every instantiation of a universally quantified sentence is enta iled by it

forallv α

Subst(vg α)

for any variable v and ground term g

Eg forallx King( x) and Greedy( x)rArr Evil( x

) yieldsKing (John ) and Greedy (John ) rArr Evil (John )

King (Richard ) and Greedy (Richard ) rArr Evil (Richard )

King (Father (John )) and Greedy (Father (John )) rArr Evil (Father (John ))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 44

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For any sentence α variable v and constant symbol k that

does not

appear elsewhere in the knowledge baseexistv α

Subst(vk α )

Eg exist x Crown( x) and OnHead ( xJohn) yields

Crown(C 1) and OnHead (C 1 John )

provided C 1 is a new constant symbol called a Skolemconstant

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 45

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Suppose the KB contains just the followingforall

x King(x)and

Greedy(x)rArr

Evil(x)King(John)Greedy(John)Brother(RichardJohn)

Instantiating the universal sentence in all possible ways we haveKing(John) and Greedy(John) rArr Evil(John)King(Richard) and Greedy(Richard) rArr Evil(Richard)King(John)Greedy(John)Brother(RichardJohn

)

The new KB is propositionalized proposition symbols are

King(John) Greedy(John) Evil(John) King(Richard

) etc

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 46

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Every FOL KB can be propositionalized so as to preserve entailment

(A ground sentence is entailed by new KB iff entailed by originalKB)

Idea propositionalize KB and query apply resolution return result

Problem with function symbols there are infinitely many ground

terms

eg Father (Father (Father (John

)))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 47

983122983141983155983151983148983157983156983145983151983150

Convert to clause form exampleldquoAll Romans who know Marcus either hate

Caesar or think that anyone who hates anyone is

crazyrdquo

forallX [roman(X) ^ know(X Marcus)]

[hate(X Ceasar) v(forallY existZ hate(YZ)

thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 48

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 49

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

forallX [roman(X) ^ know(X Marcus)][hate(X Ceasar) v

(forallY existZ hate(YZ) thinkcrazy(XY))]

=

forallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 50

983122983141983155983151983148983157983156983145983151983150

2 Move not next to terms

Using the following equivalence

a Double negative

not(notp) = p

b DeMorgan

not(a v b) = nota ^ notb

not(a ^ b) = nota v notb

c Equivalence of qualifiers notforallX P(X) = existX notP(X)

notexistX P(X) = forallX notP(X)

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Artificial Intelligence Predicate Logic Slide 51

983122983141983155983151983148983157983156983145983151983150

2 Move not next to termsforallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

=

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) vthinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 52

983122983141983155983151983148983157983156983145983151983150

3 Change name of variables as follows

forallX P(X) v forallX Q(X)

=forallX P(X) v forallY Q(Y)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 53

983122983141983155983151983148983157983156983145983151983150

4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 54

983122983141983155983151983148983157983156983145983151983150

5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

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Artificial Intelligence Predicate Logic Slide 55

983122983141983155983151983148983157983156983145983151983150

6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 56

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

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Artificial Intelligence Predicate Logic Slide 57

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 58

983122983141983155983151983148983157983156983145983151983150

8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

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Artificial Intelligence Predicate Logic Slide 59

983122983141983155983151983148983157983156983145983151983150

9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

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Artificial Intelligence Predicate Logic Slide 60

983122983141983155983151983148983157983156983145983151983150

Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

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Artificial Intelligence Predicate Logic Slide 61

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 62

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

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Artificial Intelligence Predicate Logic Slide 63

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 64

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 65

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 66

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

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Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

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Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

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Artificial Intelligence Predicate Logic Slide 69

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 71

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

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Artificial Intelligence Predicate Logic Slide 72

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 74

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 75

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

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Artificial Intelligence Predicate Logic Slide 76

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

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892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

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Artificial Intelligence Predicate Logic Slide 19

983120983154983151983152983141983154983156983145983141983155 983151983142 983153983157983137983150983156983145983142983145983141983154983155

forallx forally is the same as forally forallx

existx existy is the same as existy existx

existx forally is not the same as forally existx

existx forally Loves(xy) ldquoThere is a person who loves everyone in the worldrdquo

forally existx Loves(xy) ldquoEveryone in the world is loved by at least one personrdquo

Quantifier duality each can be expressed using the other

forallx Likes(xIceCream) notexistx notLikes(xIceCream )

existx Likes(xBroccoli) notforallx notLikes(xBroccoli

)

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Artificial Intelligence Predicate Logic Slide 20

983109983153983157983137983148983145983156983161

term1 = term2 is true under a given interpretation if andonly if term1 and term2 refer to the same object

Eg definition of Sibling in terms of Parent

forallxy Sibling(xy) hArr [not(x = y) and existmf not (m = f) and Parent(mx)and Parent(fx) and Parent(my) and Parent(fy)]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 21

983125983155983145983150983143 983110983119983116

The kinship domain Brothers are siblings

forallxy Brother(xy) hArr Sibling(xy)

Ones mother is ones female parent

forallmc Mother(c) = m hArr (Female(m) and Parent(mc))

ldquoSiblingrdquo is symmetric

forallxy Sibling(xy) hArr Sibling(yx)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 22

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

Representing the following sentences in Predicate Logic

1 Marcus was a man

2 Macus was a Pompeian

3 All Pompians were Romans4 Caesar was a ruler

5 All Romans were either loyal to Caesar or hated hime

6 Everyone is loyal to someone

7 People only try to assassinate rulers they are not loyal to

8 Marcus tried to assassinate Caesar

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 23

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

1 Marcus was a man

man(Marcus)

2 Macus was a Pompeian

Pompeian(Marcus)

3 All Pompeians were Romans

forallX Pompeian(X) Roman(X)

4 Caesar was a ruler

ruler(Caesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 24

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

5 All Romans were either loyal to Caesar or hated hime

ldquoorrdquo mean ldquoinclusive orrdquo - OR

forallX Roman(X) loyalto(X Caesar) v hate(X Caesar)

ldquoorrdquo mean ldquoexclusive orrdquo - XOR

forallX Roman(X) [ (loyalto(X Caesar) v hate(X Caesar)) ^

not(loyalto(X Caesar) ^ hate(X Caesar))]

or

forallX Roman(X) [ (loyalto(X Caesar) ^ nothate(X Caesar)) v

(notloyalto(X Caesar) ^ hate(X Caesar))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 25

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

6 Everyone is loyal to someone

forallX exist Y loyalto(XY)

7 People only try to assassinate rulers they are not loyal toforallX forall Y person(X) ^ ruler(Y) ^ tryassassinate(XY)notloyalto(XY)

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Artificial Intelligence Predicate Logic Slide 26

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

8 Marcus tried to assassinate Caesartryassassinate(Marcus Caesar)

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Artificial Intelligence Predicate Logic Slide 27

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

Given the above sentences can we make aconclusion as follows

ldquoMarcus was not loyal to Caesar rdquo

or

notloyalto(MarcusCaesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 28

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)

(Rule What)

Add an implicit sentence

9 ldquoAll men are peoplerdquo

or

forallforallforallforallX man(X) X man(X) person(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 29

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)(9 Substitute)

man(Marcus)

(1)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 30

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

How to represent exceptions example

ldquoPaulus was a Pompeian Paulus was neitherloyal nor hated Caesarrdquo

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 31

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation Semantic Net Simple by adding links

(properties) to subclasses or instances

Predicate Logic Not so simple like

983120983151983149983152983141983145983137983150(983120983137983157983148983157983155) 983134 983212983131983148983151983161983137983148983156983151(983120983137983157983148983157983155 983107983137983141983155983137983154) 983158

983144983137983156983141(983120983157983137983148983157983155 983107983137983141983155983137983154)983133

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 32

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation983118983151983156 983155983151 983155983145983149983152983148983141 983106983141983139983137983157983155983141 983156983144983141983154983141 983145983155 983139983151983150983142983148983145983139983156 983138983141983156983159983141983141983150983098

Pompeian(Paulus) ^ not[loyalto(Paulus Caesar) vhate(Pualus Caesar)]

983137983150983140

forallX Pompeian(X) loyalto(X Caesar) v hate(XCaesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 33

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation A Better way

forallX Roman(X) ^ noteq(X Paulus) loyalto(X Caesar)v hate(X Caesar

Exception

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Artificial Intelligence Predicate Logic Slide 34

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

In the case of describing relations betweennumbers example 1 983100 2

2 983100983091

983095 983102 983091 +2 983091 983102 1

991270

rArr

983123983144983151983157983148983140 983150983151983156 983141983160983152983148983145983139983145983156983148983161 983140983141983155983139983154983145983138983141983098983148983156(12) 983148983156(2983091) 991270

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 35

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

rArr Need functions and computable predicates

rArr For example 983107983137983148983148 983137 983142983157983150983139983156983145983151983150 983156983151 983139983151983149983152983157983156983141 (983091+2)

983120983137983155983155 983156983144983141 983154983141983155983157983148983156 983156983151 983156983144983141 983152983154983141983140983145983139983137983156983141 991260983143983156991261983098 983143983156(983095 983091+2)

983111983141983156 983156983144983141 983142983145983150983137983148 983154983141983155983157983148983156 (983124983154983157983141)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 36

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

1 Marcus was a man

man(Marcus)

2 Marcus was a PompeianPompeian(Marcus)

3 Marcus was born in 40 AD

born(Marcus40)

4 All men are mortal

forallX man(X) mortal(X)

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Artificial Intelligence Predicate Logic Slide 37

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

5 All Pompeian died when the vocano erupted in 79 AD

erupted(vocano 79) ^ forallX [Pompeian(X) died(X 79)]

6 No mortal lives longer then 150 yearsforallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

7 It is now 1991now = 1991

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Artificial Intelligence Predicate Logic Slide 38

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Question

Is Marcus alive

Or

alive(Marcus now)

ornotnotnotnotalive(Marcus now)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 39

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge

Relation between ldquodeadrdquo and ldquoaliverdquo

8 Alive means not deadforallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 40

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge Time properties of an event for example ldquodeadrdquo

9 Is someone dies the he is dead at all later times

forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

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Artificial Intelligence Predicate Logic Slide 41

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Summary of facts1 man(Marcus)

2 Pompeian(Marcus)

3 born(Marcus 40)

4 forallX man(X) mortal(X)

5 forallX Pompeian(X) died(X 79)

6 erupted(volcano 79)

7 forallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

8 now = 19919 forallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

10 forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 42

notalive(Marcus now)

dead(Marcus now)

(9 Substitute)

died(Marcus T1) gt(now T1)

(10 Substitute)

Pompeian(Marcus) gt(now 79)

(5 Substitute)(5 Substitute)

(2)gt(1991 79)

(8 Substitute)

(computation)

ss

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 43

983125983150983145983158983141983154983155983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983125983113)

Every instantiation of a universally quantified sentence is enta iled by it

forallv α

Subst(vg α)

for any variable v and ground term g

Eg forallx King( x) and Greedy( x)rArr Evil( x

) yieldsKing (John ) and Greedy (John ) rArr Evil (John )

King (Richard ) and Greedy (Richard ) rArr Evil (Richard )

King (Father (John )) and Greedy (Father (John )) rArr Evil (Father (John ))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 44

983109983160983145983155983156983141983150983156983145983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983109983113)

For any sentence α variable v and constant symbol k that

does not

appear elsewhere in the knowledge baseexistv α

Subst(vk α )

Eg exist x Crown( x) and OnHead ( xJohn) yields

Crown(C 1) and OnHead (C 1 John )

provided C 1 is a new constant symbol called a Skolemconstant

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 45

983122983141983140983157983139983156983145983151983150 983156983151 983152983154983151983152983151983155983145983156983145983151983150983137983148 983145983150983142983141983154983141983150983139983141

Suppose the KB contains just the followingforall

x King(x)and

Greedy(x)rArr

Evil(x)King(John)Greedy(John)Brother(RichardJohn)

Instantiating the universal sentence in all possible ways we haveKing(John) and Greedy(John) rArr Evil(John)King(Richard) and Greedy(Richard) rArr Evil(Richard)King(John)Greedy(John)Brother(RichardJohn

)

The new KB is propositionalized proposition symbols are

King(John) Greedy(John) Evil(John) King(Richard

) etc

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 46

983122983141983140983157983139983156983145983151983150 983139983151983150983156983140983086

Every FOL KB can be propositionalized so as to preserve entailment

(A ground sentence is entailed by new KB iff entailed by originalKB)

Idea propositionalize KB and query apply resolution return result

Problem with function symbols there are infinitely many ground

terms

eg Father (Father (Father (John

)))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 47

983122983141983155983151983148983157983156983145983151983150

Convert to clause form exampleldquoAll Romans who know Marcus either hate

Caesar or think that anyone who hates anyone is

crazyrdquo

forallX [roman(X) ^ know(X Marcus)]

[hate(X Ceasar) v(forallY existZ hate(YZ)

thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 48

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 49

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

forallX [roman(X) ^ know(X Marcus)][hate(X Ceasar) v

(forallY existZ hate(YZ) thinkcrazy(XY))]

=

forallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 50

983122983141983155983151983148983157983156983145983151983150

2 Move not next to terms

Using the following equivalence

a Double negative

not(notp) = p

b DeMorgan

not(a v b) = nota ^ notb

not(a ^ b) = nota v notb

c Equivalence of qualifiers notforallX P(X) = existX notP(X)

notexistX P(X) = forallX notP(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 51

983122983141983155983151983148983157983156983145983151983150

2 Move not next to termsforallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

=

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) vthinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 52

983122983141983155983151983148983157983156983145983151983150

3 Change name of variables as follows

forallX P(X) v forallX Q(X)

=forallX P(X) v forallY Q(Y)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 53

983122983141983155983151983148983157983156983145983151983150

4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 54

983122983141983155983151983148983157983156983145983151983150

5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 55

983122983141983155983151983148983157983156983145983151983150

6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 56

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 57

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 58

983122983141983155983151983148983157983156983145983151983150

8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 59

983122983141983155983151983148983157983156983145983151983150

9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 60

983122983141983155983151983148983157983156983145983151983150

Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 61

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 62

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 63

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 64

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 65

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 66

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 69

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 71

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

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Artificial Intelligence Predicate Logic Slide 72

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 74

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 75

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

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Artificial Intelligence Predicate Logic Slide 76

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

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892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

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Artificial Intelligence Predicate Logic Slide 20

983109983153983157983137983148983145983156983161

term1 = term2 is true under a given interpretation if andonly if term1 and term2 refer to the same object

Eg definition of Sibling in terms of Parent

forallxy Sibling(xy) hArr [not(x = y) and existmf not (m = f) and Parent(mx)and Parent(fx) and Parent(my) and Parent(fy)]

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Artificial Intelligence Predicate Logic Slide 21

983125983155983145983150983143 983110983119983116

The kinship domain Brothers are siblings

forallxy Brother(xy) hArr Sibling(xy)

Ones mother is ones female parent

forallmc Mother(c) = m hArr (Female(m) and Parent(mc))

ldquoSiblingrdquo is symmetric

forallxy Sibling(xy) hArr Sibling(yx)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 22

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

Representing the following sentences in Predicate Logic

1 Marcus was a man

2 Macus was a Pompeian

3 All Pompians were Romans4 Caesar was a ruler

5 All Romans were either loyal to Caesar or hated hime

6 Everyone is loyal to someone

7 People only try to assassinate rulers they are not loyal to

8 Marcus tried to assassinate Caesar

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Artificial Intelligence Predicate Logic Slide 23

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

1 Marcus was a man

man(Marcus)

2 Macus was a Pompeian

Pompeian(Marcus)

3 All Pompeians were Romans

forallX Pompeian(X) Roman(X)

4 Caesar was a ruler

ruler(Caesar)

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Artificial Intelligence Predicate Logic Slide 24

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

5 All Romans were either loyal to Caesar or hated hime

ldquoorrdquo mean ldquoinclusive orrdquo - OR

forallX Roman(X) loyalto(X Caesar) v hate(X Caesar)

ldquoorrdquo mean ldquoexclusive orrdquo - XOR

forallX Roman(X) [ (loyalto(X Caesar) v hate(X Caesar)) ^

not(loyalto(X Caesar) ^ hate(X Caesar))]

or

forallX Roman(X) [ (loyalto(X Caesar) ^ nothate(X Caesar)) v

(notloyalto(X Caesar) ^ hate(X Caesar))]

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Artificial Intelligence Predicate Logic Slide 25

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

6 Everyone is loyal to someone

forallX exist Y loyalto(XY)

7 People only try to assassinate rulers they are not loyal toforallX forall Y person(X) ^ ruler(Y) ^ tryassassinate(XY)notloyalto(XY)

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Artificial Intelligence Predicate Logic Slide 26

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

8 Marcus tried to assassinate Caesartryassassinate(Marcus Caesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 27

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

Given the above sentences can we make aconclusion as follows

ldquoMarcus was not loyal to Caesar rdquo

or

notloyalto(MarcusCaesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 28

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)

(Rule What)

Add an implicit sentence

9 ldquoAll men are peoplerdquo

or

forallforallforallforallX man(X) X man(X) person(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 29

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)(9 Substitute)

man(Marcus)

(1)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 30

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

How to represent exceptions example

ldquoPaulus was a Pompeian Paulus was neitherloyal nor hated Caesarrdquo

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Artificial Intelligence Predicate Logic Slide 31

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation Semantic Net Simple by adding links

(properties) to subclasses or instances

Predicate Logic Not so simple like

983120983151983149983152983141983145983137983150(983120983137983157983148983157983155) 983134 983212983131983148983151983161983137983148983156983151(983120983137983157983148983157983155 983107983137983141983155983137983154) 983158

983144983137983156983141(983120983157983137983148983157983155 983107983137983141983155983137983154)983133

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 32

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation983118983151983156 983155983151 983155983145983149983152983148983141 983106983141983139983137983157983155983141 983156983144983141983154983141 983145983155 983139983151983150983142983148983145983139983156 983138983141983156983159983141983141983150983098

Pompeian(Paulus) ^ not[loyalto(Paulus Caesar) vhate(Pualus Caesar)]

983137983150983140

forallX Pompeian(X) loyalto(X Caesar) v hate(XCaesar)

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Artificial Intelligence Predicate Logic Slide 33

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation A Better way

forallX Roman(X) ^ noteq(X Paulus) loyalto(X Caesar)v hate(X Caesar

Exception

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Artificial Intelligence Predicate Logic Slide 34

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

In the case of describing relations betweennumbers example 1 983100 2

2 983100983091

983095 983102 983091 +2 983091 983102 1

991270

rArr

983123983144983151983157983148983140 983150983151983156 983141983160983152983148983145983139983145983156983148983161 983140983141983155983139983154983145983138983141983098983148983156(12) 983148983156(2983091) 991270

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 35

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

rArr Need functions and computable predicates

rArr For example 983107983137983148983148 983137 983142983157983150983139983156983145983151983150 983156983151 983139983151983149983152983157983156983141 (983091+2)

983120983137983155983155 983156983144983141 983154983141983155983157983148983156 983156983151 983156983144983141 983152983154983141983140983145983139983137983156983141 991260983143983156991261983098 983143983156(983095 983091+2)

983111983141983156 983156983144983141 983142983145983150983137983148 983154983141983155983157983148983156 (983124983154983157983141)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 36

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

1 Marcus was a man

man(Marcus)

2 Marcus was a PompeianPompeian(Marcus)

3 Marcus was born in 40 AD

born(Marcus40)

4 All men are mortal

forallX man(X) mortal(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 37

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

5 All Pompeian died when the vocano erupted in 79 AD

erupted(vocano 79) ^ forallX [Pompeian(X) died(X 79)]

6 No mortal lives longer then 150 yearsforallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

7 It is now 1991now = 1991

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 38

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Question

Is Marcus alive

Or

alive(Marcus now)

ornotnotnotnotalive(Marcus now)

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Artificial Intelligence Predicate Logic Slide 39

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Add implicit knowledge

Relation between ldquodeadrdquo and ldquoaliverdquo

8 Alive means not deadforallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

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Artificial Intelligence Predicate Logic Slide 40

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Add implicit knowledge Time properties of an event for example ldquodeadrdquo

9 Is someone dies the he is dead at all later times

forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

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Artificial Intelligence Predicate Logic Slide 41

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Summary of facts1 man(Marcus)

2 Pompeian(Marcus)

3 born(Marcus 40)

4 forallX man(X) mortal(X)

5 forallX Pompeian(X) died(X 79)

6 erupted(volcano 79)

7 forallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

8 now = 19919 forallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

10 forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

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Artificial Intelligence Predicate Logic Slide 42

notalive(Marcus now)

dead(Marcus now)

(9 Substitute)

died(Marcus T1) gt(now T1)

(10 Substitute)

Pompeian(Marcus) gt(now 79)

(5 Substitute)(5 Substitute)

(2)gt(1991 79)

(8 Substitute)

(computation)

ss

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Artificial Intelligence Predicate Logic Slide 43

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Every instantiation of a universally quantified sentence is enta iled by it

forallv α

Subst(vg α)

for any variable v and ground term g

Eg forallx King( x) and Greedy( x)rArr Evil( x

) yieldsKing (John ) and Greedy (John ) rArr Evil (John )

King (Richard ) and Greedy (Richard ) rArr Evil (Richard )

King (Father (John )) and Greedy (Father (John )) rArr Evil (Father (John ))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 44

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For any sentence α variable v and constant symbol k that

does not

appear elsewhere in the knowledge baseexistv α

Subst(vk α )

Eg exist x Crown( x) and OnHead ( xJohn) yields

Crown(C 1) and OnHead (C 1 John )

provided C 1 is a new constant symbol called a Skolemconstant

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 45

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Suppose the KB contains just the followingforall

x King(x)and

Greedy(x)rArr

Evil(x)King(John)Greedy(John)Brother(RichardJohn)

Instantiating the universal sentence in all possible ways we haveKing(John) and Greedy(John) rArr Evil(John)King(Richard) and Greedy(Richard) rArr Evil(Richard)King(John)Greedy(John)Brother(RichardJohn

)

The new KB is propositionalized proposition symbols are

King(John) Greedy(John) Evil(John) King(Richard

) etc

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Artificial Intelligence Predicate Logic Slide 46

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Every FOL KB can be propositionalized so as to preserve entailment

(A ground sentence is entailed by new KB iff entailed by originalKB)

Idea propositionalize KB and query apply resolution return result

Problem with function symbols there are infinitely many ground

terms

eg Father (Father (Father (John

)))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 47

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Convert to clause form exampleldquoAll Romans who know Marcus either hate

Caesar or think that anyone who hates anyone is

crazyrdquo

forallX [roman(X) ^ know(X Marcus)]

[hate(X Ceasar) v(forallY existZ hate(YZ)

thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 48

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 49

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

forallX [roman(X) ^ know(X Marcus)][hate(X Ceasar) v

(forallY existZ hate(YZ) thinkcrazy(XY))]

=

forallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 50

983122983141983155983151983148983157983156983145983151983150

2 Move not next to terms

Using the following equivalence

a Double negative

not(notp) = p

b DeMorgan

not(a v b) = nota ^ notb

not(a ^ b) = nota v notb

c Equivalence of qualifiers notforallX P(X) = existX notP(X)

notexistX P(X) = forallX notP(X)

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Artificial Intelligence Predicate Logic Slide 51

983122983141983155983151983148983157983156983145983151983150

2 Move not next to termsforallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

=

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) vthinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 52

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3 Change name of variables as follows

forallX P(X) v forallX Q(X)

=forallX P(X) v forallY Q(Y)

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Artificial Intelligence Predicate Logic Slide 53

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4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 54

983122983141983155983151983148983157983156983145983151983150

5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

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Artificial Intelligence Predicate Logic Slide 55

983122983141983155983151983148983157983156983145983151983150

6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 56

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

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Artificial Intelligence Predicate Logic Slide 57

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

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Artificial Intelligence Predicate Logic Slide 58

983122983141983155983151983148983157983156983145983151983150

8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

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Artificial Intelligence Predicate Logic Slide 59

983122983141983155983151983148983157983156983145983151983150

9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

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Artificial Intelligence Predicate Logic Slide 60

983122983141983155983151983148983157983156983145983151983150

Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

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Artificial Intelligence Predicate Logic Slide 61

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X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

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Artificial Intelligence Predicate Logic Slide 62

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θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

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Artificial Intelligence Predicate Logic Slide 63

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

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Artificial Intelligence Predicate Logic Slide 64

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

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Artificial Intelligence Predicate Logic Slide 65

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

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Artificial Intelligence Predicate Logic Slide 66

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

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Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

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Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

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Artificial Intelligence Predicate Logic Slide 69

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 71

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

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Artificial Intelligence Predicate Logic Slide 72

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 74

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

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Artificial Intelligence Predicate Logic Slide 75

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

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Artificial Intelligence Predicate Logic Slide 76

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

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892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 21

983125983155983145983150983143 983110983119983116

The kinship domain Brothers are siblings

forallxy Brother(xy) hArr Sibling(xy)

Ones mother is ones female parent

forallmc Mother(c) = m hArr (Female(m) and Parent(mc))

ldquoSiblingrdquo is symmetric

forallxy Sibling(xy) hArr Sibling(yx)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 22

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

Representing the following sentences in Predicate Logic

1 Marcus was a man

2 Macus was a Pompeian

3 All Pompians were Romans4 Caesar was a ruler

5 All Romans were either loyal to Caesar or hated hime

6 Everyone is loyal to someone

7 People only try to assassinate rulers they are not loyal to

8 Marcus tried to assassinate Caesar

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 23

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

1 Marcus was a man

man(Marcus)

2 Macus was a Pompeian

Pompeian(Marcus)

3 All Pompeians were Romans

forallX Pompeian(X) Roman(X)

4 Caesar was a ruler

ruler(Caesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 24

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

5 All Romans were either loyal to Caesar or hated hime

ldquoorrdquo mean ldquoinclusive orrdquo - OR

forallX Roman(X) loyalto(X Caesar) v hate(X Caesar)

ldquoorrdquo mean ldquoexclusive orrdquo - XOR

forallX Roman(X) [ (loyalto(X Caesar) v hate(X Caesar)) ^

not(loyalto(X Caesar) ^ hate(X Caesar))]

or

forallX Roman(X) [ (loyalto(X Caesar) ^ nothate(X Caesar)) v

(notloyalto(X Caesar) ^ hate(X Caesar))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 25

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

6 Everyone is loyal to someone

forallX exist Y loyalto(XY)

7 People only try to assassinate rulers they are not loyal toforallX forall Y person(X) ^ ruler(Y) ^ tryassassinate(XY)notloyalto(XY)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 26

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

8 Marcus tried to assassinate Caesartryassassinate(Marcus Caesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 27

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

Given the above sentences can we make aconclusion as follows

ldquoMarcus was not loyal to Caesar rdquo

or

notloyalto(MarcusCaesar)

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 2880

Artificial Intelligence Predicate Logic Slide 28

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)

(Rule What)

Add an implicit sentence

9 ldquoAll men are peoplerdquo

or

forallforallforallforallX man(X) X man(X) person(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 29

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)(9 Substitute)

man(Marcus)

(1)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 30

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

How to represent exceptions example

ldquoPaulus was a Pompeian Paulus was neitherloyal nor hated Caesarrdquo

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 31

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation Semantic Net Simple by adding links

(properties) to subclasses or instances

Predicate Logic Not so simple like

983120983151983149983152983141983145983137983150(983120983137983157983148983157983155) 983134 983212983131983148983151983161983137983148983156983151(983120983137983157983148983157983155 983107983137983141983155983137983154) 983158

983144983137983156983141(983120983157983137983148983157983155 983107983137983141983155983137983154)983133

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 32

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation983118983151983156 983155983151 983155983145983149983152983148983141 983106983141983139983137983157983155983141 983156983144983141983154983141 983145983155 983139983151983150983142983148983145983139983156 983138983141983156983159983141983141983150983098

Pompeian(Paulus) ^ not[loyalto(Paulus Caesar) vhate(Pualus Caesar)]

983137983150983140

forallX Pompeian(X) loyalto(X Caesar) v hate(XCaesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 33

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation A Better way

forallX Roman(X) ^ noteq(X Paulus) loyalto(X Caesar)v hate(X Caesar

Exception

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 34

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

In the case of describing relations betweennumbers example 1 983100 2

2 983100983091

983095 983102 983091 +2 983091 983102 1

991270

rArr

983123983144983151983157983148983140 983150983151983156 983141983160983152983148983145983139983145983156983148983161 983140983141983155983139983154983145983138983141983098983148983156(12) 983148983156(2983091) 991270

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 35

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

rArr Need functions and computable predicates

rArr For example 983107983137983148983148 983137 983142983157983150983139983156983145983151983150 983156983151 983139983151983149983152983157983156983141 (983091+2)

983120983137983155983155 983156983144983141 983154983141983155983157983148983156 983156983151 983156983144983141 983152983154983141983140983145983139983137983156983141 991260983143983156991261983098 983143983156(983095 983091+2)

983111983141983156 983156983144983141 983142983145983150983137983148 983154983141983155983157983148983156 (983124983154983157983141)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 36

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

1 Marcus was a man

man(Marcus)

2 Marcus was a PompeianPompeian(Marcus)

3 Marcus was born in 40 AD

born(Marcus40)

4 All men are mortal

forallX man(X) mortal(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 37

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

5 All Pompeian died when the vocano erupted in 79 AD

erupted(vocano 79) ^ forallX [Pompeian(X) died(X 79)]

6 No mortal lives longer then 150 yearsforallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

7 It is now 1991now = 1991

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 38

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Question

Is Marcus alive

Or

alive(Marcus now)

ornotnotnotnotalive(Marcus now)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 39

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge

Relation between ldquodeadrdquo and ldquoaliverdquo

8 Alive means not deadforallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 40

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge Time properties of an event for example ldquodeadrdquo

9 Is someone dies the he is dead at all later times

forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4180

Artificial Intelligence Predicate Logic Slide 41

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Summary of facts1 man(Marcus)

2 Pompeian(Marcus)

3 born(Marcus 40)

4 forallX man(X) mortal(X)

5 forallX Pompeian(X) died(X 79)

6 erupted(volcano 79)

7 forallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

8 now = 19919 forallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

10 forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4280

Artificial Intelligence Predicate Logic Slide 42

notalive(Marcus now)

dead(Marcus now)

(9 Substitute)

died(Marcus T1) gt(now T1)

(10 Substitute)

Pompeian(Marcus) gt(now 79)

(5 Substitute)(5 Substitute)

(2)gt(1991 79)

(8 Substitute)

(computation)

ss

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4380

Artificial Intelligence Predicate Logic Slide 43

983125983150983145983158983141983154983155983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983125983113)

Every instantiation of a universally quantified sentence is enta iled by it

forallv α

Subst(vg α)

for any variable v and ground term g

Eg forallx King( x) and Greedy( x)rArr Evil( x

) yieldsKing (John ) and Greedy (John ) rArr Evil (John )

King (Richard ) and Greedy (Richard ) rArr Evil (Richard )

King (Father (John )) and Greedy (Father (John )) rArr Evil (Father (John ))

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4480

Artificial Intelligence Predicate Logic Slide 44

983109983160983145983155983156983141983150983156983145983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983109983113)

For any sentence α variable v and constant symbol k that

does not

appear elsewhere in the knowledge baseexistv α

Subst(vk α )

Eg exist x Crown( x) and OnHead ( xJohn) yields

Crown(C 1) and OnHead (C 1 John )

provided C 1 is a new constant symbol called a Skolemconstant

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4580

Artificial Intelligence Predicate Logic Slide 45

983122983141983140983157983139983156983145983151983150 983156983151 983152983154983151983152983151983155983145983156983145983151983150983137983148 983145983150983142983141983154983141983150983139983141

Suppose the KB contains just the followingforall

x King(x)and

Greedy(x)rArr

Evil(x)King(John)Greedy(John)Brother(RichardJohn)

Instantiating the universal sentence in all possible ways we haveKing(John) and Greedy(John) rArr Evil(John)King(Richard) and Greedy(Richard) rArr Evil(Richard)King(John)Greedy(John)Brother(RichardJohn

)

The new KB is propositionalized proposition symbols are

King(John) Greedy(John) Evil(John) King(Richard

) etc

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4680

Artificial Intelligence Predicate Logic Slide 46

983122983141983140983157983139983156983145983151983150 983139983151983150983156983140983086

Every FOL KB can be propositionalized so as to preserve entailment

(A ground sentence is entailed by new KB iff entailed by originalKB)

Idea propositionalize KB and query apply resolution return result

Problem with function symbols there are infinitely many ground

terms

eg Father (Father (Father (John

)))

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4780

Artificial Intelligence Predicate Logic Slide 47

983122983141983155983151983148983157983156983145983151983150

Convert to clause form exampleldquoAll Romans who know Marcus either hate

Caesar or think that anyone who hates anyone is

crazyrdquo

forallX [roman(X) ^ know(X Marcus)]

[hate(X Ceasar) v(forallY existZ hate(YZ)

thinkcrazy(XY))]

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4880

Artificial Intelligence Predicate Logic Slide 48

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4980

Artificial Intelligence Predicate Logic Slide 49

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

forallX [roman(X) ^ know(X Marcus)][hate(X Ceasar) v

(forallY existZ hate(YZ) thinkcrazy(XY))]

=

forallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 5080

Artificial Intelligence Predicate Logic Slide 50

983122983141983155983151983148983157983156983145983151983150

2 Move not next to terms

Using the following equivalence

a Double negative

not(notp) = p

b DeMorgan

not(a v b) = nota ^ notb

not(a ^ b) = nota v notb

c Equivalence of qualifiers notforallX P(X) = existX notP(X)

notexistX P(X) = forallX notP(X)

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 5180

Artificial Intelligence Predicate Logic Slide 51

983122983141983155983151983148983157983156983145983151983150

2 Move not next to termsforallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

=

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) vthinkcrazy(XY))]

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 5280

Artificial Intelligence Predicate Logic Slide 52

983122983141983155983151983148983157983156983145983151983150

3 Change name of variables as follows

forallX P(X) v forallX Q(X)

=forallX P(X) v forallY Q(Y)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 53

983122983141983155983151983148983157983156983145983151983150

4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 5480

Artificial Intelligence Predicate Logic Slide 54

983122983141983155983151983148983157983156983145983151983150

5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 5580

Artificial Intelligence Predicate Logic Slide 55

983122983141983155983151983148983157983156983145983151983150

6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 5680

Artificial Intelligence Predicate Logic Slide 56

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 57

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 5880

Artificial Intelligence Predicate Logic Slide 58

983122983141983155983151983148983157983156983145983151983150

8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 59

983122983141983155983151983148983157983156983145983151983150

9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 60

983122983141983155983151983148983157983156983145983151983150

Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 61

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 62

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 63

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 64

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 65

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 66

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 69

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 71

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 72

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 74

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 75

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 76

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

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892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 22

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

Representing the following sentences in Predicate Logic

1 Marcus was a man

2 Macus was a Pompeian

3 All Pompians were Romans4 Caesar was a ruler

5 All Romans were either loyal to Caesar or hated hime

6 Everyone is loyal to someone

7 People only try to assassinate rulers they are not loyal to

8 Marcus tried to assassinate Caesar

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Artificial Intelligence Predicate Logic Slide 23

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

1 Marcus was a man

man(Marcus)

2 Macus was a Pompeian

Pompeian(Marcus)

3 All Pompeians were Romans

forallX Pompeian(X) Roman(X)

4 Caesar was a ruler

ruler(Caesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 24

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

5 All Romans were either loyal to Caesar or hated hime

ldquoorrdquo mean ldquoinclusive orrdquo - OR

forallX Roman(X) loyalto(X Caesar) v hate(X Caesar)

ldquoorrdquo mean ldquoexclusive orrdquo - XOR

forallX Roman(X) [ (loyalto(X Caesar) v hate(X Caesar)) ^

not(loyalto(X Caesar) ^ hate(X Caesar))]

or

forallX Roman(X) [ (loyalto(X Caesar) ^ nothate(X Caesar)) v

(notloyalto(X Caesar) ^ hate(X Caesar))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 25

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

6 Everyone is loyal to someone

forallX exist Y loyalto(XY)

7 People only try to assassinate rulers they are not loyal toforallX forall Y person(X) ^ ruler(Y) ^ tryassassinate(XY)notloyalto(XY)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 26

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

8 Marcus tried to assassinate Caesartryassassinate(Marcus Caesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 27

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

Given the above sentences can we make aconclusion as follows

ldquoMarcus was not loyal to Caesar rdquo

or

notloyalto(MarcusCaesar)

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 2880

Artificial Intelligence Predicate Logic Slide 28

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)

(Rule What)

Add an implicit sentence

9 ldquoAll men are peoplerdquo

or

forallforallforallforallX man(X) X man(X) person(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 29

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)(9 Substitute)

man(Marcus)

(1)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 30

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

How to represent exceptions example

ldquoPaulus was a Pompeian Paulus was neitherloyal nor hated Caesarrdquo

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Artificial Intelligence Predicate Logic Slide 31

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation Semantic Net Simple by adding links

(properties) to subclasses or instances

Predicate Logic Not so simple like

983120983151983149983152983141983145983137983150(983120983137983157983148983157983155) 983134 983212983131983148983151983161983137983148983156983151(983120983137983157983148983157983155 983107983137983141983155983137983154) 983158

983144983137983156983141(983120983157983137983148983157983155 983107983137983141983155983137983154)983133

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 32

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation983118983151983156 983155983151 983155983145983149983152983148983141 983106983141983139983137983157983155983141 983156983144983141983154983141 983145983155 983139983151983150983142983148983145983139983156 983138983141983156983159983141983141983150983098

Pompeian(Paulus) ^ not[loyalto(Paulus Caesar) vhate(Pualus Caesar)]

983137983150983140

forallX Pompeian(X) loyalto(X Caesar) v hate(XCaesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 33

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation A Better way

forallX Roman(X) ^ noteq(X Paulus) loyalto(X Caesar)v hate(X Caesar

Exception

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 34

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

In the case of describing relations betweennumbers example 1 983100 2

2 983100983091

983095 983102 983091 +2 983091 983102 1

991270

rArr

983123983144983151983157983148983140 983150983151983156 983141983160983152983148983145983139983145983156983148983161 983140983141983155983139983154983145983138983141983098983148983156(12) 983148983156(2983091) 991270

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 35

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

rArr Need functions and computable predicates

rArr For example 983107983137983148983148 983137 983142983157983150983139983156983145983151983150 983156983151 983139983151983149983152983157983156983141 (983091+2)

983120983137983155983155 983156983144983141 983154983141983155983157983148983156 983156983151 983156983144983141 983152983154983141983140983145983139983137983156983141 991260983143983156991261983098 983143983156(983095 983091+2)

983111983141983156 983156983144983141 983142983145983150983137983148 983154983141983155983157983148983156 (983124983154983157983141)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 36

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

1 Marcus was a man

man(Marcus)

2 Marcus was a PompeianPompeian(Marcus)

3 Marcus was born in 40 AD

born(Marcus40)

4 All men are mortal

forallX man(X) mortal(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 37

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

5 All Pompeian died when the vocano erupted in 79 AD

erupted(vocano 79) ^ forallX [Pompeian(X) died(X 79)]

6 No mortal lives longer then 150 yearsforallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

7 It is now 1991now = 1991

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Artificial Intelligence Predicate Logic Slide 38

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Question

Is Marcus alive

Or

alive(Marcus now)

ornotnotnotnotalive(Marcus now)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 39

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge

Relation between ldquodeadrdquo and ldquoaliverdquo

8 Alive means not deadforallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 40

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Add implicit knowledge Time properties of an event for example ldquodeadrdquo

9 Is someone dies the he is dead at all later times

forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

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Artificial Intelligence Predicate Logic Slide 41

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Summary of facts1 man(Marcus)

2 Pompeian(Marcus)

3 born(Marcus 40)

4 forallX man(X) mortal(X)

5 forallX Pompeian(X) died(X 79)

6 erupted(volcano 79)

7 forallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

8 now = 19919 forallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

10 forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

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Artificial Intelligence Predicate Logic Slide 42

notalive(Marcus now)

dead(Marcus now)

(9 Substitute)

died(Marcus T1) gt(now T1)

(10 Substitute)

Pompeian(Marcus) gt(now 79)

(5 Substitute)(5 Substitute)

(2)gt(1991 79)

(8 Substitute)

(computation)

ss

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Artificial Intelligence Predicate Logic Slide 43

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Every instantiation of a universally quantified sentence is enta iled by it

forallv α

Subst(vg α)

for any variable v and ground term g

Eg forallx King( x) and Greedy( x)rArr Evil( x

) yieldsKing (John ) and Greedy (John ) rArr Evil (John )

King (Richard ) and Greedy (Richard ) rArr Evil (Richard )

King (Father (John )) and Greedy (Father (John )) rArr Evil (Father (John ))

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Artificial Intelligence Predicate Logic Slide 44

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For any sentence α variable v and constant symbol k that

does not

appear elsewhere in the knowledge baseexistv α

Subst(vk α )

Eg exist x Crown( x) and OnHead ( xJohn) yields

Crown(C 1) and OnHead (C 1 John )

provided C 1 is a new constant symbol called a Skolemconstant

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 45

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Suppose the KB contains just the followingforall

x King(x)and

Greedy(x)rArr

Evil(x)King(John)Greedy(John)Brother(RichardJohn)

Instantiating the universal sentence in all possible ways we haveKing(John) and Greedy(John) rArr Evil(John)King(Richard) and Greedy(Richard) rArr Evil(Richard)King(John)Greedy(John)Brother(RichardJohn

)

The new KB is propositionalized proposition symbols are

King(John) Greedy(John) Evil(John) King(Richard

) etc

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 46

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Every FOL KB can be propositionalized so as to preserve entailment

(A ground sentence is entailed by new KB iff entailed by originalKB)

Idea propositionalize KB and query apply resolution return result

Problem with function symbols there are infinitely many ground

terms

eg Father (Father (Father (John

)))

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Artificial Intelligence Predicate Logic Slide 47

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Convert to clause form exampleldquoAll Romans who know Marcus either hate

Caesar or think that anyone who hates anyone is

crazyrdquo

forallX [roman(X) ^ know(X Marcus)]

[hate(X Ceasar) v(forallY existZ hate(YZ)

thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 48

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 49

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

forallX [roman(X) ^ know(X Marcus)][hate(X Ceasar) v

(forallY existZ hate(YZ) thinkcrazy(XY))]

=

forallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 50

983122983141983155983151983148983157983156983145983151983150

2 Move not next to terms

Using the following equivalence

a Double negative

not(notp) = p

b DeMorgan

not(a v b) = nota ^ notb

not(a ^ b) = nota v notb

c Equivalence of qualifiers notforallX P(X) = existX notP(X)

notexistX P(X) = forallX notP(X)

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Artificial Intelligence Predicate Logic Slide 51

983122983141983155983151983148983157983156983145983151983150

2 Move not next to termsforallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

=

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) vthinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 52

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3 Change name of variables as follows

forallX P(X) v forallX Q(X)

=forallX P(X) v forallY Q(Y)

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Artificial Intelligence Predicate Logic Slide 53

983122983141983155983151983148983157983156983145983151983150

4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 54

983122983141983155983151983148983157983156983145983151983150

5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

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Artificial Intelligence Predicate Logic Slide 55

983122983141983155983151983148983157983156983145983151983150

6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 56

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

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Artificial Intelligence Predicate Logic Slide 57

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

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Artificial Intelligence Predicate Logic Slide 58

983122983141983155983151983148983157983156983145983151983150

8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

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Artificial Intelligence Predicate Logic Slide 59

983122983141983155983151983148983157983156983145983151983150

9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

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Artificial Intelligence Predicate Logic Slide 60

983122983141983155983151983148983157983156983145983151983150

Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

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Artificial Intelligence Predicate Logic Slide 61

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X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

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Artificial Intelligence Predicate Logic Slide 62

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θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

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Artificial Intelligence Predicate Logic Slide 63

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Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

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Artificial Intelligence Predicate Logic Slide 64

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

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Artificial Intelligence Predicate Logic Slide 65

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

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Artificial Intelligence Predicate Logic Slide 66

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

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Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

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Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

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Artificial Intelligence Predicate Logic Slide 69

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

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Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

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Artificial Intelligence Predicate Logic Slide 71

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

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Artificial Intelligence Predicate Logic Slide 72

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

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Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

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Artificial Intelligence Predicate Logic Slide 74

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

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Artificial Intelligence Predicate Logic Slide 75

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

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Artificial Intelligence Predicate Logic Slide 76

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

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892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

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Artificial Intelligence Predicate Logic Slide 23

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

1 Marcus was a man

man(Marcus)

2 Macus was a Pompeian

Pompeian(Marcus)

3 All Pompeians were Romans

forallX Pompeian(X) Roman(X)

4 Caesar was a ruler

ruler(Caesar)

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Artificial Intelligence Predicate Logic Slide 24

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

5 All Romans were either loyal to Caesar or hated hime

ldquoorrdquo mean ldquoinclusive orrdquo - OR

forallX Roman(X) loyalto(X Caesar) v hate(X Caesar)

ldquoorrdquo mean ldquoexclusive orrdquo - XOR

forallX Roman(X) [ (loyalto(X Caesar) v hate(X Caesar)) ^

not(loyalto(X Caesar) ^ hate(X Caesar))]

or

forallX Roman(X) [ (loyalto(X Caesar) ^ nothate(X Caesar)) v

(notloyalto(X Caesar) ^ hate(X Caesar))]

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Artificial Intelligence Predicate Logic Slide 25

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

6 Everyone is loyal to someone

forallX exist Y loyalto(XY)

7 People only try to assassinate rulers they are not loyal toforallX forall Y person(X) ^ ruler(Y) ^ tryassassinate(XY)notloyalto(XY)

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Artificial Intelligence Predicate Logic Slide 26

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

8 Marcus tried to assassinate Caesartryassassinate(Marcus Caesar)

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Artificial Intelligence Predicate Logic Slide 27

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

Given the above sentences can we make aconclusion as follows

ldquoMarcus was not loyal to Caesar rdquo

or

notloyalto(MarcusCaesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 28

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)

(Rule What)

Add an implicit sentence

9 ldquoAll men are peoplerdquo

or

forallforallforallforallX man(X) X man(X) person(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 29

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)(9 Substitute)

man(Marcus)

(1)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 30

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

How to represent exceptions example

ldquoPaulus was a Pompeian Paulus was neitherloyal nor hated Caesarrdquo

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Artificial Intelligence Predicate Logic Slide 31

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation Semantic Net Simple by adding links

(properties) to subclasses or instances

Predicate Logic Not so simple like

983120983151983149983152983141983145983137983150(983120983137983157983148983157983155) 983134 983212983131983148983151983161983137983148983156983151(983120983137983157983148983157983155 983107983137983141983155983137983154) 983158

983144983137983156983141(983120983157983137983148983157983155 983107983137983141983155983137983154)983133

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 32

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation983118983151983156 983155983151 983155983145983149983152983148983141 983106983141983139983137983157983155983141 983156983144983141983154983141 983145983155 983139983151983150983142983148983145983139983156 983138983141983156983159983141983141983150983098

Pompeian(Paulus) ^ not[loyalto(Paulus Caesar) vhate(Pualus Caesar)]

983137983150983140

forallX Pompeian(X) loyalto(X Caesar) v hate(XCaesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 33

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation A Better way

forallX Roman(X) ^ noteq(X Paulus) loyalto(X Caesar)v hate(X Caesar

Exception

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 34

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

In the case of describing relations betweennumbers example 1 983100 2

2 983100983091

983095 983102 983091 +2 983091 983102 1

991270

rArr

983123983144983151983157983148983140 983150983151983156 983141983160983152983148983145983139983145983156983148983161 983140983141983155983139983154983145983138983141983098983148983156(12) 983148983156(2983091) 991270

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 35

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

rArr Need functions and computable predicates

rArr For example 983107983137983148983148 983137 983142983157983150983139983156983145983151983150 983156983151 983139983151983149983152983157983156983141 (983091+2)

983120983137983155983155 983156983144983141 983154983141983155983157983148983156 983156983151 983156983144983141 983152983154983141983140983145983139983137983156983141 991260983143983156991261983098 983143983156(983095 983091+2)

983111983141983156 983156983144983141 983142983145983150983137983148 983154983141983155983157983148983156 (983124983154983157983141)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 36

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

1 Marcus was a man

man(Marcus)

2 Marcus was a PompeianPompeian(Marcus)

3 Marcus was born in 40 AD

born(Marcus40)

4 All men are mortal

forallX man(X) mortal(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 37

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

5 All Pompeian died when the vocano erupted in 79 AD

erupted(vocano 79) ^ forallX [Pompeian(X) died(X 79)]

6 No mortal lives longer then 150 yearsforallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

7 It is now 1991now = 1991

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Artificial Intelligence Predicate Logic Slide 38

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Question

Is Marcus alive

Or

alive(Marcus now)

ornotnotnotnotalive(Marcus now)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 39

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge

Relation between ldquodeadrdquo and ldquoaliverdquo

8 Alive means not deadforallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

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Artificial Intelligence Predicate Logic Slide 40

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge Time properties of an event for example ldquodeadrdquo

9 Is someone dies the he is dead at all later times

forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

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Artificial Intelligence Predicate Logic Slide 41

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Summary of facts1 man(Marcus)

2 Pompeian(Marcus)

3 born(Marcus 40)

4 forallX man(X) mortal(X)

5 forallX Pompeian(X) died(X 79)

6 erupted(volcano 79)

7 forallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

8 now = 19919 forallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

10 forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 42

notalive(Marcus now)

dead(Marcus now)

(9 Substitute)

died(Marcus T1) gt(now T1)

(10 Substitute)

Pompeian(Marcus) gt(now 79)

(5 Substitute)(5 Substitute)

(2)gt(1991 79)

(8 Substitute)

(computation)

ss

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 43

983125983150983145983158983141983154983155983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983125983113)

Every instantiation of a universally quantified sentence is enta iled by it

forallv α

Subst(vg α)

for any variable v and ground term g

Eg forallx King( x) and Greedy( x)rArr Evil( x

) yieldsKing (John ) and Greedy (John ) rArr Evil (John )

King (Richard ) and Greedy (Richard ) rArr Evil (Richard )

King (Father (John )) and Greedy (Father (John )) rArr Evil (Father (John ))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 44

983109983160983145983155983156983141983150983156983145983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983109983113)

For any sentence α variable v and constant symbol k that

does not

appear elsewhere in the knowledge baseexistv α

Subst(vk α )

Eg exist x Crown( x) and OnHead ( xJohn) yields

Crown(C 1) and OnHead (C 1 John )

provided C 1 is a new constant symbol called a Skolemconstant

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 45

983122983141983140983157983139983156983145983151983150 983156983151 983152983154983151983152983151983155983145983156983145983151983150983137983148 983145983150983142983141983154983141983150983139983141

Suppose the KB contains just the followingforall

x King(x)and

Greedy(x)rArr

Evil(x)King(John)Greedy(John)Brother(RichardJohn)

Instantiating the universal sentence in all possible ways we haveKing(John) and Greedy(John) rArr Evil(John)King(Richard) and Greedy(Richard) rArr Evil(Richard)King(John)Greedy(John)Brother(RichardJohn

)

The new KB is propositionalized proposition symbols are

King(John) Greedy(John) Evil(John) King(Richard

) etc

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4680

Artificial Intelligence Predicate Logic Slide 46

983122983141983140983157983139983156983145983151983150 983139983151983150983156983140983086

Every FOL KB can be propositionalized so as to preserve entailment

(A ground sentence is entailed by new KB iff entailed by originalKB)

Idea propositionalize KB and query apply resolution return result

Problem with function symbols there are infinitely many ground

terms

eg Father (Father (Father (John

)))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 47

983122983141983155983151983148983157983156983145983151983150

Convert to clause form exampleldquoAll Romans who know Marcus either hate

Caesar or think that anyone who hates anyone is

crazyrdquo

forallX [roman(X) ^ know(X Marcus)]

[hate(X Ceasar) v(forallY existZ hate(YZ)

thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 48

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 49

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

forallX [roman(X) ^ know(X Marcus)][hate(X Ceasar) v

(forallY existZ hate(YZ) thinkcrazy(XY))]

=

forallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 50

983122983141983155983151983148983157983156983145983151983150

2 Move not next to terms

Using the following equivalence

a Double negative

not(notp) = p

b DeMorgan

not(a v b) = nota ^ notb

not(a ^ b) = nota v notb

c Equivalence of qualifiers notforallX P(X) = existX notP(X)

notexistX P(X) = forallX notP(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 51

983122983141983155983151983148983157983156983145983151983150

2 Move not next to termsforallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

=

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) vthinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 52

983122983141983155983151983148983157983156983145983151983150

3 Change name of variables as follows

forallX P(X) v forallX Q(X)

=forallX P(X) v forallY Q(Y)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 53

983122983141983155983151983148983157983156983145983151983150

4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 54

983122983141983155983151983148983157983156983145983151983150

5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 55

983122983141983155983151983148983157983156983145983151983150

6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 56

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 57

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 58

983122983141983155983151983148983157983156983145983151983150

8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 59

983122983141983155983151983148983157983156983145983151983150

9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 60

983122983141983155983151983148983157983156983145983151983150

Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 61

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 62

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 63

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 64

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 65

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 66

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 69

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 71

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 72

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 74

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 75

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 76

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

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892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

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Artificial Intelligence Predicate Logic Slide 24

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

5 All Romans were either loyal to Caesar or hated hime

ldquoorrdquo mean ldquoinclusive orrdquo - OR

forallX Roman(X) loyalto(X Caesar) v hate(X Caesar)

ldquoorrdquo mean ldquoexclusive orrdquo - XOR

forallX Roman(X) [ (loyalto(X Caesar) v hate(X Caesar)) ^

not(loyalto(X Caesar) ^ hate(X Caesar))]

or

forallX Roman(X) [ (loyalto(X Caesar) ^ nothate(X Caesar)) v

(notloyalto(X Caesar) ^ hate(X Caesar))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 25

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

6 Everyone is loyal to someone

forallX exist Y loyalto(XY)

7 People only try to assassinate rulers they are not loyal toforallX forall Y person(X) ^ ruler(Y) ^ tryassassinate(XY)notloyalto(XY)

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Artificial Intelligence Predicate Logic Slide 26

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

8 Marcus tried to assassinate Caesartryassassinate(Marcus Caesar)

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Artificial Intelligence Predicate Logic Slide 27

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

Given the above sentences can we make aconclusion as follows

ldquoMarcus was not loyal to Caesar rdquo

or

notloyalto(MarcusCaesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 28

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)

(Rule What)

Add an implicit sentence

9 ldquoAll men are peoplerdquo

or

forallforallforallforallX man(X) X man(X) person(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 29

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)(9 Substitute)

man(Marcus)

(1)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 30

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

How to represent exceptions example

ldquoPaulus was a Pompeian Paulus was neitherloyal nor hated Caesarrdquo

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Artificial Intelligence Predicate Logic Slide 31

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation Semantic Net Simple by adding links

(properties) to subclasses or instances

Predicate Logic Not so simple like

983120983151983149983152983141983145983137983150(983120983137983157983148983157983155) 983134 983212983131983148983151983161983137983148983156983151(983120983137983157983148983157983155 983107983137983141983155983137983154) 983158

983144983137983156983141(983120983157983137983148983157983155 983107983137983141983155983137983154)983133

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 32

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation983118983151983156 983155983151 983155983145983149983152983148983141 983106983141983139983137983157983155983141 983156983144983141983154983141 983145983155 983139983151983150983142983148983145983139983156 983138983141983156983159983141983141983150983098

Pompeian(Paulus) ^ not[loyalto(Paulus Caesar) vhate(Pualus Caesar)]

983137983150983140

forallX Pompeian(X) loyalto(X Caesar) v hate(XCaesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 33

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation A Better way

forallX Roman(X) ^ noteq(X Paulus) loyalto(X Caesar)v hate(X Caesar

Exception

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Artificial Intelligence Predicate Logic Slide 34

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

In the case of describing relations betweennumbers example 1 983100 2

2 983100983091

983095 983102 983091 +2 983091 983102 1

991270

rArr

983123983144983151983157983148983140 983150983151983156 983141983160983152983148983145983139983145983156983148983161 983140983141983155983139983154983145983138983141983098983148983156(12) 983148983156(2983091) 991270

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 35

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

rArr Need functions and computable predicates

rArr For example 983107983137983148983148 983137 983142983157983150983139983156983145983151983150 983156983151 983139983151983149983152983157983156983141 (983091+2)

983120983137983155983155 983156983144983141 983154983141983155983157983148983156 983156983151 983156983144983141 983152983154983141983140983145983139983137983156983141 991260983143983156991261983098 983143983156(983095 983091+2)

983111983141983156 983156983144983141 983142983145983150983137983148 983154983141983155983157983148983156 (983124983154983157983141)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 36

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

1 Marcus was a man

man(Marcus)

2 Marcus was a PompeianPompeian(Marcus)

3 Marcus was born in 40 AD

born(Marcus40)

4 All men are mortal

forallX man(X) mortal(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 37

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

5 All Pompeian died when the vocano erupted in 79 AD

erupted(vocano 79) ^ forallX [Pompeian(X) died(X 79)]

6 No mortal lives longer then 150 yearsforallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

7 It is now 1991now = 1991

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Artificial Intelligence Predicate Logic Slide 38

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Question

Is Marcus alive

Or

alive(Marcus now)

ornotnotnotnotalive(Marcus now)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 39

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge

Relation between ldquodeadrdquo and ldquoaliverdquo

8 Alive means not deadforallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4080

Artificial Intelligence Predicate Logic Slide 40

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge Time properties of an event for example ldquodeadrdquo

9 Is someone dies the he is dead at all later times

forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4180

Artificial Intelligence Predicate Logic Slide 41

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Summary of facts1 man(Marcus)

2 Pompeian(Marcus)

3 born(Marcus 40)

4 forallX man(X) mortal(X)

5 forallX Pompeian(X) died(X 79)

6 erupted(volcano 79)

7 forallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

8 now = 19919 forallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

10 forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4280

Artificial Intelligence Predicate Logic Slide 42

notalive(Marcus now)

dead(Marcus now)

(9 Substitute)

died(Marcus T1) gt(now T1)

(10 Substitute)

Pompeian(Marcus) gt(now 79)

(5 Substitute)(5 Substitute)

(2)gt(1991 79)

(8 Substitute)

(computation)

ss

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4380

Artificial Intelligence Predicate Logic Slide 43

983125983150983145983158983141983154983155983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983125983113)

Every instantiation of a universally quantified sentence is enta iled by it

forallv α

Subst(vg α)

for any variable v and ground term g

Eg forallx King( x) and Greedy( x)rArr Evil( x

) yieldsKing (John ) and Greedy (John ) rArr Evil (John )

King (Richard ) and Greedy (Richard ) rArr Evil (Richard )

King (Father (John )) and Greedy (Father (John )) rArr Evil (Father (John ))

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4480

Artificial Intelligence Predicate Logic Slide 44

983109983160983145983155983156983141983150983156983145983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983109983113)

For any sentence α variable v and constant symbol k that

does not

appear elsewhere in the knowledge baseexistv α

Subst(vk α )

Eg exist x Crown( x) and OnHead ( xJohn) yields

Crown(C 1) and OnHead (C 1 John )

provided C 1 is a new constant symbol called a Skolemconstant

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4580

Artificial Intelligence Predicate Logic Slide 45

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Suppose the KB contains just the followingforall

x King(x)and

Greedy(x)rArr

Evil(x)King(John)Greedy(John)Brother(RichardJohn)

Instantiating the universal sentence in all possible ways we haveKing(John) and Greedy(John) rArr Evil(John)King(Richard) and Greedy(Richard) rArr Evil(Richard)King(John)Greedy(John)Brother(RichardJohn

)

The new KB is propositionalized proposition symbols are

King(John) Greedy(John) Evil(John) King(Richard

) etc

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Artificial Intelligence Predicate Logic Slide 46

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Every FOL KB can be propositionalized so as to preserve entailment

(A ground sentence is entailed by new KB iff entailed by originalKB)

Idea propositionalize KB and query apply resolution return result

Problem with function symbols there are infinitely many ground

terms

eg Father (Father (Father (John

)))

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Artificial Intelligence Predicate Logic Slide 47

983122983141983155983151983148983157983156983145983151983150

Convert to clause form exampleldquoAll Romans who know Marcus either hate

Caesar or think that anyone who hates anyone is

crazyrdquo

forallX [roman(X) ^ know(X Marcus)]

[hate(X Ceasar) v(forallY existZ hate(YZ)

thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 48

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

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Artificial Intelligence Predicate Logic Slide 49

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

forallX [roman(X) ^ know(X Marcus)][hate(X Ceasar) v

(forallY existZ hate(YZ) thinkcrazy(XY))]

=

forallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 50

983122983141983155983151983148983157983156983145983151983150

2 Move not next to terms

Using the following equivalence

a Double negative

not(notp) = p

b DeMorgan

not(a v b) = nota ^ notb

not(a ^ b) = nota v notb

c Equivalence of qualifiers notforallX P(X) = existX notP(X)

notexistX P(X) = forallX notP(X)

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Artificial Intelligence Predicate Logic Slide 51

983122983141983155983151983148983157983156983145983151983150

2 Move not next to termsforallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

=

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) vthinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 52

983122983141983155983151983148983157983156983145983151983150

3 Change name of variables as follows

forallX P(X) v forallX Q(X)

=forallX P(X) v forallY Q(Y)

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Artificial Intelligence Predicate Logic Slide 53

983122983141983155983151983148983157983156983145983151983150

4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 54

983122983141983155983151983148983157983156983145983151983150

5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

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Artificial Intelligence Predicate Logic Slide 55

983122983141983155983151983148983157983156983145983151983150

6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 56

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

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Artificial Intelligence Predicate Logic Slide 57

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

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Artificial Intelligence Predicate Logic Slide 58

983122983141983155983151983148983157983156983145983151983150

8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

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Artificial Intelligence Predicate Logic Slide 59

983122983141983155983151983148983157983156983145983151983150

9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

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Artificial Intelligence Predicate Logic Slide 60

983122983141983155983151983148983157983156983145983151983150

Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

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Artificial Intelligence Predicate Logic Slide 61

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

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Artificial Intelligence Predicate Logic Slide 62

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

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Artificial Intelligence Predicate Logic Slide 63

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

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Artificial Intelligence Predicate Logic Slide 64

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

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Artificial Intelligence Predicate Logic Slide 65

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

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Artificial Intelligence Predicate Logic Slide 66

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

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Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

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Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

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Artificial Intelligence Predicate Logic Slide 69

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

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Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

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Artificial Intelligence Predicate Logic Slide 71

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

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Artificial Intelligence Predicate Logic Slide 72

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

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Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

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Artificial Intelligence Predicate Logic Slide 74

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Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

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Artificial Intelligence Predicate Logic Slide 75

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Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

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Artificial Intelligence Predicate Logic Slide 76

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Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

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892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

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Artificial Intelligence Predicate Logic Slide 25

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

6 Everyone is loyal to someone

forallX exist Y loyalto(XY)

7 People only try to assassinate rulers they are not loyal toforallX forall Y person(X) ^ ruler(Y) ^ tryassassinate(XY)notloyalto(XY)

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Artificial Intelligence Predicate Logic Slide 26

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

8 Marcus tried to assassinate Caesartryassassinate(Marcus Caesar)

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Artificial Intelligence Predicate Logic Slide 27

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Given the above sentences can we make aconclusion as follows

ldquoMarcus was not loyal to Caesar rdquo

or

notloyalto(MarcusCaesar)

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Artificial Intelligence Predicate Logic Slide 28

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)

(Rule What)

Add an implicit sentence

9 ldquoAll men are peoplerdquo

or

forallforallforallforallX man(X) X man(X) person(X)

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Artificial Intelligence Predicate Logic Slide 29

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notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)(9 Substitute)

man(Marcus)

(1)

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Artificial Intelligence Predicate Logic Slide 30

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How to represent exceptions example

ldquoPaulus was a Pompeian Paulus was neitherloyal nor hated Caesarrdquo

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Artificial Intelligence Predicate Logic Slide 31

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Exception representation Semantic Net Simple by adding links

(properties) to subclasses or instances

Predicate Logic Not so simple like

983120983151983149983152983141983145983137983150(983120983137983157983148983157983155) 983134 983212983131983148983151983161983137983148983156983151(983120983137983157983148983157983155 983107983137983141983155983137983154) 983158

983144983137983156983141(983120983157983137983148983157983155 983107983137983141983155983137983154)983133

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Artificial Intelligence Predicate Logic Slide 32

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation983118983151983156 983155983151 983155983145983149983152983148983141 983106983141983139983137983157983155983141 983156983144983141983154983141 983145983155 983139983151983150983142983148983145983139983156 983138983141983156983159983141983141983150983098

Pompeian(Paulus) ^ not[loyalto(Paulus Caesar) vhate(Pualus Caesar)]

983137983150983140

forallX Pompeian(X) loyalto(X Caesar) v hate(XCaesar)

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Artificial Intelligence Predicate Logic Slide 33

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation A Better way

forallX Roman(X) ^ noteq(X Paulus) loyalto(X Caesar)v hate(X Caesar

Exception

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 34

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

In the case of describing relations betweennumbers example 1 983100 2

2 983100983091

983095 983102 983091 +2 983091 983102 1

991270

rArr

983123983144983151983157983148983140 983150983151983156 983141983160983152983148983145983139983145983156983148983161 983140983141983155983139983154983145983138983141983098983148983156(12) 983148983156(2983091) 991270

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Artificial Intelligence Predicate Logic Slide 35

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

rArr Need functions and computable predicates

rArr For example 983107983137983148983148 983137 983142983157983150983139983156983145983151983150 983156983151 983139983151983149983152983157983156983141 (983091+2)

983120983137983155983155 983156983144983141 983154983141983155983157983148983156 983156983151 983156983144983141 983152983154983141983140983145983139983137983156983141 991260983143983156991261983098 983143983156(983095 983091+2)

983111983141983156 983156983144983141 983142983145983150983137983148 983154983141983155983157983148983156 (983124983154983157983141)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 36

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

1 Marcus was a man

man(Marcus)

2 Marcus was a PompeianPompeian(Marcus)

3 Marcus was born in 40 AD

born(Marcus40)

4 All men are mortal

forallX man(X) mortal(X)

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Artificial Intelligence Predicate Logic Slide 37

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

5 All Pompeian died when the vocano erupted in 79 AD

erupted(vocano 79) ^ forallX [Pompeian(X) died(X 79)]

6 No mortal lives longer then 150 yearsforallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

7 It is now 1991now = 1991

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Artificial Intelligence Predicate Logic Slide 38

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Question

Is Marcus alive

Or

alive(Marcus now)

ornotnotnotnotalive(Marcus now)

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Artificial Intelligence Predicate Logic Slide 39

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge

Relation between ldquodeadrdquo and ldquoaliverdquo

8 Alive means not deadforallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

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Artificial Intelligence Predicate Logic Slide 40

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge Time properties of an event for example ldquodeadrdquo

9 Is someone dies the he is dead at all later times

forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

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Artificial Intelligence Predicate Logic Slide 41

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Summary of facts1 man(Marcus)

2 Pompeian(Marcus)

3 born(Marcus 40)

4 forallX man(X) mortal(X)

5 forallX Pompeian(X) died(X 79)

6 erupted(volcano 79)

7 forallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

8 now = 19919 forallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

10 forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

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Artificial Intelligence Predicate Logic Slide 42

notalive(Marcus now)

dead(Marcus now)

(9 Substitute)

died(Marcus T1) gt(now T1)

(10 Substitute)

Pompeian(Marcus) gt(now 79)

(5 Substitute)(5 Substitute)

(2)gt(1991 79)

(8 Substitute)

(computation)

ss

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 43

983125983150983145983158983141983154983155983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983125983113)

Every instantiation of a universally quantified sentence is enta iled by it

forallv α

Subst(vg α)

for any variable v and ground term g

Eg forallx King( x) and Greedy( x)rArr Evil( x

) yieldsKing (John ) and Greedy (John ) rArr Evil (John )

King (Richard ) and Greedy (Richard ) rArr Evil (Richard )

King (Father (John )) and Greedy (Father (John )) rArr Evil (Father (John ))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 44

983109983160983145983155983156983141983150983156983145983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983109983113)

For any sentence α variable v and constant symbol k that

does not

appear elsewhere in the knowledge baseexistv α

Subst(vk α )

Eg exist x Crown( x) and OnHead ( xJohn) yields

Crown(C 1) and OnHead (C 1 John )

provided C 1 is a new constant symbol called a Skolemconstant

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 45

983122983141983140983157983139983156983145983151983150 983156983151 983152983154983151983152983151983155983145983156983145983151983150983137983148 983145983150983142983141983154983141983150983139983141

Suppose the KB contains just the followingforall

x King(x)and

Greedy(x)rArr

Evil(x)King(John)Greedy(John)Brother(RichardJohn)

Instantiating the universal sentence in all possible ways we haveKing(John) and Greedy(John) rArr Evil(John)King(Richard) and Greedy(Richard) rArr Evil(Richard)King(John)Greedy(John)Brother(RichardJohn

)

The new KB is propositionalized proposition symbols are

King(John) Greedy(John) Evil(John) King(Richard

) etc

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 46

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Every FOL KB can be propositionalized so as to preserve entailment

(A ground sentence is entailed by new KB iff entailed by originalKB)

Idea propositionalize KB and query apply resolution return result

Problem with function symbols there are infinitely many ground

terms

eg Father (Father (Father (John

)))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 47

983122983141983155983151983148983157983156983145983151983150

Convert to clause form exampleldquoAll Romans who know Marcus either hate

Caesar or think that anyone who hates anyone is

crazyrdquo

forallX [roman(X) ^ know(X Marcus)]

[hate(X Ceasar) v(forallY existZ hate(YZ)

thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 48

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 49

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

forallX [roman(X) ^ know(X Marcus)][hate(X Ceasar) v

(forallY existZ hate(YZ) thinkcrazy(XY))]

=

forallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 50

983122983141983155983151983148983157983156983145983151983150

2 Move not next to terms

Using the following equivalence

a Double negative

not(notp) = p

b DeMorgan

not(a v b) = nota ^ notb

not(a ^ b) = nota v notb

c Equivalence of qualifiers notforallX P(X) = existX notP(X)

notexistX P(X) = forallX notP(X)

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Artificial Intelligence Predicate Logic Slide 51

983122983141983155983151983148983157983156983145983151983150

2 Move not next to termsforallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

=

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) vthinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 52

983122983141983155983151983148983157983156983145983151983150

3 Change name of variables as follows

forallX P(X) v forallX Q(X)

=forallX P(X) v forallY Q(Y)

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Artificial Intelligence Predicate Logic Slide 53

983122983141983155983151983148983157983156983145983151983150

4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 54

983122983141983155983151983148983157983156983145983151983150

5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

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Artificial Intelligence Predicate Logic Slide 55

983122983141983155983151983148983157983156983145983151983150

6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 56

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

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Artificial Intelligence Predicate Logic Slide 57

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

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Artificial Intelligence Predicate Logic Slide 58

983122983141983155983151983148983157983156983145983151983150

8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

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Artificial Intelligence Predicate Logic Slide 59

983122983141983155983151983148983157983156983145983151983150

9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

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Artificial Intelligence Predicate Logic Slide 60

983122983141983155983151983148983157983156983145983151983150

Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

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Artificial Intelligence Predicate Logic Slide 61

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

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Artificial Intelligence Predicate Logic Slide 62

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

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Artificial Intelligence Predicate Logic Slide 63

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

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Artificial Intelligence Predicate Logic Slide 64

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

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Artificial Intelligence Predicate Logic Slide 65

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

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Artificial Intelligence Predicate Logic Slide 66

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

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Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

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Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

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Artificial Intelligence Predicate Logic Slide 69

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

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Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

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Artificial Intelligence Predicate Logic Slide 71

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

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Artificial Intelligence Predicate Logic Slide 72

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

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Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 74

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 75

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 76

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

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892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 26

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

8 Marcus tried to assassinate Caesartryassassinate(Marcus Caesar)

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Artificial Intelligence Predicate Logic Slide 27

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

Given the above sentences can we make aconclusion as follows

ldquoMarcus was not loyal to Caesar rdquo

or

notloyalto(MarcusCaesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 28

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)

(Rule What)

Add an implicit sentence

9 ldquoAll men are peoplerdquo

or

forallforallforallforallX man(X) X man(X) person(X)

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Artificial Intelligence Predicate Logic Slide 29

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)(9 Substitute)

man(Marcus)

(1)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 30

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

How to represent exceptions example

ldquoPaulus was a Pompeian Paulus was neitherloyal nor hated Caesarrdquo

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Artificial Intelligence Predicate Logic Slide 31

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation Semantic Net Simple by adding links

(properties) to subclasses or instances

Predicate Logic Not so simple like

983120983151983149983152983141983145983137983150(983120983137983157983148983157983155) 983134 983212983131983148983151983161983137983148983156983151(983120983137983157983148983157983155 983107983137983141983155983137983154) 983158

983144983137983156983141(983120983157983137983148983157983155 983107983137983141983155983137983154)983133

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 32

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation983118983151983156 983155983151 983155983145983149983152983148983141 983106983141983139983137983157983155983141 983156983144983141983154983141 983145983155 983139983151983150983142983148983145983139983156 983138983141983156983159983141983141983150983098

Pompeian(Paulus) ^ not[loyalto(Paulus Caesar) vhate(Pualus Caesar)]

983137983150983140

forallX Pompeian(X) loyalto(X Caesar) v hate(XCaesar)

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Artificial Intelligence Predicate Logic Slide 33

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation A Better way

forallX Roman(X) ^ noteq(X Paulus) loyalto(X Caesar)v hate(X Caesar

Exception

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Artificial Intelligence Predicate Logic Slide 34

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

In the case of describing relations betweennumbers example 1 983100 2

2 983100983091

983095 983102 983091 +2 983091 983102 1

991270

rArr

983123983144983151983157983148983140 983150983151983156 983141983160983152983148983145983139983145983156983148983161 983140983141983155983139983154983145983138983141983098983148983156(12) 983148983156(2983091) 991270

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Artificial Intelligence Predicate Logic Slide 35

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

rArr Need functions and computable predicates

rArr For example 983107983137983148983148 983137 983142983157983150983139983156983145983151983150 983156983151 983139983151983149983152983157983156983141 (983091+2)

983120983137983155983155 983156983144983141 983154983141983155983157983148983156 983156983151 983156983144983141 983152983154983141983140983145983139983137983156983141 991260983143983156991261983098 983143983156(983095 983091+2)

983111983141983156 983156983144983141 983142983145983150983137983148 983154983141983155983157983148983156 (983124983154983157983141)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 36

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

1 Marcus was a man

man(Marcus)

2 Marcus was a PompeianPompeian(Marcus)

3 Marcus was born in 40 AD

born(Marcus40)

4 All men are mortal

forallX man(X) mortal(X)

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Artificial Intelligence Predicate Logic Slide 37

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

5 All Pompeian died when the vocano erupted in 79 AD

erupted(vocano 79) ^ forallX [Pompeian(X) died(X 79)]

6 No mortal lives longer then 150 yearsforallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

7 It is now 1991now = 1991

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Artificial Intelligence Predicate Logic Slide 38

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Question

Is Marcus alive

Or

alive(Marcus now)

ornotnotnotnotalive(Marcus now)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 39

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge

Relation between ldquodeadrdquo and ldquoaliverdquo

8 Alive means not deadforallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 40

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge Time properties of an event for example ldquodeadrdquo

9 Is someone dies the he is dead at all later times

forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

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Artificial Intelligence Predicate Logic Slide 41

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Summary of facts1 man(Marcus)

2 Pompeian(Marcus)

3 born(Marcus 40)

4 forallX man(X) mortal(X)

5 forallX Pompeian(X) died(X 79)

6 erupted(volcano 79)

7 forallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

8 now = 19919 forallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

10 forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 42

notalive(Marcus now)

dead(Marcus now)

(9 Substitute)

died(Marcus T1) gt(now T1)

(10 Substitute)

Pompeian(Marcus) gt(now 79)

(5 Substitute)(5 Substitute)

(2)gt(1991 79)

(8 Substitute)

(computation)

ss

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 43

983125983150983145983158983141983154983155983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983125983113)

Every instantiation of a universally quantified sentence is enta iled by it

forallv α

Subst(vg α)

for any variable v and ground term g

Eg forallx King( x) and Greedy( x)rArr Evil( x

) yieldsKing (John ) and Greedy (John ) rArr Evil (John )

King (Richard ) and Greedy (Richard ) rArr Evil (Richard )

King (Father (John )) and Greedy (Father (John )) rArr Evil (Father (John ))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 44

983109983160983145983155983156983141983150983156983145983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983109983113)

For any sentence α variable v and constant symbol k that

does not

appear elsewhere in the knowledge baseexistv α

Subst(vk α )

Eg exist x Crown( x) and OnHead ( xJohn) yields

Crown(C 1) and OnHead (C 1 John )

provided C 1 is a new constant symbol called a Skolemconstant

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4580

Artificial Intelligence Predicate Logic Slide 45

983122983141983140983157983139983156983145983151983150 983156983151 983152983154983151983152983151983155983145983156983145983151983150983137983148 983145983150983142983141983154983141983150983139983141

Suppose the KB contains just the followingforall

x King(x)and

Greedy(x)rArr

Evil(x)King(John)Greedy(John)Brother(RichardJohn)

Instantiating the universal sentence in all possible ways we haveKing(John) and Greedy(John) rArr Evil(John)King(Richard) and Greedy(Richard) rArr Evil(Richard)King(John)Greedy(John)Brother(RichardJohn

)

The new KB is propositionalized proposition symbols are

King(John) Greedy(John) Evil(John) King(Richard

) etc

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 46

983122983141983140983157983139983156983145983151983150 983139983151983150983156983140983086

Every FOL KB can be propositionalized so as to preserve entailment

(A ground sentence is entailed by new KB iff entailed by originalKB)

Idea propositionalize KB and query apply resolution return result

Problem with function symbols there are infinitely many ground

terms

eg Father (Father (Father (John

)))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 47

983122983141983155983151983148983157983156983145983151983150

Convert to clause form exampleldquoAll Romans who know Marcus either hate

Caesar or think that anyone who hates anyone is

crazyrdquo

forallX [roman(X) ^ know(X Marcus)]

[hate(X Ceasar) v(forallY existZ hate(YZ)

thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 48

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 49

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

forallX [roman(X) ^ know(X Marcus)][hate(X Ceasar) v

(forallY existZ hate(YZ) thinkcrazy(XY))]

=

forallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 50

983122983141983155983151983148983157983156983145983151983150

2 Move not next to terms

Using the following equivalence

a Double negative

not(notp) = p

b DeMorgan

not(a v b) = nota ^ notb

not(a ^ b) = nota v notb

c Equivalence of qualifiers notforallX P(X) = existX notP(X)

notexistX P(X) = forallX notP(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 51

983122983141983155983151983148983157983156983145983151983150

2 Move not next to termsforallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

=

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) vthinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 52

983122983141983155983151983148983157983156983145983151983150

3 Change name of variables as follows

forallX P(X) v forallX Q(X)

=forallX P(X) v forallY Q(Y)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 53

983122983141983155983151983148983157983156983145983151983150

4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 54

983122983141983155983151983148983157983156983145983151983150

5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 55

983122983141983155983151983148983157983156983145983151983150

6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 56

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 57

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 58

983122983141983155983151983148983157983156983145983151983150

8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 59

983122983141983155983151983148983157983156983145983151983150

9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 60

983122983141983155983151983148983157983156983145983151983150

Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 61

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 62

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 63

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 64

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 65

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 66

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 69

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 71

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 72

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 74

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 75

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 76

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

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892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 27

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983122983141983152983154983141983155983141983150983156983137983156983145983151983150

Given the above sentences can we make aconclusion as follows

ldquoMarcus was not loyal to Caesar rdquo

or

notloyalto(MarcusCaesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 28

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)

(Rule What)

Add an implicit sentence

9 ldquoAll men are peoplerdquo

or

forallforallforallforallX man(X) X man(X) person(X)

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Artificial Intelligence Predicate Logic Slide 29

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)(9 Substitute)

man(Marcus)

(1)

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Artificial Intelligence Predicate Logic Slide 30

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

How to represent exceptions example

ldquoPaulus was a Pompeian Paulus was neitherloyal nor hated Caesarrdquo

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Artificial Intelligence Predicate Logic Slide 31

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation Semantic Net Simple by adding links

(properties) to subclasses or instances

Predicate Logic Not so simple like

983120983151983149983152983141983145983137983150(983120983137983157983148983157983155) 983134 983212983131983148983151983161983137983148983156983151(983120983137983157983148983157983155 983107983137983141983155983137983154) 983158

983144983137983156983141(983120983157983137983148983157983155 983107983137983141983155983137983154)983133

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 32

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation983118983151983156 983155983151 983155983145983149983152983148983141 983106983141983139983137983157983155983141 983156983144983141983154983141 983145983155 983139983151983150983142983148983145983139983156 983138983141983156983159983141983141983150983098

Pompeian(Paulus) ^ not[loyalto(Paulus Caesar) vhate(Pualus Caesar)]

983137983150983140

forallX Pompeian(X) loyalto(X Caesar) v hate(XCaesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 33

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation A Better way

forallX Roman(X) ^ noteq(X Paulus) loyalto(X Caesar)v hate(X Caesar

Exception

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Artificial Intelligence Predicate Logic Slide 34

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

In the case of describing relations betweennumbers example 1 983100 2

2 983100983091

983095 983102 983091 +2 983091 983102 1

991270

rArr

983123983144983151983157983148983140 983150983151983156 983141983160983152983148983145983139983145983156983148983161 983140983141983155983139983154983145983138983141983098983148983156(12) 983148983156(2983091) 991270

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 35

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

rArr Need functions and computable predicates

rArr For example 983107983137983148983148 983137 983142983157983150983139983156983145983151983150 983156983151 983139983151983149983152983157983156983141 (983091+2)

983120983137983155983155 983156983144983141 983154983141983155983157983148983156 983156983151 983156983144983141 983152983154983141983140983145983139983137983156983141 991260983143983156991261983098 983143983156(983095 983091+2)

983111983141983156 983156983144983141 983142983145983150983137983148 983154983141983155983157983148983156 (983124983154983157983141)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 36

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

1 Marcus was a man

man(Marcus)

2 Marcus was a PompeianPompeian(Marcus)

3 Marcus was born in 40 AD

born(Marcus40)

4 All men are mortal

forallX man(X) mortal(X)

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Artificial Intelligence Predicate Logic Slide 37

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

5 All Pompeian died when the vocano erupted in 79 AD

erupted(vocano 79) ^ forallX [Pompeian(X) died(X 79)]

6 No mortal lives longer then 150 yearsforallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

7 It is now 1991now = 1991

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Artificial Intelligence Predicate Logic Slide 38

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Question

Is Marcus alive

Or

alive(Marcus now)

ornotnotnotnotalive(Marcus now)

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Artificial Intelligence Predicate Logic Slide 39

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge

Relation between ldquodeadrdquo and ldquoaliverdquo

8 Alive means not deadforallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 40

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge Time properties of an event for example ldquodeadrdquo

9 Is someone dies the he is dead at all later times

forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4180

Artificial Intelligence Predicate Logic Slide 41

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Summary of facts1 man(Marcus)

2 Pompeian(Marcus)

3 born(Marcus 40)

4 forallX man(X) mortal(X)

5 forallX Pompeian(X) died(X 79)

6 erupted(volcano 79)

7 forallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

8 now = 19919 forallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

10 forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4280

Artificial Intelligence Predicate Logic Slide 42

notalive(Marcus now)

dead(Marcus now)

(9 Substitute)

died(Marcus T1) gt(now T1)

(10 Substitute)

Pompeian(Marcus) gt(now 79)

(5 Substitute)(5 Substitute)

(2)gt(1991 79)

(8 Substitute)

(computation)

ss

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 43

983125983150983145983158983141983154983155983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983125983113)

Every instantiation of a universally quantified sentence is enta iled by it

forallv α

Subst(vg α)

for any variable v and ground term g

Eg forallx King( x) and Greedy( x)rArr Evil( x

) yieldsKing (John ) and Greedy (John ) rArr Evil (John )

King (Richard ) and Greedy (Richard ) rArr Evil (Richard )

King (Father (John )) and Greedy (Father (John )) rArr Evil (Father (John ))

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Artificial Intelligence Predicate Logic Slide 44

983109983160983145983155983156983141983150983156983145983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983109983113)

For any sentence α variable v and constant symbol k that

does not

appear elsewhere in the knowledge baseexistv α

Subst(vk α )

Eg exist x Crown( x) and OnHead ( xJohn) yields

Crown(C 1) and OnHead (C 1 John )

provided C 1 is a new constant symbol called a Skolemconstant

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 45

983122983141983140983157983139983156983145983151983150 983156983151 983152983154983151983152983151983155983145983156983145983151983150983137983148 983145983150983142983141983154983141983150983139983141

Suppose the KB contains just the followingforall

x King(x)and

Greedy(x)rArr

Evil(x)King(John)Greedy(John)Brother(RichardJohn)

Instantiating the universal sentence in all possible ways we haveKing(John) and Greedy(John) rArr Evil(John)King(Richard) and Greedy(Richard) rArr Evil(Richard)King(John)Greedy(John)Brother(RichardJohn

)

The new KB is propositionalized proposition symbols are

King(John) Greedy(John) Evil(John) King(Richard

) etc

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 46

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Every FOL KB can be propositionalized so as to preserve entailment

(A ground sentence is entailed by new KB iff entailed by originalKB)

Idea propositionalize KB and query apply resolution return result

Problem with function symbols there are infinitely many ground

terms

eg Father (Father (Father (John

)))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 47

983122983141983155983151983148983157983156983145983151983150

Convert to clause form exampleldquoAll Romans who know Marcus either hate

Caesar or think that anyone who hates anyone is

crazyrdquo

forallX [roman(X) ^ know(X Marcus)]

[hate(X Ceasar) v(forallY existZ hate(YZ)

thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 48

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

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Artificial Intelligence Predicate Logic Slide 49

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

forallX [roman(X) ^ know(X Marcus)][hate(X Ceasar) v

(forallY existZ hate(YZ) thinkcrazy(XY))]

=

forallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 50

983122983141983155983151983148983157983156983145983151983150

2 Move not next to terms

Using the following equivalence

a Double negative

not(notp) = p

b DeMorgan

not(a v b) = nota ^ notb

not(a ^ b) = nota v notb

c Equivalence of qualifiers notforallX P(X) = existX notP(X)

notexistX P(X) = forallX notP(X)

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Artificial Intelligence Predicate Logic Slide 51

983122983141983155983151983148983157983156983145983151983150

2 Move not next to termsforallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

=

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) vthinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 52

983122983141983155983151983148983157983156983145983151983150

3 Change name of variables as follows

forallX P(X) v forallX Q(X)

=forallX P(X) v forallY Q(Y)

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Artificial Intelligence Predicate Logic Slide 53

983122983141983155983151983148983157983156983145983151983150

4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 54

983122983141983155983151983148983157983156983145983151983150

5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

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Artificial Intelligence Predicate Logic Slide 55

983122983141983155983151983148983157983156983145983151983150

6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 56

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

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Artificial Intelligence Predicate Logic Slide 57

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 58

983122983141983155983151983148983157983156983145983151983150

8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

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Artificial Intelligence Predicate Logic Slide 59

983122983141983155983151983148983157983156983145983151983150

9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

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Artificial Intelligence Predicate Logic Slide 60

983122983141983155983151983148983157983156983145983151983150

Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

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Artificial Intelligence Predicate Logic Slide 61

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X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

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Artificial Intelligence Predicate Logic Slide 62

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θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

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Artificial Intelligence Predicate Logic Slide 63

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Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

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Artificial Intelligence Predicate Logic Slide 64

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

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Artificial Intelligence Predicate Logic Slide 65

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Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

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Artificial Intelligence Predicate Logic Slide 66

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Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

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Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

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Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

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Artificial Intelligence Predicate Logic Slide 69

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Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

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Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

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Artificial Intelligence Predicate Logic Slide 71

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Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

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Artificial Intelligence Predicate Logic Slide 72

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

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Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

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Artificial Intelligence Predicate Logic Slide 74

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

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Artificial Intelligence Predicate Logic Slide 75

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Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

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Artificial Intelligence Predicate Logic Slide 76

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

892019 First Order(Predicate)Logic

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892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

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Artificial Intelligence Predicate Logic Slide 28

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)

(Rule What)

Add an implicit sentence

9 ldquoAll men are peoplerdquo

or

forallforallforallforallX man(X) X man(X) person(X)

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Artificial Intelligence Predicate Logic Slide 29

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notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)(9 Substitute)

man(Marcus)

(1)

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Artificial Intelligence Predicate Logic Slide 30

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

How to represent exceptions example

ldquoPaulus was a Pompeian Paulus was neitherloyal nor hated Caesarrdquo

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Artificial Intelligence Predicate Logic Slide 31

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation Semantic Net Simple by adding links

(properties) to subclasses or instances

Predicate Logic Not so simple like

983120983151983149983152983141983145983137983150(983120983137983157983148983157983155) 983134 983212983131983148983151983161983137983148983156983151(983120983137983157983148983157983155 983107983137983141983155983137983154) 983158

983144983137983156983141(983120983157983137983148983157983155 983107983137983141983155983137983154)983133

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 32

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation983118983151983156 983155983151 983155983145983149983152983148983141 983106983141983139983137983157983155983141 983156983144983141983154983141 983145983155 983139983151983150983142983148983145983139983156 983138983141983156983159983141983141983150983098

Pompeian(Paulus) ^ not[loyalto(Paulus Caesar) vhate(Pualus Caesar)]

983137983150983140

forallX Pompeian(X) loyalto(X Caesar) v hate(XCaesar)

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Artificial Intelligence Predicate Logic Slide 33

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation A Better way

forallX Roman(X) ^ noteq(X Paulus) loyalto(X Caesar)v hate(X Caesar

Exception

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Artificial Intelligence Predicate Logic Slide 34

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

In the case of describing relations betweennumbers example 1 983100 2

2 983100983091

983095 983102 983091 +2 983091 983102 1

991270

rArr

983123983144983151983157983148983140 983150983151983156 983141983160983152983148983145983139983145983156983148983161 983140983141983155983139983154983145983138983141983098983148983156(12) 983148983156(2983091) 991270

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Artificial Intelligence Predicate Logic Slide 35

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

rArr Need functions and computable predicates

rArr For example 983107983137983148983148 983137 983142983157983150983139983156983145983151983150 983156983151 983139983151983149983152983157983156983141 (983091+2)

983120983137983155983155 983156983144983141 983154983141983155983157983148983156 983156983151 983156983144983141 983152983154983141983140983145983139983137983156983141 991260983143983156991261983098 983143983156(983095 983091+2)

983111983141983156 983156983144983141 983142983145983150983137983148 983154983141983155983157983148983156 (983124983154983157983141)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 36

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

1 Marcus was a man

man(Marcus)

2 Marcus was a PompeianPompeian(Marcus)

3 Marcus was born in 40 AD

born(Marcus40)

4 All men are mortal

forallX man(X) mortal(X)

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Artificial Intelligence Predicate Logic Slide 37

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

5 All Pompeian died when the vocano erupted in 79 AD

erupted(vocano 79) ^ forallX [Pompeian(X) died(X 79)]

6 No mortal lives longer then 150 yearsforallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

7 It is now 1991now = 1991

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Artificial Intelligence Predicate Logic Slide 38

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Question

Is Marcus alive

Or

alive(Marcus now)

ornotnotnotnotalive(Marcus now)

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Artificial Intelligence Predicate Logic Slide 39

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge

Relation between ldquodeadrdquo and ldquoaliverdquo

8 Alive means not deadforallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

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Artificial Intelligence Predicate Logic Slide 40

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge Time properties of an event for example ldquodeadrdquo

9 Is someone dies the he is dead at all later times

forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

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Artificial Intelligence Predicate Logic Slide 41

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Summary of facts1 man(Marcus)

2 Pompeian(Marcus)

3 born(Marcus 40)

4 forallX man(X) mortal(X)

5 forallX Pompeian(X) died(X 79)

6 erupted(volcano 79)

7 forallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

8 now = 19919 forallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

10 forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

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Artificial Intelligence Predicate Logic Slide 42

notalive(Marcus now)

dead(Marcus now)

(9 Substitute)

died(Marcus T1) gt(now T1)

(10 Substitute)

Pompeian(Marcus) gt(now 79)

(5 Substitute)(5 Substitute)

(2)gt(1991 79)

(8 Substitute)

(computation)

ss

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Artificial Intelligence Predicate Logic Slide 43

983125983150983145983158983141983154983155983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983125983113)

Every instantiation of a universally quantified sentence is enta iled by it

forallv α

Subst(vg α)

for any variable v and ground term g

Eg forallx King( x) and Greedy( x)rArr Evil( x

) yieldsKing (John ) and Greedy (John ) rArr Evil (John )

King (Richard ) and Greedy (Richard ) rArr Evil (Richard )

King (Father (John )) and Greedy (Father (John )) rArr Evil (Father (John ))

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Artificial Intelligence Predicate Logic Slide 44

983109983160983145983155983156983141983150983156983145983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983109983113)

For any sentence α variable v and constant symbol k that

does not

appear elsewhere in the knowledge baseexistv α

Subst(vk α )

Eg exist x Crown( x) and OnHead ( xJohn) yields

Crown(C 1) and OnHead (C 1 John )

provided C 1 is a new constant symbol called a Skolemconstant

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Artificial Intelligence Predicate Logic Slide 45

983122983141983140983157983139983156983145983151983150 983156983151 983152983154983151983152983151983155983145983156983145983151983150983137983148 983145983150983142983141983154983141983150983139983141

Suppose the KB contains just the followingforall

x King(x)and

Greedy(x)rArr

Evil(x)King(John)Greedy(John)Brother(RichardJohn)

Instantiating the universal sentence in all possible ways we haveKing(John) and Greedy(John) rArr Evil(John)King(Richard) and Greedy(Richard) rArr Evil(Richard)King(John)Greedy(John)Brother(RichardJohn

)

The new KB is propositionalized proposition symbols are

King(John) Greedy(John) Evil(John) King(Richard

) etc

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 46

983122983141983140983157983139983156983145983151983150 983139983151983150983156983140983086

Every FOL KB can be propositionalized so as to preserve entailment

(A ground sentence is entailed by new KB iff entailed by originalKB)

Idea propositionalize KB and query apply resolution return result

Problem with function symbols there are infinitely many ground

terms

eg Father (Father (Father (John

)))

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Artificial Intelligence Predicate Logic Slide 47

983122983141983155983151983148983157983156983145983151983150

Convert to clause form exampleldquoAll Romans who know Marcus either hate

Caesar or think that anyone who hates anyone is

crazyrdquo

forallX [roman(X) ^ know(X Marcus)]

[hate(X Ceasar) v(forallY existZ hate(YZ)

thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 48

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 49

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

forallX [roman(X) ^ know(X Marcus)][hate(X Ceasar) v

(forallY existZ hate(YZ) thinkcrazy(XY))]

=

forallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 50

983122983141983155983151983148983157983156983145983151983150

2 Move not next to terms

Using the following equivalence

a Double negative

not(notp) = p

b DeMorgan

not(a v b) = nota ^ notb

not(a ^ b) = nota v notb

c Equivalence of qualifiers notforallX P(X) = existX notP(X)

notexistX P(X) = forallX notP(X)

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Artificial Intelligence Predicate Logic Slide 51

983122983141983155983151983148983157983156983145983151983150

2 Move not next to termsforallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

=

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) vthinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 52

983122983141983155983151983148983157983156983145983151983150

3 Change name of variables as follows

forallX P(X) v forallX Q(X)

=forallX P(X) v forallY Q(Y)

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Artificial Intelligence Predicate Logic Slide 53

983122983141983155983151983148983157983156983145983151983150

4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 54

983122983141983155983151983148983157983156983145983151983150

5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

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Artificial Intelligence Predicate Logic Slide 55

983122983141983155983151983148983157983156983145983151983150

6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 56

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

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Artificial Intelligence Predicate Logic Slide 57

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

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Artificial Intelligence Predicate Logic Slide 58

983122983141983155983151983148983157983156983145983151983150

8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

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Artificial Intelligence Predicate Logic Slide 59

983122983141983155983151983148983157983156983145983151983150

9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

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Artificial Intelligence Predicate Logic Slide 60

983122983141983155983151983148983157983156983145983151983150

Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

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Artificial Intelligence Predicate Logic Slide 61

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

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Artificial Intelligence Predicate Logic Slide 62

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

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Artificial Intelligence Predicate Logic Slide 63

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

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Artificial Intelligence Predicate Logic Slide 64

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

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Artificial Intelligence Predicate Logic Slide 65

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

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Artificial Intelligence Predicate Logic Slide 66

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

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Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

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Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

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Artificial Intelligence Predicate Logic Slide 69

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

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Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

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Artificial Intelligence Predicate Logic Slide 71

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

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Artificial Intelligence Predicate Logic Slide 72

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 74

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 75

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 76

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

892019 First Order(Predicate)Logic

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892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 29

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983120983154983151983151983142

notloyalto(MarcusCaesar)

person(Marcus) ruler(Caesar) tryassassinate(Marcus Caesar)

(7 Substitute)

(4) (8)(9 Substitute)

man(Marcus)

(1)

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Artificial Intelligence Predicate Logic Slide 30

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

How to represent exceptions example

ldquoPaulus was a Pompeian Paulus was neitherloyal nor hated Caesarrdquo

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Artificial Intelligence Predicate Logic Slide 31

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation Semantic Net Simple by adding links

(properties) to subclasses or instances

Predicate Logic Not so simple like

983120983151983149983152983141983145983137983150(983120983137983157983148983157983155) 983134 983212983131983148983151983161983137983148983156983151(983120983137983157983148983157983155 983107983137983141983155983137983154) 983158

983144983137983156983141(983120983157983137983148983157983155 983107983137983141983155983137983154)983133

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 32

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation983118983151983156 983155983151 983155983145983149983152983148983141 983106983141983139983137983157983155983141 983156983144983141983154983141 983145983155 983139983151983150983142983148983145983139983156 983138983141983156983159983141983141983150983098

Pompeian(Paulus) ^ not[loyalto(Paulus Caesar) vhate(Pualus Caesar)]

983137983150983140

forallX Pompeian(X) loyalto(X Caesar) v hate(XCaesar)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 33

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation A Better way

forallX Roman(X) ^ noteq(X Paulus) loyalto(X Caesar)v hate(X Caesar

Exception

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Artificial Intelligence Predicate Logic Slide 34

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

In the case of describing relations betweennumbers example 1 983100 2

2 983100983091

983095 983102 983091 +2 983091 983102 1

991270

rArr

983123983144983151983157983148983140 983150983151983156 983141983160983152983148983145983139983145983156983148983161 983140983141983155983139983154983145983138983141983098983148983156(12) 983148983156(2983091) 991270

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 35

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

rArr Need functions and computable predicates

rArr For example 983107983137983148983148 983137 983142983157983150983139983156983145983151983150 983156983151 983139983151983149983152983157983156983141 (983091+2)

983120983137983155983155 983156983144983141 983154983141983155983157983148983156 983156983151 983156983144983141 983152983154983141983140983145983139983137983156983141 991260983143983156991261983098 983143983156(983095 983091+2)

983111983141983156 983156983144983141 983142983145983150983137983148 983154983141983155983157983148983156 (983124983154983157983141)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 36

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

1 Marcus was a man

man(Marcus)

2 Marcus was a PompeianPompeian(Marcus)

3 Marcus was born in 40 AD

born(Marcus40)

4 All men are mortal

forallX man(X) mortal(X)

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Artificial Intelligence Predicate Logic Slide 37

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

5 All Pompeian died when the vocano erupted in 79 AD

erupted(vocano 79) ^ forallX [Pompeian(X) died(X 79)]

6 No mortal lives longer then 150 yearsforallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

7 It is now 1991now = 1991

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Artificial Intelligence Predicate Logic Slide 38

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Question

Is Marcus alive

Or

alive(Marcus now)

ornotnotnotnotalive(Marcus now)

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Artificial Intelligence Predicate Logic Slide 39

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge

Relation between ldquodeadrdquo and ldquoaliverdquo

8 Alive means not deadforallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

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Artificial Intelligence Predicate Logic Slide 40

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge Time properties of an event for example ldquodeadrdquo

9 Is someone dies the he is dead at all later times

forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

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Artificial Intelligence Predicate Logic Slide 41

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Summary of facts1 man(Marcus)

2 Pompeian(Marcus)

3 born(Marcus 40)

4 forallX man(X) mortal(X)

5 forallX Pompeian(X) died(X 79)

6 erupted(volcano 79)

7 forallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

8 now = 19919 forallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

10 forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 42

notalive(Marcus now)

dead(Marcus now)

(9 Substitute)

died(Marcus T1) gt(now T1)

(10 Substitute)

Pompeian(Marcus) gt(now 79)

(5 Substitute)(5 Substitute)

(2)gt(1991 79)

(8 Substitute)

(computation)

ss

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 43

983125983150983145983158983141983154983155983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983125983113)

Every instantiation of a universally quantified sentence is enta iled by it

forallv α

Subst(vg α)

for any variable v and ground term g

Eg forallx King( x) and Greedy( x)rArr Evil( x

) yieldsKing (John ) and Greedy (John ) rArr Evil (John )

King (Richard ) and Greedy (Richard ) rArr Evil (Richard )

King (Father (John )) and Greedy (Father (John )) rArr Evil (Father (John ))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 44

983109983160983145983155983156983141983150983156983145983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983109983113)

For any sentence α variable v and constant symbol k that

does not

appear elsewhere in the knowledge baseexistv α

Subst(vk α )

Eg exist x Crown( x) and OnHead ( xJohn) yields

Crown(C 1) and OnHead (C 1 John )

provided C 1 is a new constant symbol called a Skolemconstant

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 45

983122983141983140983157983139983156983145983151983150 983156983151 983152983154983151983152983151983155983145983156983145983151983150983137983148 983145983150983142983141983154983141983150983139983141

Suppose the KB contains just the followingforall

x King(x)and

Greedy(x)rArr

Evil(x)King(John)Greedy(John)Brother(RichardJohn)

Instantiating the universal sentence in all possible ways we haveKing(John) and Greedy(John) rArr Evil(John)King(Richard) and Greedy(Richard) rArr Evil(Richard)King(John)Greedy(John)Brother(RichardJohn

)

The new KB is propositionalized proposition symbols are

King(John) Greedy(John) Evil(John) King(Richard

) etc

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 46

983122983141983140983157983139983156983145983151983150 983139983151983150983156983140983086

Every FOL KB can be propositionalized so as to preserve entailment

(A ground sentence is entailed by new KB iff entailed by originalKB)

Idea propositionalize KB and query apply resolution return result

Problem with function symbols there are infinitely many ground

terms

eg Father (Father (Father (John

)))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 47

983122983141983155983151983148983157983156983145983151983150

Convert to clause form exampleldquoAll Romans who know Marcus either hate

Caesar or think that anyone who hates anyone is

crazyrdquo

forallX [roman(X) ^ know(X Marcus)]

[hate(X Ceasar) v(forallY existZ hate(YZ)

thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 48

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 49

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

forallX [roman(X) ^ know(X Marcus)][hate(X Ceasar) v

(forallY existZ hate(YZ) thinkcrazy(XY))]

=

forallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 50

983122983141983155983151983148983157983156983145983151983150

2 Move not next to terms

Using the following equivalence

a Double negative

not(notp) = p

b DeMorgan

not(a v b) = nota ^ notb

not(a ^ b) = nota v notb

c Equivalence of qualifiers notforallX P(X) = existX notP(X)

notexistX P(X) = forallX notP(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 51

983122983141983155983151983148983157983156983145983151983150

2 Move not next to termsforallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

=

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) vthinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 52

983122983141983155983151983148983157983156983145983151983150

3 Change name of variables as follows

forallX P(X) v forallX Q(X)

=forallX P(X) v forallY Q(Y)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 53

983122983141983155983151983148983157983156983145983151983150

4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 54

983122983141983155983151983148983157983156983145983151983150

5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 55

983122983141983155983151983148983157983156983145983151983150

6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 56

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 57

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 58

983122983141983155983151983148983157983156983145983151983150

8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 59

983122983141983155983151983148983157983156983145983151983150

9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 60

983122983141983155983151983148983157983156983145983151983150

Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 61

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 62

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 63

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 64

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 65

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

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Artificial Intelligence Predicate Logic Slide 66

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

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Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

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Artificial Intelligence Predicate Logic Slide 69

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

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Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

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Artificial Intelligence Predicate Logic Slide 71

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

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Artificial Intelligence Predicate Logic Slide 72

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

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Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 74

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

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Artificial Intelligence Predicate Logic Slide 75

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

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Artificial Intelligence Predicate Logic Slide 76

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

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892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

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Artificial Intelligence Predicate Logic Slide 30

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

How to represent exceptions example

ldquoPaulus was a Pompeian Paulus was neitherloyal nor hated Caesarrdquo

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Artificial Intelligence Predicate Logic Slide 31

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation Semantic Net Simple by adding links

(properties) to subclasses or instances

Predicate Logic Not so simple like

983120983151983149983152983141983145983137983150(983120983137983157983148983157983155) 983134 983212983131983148983151983161983137983148983156983151(983120983137983157983148983157983155 983107983137983141983155983137983154) 983158

983144983137983156983141(983120983157983137983148983157983155 983107983137983141983155983137983154)983133

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Artificial Intelligence Predicate Logic Slide 32

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation983118983151983156 983155983151 983155983145983149983152983148983141 983106983141983139983137983157983155983141 983156983144983141983154983141 983145983155 983139983151983150983142983148983145983139983156 983138983141983156983159983141983141983150983098

Pompeian(Paulus) ^ not[loyalto(Paulus Caesar) vhate(Pualus Caesar)]

983137983150983140

forallX Pompeian(X) loyalto(X Caesar) v hate(XCaesar)

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Artificial Intelligence Predicate Logic Slide 33

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation A Better way

forallX Roman(X) ^ noteq(X Paulus) loyalto(X Caesar)v hate(X Caesar

Exception

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Artificial Intelligence Predicate Logic Slide 34

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

In the case of describing relations betweennumbers example 1 983100 2

2 983100983091

983095 983102 983091 +2 983091 983102 1

991270

rArr

983123983144983151983157983148983140 983150983151983156 983141983160983152983148983145983139983145983156983148983161 983140983141983155983139983154983145983138983141983098983148983156(12) 983148983156(2983091) 991270

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Artificial Intelligence Predicate Logic Slide 35

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

rArr Need functions and computable predicates

rArr For example 983107983137983148983148 983137 983142983157983150983139983156983145983151983150 983156983151 983139983151983149983152983157983156983141 (983091+2)

983120983137983155983155 983156983144983141 983154983141983155983157983148983156 983156983151 983156983144983141 983152983154983141983140983145983139983137983156983141 991260983143983156991261983098 983143983156(983095 983091+2)

983111983141983156 983156983144983141 983142983145983150983137983148 983154983141983155983157983148983156 (983124983154983157983141)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 36

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

1 Marcus was a man

man(Marcus)

2 Marcus was a PompeianPompeian(Marcus)

3 Marcus was born in 40 AD

born(Marcus40)

4 All men are mortal

forallX man(X) mortal(X)

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Artificial Intelligence Predicate Logic Slide 37

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

5 All Pompeian died when the vocano erupted in 79 AD

erupted(vocano 79) ^ forallX [Pompeian(X) died(X 79)]

6 No mortal lives longer then 150 yearsforallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

7 It is now 1991now = 1991

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Artificial Intelligence Predicate Logic Slide 38

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Question

Is Marcus alive

Or

alive(Marcus now)

ornotnotnotnotalive(Marcus now)

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Artificial Intelligence Predicate Logic Slide 39

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge

Relation between ldquodeadrdquo and ldquoaliverdquo

8 Alive means not deadforallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

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Artificial Intelligence Predicate Logic Slide 40

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge Time properties of an event for example ldquodeadrdquo

9 Is someone dies the he is dead at all later times

forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

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Artificial Intelligence Predicate Logic Slide 41

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Summary of facts1 man(Marcus)

2 Pompeian(Marcus)

3 born(Marcus 40)

4 forallX man(X) mortal(X)

5 forallX Pompeian(X) died(X 79)

6 erupted(volcano 79)

7 forallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

8 now = 19919 forallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

10 forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

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Artificial Intelligence Predicate Logic Slide 42

notalive(Marcus now)

dead(Marcus now)

(9 Substitute)

died(Marcus T1) gt(now T1)

(10 Substitute)

Pompeian(Marcus) gt(now 79)

(5 Substitute)(5 Substitute)

(2)gt(1991 79)

(8 Substitute)

(computation)

ss

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 43

983125983150983145983158983141983154983155983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983125983113)

Every instantiation of a universally quantified sentence is enta iled by it

forallv α

Subst(vg α)

for any variable v and ground term g

Eg forallx King( x) and Greedy( x)rArr Evil( x

) yieldsKing (John ) and Greedy (John ) rArr Evil (John )

King (Richard ) and Greedy (Richard ) rArr Evil (Richard )

King (Father (John )) and Greedy (Father (John )) rArr Evil (Father (John ))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 44

983109983160983145983155983156983141983150983156983145983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983109983113)

For any sentence α variable v and constant symbol k that

does not

appear elsewhere in the knowledge baseexistv α

Subst(vk α )

Eg exist x Crown( x) and OnHead ( xJohn) yields

Crown(C 1) and OnHead (C 1 John )

provided C 1 is a new constant symbol called a Skolemconstant

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 45

983122983141983140983157983139983156983145983151983150 983156983151 983152983154983151983152983151983155983145983156983145983151983150983137983148 983145983150983142983141983154983141983150983139983141

Suppose the KB contains just the followingforall

x King(x)and

Greedy(x)rArr

Evil(x)King(John)Greedy(John)Brother(RichardJohn)

Instantiating the universal sentence in all possible ways we haveKing(John) and Greedy(John) rArr Evil(John)King(Richard) and Greedy(Richard) rArr Evil(Richard)King(John)Greedy(John)Brother(RichardJohn

)

The new KB is propositionalized proposition symbols are

King(John) Greedy(John) Evil(John) King(Richard

) etc

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 46

983122983141983140983157983139983156983145983151983150 983139983151983150983156983140983086

Every FOL KB can be propositionalized so as to preserve entailment

(A ground sentence is entailed by new KB iff entailed by originalKB)

Idea propositionalize KB and query apply resolution return result

Problem with function symbols there are infinitely many ground

terms

eg Father (Father (Father (John

)))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 47

983122983141983155983151983148983157983156983145983151983150

Convert to clause form exampleldquoAll Romans who know Marcus either hate

Caesar or think that anyone who hates anyone is

crazyrdquo

forallX [roman(X) ^ know(X Marcus)]

[hate(X Ceasar) v(forallY existZ hate(YZ)

thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 48

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 49

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

forallX [roman(X) ^ know(X Marcus)][hate(X Ceasar) v

(forallY existZ hate(YZ) thinkcrazy(XY))]

=

forallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 50

983122983141983155983151983148983157983156983145983151983150

2 Move not next to terms

Using the following equivalence

a Double negative

not(notp) = p

b DeMorgan

not(a v b) = nota ^ notb

not(a ^ b) = nota v notb

c Equivalence of qualifiers notforallX P(X) = existX notP(X)

notexistX P(X) = forallX notP(X)

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Artificial Intelligence Predicate Logic Slide 51

983122983141983155983151983148983157983156983145983151983150

2 Move not next to termsforallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

=

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) vthinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 52

983122983141983155983151983148983157983156983145983151983150

3 Change name of variables as follows

forallX P(X) v forallX Q(X)

=forallX P(X) v forallY Q(Y)

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Artificial Intelligence Predicate Logic Slide 53

983122983141983155983151983148983157983156983145983151983150

4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 54

983122983141983155983151983148983157983156983145983151983150

5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

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Artificial Intelligence Predicate Logic Slide 55

983122983141983155983151983148983157983156983145983151983150

6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 56

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

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Artificial Intelligence Predicate Logic Slide 57

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

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Artificial Intelligence Predicate Logic Slide 58

983122983141983155983151983148983157983156983145983151983150

8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

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Artificial Intelligence Predicate Logic Slide 59

983122983141983155983151983148983157983156983145983151983150

9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

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Artificial Intelligence Predicate Logic Slide 60

983122983141983155983151983148983157983156983145983151983150

Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

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Artificial Intelligence Predicate Logic Slide 61

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

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Artificial Intelligence Predicate Logic Slide 62

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θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

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Artificial Intelligence Predicate Logic Slide 63

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

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Artificial Intelligence Predicate Logic Slide 64

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

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Artificial Intelligence Predicate Logic Slide 65

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

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Artificial Intelligence Predicate Logic Slide 66

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

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Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

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Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

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Artificial Intelligence Predicate Logic Slide 69

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Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

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Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

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Artificial Intelligence Predicate Logic Slide 71

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Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

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Artificial Intelligence Predicate Logic Slide 72

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Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

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Artificial Intelligence Predicate Logic Slide 73

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Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

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Artificial Intelligence Predicate Logic Slide 74

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Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

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Artificial Intelligence Predicate Logic Slide 75

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Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

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Artificial Intelligence Predicate Logic Slide 76

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Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

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892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

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Artificial Intelligence Predicate Logic Slide 31

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation Semantic Net Simple by adding links

(properties) to subclasses or instances

Predicate Logic Not so simple like

983120983151983149983152983141983145983137983150(983120983137983157983148983157983155) 983134 983212983131983148983151983161983137983148983156983151(983120983137983157983148983157983155 983107983137983141983155983137983154) 983158

983144983137983156983141(983120983157983137983148983157983155 983107983137983141983155983137983154)983133

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Artificial Intelligence Predicate Logic Slide 32

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation983118983151983156 983155983151 983155983145983149983152983148983141 983106983141983139983137983157983155983141 983156983144983141983154983141 983145983155 983139983151983150983142983148983145983139983156 983138983141983156983159983141983141983150983098

Pompeian(Paulus) ^ not[loyalto(Paulus Caesar) vhate(Pualus Caesar)]

983137983150983140

forallX Pompeian(X) loyalto(X Caesar) v hate(XCaesar)

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Artificial Intelligence Predicate Logic Slide 33

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation A Better way

forallX Roman(X) ^ noteq(X Paulus) loyalto(X Caesar)v hate(X Caesar

Exception

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Artificial Intelligence Predicate Logic Slide 34

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

In the case of describing relations betweennumbers example 1 983100 2

2 983100983091

983095 983102 983091 +2 983091 983102 1

991270

rArr

983123983144983151983157983148983140 983150983151983156 983141983160983152983148983145983139983145983156983148983161 983140983141983155983139983154983145983138983141983098983148983156(12) 983148983156(2983091) 991270

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Artificial Intelligence Predicate Logic Slide 35

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

rArr Need functions and computable predicates

rArr For example 983107983137983148983148 983137 983142983157983150983139983156983145983151983150 983156983151 983139983151983149983152983157983156983141 (983091+2)

983120983137983155983155 983156983144983141 983154983141983155983157983148983156 983156983151 983156983144983141 983152983154983141983140983145983139983137983156983141 991260983143983156991261983098 983143983156(983095 983091+2)

983111983141983156 983156983144983141 983142983145983150983137983148 983154983141983155983157983148983156 (983124983154983157983141)

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Artificial Intelligence Predicate Logic Slide 36

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

1 Marcus was a man

man(Marcus)

2 Marcus was a PompeianPompeian(Marcus)

3 Marcus was born in 40 AD

born(Marcus40)

4 All men are mortal

forallX man(X) mortal(X)

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Artificial Intelligence Predicate Logic Slide 37

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

5 All Pompeian died when the vocano erupted in 79 AD

erupted(vocano 79) ^ forallX [Pompeian(X) died(X 79)]

6 No mortal lives longer then 150 yearsforallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

7 It is now 1991now = 1991

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Artificial Intelligence Predicate Logic Slide 38

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Question

Is Marcus alive

Or

alive(Marcus now)

ornotnotnotnotalive(Marcus now)

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Artificial Intelligence Predicate Logic Slide 39

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge

Relation between ldquodeadrdquo and ldquoaliverdquo

8 Alive means not deadforallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

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Artificial Intelligence Predicate Logic Slide 40

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Add implicit knowledge Time properties of an event for example ldquodeadrdquo

9 Is someone dies the he is dead at all later times

forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

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Artificial Intelligence Predicate Logic Slide 41

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Summary of facts1 man(Marcus)

2 Pompeian(Marcus)

3 born(Marcus 40)

4 forallX man(X) mortal(X)

5 forallX Pompeian(X) died(X 79)

6 erupted(volcano 79)

7 forallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

8 now = 19919 forallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

10 forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 42

notalive(Marcus now)

dead(Marcus now)

(9 Substitute)

died(Marcus T1) gt(now T1)

(10 Substitute)

Pompeian(Marcus) gt(now 79)

(5 Substitute)(5 Substitute)

(2)gt(1991 79)

(8 Substitute)

(computation)

ss

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 43

983125983150983145983158983141983154983155983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983125983113)

Every instantiation of a universally quantified sentence is enta iled by it

forallv α

Subst(vg α)

for any variable v and ground term g

Eg forallx King( x) and Greedy( x)rArr Evil( x

) yieldsKing (John ) and Greedy (John ) rArr Evil (John )

King (Richard ) and Greedy (Richard ) rArr Evil (Richard )

King (Father (John )) and Greedy (Father (John )) rArr Evil (Father (John ))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 44

983109983160983145983155983156983141983150983156983145983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983109983113)

For any sentence α variable v and constant symbol k that

does not

appear elsewhere in the knowledge baseexistv α

Subst(vk α )

Eg exist x Crown( x) and OnHead ( xJohn) yields

Crown(C 1) and OnHead (C 1 John )

provided C 1 is a new constant symbol called a Skolemconstant

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 45

983122983141983140983157983139983156983145983151983150 983156983151 983152983154983151983152983151983155983145983156983145983151983150983137983148 983145983150983142983141983154983141983150983139983141

Suppose the KB contains just the followingforall

x King(x)and

Greedy(x)rArr

Evil(x)King(John)Greedy(John)Brother(RichardJohn)

Instantiating the universal sentence in all possible ways we haveKing(John) and Greedy(John) rArr Evil(John)King(Richard) and Greedy(Richard) rArr Evil(Richard)King(John)Greedy(John)Brother(RichardJohn

)

The new KB is propositionalized proposition symbols are

King(John) Greedy(John) Evil(John) King(Richard

) etc

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 46

983122983141983140983157983139983156983145983151983150 983139983151983150983156983140983086

Every FOL KB can be propositionalized so as to preserve entailment

(A ground sentence is entailed by new KB iff entailed by originalKB)

Idea propositionalize KB and query apply resolution return result

Problem with function symbols there are infinitely many ground

terms

eg Father (Father (Father (John

)))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 47

983122983141983155983151983148983157983156983145983151983150

Convert to clause form exampleldquoAll Romans who know Marcus either hate

Caesar or think that anyone who hates anyone is

crazyrdquo

forallX [roman(X) ^ know(X Marcus)]

[hate(X Ceasar) v(forallY existZ hate(YZ)

thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 48

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 49

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

forallX [roman(X) ^ know(X Marcus)][hate(X Ceasar) v

(forallY existZ hate(YZ) thinkcrazy(XY))]

=

forallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 50

983122983141983155983151983148983157983156983145983151983150

2 Move not next to terms

Using the following equivalence

a Double negative

not(notp) = p

b DeMorgan

not(a v b) = nota ^ notb

not(a ^ b) = nota v notb

c Equivalence of qualifiers notforallX P(X) = existX notP(X)

notexistX P(X) = forallX notP(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 51

983122983141983155983151983148983157983156983145983151983150

2 Move not next to termsforallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

=

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) vthinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 52

983122983141983155983151983148983157983156983145983151983150

3 Change name of variables as follows

forallX P(X) v forallX Q(X)

=forallX P(X) v forallY Q(Y)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 53

983122983141983155983151983148983157983156983145983151983150

4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 54

983122983141983155983151983148983157983156983145983151983150

5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

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Artificial Intelligence Predicate Logic Slide 55

983122983141983155983151983148983157983156983145983151983150

6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 56

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

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Artificial Intelligence Predicate Logic Slide 57

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 58

983122983141983155983151983148983157983156983145983151983150

8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

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Artificial Intelligence Predicate Logic Slide 59

983122983141983155983151983148983157983156983145983151983150

9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

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Artificial Intelligence Predicate Logic Slide 60

983122983141983155983151983148983157983156983145983151983150

Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

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Artificial Intelligence Predicate Logic Slide 61

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 62

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 63

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 64

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 65

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 66

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

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Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

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Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 69

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 71

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 72

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 74

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 75

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

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Artificial Intelligence Predicate Logic Slide 76

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

892019 First Order(Predicate)Logic

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892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 32

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation983118983151983156 983155983151 983155983145983149983152983148983141 983106983141983139983137983157983155983141 983156983144983141983154983141 983145983155 983139983151983150983142983148983145983139983156 983138983141983156983159983141983141983150983098

Pompeian(Paulus) ^ not[loyalto(Paulus Caesar) vhate(Pualus Caesar)]

983137983150983140

forallX Pompeian(X) loyalto(X Caesar) v hate(XCaesar)

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Artificial Intelligence Predicate Logic Slide 33

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation A Better way

forallX Roman(X) ^ noteq(X Paulus) loyalto(X Caesar)v hate(X Caesar

Exception

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Artificial Intelligence Predicate Logic Slide 34

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

In the case of describing relations betweennumbers example 1 983100 2

2 983100983091

983095 983102 983091 +2 983091 983102 1

991270

rArr

983123983144983151983157983148983140 983150983151983156 983141983160983152983148983145983139983145983156983148983161 983140983141983155983139983154983145983138983141983098983148983156(12) 983148983156(2983091) 991270

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Artificial Intelligence Predicate Logic Slide 35

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

rArr Need functions and computable predicates

rArr For example 983107983137983148983148 983137 983142983157983150983139983156983145983151983150 983156983151 983139983151983149983152983157983156983141 (983091+2)

983120983137983155983155 983156983144983141 983154983141983155983157983148983156 983156983151 983156983144983141 983152983154983141983140983145983139983137983156983141 991260983143983156991261983098 983143983156(983095 983091+2)

983111983141983156 983156983144983141 983142983145983150983137983148 983154983141983155983157983148983156 (983124983154983157983141)

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Artificial Intelligence Predicate Logic Slide 36

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Facts

1 Marcus was a man

man(Marcus)

2 Marcus was a PompeianPompeian(Marcus)

3 Marcus was born in 40 AD

born(Marcus40)

4 All men are mortal

forallX man(X) mortal(X)

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Artificial Intelligence Predicate Logic Slide 37

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Facts

5 All Pompeian died when the vocano erupted in 79 AD

erupted(vocano 79) ^ forallX [Pompeian(X) died(X 79)]

6 No mortal lives longer then 150 yearsforallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

7 It is now 1991now = 1991

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Artificial Intelligence Predicate Logic Slide 38

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Question

Is Marcus alive

Or

alive(Marcus now)

ornotnotnotnotalive(Marcus now)

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Artificial Intelligence Predicate Logic Slide 39

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge

Relation between ldquodeadrdquo and ldquoaliverdquo

8 Alive means not deadforallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

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Artificial Intelligence Predicate Logic Slide 40

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge Time properties of an event for example ldquodeadrdquo

9 Is someone dies the he is dead at all later times

forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

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Artificial Intelligence Predicate Logic Slide 41

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Summary of facts1 man(Marcus)

2 Pompeian(Marcus)

3 born(Marcus 40)

4 forallX man(X) mortal(X)

5 forallX Pompeian(X) died(X 79)

6 erupted(volcano 79)

7 forallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

8 now = 19919 forallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

10 forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

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Artificial Intelligence Predicate Logic Slide 42

notalive(Marcus now)

dead(Marcus now)

(9 Substitute)

died(Marcus T1) gt(now T1)

(10 Substitute)

Pompeian(Marcus) gt(now 79)

(5 Substitute)(5 Substitute)

(2)gt(1991 79)

(8 Substitute)

(computation)

ss

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Artificial Intelligence Predicate Logic Slide 43

983125983150983145983158983141983154983155983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983125983113)

Every instantiation of a universally quantified sentence is enta iled by it

forallv α

Subst(vg α)

for any variable v and ground term g

Eg forallx King( x) and Greedy( x)rArr Evil( x

) yieldsKing (John ) and Greedy (John ) rArr Evil (John )

King (Richard ) and Greedy (Richard ) rArr Evil (Richard )

King (Father (John )) and Greedy (Father (John )) rArr Evil (Father (John ))

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Artificial Intelligence Predicate Logic Slide 44

983109983160983145983155983156983141983150983156983145983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983109983113)

For any sentence α variable v and constant symbol k that

does not

appear elsewhere in the knowledge baseexistv α

Subst(vk α )

Eg exist x Crown( x) and OnHead ( xJohn) yields

Crown(C 1) and OnHead (C 1 John )

provided C 1 is a new constant symbol called a Skolemconstant

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Artificial Intelligence Predicate Logic Slide 45

983122983141983140983157983139983156983145983151983150 983156983151 983152983154983151983152983151983155983145983156983145983151983150983137983148 983145983150983142983141983154983141983150983139983141

Suppose the KB contains just the followingforall

x King(x)and

Greedy(x)rArr

Evil(x)King(John)Greedy(John)Brother(RichardJohn)

Instantiating the universal sentence in all possible ways we haveKing(John) and Greedy(John) rArr Evil(John)King(Richard) and Greedy(Richard) rArr Evil(Richard)King(John)Greedy(John)Brother(RichardJohn

)

The new KB is propositionalized proposition symbols are

King(John) Greedy(John) Evil(John) King(Richard

) etc

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Artificial Intelligence Predicate Logic Slide 46

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Every FOL KB can be propositionalized so as to preserve entailment

(A ground sentence is entailed by new KB iff entailed by originalKB)

Idea propositionalize KB and query apply resolution return result

Problem with function symbols there are infinitely many ground

terms

eg Father (Father (Father (John

)))

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Artificial Intelligence Predicate Logic Slide 47

983122983141983155983151983148983157983156983145983151983150

Convert to clause form exampleldquoAll Romans who know Marcus either hate

Caesar or think that anyone who hates anyone is

crazyrdquo

forallX [roman(X) ^ know(X Marcus)]

[hate(X Ceasar) v(forallY existZ hate(YZ)

thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 48

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 49

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

forallX [roman(X) ^ know(X Marcus)][hate(X Ceasar) v

(forallY existZ hate(YZ) thinkcrazy(XY))]

=

forallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 50

983122983141983155983151983148983157983156983145983151983150

2 Move not next to terms

Using the following equivalence

a Double negative

not(notp) = p

b DeMorgan

not(a v b) = nota ^ notb

not(a ^ b) = nota v notb

c Equivalence of qualifiers notforallX P(X) = existX notP(X)

notexistX P(X) = forallX notP(X)

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Artificial Intelligence Predicate Logic Slide 51

983122983141983155983151983148983157983156983145983151983150

2 Move not next to termsforallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

=

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) vthinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 52

983122983141983155983151983148983157983156983145983151983150

3 Change name of variables as follows

forallX P(X) v forallX Q(X)

=forallX P(X) v forallY Q(Y)

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Artificial Intelligence Predicate Logic Slide 53

983122983141983155983151983148983157983156983145983151983150

4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 54

983122983141983155983151983148983157983156983145983151983150

5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

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Artificial Intelligence Predicate Logic Slide 55

983122983141983155983151983148983157983156983145983151983150

6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 56

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

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Artificial Intelligence Predicate Logic Slide 57

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

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Artificial Intelligence Predicate Logic Slide 58

983122983141983155983151983148983157983156983145983151983150

8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

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Artificial Intelligence Predicate Logic Slide 59

983122983141983155983151983148983157983156983145983151983150

9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

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Artificial Intelligence Predicate Logic Slide 60

983122983141983155983151983148983157983156983145983151983150

Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

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Artificial Intelligence Predicate Logic Slide 61

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

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Artificial Intelligence Predicate Logic Slide 62

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

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Artificial Intelligence Predicate Logic Slide 63

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

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Artificial Intelligence Predicate Logic Slide 64

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

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Artificial Intelligence Predicate Logic Slide 65

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

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Artificial Intelligence Predicate Logic Slide 66

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

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Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

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Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

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Artificial Intelligence Predicate Logic Slide 69

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

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Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

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Artificial Intelligence Predicate Logic Slide 71

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

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Artificial Intelligence Predicate Logic Slide 72

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 74

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

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Artificial Intelligence Predicate Logic Slide 75

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

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Artificial Intelligence Predicate Logic Slide 76

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

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Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

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Artificial Intelligence Predicate Logic Slide 33

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098 983109983160983139983141983152983156983145983151983150

Exception representation A Better way

forallX Roman(X) ^ noteq(X Paulus) loyalto(X Caesar)v hate(X Caesar

Exception

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Artificial Intelligence Predicate Logic Slide 34

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

In the case of describing relations betweennumbers example 1 983100 2

2 983100983091

983095 983102 983091 +2 983091 983102 1

991270

rArr

983123983144983151983157983148983140 983150983151983156 983141983160983152983148983145983139983145983156983148983161 983140983141983155983139983154983145983138983141983098983148983156(12) 983148983156(2983091) 991270

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Artificial Intelligence Predicate Logic Slide 35

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

rArr Need functions and computable predicates

rArr For example 983107983137983148983148 983137 983142983157983150983139983156983145983151983150 983156983151 983139983151983149983152983157983156983141 (983091+2)

983120983137983155983155 983156983144983141 983154983141983155983157983148983156 983156983151 983156983144983141 983152983154983141983140983145983139983137983156983141 991260983143983156991261983098 983143983156(983095 983091+2)

983111983141983156 983156983144983141 983142983145983150983137983148 983154983141983155983157983148983156 (983124983154983157983141)

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Artificial Intelligence Predicate Logic Slide 36

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

1 Marcus was a man

man(Marcus)

2 Marcus was a PompeianPompeian(Marcus)

3 Marcus was born in 40 AD

born(Marcus40)

4 All men are mortal

forallX man(X) mortal(X)

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Artificial Intelligence Predicate Logic Slide 37

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Facts

5 All Pompeian died when the vocano erupted in 79 AD

erupted(vocano 79) ^ forallX [Pompeian(X) died(X 79)]

6 No mortal lives longer then 150 yearsforallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

7 It is now 1991now = 1991

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Artificial Intelligence Predicate Logic Slide 38

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Question

Is Marcus alive

Or

alive(Marcus now)

ornotnotnotnotalive(Marcus now)

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Artificial Intelligence Predicate Logic Slide 39

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge

Relation between ldquodeadrdquo and ldquoaliverdquo

8 Alive means not deadforallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

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Artificial Intelligence Predicate Logic Slide 40

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge Time properties of an event for example ldquodeadrdquo

9 Is someone dies the he is dead at all later times

forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

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Artificial Intelligence Predicate Logic Slide 41

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Summary of facts1 man(Marcus)

2 Pompeian(Marcus)

3 born(Marcus 40)

4 forallX man(X) mortal(X)

5 forallX Pompeian(X) died(X 79)

6 erupted(volcano 79)

7 forallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

8 now = 19919 forallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

10 forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

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Artificial Intelligence Predicate Logic Slide 42

notalive(Marcus now)

dead(Marcus now)

(9 Substitute)

died(Marcus T1) gt(now T1)

(10 Substitute)

Pompeian(Marcus) gt(now 79)

(5 Substitute)(5 Substitute)

(2)gt(1991 79)

(8 Substitute)

(computation)

ss

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Artificial Intelligence Predicate Logic Slide 43

983125983150983145983158983141983154983155983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983125983113)

Every instantiation of a universally quantified sentence is enta iled by it

forallv α

Subst(vg α)

for any variable v and ground term g

Eg forallx King( x) and Greedy( x)rArr Evil( x

) yieldsKing (John ) and Greedy (John ) rArr Evil (John )

King (Richard ) and Greedy (Richard ) rArr Evil (Richard )

King (Father (John )) and Greedy (Father (John )) rArr Evil (Father (John ))

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Artificial Intelligence Predicate Logic Slide 44

983109983160983145983155983156983141983150983156983145983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983109983113)

For any sentence α variable v and constant symbol k that

does not

appear elsewhere in the knowledge baseexistv α

Subst(vk α )

Eg exist x Crown( x) and OnHead ( xJohn) yields

Crown(C 1) and OnHead (C 1 John )

provided C 1 is a new constant symbol called a Skolemconstant

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 45

983122983141983140983157983139983156983145983151983150 983156983151 983152983154983151983152983151983155983145983156983145983151983150983137983148 983145983150983142983141983154983141983150983139983141

Suppose the KB contains just the followingforall

x King(x)and

Greedy(x)rArr

Evil(x)King(John)Greedy(John)Brother(RichardJohn)

Instantiating the universal sentence in all possible ways we haveKing(John) and Greedy(John) rArr Evil(John)King(Richard) and Greedy(Richard) rArr Evil(Richard)King(John)Greedy(John)Brother(RichardJohn

)

The new KB is propositionalized proposition symbols are

King(John) Greedy(John) Evil(John) King(Richard

) etc

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 46

983122983141983140983157983139983156983145983151983150 983139983151983150983156983140983086

Every FOL KB can be propositionalized so as to preserve entailment

(A ground sentence is entailed by new KB iff entailed by originalKB)

Idea propositionalize KB and query apply resolution return result

Problem with function symbols there are infinitely many ground

terms

eg Father (Father (Father (John

)))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 47

983122983141983155983151983148983157983156983145983151983150

Convert to clause form exampleldquoAll Romans who know Marcus either hate

Caesar or think that anyone who hates anyone is

crazyrdquo

forallX [roman(X) ^ know(X Marcus)]

[hate(X Ceasar) v(forallY existZ hate(YZ)

thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 48

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 49

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

forallX [roman(X) ^ know(X Marcus)][hate(X Ceasar) v

(forallY existZ hate(YZ) thinkcrazy(XY))]

=

forallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 50

983122983141983155983151983148983157983156983145983151983150

2 Move not next to terms

Using the following equivalence

a Double negative

not(notp) = p

b DeMorgan

not(a v b) = nota ^ notb

not(a ^ b) = nota v notb

c Equivalence of qualifiers notforallX P(X) = existX notP(X)

notexistX P(X) = forallX notP(X)

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Artificial Intelligence Predicate Logic Slide 51

983122983141983155983151983148983157983156983145983151983150

2 Move not next to termsforallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

=

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) vthinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 52

983122983141983155983151983148983157983156983145983151983150

3 Change name of variables as follows

forallX P(X) v forallX Q(X)

=forallX P(X) v forallY Q(Y)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 53

983122983141983155983151983148983157983156983145983151983150

4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 54

983122983141983155983151983148983157983156983145983151983150

5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 55

983122983141983155983151983148983157983156983145983151983150

6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 56

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 57

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 58

983122983141983155983151983148983157983156983145983151983150

8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 59

983122983141983155983151983148983157983156983145983151983150

9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 60

983122983141983155983151983148983157983156983145983151983150

Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 61

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

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Artificial Intelligence Predicate Logic Slide 62

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 63

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

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Artificial Intelligence Predicate Logic Slide 64

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 65

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 66

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 69

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

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Artificial Intelligence Predicate Logic Slide 71

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 72

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 74

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

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Artificial Intelligence Predicate Logic Slide 75

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 76

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

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892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 34

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

In the case of describing relations betweennumbers example 1 983100 2

2 983100983091

983095 983102 983091 +2 983091 983102 1

991270

rArr

983123983144983151983157983148983140 983150983151983156 983141983160983152983148983145983139983145983156983148983161 983140983141983155983139983154983145983138983141983098983148983156(12) 983148983156(2983091) 991270

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 35

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

rArr Need functions and computable predicates

rArr For example 983107983137983148983148 983137 983142983157983150983139983156983145983151983150 983156983151 983139983151983149983152983157983156983141 (983091+2)

983120983137983155983155 983156983144983141 983154983141983155983157983148983156 983156983151 983156983144983141 983152983154983141983140983145983139983137983156983141 991260983143983156991261983098 983143983156(983095 983091+2)

983111983141983156 983156983144983141 983142983145983150983137983148 983154983141983155983157983148983156 (983124983154983157983141)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 36

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

1 Marcus was a man

man(Marcus)

2 Marcus was a PompeianPompeian(Marcus)

3 Marcus was born in 40 AD

born(Marcus40)

4 All men are mortal

forallX man(X) mortal(X)

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Artificial Intelligence Predicate Logic Slide 37

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

5 All Pompeian died when the vocano erupted in 79 AD

erupted(vocano 79) ^ forallX [Pompeian(X) died(X 79)]

6 No mortal lives longer then 150 yearsforallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

7 It is now 1991now = 1991

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Artificial Intelligence Predicate Logic Slide 38

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Question

Is Marcus alive

Or

alive(Marcus now)

ornotnotnotnotalive(Marcus now)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 39

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge

Relation between ldquodeadrdquo and ldquoaliverdquo

8 Alive means not deadforallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 40

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge Time properties of an event for example ldquodeadrdquo

9 Is someone dies the he is dead at all later times

forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

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Artificial Intelligence Predicate Logic Slide 41

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Summary of facts1 man(Marcus)

2 Pompeian(Marcus)

3 born(Marcus 40)

4 forallX man(X) mortal(X)

5 forallX Pompeian(X) died(X 79)

6 erupted(volcano 79)

7 forallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

8 now = 19919 forallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

10 forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 42

notalive(Marcus now)

dead(Marcus now)

(9 Substitute)

died(Marcus T1) gt(now T1)

(10 Substitute)

Pompeian(Marcus) gt(now 79)

(5 Substitute)(5 Substitute)

(2)gt(1991 79)

(8 Substitute)

(computation)

ss

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 43

983125983150983145983158983141983154983155983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983125983113)

Every instantiation of a universally quantified sentence is enta iled by it

forallv α

Subst(vg α)

for any variable v and ground term g

Eg forallx King( x) and Greedy( x)rArr Evil( x

) yieldsKing (John ) and Greedy (John ) rArr Evil (John )

King (Richard ) and Greedy (Richard ) rArr Evil (Richard )

King (Father (John )) and Greedy (Father (John )) rArr Evil (Father (John ))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 44

983109983160983145983155983156983141983150983156983145983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983109983113)

For any sentence α variable v and constant symbol k that

does not

appear elsewhere in the knowledge baseexistv α

Subst(vk α )

Eg exist x Crown( x) and OnHead ( xJohn) yields

Crown(C 1) and OnHead (C 1 John )

provided C 1 is a new constant symbol called a Skolemconstant

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 45

983122983141983140983157983139983156983145983151983150 983156983151 983152983154983151983152983151983155983145983156983145983151983150983137983148 983145983150983142983141983154983141983150983139983141

Suppose the KB contains just the followingforall

x King(x)and

Greedy(x)rArr

Evil(x)King(John)Greedy(John)Brother(RichardJohn)

Instantiating the universal sentence in all possible ways we haveKing(John) and Greedy(John) rArr Evil(John)King(Richard) and Greedy(Richard) rArr Evil(Richard)King(John)Greedy(John)Brother(RichardJohn

)

The new KB is propositionalized proposition symbols are

King(John) Greedy(John) Evil(John) King(Richard

) etc

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 46

983122983141983140983157983139983156983145983151983150 983139983151983150983156983140983086

Every FOL KB can be propositionalized so as to preserve entailment

(A ground sentence is entailed by new KB iff entailed by originalKB)

Idea propositionalize KB and query apply resolution return result

Problem with function symbols there are infinitely many ground

terms

eg Father (Father (Father (John

)))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 47

983122983141983155983151983148983157983156983145983151983150

Convert to clause form exampleldquoAll Romans who know Marcus either hate

Caesar or think that anyone who hates anyone is

crazyrdquo

forallX [roman(X) ^ know(X Marcus)]

[hate(X Ceasar) v(forallY existZ hate(YZ)

thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 48

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 49

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

forallX [roman(X) ^ know(X Marcus)][hate(X Ceasar) v

(forallY existZ hate(YZ) thinkcrazy(XY))]

=

forallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 50

983122983141983155983151983148983157983156983145983151983150

2 Move not next to terms

Using the following equivalence

a Double negative

not(notp) = p

b DeMorgan

not(a v b) = nota ^ notb

not(a ^ b) = nota v notb

c Equivalence of qualifiers notforallX P(X) = existX notP(X)

notexistX P(X) = forallX notP(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 51

983122983141983155983151983148983157983156983145983151983150

2 Move not next to termsforallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

=

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) vthinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 52

983122983141983155983151983148983157983156983145983151983150

3 Change name of variables as follows

forallX P(X) v forallX Q(X)

=forallX P(X) v forallY Q(Y)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 53

983122983141983155983151983148983157983156983145983151983150

4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 54

983122983141983155983151983148983157983156983145983151983150

5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 55

983122983141983155983151983148983157983156983145983151983150

6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 56

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 57

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 58

983122983141983155983151983148983157983156983145983151983150

8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 59

983122983141983155983151983148983157983156983145983151983150

9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

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Artificial Intelligence Predicate Logic Slide 60

983122983141983155983151983148983157983156983145983151983150

Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

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Artificial Intelligence Predicate Logic Slide 61

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 62

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

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Artificial Intelligence Predicate Logic Slide 63

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

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Artificial Intelligence Predicate Logic Slide 64

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 65

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

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Artificial Intelligence Predicate Logic Slide 66

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

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Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

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Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

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Artificial Intelligence Predicate Logic Slide 69

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

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Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

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Artificial Intelligence Predicate Logic Slide 71

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

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Artificial Intelligence Predicate Logic Slide 72

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

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Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

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Artificial Intelligence Predicate Logic Slide 74

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

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Artificial Intelligence Predicate Logic Slide 75

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

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Artificial Intelligence Predicate Logic Slide 76

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Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

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892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

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Artificial Intelligence Predicate Logic Slide 35

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141

rArr Need functions and computable predicates

rArr For example 983107983137983148983148 983137 983142983157983150983139983156983145983151983150 983156983151 983139983151983149983152983157983156983141 (983091+2)

983120983137983155983155 983156983144983141 983154983141983155983157983148983156 983156983151 983156983144983141 983152983154983141983140983145983139983137983156983141 991260983143983156991261983098 983143983156(983095 983091+2)

983111983141983156 983156983144983141 983142983145983150983137983148 983154983141983155983157983148983156 (983124983154983157983141)

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Artificial Intelligence Predicate Logic Slide 36

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

1 Marcus was a man

man(Marcus)

2 Marcus was a PompeianPompeian(Marcus)

3 Marcus was born in 40 AD

born(Marcus40)

4 All men are mortal

forallX man(X) mortal(X)

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Artificial Intelligence Predicate Logic Slide 37

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Facts

5 All Pompeian died when the vocano erupted in 79 AD

erupted(vocano 79) ^ forallX [Pompeian(X) died(X 79)]

6 No mortal lives longer then 150 yearsforallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

7 It is now 1991now = 1991

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Artificial Intelligence Predicate Logic Slide 38

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Question

Is Marcus alive

Or

alive(Marcus now)

ornotnotnotnotalive(Marcus now)

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Artificial Intelligence Predicate Logic Slide 39

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge

Relation between ldquodeadrdquo and ldquoaliverdquo

8 Alive means not deadforallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

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Artificial Intelligence Predicate Logic Slide 40

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge Time properties of an event for example ldquodeadrdquo

9 Is someone dies the he is dead at all later times

forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

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Artificial Intelligence Predicate Logic Slide 41

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Summary of facts1 man(Marcus)

2 Pompeian(Marcus)

3 born(Marcus 40)

4 forallX man(X) mortal(X)

5 forallX Pompeian(X) died(X 79)

6 erupted(volcano 79)

7 forallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

8 now = 19919 forallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

10 forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

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Artificial Intelligence Predicate Logic Slide 42

notalive(Marcus now)

dead(Marcus now)

(9 Substitute)

died(Marcus T1) gt(now T1)

(10 Substitute)

Pompeian(Marcus) gt(now 79)

(5 Substitute)(5 Substitute)

(2)gt(1991 79)

(8 Substitute)

(computation)

ss

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Artificial Intelligence Predicate Logic Slide 43

983125983150983145983158983141983154983155983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983125983113)

Every instantiation of a universally quantified sentence is enta iled by it

forallv α

Subst(vg α)

for any variable v and ground term g

Eg forallx King( x) and Greedy( x)rArr Evil( x

) yieldsKing (John ) and Greedy (John ) rArr Evil (John )

King (Richard ) and Greedy (Richard ) rArr Evil (Richard )

King (Father (John )) and Greedy (Father (John )) rArr Evil (Father (John ))

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Artificial Intelligence Predicate Logic Slide 44

983109983160983145983155983156983141983150983156983145983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983109983113)

For any sentence α variable v and constant symbol k that

does not

appear elsewhere in the knowledge baseexistv α

Subst(vk α )

Eg exist x Crown( x) and OnHead ( xJohn) yields

Crown(C 1) and OnHead (C 1 John )

provided C 1 is a new constant symbol called a Skolemconstant

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Artificial Intelligence Predicate Logic Slide 45

983122983141983140983157983139983156983145983151983150 983156983151 983152983154983151983152983151983155983145983156983145983151983150983137983148 983145983150983142983141983154983141983150983139983141

Suppose the KB contains just the followingforall

x King(x)and

Greedy(x)rArr

Evil(x)King(John)Greedy(John)Brother(RichardJohn)

Instantiating the universal sentence in all possible ways we haveKing(John) and Greedy(John) rArr Evil(John)King(Richard) and Greedy(Richard) rArr Evil(Richard)King(John)Greedy(John)Brother(RichardJohn

)

The new KB is propositionalized proposition symbols are

King(John) Greedy(John) Evil(John) King(Richard

) etc

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 46

983122983141983140983157983139983156983145983151983150 983139983151983150983156983140983086

Every FOL KB can be propositionalized so as to preserve entailment

(A ground sentence is entailed by new KB iff entailed by originalKB)

Idea propositionalize KB and query apply resolution return result

Problem with function symbols there are infinitely many ground

terms

eg Father (Father (Father (John

)))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 47

983122983141983155983151983148983157983156983145983151983150

Convert to clause form exampleldquoAll Romans who know Marcus either hate

Caesar or think that anyone who hates anyone is

crazyrdquo

forallX [roman(X) ^ know(X Marcus)]

[hate(X Ceasar) v(forallY existZ hate(YZ)

thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 48

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 49

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

forallX [roman(X) ^ know(X Marcus)][hate(X Ceasar) v

(forallY existZ hate(YZ) thinkcrazy(XY))]

=

forallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 50

983122983141983155983151983148983157983156983145983151983150

2 Move not next to terms

Using the following equivalence

a Double negative

not(notp) = p

b DeMorgan

not(a v b) = nota ^ notb

not(a ^ b) = nota v notb

c Equivalence of qualifiers notforallX P(X) = existX notP(X)

notexistX P(X) = forallX notP(X)

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Artificial Intelligence Predicate Logic Slide 51

983122983141983155983151983148983157983156983145983151983150

2 Move not next to termsforallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

=

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) vthinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 52

983122983141983155983151983148983157983156983145983151983150

3 Change name of variables as follows

forallX P(X) v forallX Q(X)

=forallX P(X) v forallY Q(Y)

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Artificial Intelligence Predicate Logic Slide 53

983122983141983155983151983148983157983156983145983151983150

4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 5480

Artificial Intelligence Predicate Logic Slide 54

983122983141983155983151983148983157983156983145983151983150

5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 55

983122983141983155983151983148983157983156983145983151983150

6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 56

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 57

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 58

983122983141983155983151983148983157983156983145983151983150

8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 59

983122983141983155983151983148983157983156983145983151983150

9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

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Artificial Intelligence Predicate Logic Slide 60

983122983141983155983151983148983157983156983145983151983150

Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 61

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

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Artificial Intelligence Predicate Logic Slide 62

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

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Artificial Intelligence Predicate Logic Slide 63

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 64

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 65

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

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Artificial Intelligence Predicate Logic Slide 66

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

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Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

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Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

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Artificial Intelligence Predicate Logic Slide 69

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

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Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

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Artificial Intelligence Predicate Logic Slide 71

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

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Artificial Intelligence Predicate Logic Slide 72

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

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Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

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Artificial Intelligence Predicate Logic Slide 74

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

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Artificial Intelligence Predicate Logic Slide 75

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

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Artificial Intelligence Predicate Logic Slide 76

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Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

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892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

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Artificial Intelligence Predicate Logic Slide 36

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Facts

1 Marcus was a man

man(Marcus)

2 Marcus was a PompeianPompeian(Marcus)

3 Marcus was born in 40 AD

born(Marcus40)

4 All men are mortal

forallX man(X) mortal(X)

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Artificial Intelligence Predicate Logic Slide 37

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Facts

5 All Pompeian died when the vocano erupted in 79 AD

erupted(vocano 79) ^ forallX [Pompeian(X) died(X 79)]

6 No mortal lives longer then 150 yearsforallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

7 It is now 1991now = 1991

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Artificial Intelligence Predicate Logic Slide 38

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Question

Is Marcus alive

Or

alive(Marcus now)

ornotnotnotnotalive(Marcus now)

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Artificial Intelligence Predicate Logic Slide 39

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Add implicit knowledge

Relation between ldquodeadrdquo and ldquoaliverdquo

8 Alive means not deadforallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

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Artificial Intelligence Predicate Logic Slide 40

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Add implicit knowledge Time properties of an event for example ldquodeadrdquo

9 Is someone dies the he is dead at all later times

forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

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Artificial Intelligence Predicate Logic Slide 41

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Summary of facts1 man(Marcus)

2 Pompeian(Marcus)

3 born(Marcus 40)

4 forallX man(X) mortal(X)

5 forallX Pompeian(X) died(X 79)

6 erupted(volcano 79)

7 forallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

8 now = 19919 forallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

10 forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

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Artificial Intelligence Predicate Logic Slide 42

notalive(Marcus now)

dead(Marcus now)

(9 Substitute)

died(Marcus T1) gt(now T1)

(10 Substitute)

Pompeian(Marcus) gt(now 79)

(5 Substitute)(5 Substitute)

(2)gt(1991 79)

(8 Substitute)

(computation)

ss

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Artificial Intelligence Predicate Logic Slide 43

983125983150983145983158983141983154983155983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983125983113)

Every instantiation of a universally quantified sentence is enta iled by it

forallv α

Subst(vg α)

for any variable v and ground term g

Eg forallx King( x) and Greedy( x)rArr Evil( x

) yieldsKing (John ) and Greedy (John ) rArr Evil (John )

King (Richard ) and Greedy (Richard ) rArr Evil (Richard )

King (Father (John )) and Greedy (Father (John )) rArr Evil (Father (John ))

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Artificial Intelligence Predicate Logic Slide 44

983109983160983145983155983156983141983150983156983145983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983109983113)

For any sentence α variable v and constant symbol k that

does not

appear elsewhere in the knowledge baseexistv α

Subst(vk α )

Eg exist x Crown( x) and OnHead ( xJohn) yields

Crown(C 1) and OnHead (C 1 John )

provided C 1 is a new constant symbol called a Skolemconstant

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Artificial Intelligence Predicate Logic Slide 45

983122983141983140983157983139983156983145983151983150 983156983151 983152983154983151983152983151983155983145983156983145983151983150983137983148 983145983150983142983141983154983141983150983139983141

Suppose the KB contains just the followingforall

x King(x)and

Greedy(x)rArr

Evil(x)King(John)Greedy(John)Brother(RichardJohn)

Instantiating the universal sentence in all possible ways we haveKing(John) and Greedy(John) rArr Evil(John)King(Richard) and Greedy(Richard) rArr Evil(Richard)King(John)Greedy(John)Brother(RichardJohn

)

The new KB is propositionalized proposition symbols are

King(John) Greedy(John) Evil(John) King(Richard

) etc

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Artificial Intelligence Predicate Logic Slide 46

983122983141983140983157983139983156983145983151983150 983139983151983150983156983140983086

Every FOL KB can be propositionalized so as to preserve entailment

(A ground sentence is entailed by new KB iff entailed by originalKB)

Idea propositionalize KB and query apply resolution return result

Problem with function symbols there are infinitely many ground

terms

eg Father (Father (Father (John

)))

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Artificial Intelligence Predicate Logic Slide 47

983122983141983155983151983148983157983156983145983151983150

Convert to clause form exampleldquoAll Romans who know Marcus either hate

Caesar or think that anyone who hates anyone is

crazyrdquo

forallX [roman(X) ^ know(X Marcus)]

[hate(X Ceasar) v(forallY existZ hate(YZ)

thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 48

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 49

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

forallX [roman(X) ^ know(X Marcus)][hate(X Ceasar) v

(forallY existZ hate(YZ) thinkcrazy(XY))]

=

forallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 50

983122983141983155983151983148983157983156983145983151983150

2 Move not next to terms

Using the following equivalence

a Double negative

not(notp) = p

b DeMorgan

not(a v b) = nota ^ notb

not(a ^ b) = nota v notb

c Equivalence of qualifiers notforallX P(X) = existX notP(X)

notexistX P(X) = forallX notP(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 51

983122983141983155983151983148983157983156983145983151983150

2 Move not next to termsforallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

=

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) vthinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 52

983122983141983155983151983148983157983156983145983151983150

3 Change name of variables as follows

forallX P(X) v forallX Q(X)

=forallX P(X) v forallY Q(Y)

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Artificial Intelligence Predicate Logic Slide 53

983122983141983155983151983148983157983156983145983151983150

4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 54

983122983141983155983151983148983157983156983145983151983150

5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

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Artificial Intelligence Predicate Logic Slide 55

983122983141983155983151983148983157983156983145983151983150

6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 56

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

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Artificial Intelligence Predicate Logic Slide 57

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

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Artificial Intelligence Predicate Logic Slide 58

983122983141983155983151983148983157983156983145983151983150

8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

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Artificial Intelligence Predicate Logic Slide 59

983122983141983155983151983148983157983156983145983151983150

9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

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Artificial Intelligence Predicate Logic Slide 60

983122983141983155983151983148983157983156983145983151983150

Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

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Artificial Intelligence Predicate Logic Slide 61

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

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Artificial Intelligence Predicate Logic Slide 62

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

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Artificial Intelligence Predicate Logic Slide 63

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

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Artificial Intelligence Predicate Logic Slide 64

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

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Artificial Intelligence Predicate Logic Slide 65

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

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Artificial Intelligence Predicate Logic Slide 66

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

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Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

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Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

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Artificial Intelligence Predicate Logic Slide 69

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

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Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

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Artificial Intelligence Predicate Logic Slide 71

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Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

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Artificial Intelligence Predicate Logic Slide 72

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Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

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Artificial Intelligence Predicate Logic Slide 73

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Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

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Artificial Intelligence Predicate Logic Slide 74

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Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

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Artificial Intelligence Predicate Logic Slide 75

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Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

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Artificial Intelligence Predicate Logic Slide 76

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Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

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892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

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Artificial Intelligence Predicate Logic Slide 37

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Facts

5 All Pompeian died when the vocano erupted in 79 AD

erupted(vocano 79) ^ forallX [Pompeian(X) died(X 79)]

6 No mortal lives longer then 150 yearsforallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

7 It is now 1991now = 1991

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Artificial Intelligence Predicate Logic Slide 38

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Question

Is Marcus alive

Or

alive(Marcus now)

ornotnotnotnotalive(Marcus now)

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Artificial Intelligence Predicate Logic Slide 39

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Add implicit knowledge

Relation between ldquodeadrdquo and ldquoaliverdquo

8 Alive means not deadforallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

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Artificial Intelligence Predicate Logic Slide 40

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Add implicit knowledge Time properties of an event for example ldquodeadrdquo

9 Is someone dies the he is dead at all later times

forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

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Artificial Intelligence Predicate Logic Slide 41

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Summary of facts1 man(Marcus)

2 Pompeian(Marcus)

3 born(Marcus 40)

4 forallX man(X) mortal(X)

5 forallX Pompeian(X) died(X 79)

6 erupted(volcano 79)

7 forallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

8 now = 19919 forallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

10 forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

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Artificial Intelligence Predicate Logic Slide 42

notalive(Marcus now)

dead(Marcus now)

(9 Substitute)

died(Marcus T1) gt(now T1)

(10 Substitute)

Pompeian(Marcus) gt(now 79)

(5 Substitute)(5 Substitute)

(2)gt(1991 79)

(8 Substitute)

(computation)

ss

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Artificial Intelligence Predicate Logic Slide 43

983125983150983145983158983141983154983155983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983125983113)

Every instantiation of a universally quantified sentence is enta iled by it

forallv α

Subst(vg α)

for any variable v and ground term g

Eg forallx King( x) and Greedy( x)rArr Evil( x

) yieldsKing (John ) and Greedy (John ) rArr Evil (John )

King (Richard ) and Greedy (Richard ) rArr Evil (Richard )

King (Father (John )) and Greedy (Father (John )) rArr Evil (Father (John ))

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Artificial Intelligence Predicate Logic Slide 44

983109983160983145983155983156983141983150983156983145983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983109983113)

For any sentence α variable v and constant symbol k that

does not

appear elsewhere in the knowledge baseexistv α

Subst(vk α )

Eg exist x Crown( x) and OnHead ( xJohn) yields

Crown(C 1) and OnHead (C 1 John )

provided C 1 is a new constant symbol called a Skolemconstant

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Artificial Intelligence Predicate Logic Slide 45

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Suppose the KB contains just the followingforall

x King(x)and

Greedy(x)rArr

Evil(x)King(John)Greedy(John)Brother(RichardJohn)

Instantiating the universal sentence in all possible ways we haveKing(John) and Greedy(John) rArr Evil(John)King(Richard) and Greedy(Richard) rArr Evil(Richard)King(John)Greedy(John)Brother(RichardJohn

)

The new KB is propositionalized proposition symbols are

King(John) Greedy(John) Evil(John) King(Richard

) etc

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Artificial Intelligence Predicate Logic Slide 46

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Every FOL KB can be propositionalized so as to preserve entailment

(A ground sentence is entailed by new KB iff entailed by originalKB)

Idea propositionalize KB and query apply resolution return result

Problem with function symbols there are infinitely many ground

terms

eg Father (Father (Father (John

)))

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Artificial Intelligence Predicate Logic Slide 47

983122983141983155983151983148983157983156983145983151983150

Convert to clause form exampleldquoAll Romans who know Marcus either hate

Caesar or think that anyone who hates anyone is

crazyrdquo

forallX [roman(X) ^ know(X Marcus)]

[hate(X Ceasar) v(forallY existZ hate(YZ)

thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 48

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

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Artificial Intelligence Predicate Logic Slide 49

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

forallX [roman(X) ^ know(X Marcus)][hate(X Ceasar) v

(forallY existZ hate(YZ) thinkcrazy(XY))]

=

forallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 50

983122983141983155983151983148983157983156983145983151983150

2 Move not next to terms

Using the following equivalence

a Double negative

not(notp) = p

b DeMorgan

not(a v b) = nota ^ notb

not(a ^ b) = nota v notb

c Equivalence of qualifiers notforallX P(X) = existX notP(X)

notexistX P(X) = forallX notP(X)

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Artificial Intelligence Predicate Logic Slide 51

983122983141983155983151983148983157983156983145983151983150

2 Move not next to termsforallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

=

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) vthinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 52

983122983141983155983151983148983157983156983145983151983150

3 Change name of variables as follows

forallX P(X) v forallX Q(X)

=forallX P(X) v forallY Q(Y)

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Artificial Intelligence Predicate Logic Slide 53

983122983141983155983151983148983157983156983145983151983150

4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 54

983122983141983155983151983148983157983156983145983151983150

5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

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Artificial Intelligence Predicate Logic Slide 55

983122983141983155983151983148983157983156983145983151983150

6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 56

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

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Artificial Intelligence Predicate Logic Slide 57

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

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Artificial Intelligence Predicate Logic Slide 58

983122983141983155983151983148983157983156983145983151983150

8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

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Artificial Intelligence Predicate Logic Slide 59

983122983141983155983151983148983157983156983145983151983150

9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

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Artificial Intelligence Predicate Logic Slide 60

983122983141983155983151983148983157983156983145983151983150

Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

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Artificial Intelligence Predicate Logic Slide 61

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

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Artificial Intelligence Predicate Logic Slide 62

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

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Artificial Intelligence Predicate Logic Slide 63

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

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Artificial Intelligence Predicate Logic Slide 64

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

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Artificial Intelligence Predicate Logic Slide 65

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

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Artificial Intelligence Predicate Logic Slide 66

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

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Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

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Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

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Artificial Intelligence Predicate Logic Slide 69

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

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Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

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Artificial Intelligence Predicate Logic Slide 71

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Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

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Artificial Intelligence Predicate Logic Slide 72

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

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Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

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Artificial Intelligence Predicate Logic Slide 74

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Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

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Artificial Intelligence Predicate Logic Slide 75

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

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Artificial Intelligence Predicate Logic Slide 76

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Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

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892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

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Artificial Intelligence Predicate Logic Slide 38

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Question

Is Marcus alive

Or

alive(Marcus now)

ornotnotnotnotalive(Marcus now)

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Artificial Intelligence Predicate Logic Slide 39

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Add implicit knowledge

Relation between ldquodeadrdquo and ldquoaliverdquo

8 Alive means not deadforallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

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Artificial Intelligence Predicate Logic Slide 40

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Add implicit knowledge Time properties of an event for example ldquodeadrdquo

9 Is someone dies the he is dead at all later times

forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

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Artificial Intelligence Predicate Logic Slide 41

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Summary of facts1 man(Marcus)

2 Pompeian(Marcus)

3 born(Marcus 40)

4 forallX man(X) mortal(X)

5 forallX Pompeian(X) died(X 79)

6 erupted(volcano 79)

7 forallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

8 now = 19919 forallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

10 forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

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Artificial Intelligence Predicate Logic Slide 42

notalive(Marcus now)

dead(Marcus now)

(9 Substitute)

died(Marcus T1) gt(now T1)

(10 Substitute)

Pompeian(Marcus) gt(now 79)

(5 Substitute)(5 Substitute)

(2)gt(1991 79)

(8 Substitute)

(computation)

ss

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Artificial Intelligence Predicate Logic Slide 43

983125983150983145983158983141983154983155983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983125983113)

Every instantiation of a universally quantified sentence is enta iled by it

forallv α

Subst(vg α)

for any variable v and ground term g

Eg forallx King( x) and Greedy( x)rArr Evil( x

) yieldsKing (John ) and Greedy (John ) rArr Evil (John )

King (Richard ) and Greedy (Richard ) rArr Evil (Richard )

King (Father (John )) and Greedy (Father (John )) rArr Evil (Father (John ))

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Artificial Intelligence Predicate Logic Slide 44

983109983160983145983155983156983141983150983156983145983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983109983113)

For any sentence α variable v and constant symbol k that

does not

appear elsewhere in the knowledge baseexistv α

Subst(vk α )

Eg exist x Crown( x) and OnHead ( xJohn) yields

Crown(C 1) and OnHead (C 1 John )

provided C 1 is a new constant symbol called a Skolemconstant

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Artificial Intelligence Predicate Logic Slide 45

983122983141983140983157983139983156983145983151983150 983156983151 983152983154983151983152983151983155983145983156983145983151983150983137983148 983145983150983142983141983154983141983150983139983141

Suppose the KB contains just the followingforall

x King(x)and

Greedy(x)rArr

Evil(x)King(John)Greedy(John)Brother(RichardJohn)

Instantiating the universal sentence in all possible ways we haveKing(John) and Greedy(John) rArr Evil(John)King(Richard) and Greedy(Richard) rArr Evil(Richard)King(John)Greedy(John)Brother(RichardJohn

)

The new KB is propositionalized proposition symbols are

King(John) Greedy(John) Evil(John) King(Richard

) etc

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Artificial Intelligence Predicate Logic Slide 46

983122983141983140983157983139983156983145983151983150 983139983151983150983156983140983086

Every FOL KB can be propositionalized so as to preserve entailment

(A ground sentence is entailed by new KB iff entailed by originalKB)

Idea propositionalize KB and query apply resolution return result

Problem with function symbols there are infinitely many ground

terms

eg Father (Father (Father (John

)))

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Artificial Intelligence Predicate Logic Slide 47

983122983141983155983151983148983157983156983145983151983150

Convert to clause form exampleldquoAll Romans who know Marcus either hate

Caesar or think that anyone who hates anyone is

crazyrdquo

forallX [roman(X) ^ know(X Marcus)]

[hate(X Ceasar) v(forallY existZ hate(YZ)

thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 48

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 49

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

forallX [roman(X) ^ know(X Marcus)][hate(X Ceasar) v

(forallY existZ hate(YZ) thinkcrazy(XY))]

=

forallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 50

983122983141983155983151983148983157983156983145983151983150

2 Move not next to terms

Using the following equivalence

a Double negative

not(notp) = p

b DeMorgan

not(a v b) = nota ^ notb

not(a ^ b) = nota v notb

c Equivalence of qualifiers notforallX P(X) = existX notP(X)

notexistX P(X) = forallX notP(X)

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Artificial Intelligence Predicate Logic Slide 51

983122983141983155983151983148983157983156983145983151983150

2 Move not next to termsforallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

=

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) vthinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 52

983122983141983155983151983148983157983156983145983151983150

3 Change name of variables as follows

forallX P(X) v forallX Q(X)

=forallX P(X) v forallY Q(Y)

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Artificial Intelligence Predicate Logic Slide 53

983122983141983155983151983148983157983156983145983151983150

4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 54

983122983141983155983151983148983157983156983145983151983150

5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

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Artificial Intelligence Predicate Logic Slide 55

983122983141983155983151983148983157983156983145983151983150

6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 56

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

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Artificial Intelligence Predicate Logic Slide 57

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

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Artificial Intelligence Predicate Logic Slide 58

983122983141983155983151983148983157983156983145983151983150

8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

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Artificial Intelligence Predicate Logic Slide 59

983122983141983155983151983148983157983156983145983151983150

9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

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Artificial Intelligence Predicate Logic Slide 60

983122983141983155983151983148983157983156983145983151983150

Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

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Artificial Intelligence Predicate Logic Slide 61

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

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Artificial Intelligence Predicate Logic Slide 62

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

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Artificial Intelligence Predicate Logic Slide 63

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

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Artificial Intelligence Predicate Logic Slide 64

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

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Artificial Intelligence Predicate Logic Slide 65

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

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Artificial Intelligence Predicate Logic Slide 66

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

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Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

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Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

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Artificial Intelligence Predicate Logic Slide 69

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

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Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

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Artificial Intelligence Predicate Logic Slide 71

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Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

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Artificial Intelligence Predicate Logic Slide 72

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Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

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Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

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Artificial Intelligence Predicate Logic Slide 74

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Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

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Artificial Intelligence Predicate Logic Slide 75

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Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

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Artificial Intelligence Predicate Logic Slide 76

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Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

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892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

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Artificial Intelligence Predicate Logic Slide 39

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Add implicit knowledge

Relation between ldquodeadrdquo and ldquoaliverdquo

8 Alive means not deadforallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

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Artificial Intelligence Predicate Logic Slide 40

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Add implicit knowledge Time properties of an event for example ldquodeadrdquo

9 Is someone dies the he is dead at all later times

forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

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Artificial Intelligence Predicate Logic Slide 41

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Summary of facts1 man(Marcus)

2 Pompeian(Marcus)

3 born(Marcus 40)

4 forallX man(X) mortal(X)

5 forallX Pompeian(X) died(X 79)

6 erupted(volcano 79)

7 forallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

8 now = 19919 forallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

10 forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

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Artificial Intelligence Predicate Logic Slide 42

notalive(Marcus now)

dead(Marcus now)

(9 Substitute)

died(Marcus T1) gt(now T1)

(10 Substitute)

Pompeian(Marcus) gt(now 79)

(5 Substitute)(5 Substitute)

(2)gt(1991 79)

(8 Substitute)

(computation)

ss

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Artificial Intelligence Predicate Logic Slide 43

983125983150983145983158983141983154983155983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983125983113)

Every instantiation of a universally quantified sentence is enta iled by it

forallv α

Subst(vg α)

for any variable v and ground term g

Eg forallx King( x) and Greedy( x)rArr Evil( x

) yieldsKing (John ) and Greedy (John ) rArr Evil (John )

King (Richard ) and Greedy (Richard ) rArr Evil (Richard )

King (Father (John )) and Greedy (Father (John )) rArr Evil (Father (John ))

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Artificial Intelligence Predicate Logic Slide 44

983109983160983145983155983156983141983150983156983145983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983109983113)

For any sentence α variable v and constant symbol k that

does not

appear elsewhere in the knowledge baseexistv α

Subst(vk α )

Eg exist x Crown( x) and OnHead ( xJohn) yields

Crown(C 1) and OnHead (C 1 John )

provided C 1 is a new constant symbol called a Skolemconstant

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Artificial Intelligence Predicate Logic Slide 45

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Suppose the KB contains just the followingforall

x King(x)and

Greedy(x)rArr

Evil(x)King(John)Greedy(John)Brother(RichardJohn)

Instantiating the universal sentence in all possible ways we haveKing(John) and Greedy(John) rArr Evil(John)King(Richard) and Greedy(Richard) rArr Evil(Richard)King(John)Greedy(John)Brother(RichardJohn

)

The new KB is propositionalized proposition symbols are

King(John) Greedy(John) Evil(John) King(Richard

) etc

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Artificial Intelligence Predicate Logic Slide 46

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Every FOL KB can be propositionalized so as to preserve entailment

(A ground sentence is entailed by new KB iff entailed by originalKB)

Idea propositionalize KB and query apply resolution return result

Problem with function symbols there are infinitely many ground

terms

eg Father (Father (Father (John

)))

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Artificial Intelligence Predicate Logic Slide 47

983122983141983155983151983148983157983156983145983151983150

Convert to clause form exampleldquoAll Romans who know Marcus either hate

Caesar or think that anyone who hates anyone is

crazyrdquo

forallX [roman(X) ^ know(X Marcus)]

[hate(X Ceasar) v(forallY existZ hate(YZ)

thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 48

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

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Artificial Intelligence Predicate Logic Slide 49

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

forallX [roman(X) ^ know(X Marcus)][hate(X Ceasar) v

(forallY existZ hate(YZ) thinkcrazy(XY))]

=

forallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 50

983122983141983155983151983148983157983156983145983151983150

2 Move not next to terms

Using the following equivalence

a Double negative

not(notp) = p

b DeMorgan

not(a v b) = nota ^ notb

not(a ^ b) = nota v notb

c Equivalence of qualifiers notforallX P(X) = existX notP(X)

notexistX P(X) = forallX notP(X)

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Artificial Intelligence Predicate Logic Slide 51

983122983141983155983151983148983157983156983145983151983150

2 Move not next to termsforallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

=

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) vthinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 52

983122983141983155983151983148983157983156983145983151983150

3 Change name of variables as follows

forallX P(X) v forallX Q(X)

=forallX P(X) v forallY Q(Y)

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Artificial Intelligence Predicate Logic Slide 53

983122983141983155983151983148983157983156983145983151983150

4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 54

983122983141983155983151983148983157983156983145983151983150

5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

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Artificial Intelligence Predicate Logic Slide 55

983122983141983155983151983148983157983156983145983151983150

6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 56

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

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Artificial Intelligence Predicate Logic Slide 57

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

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Artificial Intelligence Predicate Logic Slide 58

983122983141983155983151983148983157983156983145983151983150

8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

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Artificial Intelligence Predicate Logic Slide 59

983122983141983155983151983148983157983156983145983151983150

9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

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Artificial Intelligence Predicate Logic Slide 60

983122983141983155983151983148983157983156983145983151983150

Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

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Artificial Intelligence Predicate Logic Slide 61

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

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Artificial Intelligence Predicate Logic Slide 62

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θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

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Artificial Intelligence Predicate Logic Slide 63

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Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

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Artificial Intelligence Predicate Logic Slide 64

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

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Artificial Intelligence Predicate Logic Slide 65

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

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Artificial Intelligence Predicate Logic Slide 66

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

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Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

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Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

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Artificial Intelligence Predicate Logic Slide 69

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Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

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Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

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Artificial Intelligence Predicate Logic Slide 71

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Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

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Artificial Intelligence Predicate Logic Slide 72

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Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

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Artificial Intelligence Predicate Logic Slide 73

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Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

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Artificial Intelligence Predicate Logic Slide 74

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Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

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Artificial Intelligence Predicate Logic Slide 75

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Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

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Artificial Intelligence Predicate Logic Slide 76

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Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

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892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

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Artificial Intelligence Predicate Logic Slide 40

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Add implicit knowledge Time properties of an event for example ldquodeadrdquo

9 Is someone dies the he is dead at all later times

forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

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Artificial Intelligence Predicate Logic Slide 41

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Summary of facts1 man(Marcus)

2 Pompeian(Marcus)

3 born(Marcus 40)

4 forallX man(X) mortal(X)

5 forallX Pompeian(X) died(X 79)

6 erupted(volcano 79)

7 forallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

8 now = 19919 forallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

10 forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

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Artificial Intelligence Predicate Logic Slide 42

notalive(Marcus now)

dead(Marcus now)

(9 Substitute)

died(Marcus T1) gt(now T1)

(10 Substitute)

Pompeian(Marcus) gt(now 79)

(5 Substitute)(5 Substitute)

(2)gt(1991 79)

(8 Substitute)

(computation)

ss

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Artificial Intelligence Predicate Logic Slide 43

983125983150983145983158983141983154983155983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983125983113)

Every instantiation of a universally quantified sentence is enta iled by it

forallv α

Subst(vg α)

for any variable v and ground term g

Eg forallx King( x) and Greedy( x)rArr Evil( x

) yieldsKing (John ) and Greedy (John ) rArr Evil (John )

King (Richard ) and Greedy (Richard ) rArr Evil (Richard )

King (Father (John )) and Greedy (Father (John )) rArr Evil (Father (John ))

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Artificial Intelligence Predicate Logic Slide 44

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For any sentence α variable v and constant symbol k that

does not

appear elsewhere in the knowledge baseexistv α

Subst(vk α )

Eg exist x Crown( x) and OnHead ( xJohn) yields

Crown(C 1) and OnHead (C 1 John )

provided C 1 is a new constant symbol called a Skolemconstant

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Artificial Intelligence Predicate Logic Slide 45

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Suppose the KB contains just the followingforall

x King(x)and

Greedy(x)rArr

Evil(x)King(John)Greedy(John)Brother(RichardJohn)

Instantiating the universal sentence in all possible ways we haveKing(John) and Greedy(John) rArr Evil(John)King(Richard) and Greedy(Richard) rArr Evil(Richard)King(John)Greedy(John)Brother(RichardJohn

)

The new KB is propositionalized proposition symbols are

King(John) Greedy(John) Evil(John) King(Richard

) etc

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Artificial Intelligence Predicate Logic Slide 46

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Every FOL KB can be propositionalized so as to preserve entailment

(A ground sentence is entailed by new KB iff entailed by originalKB)

Idea propositionalize KB and query apply resolution return result

Problem with function symbols there are infinitely many ground

terms

eg Father (Father (Father (John

)))

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Artificial Intelligence Predicate Logic Slide 47

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Convert to clause form exampleldquoAll Romans who know Marcus either hate

Caesar or think that anyone who hates anyone is

crazyrdquo

forallX [roman(X) ^ know(X Marcus)]

[hate(X Ceasar) v(forallY existZ hate(YZ)

thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 48

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

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Artificial Intelligence Predicate Logic Slide 49

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

forallX [roman(X) ^ know(X Marcus)][hate(X Ceasar) v

(forallY existZ hate(YZ) thinkcrazy(XY))]

=

forallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 50

983122983141983155983151983148983157983156983145983151983150

2 Move not next to terms

Using the following equivalence

a Double negative

not(notp) = p

b DeMorgan

not(a v b) = nota ^ notb

not(a ^ b) = nota v notb

c Equivalence of qualifiers notforallX P(X) = existX notP(X)

notexistX P(X) = forallX notP(X)

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Artificial Intelligence Predicate Logic Slide 51

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2 Move not next to termsforallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

=

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) vthinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 52

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3 Change name of variables as follows

forallX P(X) v forallX Q(X)

=forallX P(X) v forallY Q(Y)

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Artificial Intelligence Predicate Logic Slide 53

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4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 54

983122983141983155983151983148983157983156983145983151983150

5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

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Artificial Intelligence Predicate Logic Slide 55

983122983141983155983151983148983157983156983145983151983150

6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 56

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

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Artificial Intelligence Predicate Logic Slide 57

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7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

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Artificial Intelligence Predicate Logic Slide 58

983122983141983155983151983148983157983156983145983151983150

8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

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Artificial Intelligence Predicate Logic Slide 59

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9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

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Artificial Intelligence Predicate Logic Slide 60

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Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

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Artificial Intelligence Predicate Logic Slide 61

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X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

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Artificial Intelligence Predicate Logic Slide 62

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θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

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Artificial Intelligence Predicate Logic Slide 63

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Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

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Artificial Intelligence Predicate Logic Slide 64

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

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Artificial Intelligence Predicate Logic Slide 65

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Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

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Artificial Intelligence Predicate Logic Slide 66

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Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

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Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

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Artificial Intelligence Predicate Logic Slide 68

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Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

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Artificial Intelligence Predicate Logic Slide 69

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Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

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Artificial Intelligence Predicate Logic Slide 70

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Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

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Artificial Intelligence Predicate Logic Slide 71

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Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

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Artificial Intelligence Predicate Logic Slide 72

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Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

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Artificial Intelligence Predicate Logic Slide 73

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Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

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Artificial Intelligence Predicate Logic Slide 74

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Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

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Artificial Intelligence Predicate Logic Slide 75

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Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

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Artificial Intelligence Predicate Logic Slide 76

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Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

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892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

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Artificial Intelligence Predicate Logic Slide 41

983120983154983141983140983145983139983137983156983141 983116983151983143983145983139983098983110983157983150983139983156983145983151983150 983083 983107983151983149983152983157983156983137983138983148983141 983120983154983141983140983145983139983137983156983141983084 983109983160983137983149983152983148983141

Summary of facts1 man(Marcus)

2 Pompeian(Marcus)

3 born(Marcus 40)

4 forallX man(X) mortal(X)

5 forallX Pompeian(X) died(X 79)

6 erupted(volcano 79)

7 forallX forallT1 forallT2 mortal(X) ^ born(X T1) ^ gt(T2 ndash T1 150) dead(X T2)

8 now = 19919 forallX forallT [alive(XT) notdead(XT)] ^

[notdead(XT) alive(XT)]

10 forallX forallT1 forallT2 died(XT1) ^ gt(T2 T1) dead(X T2)

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Artificial Intelligence Predicate Logic Slide 42

notalive(Marcus now)

dead(Marcus now)

(9 Substitute)

died(Marcus T1) gt(now T1)

(10 Substitute)

Pompeian(Marcus) gt(now 79)

(5 Substitute)(5 Substitute)

(2)gt(1991 79)

(8 Substitute)

(computation)

ss

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Artificial Intelligence Predicate Logic Slide 43

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Every instantiation of a universally quantified sentence is enta iled by it

forallv α

Subst(vg α)

for any variable v and ground term g

Eg forallx King( x) and Greedy( x)rArr Evil( x

) yieldsKing (John ) and Greedy (John ) rArr Evil (John )

King (Richard ) and Greedy (Richard ) rArr Evil (Richard )

King (Father (John )) and Greedy (Father (John )) rArr Evil (Father (John ))

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Artificial Intelligence Predicate Logic Slide 44

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For any sentence α variable v and constant symbol k that

does not

appear elsewhere in the knowledge baseexistv α

Subst(vk α )

Eg exist x Crown( x) and OnHead ( xJohn) yields

Crown(C 1) and OnHead (C 1 John )

provided C 1 is a new constant symbol called a Skolemconstant

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Artificial Intelligence Predicate Logic Slide 45

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Suppose the KB contains just the followingforall

x King(x)and

Greedy(x)rArr

Evil(x)King(John)Greedy(John)Brother(RichardJohn)

Instantiating the universal sentence in all possible ways we haveKing(John) and Greedy(John) rArr Evil(John)King(Richard) and Greedy(Richard) rArr Evil(Richard)King(John)Greedy(John)Brother(RichardJohn

)

The new KB is propositionalized proposition symbols are

King(John) Greedy(John) Evil(John) King(Richard

) etc

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Artificial Intelligence Predicate Logic Slide 46

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Every FOL KB can be propositionalized so as to preserve entailment

(A ground sentence is entailed by new KB iff entailed by originalKB)

Idea propositionalize KB and query apply resolution return result

Problem with function symbols there are infinitely many ground

terms

eg Father (Father (Father (John

)))

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Artificial Intelligence Predicate Logic Slide 47

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Convert to clause form exampleldquoAll Romans who know Marcus either hate

Caesar or think that anyone who hates anyone is

crazyrdquo

forallX [roman(X) ^ know(X Marcus)]

[hate(X Ceasar) v(forallY existZ hate(YZ)

thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 48

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1 Remove

using the equivalence a b = nota v b

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Artificial Intelligence Predicate Logic Slide 49

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1 Remove

using the equivalence a b = nota v b

forallX [roman(X) ^ know(X Marcus)][hate(X Ceasar) v

(forallY existZ hate(YZ) thinkcrazy(XY))]

=

forallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 50

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2 Move not next to terms

Using the following equivalence

a Double negative

not(notp) = p

b DeMorgan

not(a v b) = nota ^ notb

not(a ^ b) = nota v notb

c Equivalence of qualifiers notforallX P(X) = existX notP(X)

notexistX P(X) = forallX notP(X)

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Artificial Intelligence Predicate Logic Slide 51

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2 Move not next to termsforallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

=

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) vthinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 52

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3 Change name of variables as follows

forallX P(X) v forallX Q(X)

=forallX P(X) v forallY Q(Y)

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Artificial Intelligence Predicate Logic Slide 53

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4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 54

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5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

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Artificial Intelligence Predicate Logic Slide 55

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6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 56

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7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

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Artificial Intelligence Predicate Logic Slide 57

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7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

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Artificial Intelligence Predicate Logic Slide 58

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8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

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Artificial Intelligence Predicate Logic Slide 59

983122983141983155983151983148983157983156983145983151983150

9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

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Artificial Intelligence Predicate Logic Slide 60

983122983141983155983151983148983157983156983145983151983150

Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

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Artificial Intelligence Predicate Logic Slide 61

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X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

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Artificial Intelligence Predicate Logic Slide 62

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θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

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Artificial Intelligence Predicate Logic Slide 63

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Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

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Artificial Intelligence Predicate Logic Slide 64

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Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

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Artificial Intelligence Predicate Logic Slide 65

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Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

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Artificial Intelligence Predicate Logic Slide 66

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Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

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Artificial Intelligence Predicate Logic Slide 67

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mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

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Artificial Intelligence Predicate Logic Slide 68

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Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

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Artificial Intelligence Predicate Logic Slide 69

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Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

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Artificial Intelligence Predicate Logic Slide 70

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Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

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Artificial Intelligence Predicate Logic Slide 71

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Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

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Artificial Intelligence Predicate Logic Slide 72

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Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

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Artificial Intelligence Predicate Logic Slide 73

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Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

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Artificial Intelligence Predicate Logic Slide 74

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Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

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Artificial Intelligence Predicate Logic Slide 75

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Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

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Artificial Intelligence Predicate Logic Slide 76

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Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

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892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

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Artificial Intelligence Predicate Logic Slide 42

notalive(Marcus now)

dead(Marcus now)

(9 Substitute)

died(Marcus T1) gt(now T1)

(10 Substitute)

Pompeian(Marcus) gt(now 79)

(5 Substitute)(5 Substitute)

(2)gt(1991 79)

(8 Substitute)

(computation)

ss

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Artificial Intelligence Predicate Logic Slide 43

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Every instantiation of a universally quantified sentence is enta iled by it

forallv α

Subst(vg α)

for any variable v and ground term g

Eg forallx King( x) and Greedy( x)rArr Evil( x

) yieldsKing (John ) and Greedy (John ) rArr Evil (John )

King (Richard ) and Greedy (Richard ) rArr Evil (Richard )

King (Father (John )) and Greedy (Father (John )) rArr Evil (Father (John ))

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Artificial Intelligence Predicate Logic Slide 44

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For any sentence α variable v and constant symbol k that

does not

appear elsewhere in the knowledge baseexistv α

Subst(vk α )

Eg exist x Crown( x) and OnHead ( xJohn) yields

Crown(C 1) and OnHead (C 1 John )

provided C 1 is a new constant symbol called a Skolemconstant

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Artificial Intelligence Predicate Logic Slide 45

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Suppose the KB contains just the followingforall

x King(x)and

Greedy(x)rArr

Evil(x)King(John)Greedy(John)Brother(RichardJohn)

Instantiating the universal sentence in all possible ways we haveKing(John) and Greedy(John) rArr Evil(John)King(Richard) and Greedy(Richard) rArr Evil(Richard)King(John)Greedy(John)Brother(RichardJohn

)

The new KB is propositionalized proposition symbols are

King(John) Greedy(John) Evil(John) King(Richard

) etc

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Artificial Intelligence Predicate Logic Slide 46

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Every FOL KB can be propositionalized so as to preserve entailment

(A ground sentence is entailed by new KB iff entailed by originalKB)

Idea propositionalize KB and query apply resolution return result

Problem with function symbols there are infinitely many ground

terms

eg Father (Father (Father (John

)))

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Artificial Intelligence Predicate Logic Slide 47

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Convert to clause form exampleldquoAll Romans who know Marcus either hate

Caesar or think that anyone who hates anyone is

crazyrdquo

forallX [roman(X) ^ know(X Marcus)]

[hate(X Ceasar) v(forallY existZ hate(YZ)

thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 48

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1 Remove

using the equivalence a b = nota v b

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Artificial Intelligence Predicate Logic Slide 49

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

forallX [roman(X) ^ know(X Marcus)][hate(X Ceasar) v

(forallY existZ hate(YZ) thinkcrazy(XY))]

=

forallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 50

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2 Move not next to terms

Using the following equivalence

a Double negative

not(notp) = p

b DeMorgan

not(a v b) = nota ^ notb

not(a ^ b) = nota v notb

c Equivalence of qualifiers notforallX P(X) = existX notP(X)

notexistX P(X) = forallX notP(X)

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Artificial Intelligence Predicate Logic Slide 51

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2 Move not next to termsforallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

=

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) vthinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 52

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3 Change name of variables as follows

forallX P(X) v forallX Q(X)

=forallX P(X) v forallY Q(Y)

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Artificial Intelligence Predicate Logic Slide 53

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4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 54

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5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

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Artificial Intelligence Predicate Logic Slide 55

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6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 56

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7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

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Artificial Intelligence Predicate Logic Slide 57

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7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

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Artificial Intelligence Predicate Logic Slide 58

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8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

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Artificial Intelligence Predicate Logic Slide 59

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9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

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Artificial Intelligence Predicate Logic Slide 60

983122983141983155983151983148983157983156983145983151983150

Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

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Artificial Intelligence Predicate Logic Slide 61

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X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

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Artificial Intelligence Predicate Logic Slide 62

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θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

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Artificial Intelligence Predicate Logic Slide 63

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Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

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Artificial Intelligence Predicate Logic Slide 64

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Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

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Artificial Intelligence Predicate Logic Slide 65

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Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

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Artificial Intelligence Predicate Logic Slide 66

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Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

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Artificial Intelligence Predicate Logic Slide 67

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mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

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Artificial Intelligence Predicate Logic Slide 68

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Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

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Artificial Intelligence Predicate Logic Slide 69

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Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

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Artificial Intelligence Predicate Logic Slide 70

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Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

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Artificial Intelligence Predicate Logic Slide 71

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Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

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Artificial Intelligence Predicate Logic Slide 72

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Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

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Artificial Intelligence Predicate Logic Slide 73

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Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

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Artificial Intelligence Predicate Logic Slide 74

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Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

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Artificial Intelligence Predicate Logic Slide 75

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Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

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Artificial Intelligence Predicate Logic Slide 76

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Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

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892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 80

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You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

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Artificial Intelligence Predicate Logic Slide 43

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Every instantiation of a universally quantified sentence is enta iled by it

forallv α

Subst(vg α)

for any variable v and ground term g

Eg forallx King( x) and Greedy( x)rArr Evil( x

) yieldsKing (John ) and Greedy (John ) rArr Evil (John )

King (Richard ) and Greedy (Richard ) rArr Evil (Richard )

King (Father (John )) and Greedy (Father (John )) rArr Evil (Father (John ))

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Artificial Intelligence Predicate Logic Slide 44

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For any sentence α variable v and constant symbol k that

does not

appear elsewhere in the knowledge baseexistv α

Subst(vk α )

Eg exist x Crown( x) and OnHead ( xJohn) yields

Crown(C 1) and OnHead (C 1 John )

provided C 1 is a new constant symbol called a Skolemconstant

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Artificial Intelligence Predicate Logic Slide 45

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Suppose the KB contains just the followingforall

x King(x)and

Greedy(x)rArr

Evil(x)King(John)Greedy(John)Brother(RichardJohn)

Instantiating the universal sentence in all possible ways we haveKing(John) and Greedy(John) rArr Evil(John)King(Richard) and Greedy(Richard) rArr Evil(Richard)King(John)Greedy(John)Brother(RichardJohn

)

The new KB is propositionalized proposition symbols are

King(John) Greedy(John) Evil(John) King(Richard

) etc

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Artificial Intelligence Predicate Logic Slide 46

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Every FOL KB can be propositionalized so as to preserve entailment

(A ground sentence is entailed by new KB iff entailed by originalKB)

Idea propositionalize KB and query apply resolution return result

Problem with function symbols there are infinitely many ground

terms

eg Father (Father (Father (John

)))

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Artificial Intelligence Predicate Logic Slide 47

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Convert to clause form exampleldquoAll Romans who know Marcus either hate

Caesar or think that anyone who hates anyone is

crazyrdquo

forallX [roman(X) ^ know(X Marcus)]

[hate(X Ceasar) v(forallY existZ hate(YZ)

thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 48

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1 Remove

using the equivalence a b = nota v b

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 49

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1 Remove

using the equivalence a b = nota v b

forallX [roman(X) ^ know(X Marcus)][hate(X Ceasar) v

(forallY existZ hate(YZ) thinkcrazy(XY))]

=

forallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 50

983122983141983155983151983148983157983156983145983151983150

2 Move not next to terms

Using the following equivalence

a Double negative

not(notp) = p

b DeMorgan

not(a v b) = nota ^ notb

not(a ^ b) = nota v notb

c Equivalence of qualifiers notforallX P(X) = existX notP(X)

notexistX P(X) = forallX notP(X)

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Artificial Intelligence Predicate Logic Slide 51

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2 Move not next to termsforallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

=

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) vthinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 52

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3 Change name of variables as follows

forallX P(X) v forallX Q(X)

=forallX P(X) v forallY Q(Y)

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Artificial Intelligence Predicate Logic Slide 53

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4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 54

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5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

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Artificial Intelligence Predicate Logic Slide 55

983122983141983155983151983148983157983156983145983151983150

6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 56

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

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Artificial Intelligence Predicate Logic Slide 57

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7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

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Artificial Intelligence Predicate Logic Slide 58

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8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

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Artificial Intelligence Predicate Logic Slide 59

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9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

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Artificial Intelligence Predicate Logic Slide 60

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Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

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Artificial Intelligence Predicate Logic Slide 61

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X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

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Artificial Intelligence Predicate Logic Slide 62

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θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

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Artificial Intelligence Predicate Logic Slide 63

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Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

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Artificial Intelligence Predicate Logic Slide 64

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

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Artificial Intelligence Predicate Logic Slide 65

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Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

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Artificial Intelligence Predicate Logic Slide 66

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

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Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

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Artificial Intelligence Predicate Logic Slide 69

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

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Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

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Artificial Intelligence Predicate Logic Slide 71

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

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Artificial Intelligence Predicate Logic Slide 72

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Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

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Artificial Intelligence Predicate Logic Slide 73

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Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

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Artificial Intelligence Predicate Logic Slide 74

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Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

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Artificial Intelligence Predicate Logic Slide 75

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Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

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Artificial Intelligence Predicate Logic Slide 76

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Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

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892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

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Artificial Intelligence Predicate Logic Slide 44

983109983160983145983155983156983141983150983156983145983137983148 983145983150983155983156983137983150983156983145983137983156983145983151983150 (983109983113)

For any sentence α variable v and constant symbol k that

does not

appear elsewhere in the knowledge baseexistv α

Subst(vk α )

Eg exist x Crown( x) and OnHead ( xJohn) yields

Crown(C 1) and OnHead (C 1 John )

provided C 1 is a new constant symbol called a Skolemconstant

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 45

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Suppose the KB contains just the followingforall

x King(x)and

Greedy(x)rArr

Evil(x)King(John)Greedy(John)Brother(RichardJohn)

Instantiating the universal sentence in all possible ways we haveKing(John) and Greedy(John) rArr Evil(John)King(Richard) and Greedy(Richard) rArr Evil(Richard)King(John)Greedy(John)Brother(RichardJohn

)

The new KB is propositionalized proposition symbols are

King(John) Greedy(John) Evil(John) King(Richard

) etc

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 46

983122983141983140983157983139983156983145983151983150 983139983151983150983156983140983086

Every FOL KB can be propositionalized so as to preserve entailment

(A ground sentence is entailed by new KB iff entailed by originalKB)

Idea propositionalize KB and query apply resolution return result

Problem with function symbols there are infinitely many ground

terms

eg Father (Father (Father (John

)))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 47

983122983141983155983151983148983157983156983145983151983150

Convert to clause form exampleldquoAll Romans who know Marcus either hate

Caesar or think that anyone who hates anyone is

crazyrdquo

forallX [roman(X) ^ know(X Marcus)]

[hate(X Ceasar) v(forallY existZ hate(YZ)

thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 48

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 49

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

forallX [roman(X) ^ know(X Marcus)][hate(X Ceasar) v

(forallY existZ hate(YZ) thinkcrazy(XY))]

=

forallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 50

983122983141983155983151983148983157983156983145983151983150

2 Move not next to terms

Using the following equivalence

a Double negative

not(notp) = p

b DeMorgan

not(a v b) = nota ^ notb

not(a ^ b) = nota v notb

c Equivalence of qualifiers notforallX P(X) = existX notP(X)

notexistX P(X) = forallX notP(X)

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Artificial Intelligence Predicate Logic Slide 51

983122983141983155983151983148983157983156983145983151983150

2 Move not next to termsforallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

=

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) vthinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 52

983122983141983155983151983148983157983156983145983151983150

3 Change name of variables as follows

forallX P(X) v forallX Q(X)

=forallX P(X) v forallY Q(Y)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 53

983122983141983155983151983148983157983156983145983151983150

4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 54

983122983141983155983151983148983157983156983145983151983150

5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

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Artificial Intelligence Predicate Logic Slide 55

983122983141983155983151983148983157983156983145983151983150

6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 56

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

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Artificial Intelligence Predicate Logic Slide 57

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7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

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Artificial Intelligence Predicate Logic Slide 58

983122983141983155983151983148983157983156983145983151983150

8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

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Artificial Intelligence Predicate Logic Slide 59

983122983141983155983151983148983157983156983145983151983150

9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

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Artificial Intelligence Predicate Logic Slide 60

983122983141983155983151983148983157983156983145983151983150

Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

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Artificial Intelligence Predicate Logic Slide 61

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

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Artificial Intelligence Predicate Logic Slide 62

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θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 63

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Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

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Artificial Intelligence Predicate Logic Slide 64

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 65

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 66

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

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Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

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Artificial Intelligence Predicate Logic Slide 69

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

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Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 71

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

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Artificial Intelligence Predicate Logic Slide 72

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 74

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 75

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 76

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

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892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4580

Artificial Intelligence Predicate Logic Slide 45

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Suppose the KB contains just the followingforall

x King(x)and

Greedy(x)rArr

Evil(x)King(John)Greedy(John)Brother(RichardJohn)

Instantiating the universal sentence in all possible ways we haveKing(John) and Greedy(John) rArr Evil(John)King(Richard) and Greedy(Richard) rArr Evil(Richard)King(John)Greedy(John)Brother(RichardJohn

)

The new KB is propositionalized proposition symbols are

King(John) Greedy(John) Evil(John) King(Richard

) etc

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 46

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Every FOL KB can be propositionalized so as to preserve entailment

(A ground sentence is entailed by new KB iff entailed by originalKB)

Idea propositionalize KB and query apply resolution return result

Problem with function symbols there are infinitely many ground

terms

eg Father (Father (Father (John

)))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 47

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Convert to clause form exampleldquoAll Romans who know Marcus either hate

Caesar or think that anyone who hates anyone is

crazyrdquo

forallX [roman(X) ^ know(X Marcus)]

[hate(X Ceasar) v(forallY existZ hate(YZ)

thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 48

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1 Remove

using the equivalence a b = nota v b

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Artificial Intelligence Predicate Logic Slide 49

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

forallX [roman(X) ^ know(X Marcus)][hate(X Ceasar) v

(forallY existZ hate(YZ) thinkcrazy(XY))]

=

forallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 50

983122983141983155983151983148983157983156983145983151983150

2 Move not next to terms

Using the following equivalence

a Double negative

not(notp) = p

b DeMorgan

not(a v b) = nota ^ notb

not(a ^ b) = nota v notb

c Equivalence of qualifiers notforallX P(X) = existX notP(X)

notexistX P(X) = forallX notP(X)

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Artificial Intelligence Predicate Logic Slide 51

983122983141983155983151983148983157983156983145983151983150

2 Move not next to termsforallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

=

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) vthinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 52

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3 Change name of variables as follows

forallX P(X) v forallX Q(X)

=forallX P(X) v forallY Q(Y)

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Artificial Intelligence Predicate Logic Slide 53

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4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 54

983122983141983155983151983148983157983156983145983151983150

5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

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Artificial Intelligence Predicate Logic Slide 55

983122983141983155983151983148983157983156983145983151983150

6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 56

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

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Artificial Intelligence Predicate Logic Slide 57

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7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

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Artificial Intelligence Predicate Logic Slide 58

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8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

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Artificial Intelligence Predicate Logic Slide 59

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9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

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Artificial Intelligence Predicate Logic Slide 60

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Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

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Artificial Intelligence Predicate Logic Slide 61

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X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

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Artificial Intelligence Predicate Logic Slide 62

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θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

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Artificial Intelligence Predicate Logic Slide 63

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Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

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Artificial Intelligence Predicate Logic Slide 64

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Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

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Artificial Intelligence Predicate Logic Slide 65

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Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

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Artificial Intelligence Predicate Logic Slide 66

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Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

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Artificial Intelligence Predicate Logic Slide 67

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mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

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Artificial Intelligence Predicate Logic Slide 68

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Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

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Artificial Intelligence Predicate Logic Slide 69

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Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

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Artificial Intelligence Predicate Logic Slide 70

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Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

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Artificial Intelligence Predicate Logic Slide 71

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Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

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Artificial Intelligence Predicate Logic Slide 72

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Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

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Artificial Intelligence Predicate Logic Slide 73

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Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

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Artificial Intelligence Predicate Logic Slide 74

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Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

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Artificial Intelligence Predicate Logic Slide 75

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Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

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Artificial Intelligence Predicate Logic Slide 76

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Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

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892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4680

Artificial Intelligence Predicate Logic Slide 46

983122983141983140983157983139983156983145983151983150 983139983151983150983156983140983086

Every FOL KB can be propositionalized so as to preserve entailment

(A ground sentence is entailed by new KB iff entailed by originalKB)

Idea propositionalize KB and query apply resolution return result

Problem with function symbols there are infinitely many ground

terms

eg Father (Father (Father (John

)))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 47

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Convert to clause form exampleldquoAll Romans who know Marcus either hate

Caesar or think that anyone who hates anyone is

crazyrdquo

forallX [roman(X) ^ know(X Marcus)]

[hate(X Ceasar) v(forallY existZ hate(YZ)

thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 48

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

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Artificial Intelligence Predicate Logic Slide 49

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1 Remove

using the equivalence a b = nota v b

forallX [roman(X) ^ know(X Marcus)][hate(X Ceasar) v

(forallY existZ hate(YZ) thinkcrazy(XY))]

=

forallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 50

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2 Move not next to terms

Using the following equivalence

a Double negative

not(notp) = p

b DeMorgan

not(a v b) = nota ^ notb

not(a ^ b) = nota v notb

c Equivalence of qualifiers notforallX P(X) = existX notP(X)

notexistX P(X) = forallX notP(X)

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Artificial Intelligence Predicate Logic Slide 51

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2 Move not next to termsforallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

=

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) vthinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 52

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3 Change name of variables as follows

forallX P(X) v forallX Q(X)

=forallX P(X) v forallY Q(Y)

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Artificial Intelligence Predicate Logic Slide 53

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4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 54

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5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

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Artificial Intelligence Predicate Logic Slide 55

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6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 56

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

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Artificial Intelligence Predicate Logic Slide 57

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7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

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Artificial Intelligence Predicate Logic Slide 58

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8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

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Artificial Intelligence Predicate Logic Slide 59

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9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

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Artificial Intelligence Predicate Logic Slide 60

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Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

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Artificial Intelligence Predicate Logic Slide 61

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X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

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Artificial Intelligence Predicate Logic Slide 62

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θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

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Artificial Intelligence Predicate Logic Slide 63

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Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

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Artificial Intelligence Predicate Logic Slide 64

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Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

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Artificial Intelligence Predicate Logic Slide 65

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Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

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Artificial Intelligence Predicate Logic Slide 66

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Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

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Artificial Intelligence Predicate Logic Slide 67

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mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

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Artificial Intelligence Predicate Logic Slide 68

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Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

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Artificial Intelligence Predicate Logic Slide 69

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Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

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Artificial Intelligence Predicate Logic Slide 70

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Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

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Artificial Intelligence Predicate Logic Slide 71

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Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

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Artificial Intelligence Predicate Logic Slide 72

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Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

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Artificial Intelligence Predicate Logic Slide 73

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Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

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Artificial Intelligence Predicate Logic Slide 74

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Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

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Artificial Intelligence Predicate Logic Slide 75

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Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

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Artificial Intelligence Predicate Logic Slide 76

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Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

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892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 80

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You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

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Artificial Intelligence Predicate Logic Slide 47

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Convert to clause form exampleldquoAll Romans who know Marcus either hate

Caesar or think that anyone who hates anyone is

crazyrdquo

forallX [roman(X) ^ know(X Marcus)]

[hate(X Ceasar) v(forallY existZ hate(YZ)

thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 48

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1 Remove

using the equivalence a b = nota v b

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Artificial Intelligence Predicate Logic Slide 49

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1 Remove

using the equivalence a b = nota v b

forallX [roman(X) ^ know(X Marcus)][hate(X Ceasar) v

(forallY existZ hate(YZ) thinkcrazy(XY))]

=

forallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 50

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2 Move not next to terms

Using the following equivalence

a Double negative

not(notp) = p

b DeMorgan

not(a v b) = nota ^ notb

not(a ^ b) = nota v notb

c Equivalence of qualifiers notforallX P(X) = existX notP(X)

notexistX P(X) = forallX notP(X)

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Artificial Intelligence Predicate Logic Slide 51

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2 Move not next to termsforallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

=

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) vthinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 52

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3 Change name of variables as follows

forallX P(X) v forallX Q(X)

=forallX P(X) v forallY Q(Y)

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Artificial Intelligence Predicate Logic Slide 53

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4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 54

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5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

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Artificial Intelligence Predicate Logic Slide 55

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6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 56

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7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

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Artificial Intelligence Predicate Logic Slide 57

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7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

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Artificial Intelligence Predicate Logic Slide 58

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8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

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Artificial Intelligence Predicate Logic Slide 59

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9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

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Artificial Intelligence Predicate Logic Slide 60

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Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

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Artificial Intelligence Predicate Logic Slide 61

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X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

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Artificial Intelligence Predicate Logic Slide 62

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θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

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Artificial Intelligence Predicate Logic Slide 63

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Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

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Artificial Intelligence Predicate Logic Slide 64

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Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 65

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Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

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Artificial Intelligence Predicate Logic Slide 66

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Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

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Artificial Intelligence Predicate Logic Slide 67

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mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

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Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

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Artificial Intelligence Predicate Logic Slide 69

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Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

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Artificial Intelligence Predicate Logic Slide 70

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Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

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Artificial Intelligence Predicate Logic Slide 71

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Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

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Artificial Intelligence Predicate Logic Slide 72

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Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 73

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Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

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Artificial Intelligence Predicate Logic Slide 74

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Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

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Artificial Intelligence Predicate Logic Slide 75

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Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

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Artificial Intelligence Predicate Logic Slide 76

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Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

892019 First Order(Predicate)Logic

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892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 8080

Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 4880

Artificial Intelligence Predicate Logic Slide 48

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 49

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

forallX [roman(X) ^ know(X Marcus)][hate(X Ceasar) v

(forallY existZ hate(YZ) thinkcrazy(XY))]

=

forallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 50

983122983141983155983151983148983157983156983145983151983150

2 Move not next to terms

Using the following equivalence

a Double negative

not(notp) = p

b DeMorgan

not(a v b) = nota ^ notb

not(a ^ b) = nota v notb

c Equivalence of qualifiers notforallX P(X) = existX notP(X)

notexistX P(X) = forallX notP(X)

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Artificial Intelligence Predicate Logic Slide 51

983122983141983155983151983148983157983156983145983151983150

2 Move not next to termsforallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

=

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) vthinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 52

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3 Change name of variables as follows

forallX P(X) v forallX Q(X)

=forallX P(X) v forallY Q(Y)

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Artificial Intelligence Predicate Logic Slide 53

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4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 54

983122983141983155983151983148983157983156983145983151983150

5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

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Artificial Intelligence Predicate Logic Slide 55

983122983141983155983151983148983157983156983145983151983150

6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 56

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

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Artificial Intelligence Predicate Logic Slide 57

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7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

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Artificial Intelligence Predicate Logic Slide 58

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8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

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Artificial Intelligence Predicate Logic Slide 59

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9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

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Artificial Intelligence Predicate Logic Slide 60

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Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

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Artificial Intelligence Predicate Logic Slide 61

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X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

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Artificial Intelligence Predicate Logic Slide 62

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θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

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Artificial Intelligence Predicate Logic Slide 63

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Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

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Artificial Intelligence Predicate Logic Slide 64

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Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 65

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Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

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Artificial Intelligence Predicate Logic Slide 66

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Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 67

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mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

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Artificial Intelligence Predicate Logic Slide 68

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Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 69

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Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

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Artificial Intelligence Predicate Logic Slide 71

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Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 72

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Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 73

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Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

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Artificial Intelligence Predicate Logic Slide 74

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Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 75

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Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

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Artificial Intelligence Predicate Logic Slide 76

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Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

892019 First Order(Predicate)Logic

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892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 49

983122983141983155983151983148983157983156983145983151983150

1 Remove

using the equivalence a b = nota v b

forallX [roman(X) ^ know(X Marcus)][hate(X Ceasar) v

(forallY existZ hate(YZ) thinkcrazy(XY))]

=

forallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 50

983122983141983155983151983148983157983156983145983151983150

2 Move not next to terms

Using the following equivalence

a Double negative

not(notp) = p

b DeMorgan

not(a v b) = nota ^ notb

not(a ^ b) = nota v notb

c Equivalence of qualifiers notforallX P(X) = existX notP(X)

notexistX P(X) = forallX notP(X)

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Artificial Intelligence Predicate Logic Slide 51

983122983141983155983151983148983157983156983145983151983150

2 Move not next to termsforallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

=

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) vthinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 52

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3 Change name of variables as follows

forallX P(X) v forallX Q(X)

=forallX P(X) v forallY Q(Y)

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Artificial Intelligence Predicate Logic Slide 53

983122983141983155983151983148983157983156983145983151983150

4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 54

983122983141983155983151983148983157983156983145983151983150

5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

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Artificial Intelligence Predicate Logic Slide 55

983122983141983155983151983148983157983156983145983151983150

6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 56

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7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

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Artificial Intelligence Predicate Logic Slide 57

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7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

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Artificial Intelligence Predicate Logic Slide 58

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8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

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Artificial Intelligence Predicate Logic Slide 59

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9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

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Artificial Intelligence Predicate Logic Slide 60

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Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

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Artificial Intelligence Predicate Logic Slide 61

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X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

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Artificial Intelligence Predicate Logic Slide 62

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θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

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Artificial Intelligence Predicate Logic Slide 63

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Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

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Artificial Intelligence Predicate Logic Slide 64

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Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

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Artificial Intelligence Predicate Logic Slide 65

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Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

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Artificial Intelligence Predicate Logic Slide 66

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Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

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Artificial Intelligence Predicate Logic Slide 67

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mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

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Artificial Intelligence Predicate Logic Slide 68

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Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

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Artificial Intelligence Predicate Logic Slide 69

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Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

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Artificial Intelligence Predicate Logic Slide 70

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Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

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Artificial Intelligence Predicate Logic Slide 71

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Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

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Artificial Intelligence Predicate Logic Slide 72

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Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

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Artificial Intelligence Predicate Logic Slide 73

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Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

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Artificial Intelligence Predicate Logic Slide 74

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Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

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Artificial Intelligence Predicate Logic Slide 75

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Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

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Artificial Intelligence Predicate Logic Slide 76

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Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

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892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 80

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You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

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Artificial Intelligence Predicate Logic Slide 50

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2 Move not next to terms

Using the following equivalence

a Double negative

not(notp) = p

b DeMorgan

not(a v b) = nota ^ notb

not(a ^ b) = nota v notb

c Equivalence of qualifiers notforallX P(X) = existX notP(X)

notexistX P(X) = forallX notP(X)

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Artificial Intelligence Predicate Logic Slide 51

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2 Move not next to termsforallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

=

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) vthinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 52

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3 Change name of variables as follows

forallX P(X) v forallX Q(X)

=forallX P(X) v forallY Q(Y)

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Artificial Intelligence Predicate Logic Slide 53

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4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 54

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5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 55

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6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 56

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7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

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Artificial Intelligence Predicate Logic Slide 57

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7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

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Artificial Intelligence Predicate Logic Slide 58

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8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

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Artificial Intelligence Predicate Logic Slide 59

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9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

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Artificial Intelligence Predicate Logic Slide 60

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Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

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Artificial Intelligence Predicate Logic Slide 61

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X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

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Artificial Intelligence Predicate Logic Slide 62

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θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

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Artificial Intelligence Predicate Logic Slide 63

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Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

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Artificial Intelligence Predicate Logic Slide 64

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Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 65

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

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Artificial Intelligence Predicate Logic Slide 66

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

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Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

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Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

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Artificial Intelligence Predicate Logic Slide 69

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Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

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Artificial Intelligence Predicate Logic Slide 71

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 72

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 74

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

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Artificial Intelligence Predicate Logic Slide 75

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Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

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Artificial Intelligence Predicate Logic Slide 76

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Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

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892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

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Artificial Intelligence Predicate Logic Slide 51

983122983141983155983151983148983157983156983145983151983150

2 Move not next to termsforallX not[roman(X) ^ know(X Marcus)] v

[hate(X Ceasar) v

(forallY not(existZ hate(YZ)) v thinkcrazy(XY))]

=

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) vthinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 52

983122983141983155983151983148983157983156983145983151983150

3 Change name of variables as follows

forallX P(X) v forallX Q(X)

=forallX P(X) v forallY Q(Y)

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Artificial Intelligence Predicate Logic Slide 53

983122983141983155983151983148983157983156983145983151983150

4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 54

983122983141983155983151983148983157983156983145983151983150

5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

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Artificial Intelligence Predicate Logic Slide 55

983122983141983155983151983148983157983156983145983151983150

6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 56

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

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Artificial Intelligence Predicate Logic Slide 57

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 58

983122983141983155983151983148983157983156983145983151983150

8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 59

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9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

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Artificial Intelligence Predicate Logic Slide 60

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Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

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Artificial Intelligence Predicate Logic Slide 61

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 62

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 63

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Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 64

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Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 65

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

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Artificial Intelligence Predicate Logic Slide 66

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

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Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

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Artificial Intelligence Predicate Logic Slide 69

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Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 71

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 72

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 74

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 75

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

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Artificial Intelligence Predicate Logic Slide 76

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Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

892019 First Order(Predicate)Logic

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892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 8080

Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 5280

Artificial Intelligence Predicate Logic Slide 52

983122983141983155983151983148983157983156983145983151983150

3 Change name of variables as follows

forallX P(X) v forallX Q(X)

=forallX P(X) v forallY Q(Y)

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Artificial Intelligence Predicate Logic Slide 53

983122983141983155983151983148983157983156983145983151983150

4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 54

983122983141983155983151983148983157983156983145983151983150

5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 55

983122983141983155983151983148983157983156983145983151983150

6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 56

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 57

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 58

983122983141983155983151983148983157983156983145983151983150

8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

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Artificial Intelligence Predicate Logic Slide 59

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9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

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Artificial Intelligence Predicate Logic Slide 60

983122983141983155983151983148983157983156983145983151983150

Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

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Artificial Intelligence Predicate Logic Slide 61

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X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 62

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θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

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Artificial Intelligence Predicate Logic Slide 63

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Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

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Artificial Intelligence Predicate Logic Slide 64

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Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 65

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Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

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Artificial Intelligence Predicate Logic Slide 66

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Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

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Artificial Intelligence Predicate Logic Slide 68

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Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

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Artificial Intelligence Predicate Logic Slide 69

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Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

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Artificial Intelligence Predicate Logic Slide 70

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Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

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Artificial Intelligence Predicate Logic Slide 71

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Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

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Artificial Intelligence Predicate Logic Slide 72

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Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

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Artificial Intelligence Predicate Logic Slide 73

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Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

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Artificial Intelligence Predicate Logic Slide 74

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Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

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Artificial Intelligence Predicate Logic Slide 75

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Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

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Artificial Intelligence Predicate Logic Slide 76

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Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

892019 First Order(Predicate)Logic

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892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 53

983122983141983155983151983148983157983156983145983151983150

4 Move qualifier to left donrsquot change the their order

forallX [notroman(X) v notknow(X Marcus)]v

[hate(X Ceasar) v

(forallY forallZ nothate(YZ) v

thinkcrazy(XY))]

=

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

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Artificial Intelligence Predicate Logic Slide 54

983122983141983155983151983148983157983156983145983151983150

5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

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Artificial Intelligence Predicate Logic Slide 55

983122983141983155983151983148983157983156983145983151983150

6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 56

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7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

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Artificial Intelligence Predicate Logic Slide 57

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7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

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Artificial Intelligence Predicate Logic Slide 58

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8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

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Artificial Intelligence Predicate Logic Slide 59

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9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

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Artificial Intelligence Predicate Logic Slide 60

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Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

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Artificial Intelligence Predicate Logic Slide 61

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X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

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Artificial Intelligence Predicate Logic Slide 62

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θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

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Artificial Intelligence Predicate Logic Slide 63

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Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

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Artificial Intelligence Predicate Logic Slide 64

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Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

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Artificial Intelligence Predicate Logic Slide 65

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Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

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Artificial Intelligence Predicate Logic Slide 66

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Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

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Artificial Intelligence Predicate Logic Slide 67

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mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

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Artificial Intelligence Predicate Logic Slide 68

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Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 69

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Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 70

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Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

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Artificial Intelligence Predicate Logic Slide 71

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Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 72

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Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 73

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Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 74

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Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 75

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Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 76

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Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

892019 First Order(Predicate)Logic

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892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 5480

Artificial Intelligence Predicate Logic Slide 54

983122983141983155983151983148983157983156983145983151983150

5 Replace existence qualifier using Skolem

function as follows

Skolem Function

forallX forallY existZ P(XYZ)

=

forallX forallY P(XYf(XY)) Variable of existence qualifier = function of all

the preceding variables in the every qualifier

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 55

983122983141983155983151983148983157983156983145983151983150

6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 56

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 57

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 58

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8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 59

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9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 60

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Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

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Artificial Intelligence Predicate Logic Slide 61

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X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 62

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θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 63

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Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

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Artificial Intelligence Predicate Logic Slide 64

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Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 65

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Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

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Artificial Intelligence Predicate Logic Slide 66

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Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

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Artificial Intelligence Predicate Logic Slide 67

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mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

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Artificial Intelligence Predicate Logic Slide 68

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Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

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Artificial Intelligence Predicate Logic Slide 69

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Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

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Artificial Intelligence Predicate Logic Slide 70

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Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

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Artificial Intelligence Predicate Logic Slide 71

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Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

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Artificial Intelligence Predicate Logic Slide 72

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Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 74

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 75

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 76

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Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7780

Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

892019 First Order(Predicate)Logic

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892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 5580

Artificial Intelligence Predicate Logic Slide 55

983122983141983155983151983148983157983156983145983151983150

6 Remove every qualifiers (the default qualifier is

every)

forallXforallYforallZ [notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

=

[notroman(X) v notknow(X Marcus)] v[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 56

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 57

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 58

983122983141983155983151983148983157983156983145983151983150

8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 59

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9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 60

983122983141983155983151983148983157983156983145983151983150

Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 61

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 62

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θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

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Artificial Intelligence Predicate Logic Slide 63

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

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Artificial Intelligence Predicate Logic Slide 64

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 65

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

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Artificial Intelligence Predicate Logic Slide 66

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 69

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 71

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Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 72

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Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 74

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Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 75

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Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 76

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Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

892019 First Order(Predicate)Logic

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892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 56

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

Using distribution rule of v and ^

Examples

(a ^ b) v c = (a v c) ^ (b v c)(a ^ b) v (c ^ d) = (a v c) ^ (a v d) ^ (b v c) ^ (b v d)

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Artificial Intelligence Predicate Logic Slide 57

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 58

983122983141983155983151983148983157983156983145983151983150

8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

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Artificial Intelligence Predicate Logic Slide 59

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9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

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Artificial Intelligence Predicate Logic Slide 60

983122983141983155983151983148983157983156983145983151983150

Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 61

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 62

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θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 63

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Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

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Artificial Intelligence Predicate Logic Slide 64

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 65

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Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

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Artificial Intelligence Predicate Logic Slide 66

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Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

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Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

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Artificial Intelligence Predicate Logic Slide 69

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Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

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Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

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Artificial Intelligence Predicate Logic Slide 71

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

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Artificial Intelligence Predicate Logic Slide 72

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 74

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7580

Artificial Intelligence Predicate Logic Slide 75

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 76

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7880

Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7980

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 8080

Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 5780

Artificial Intelligence Predicate Logic Slide 57

983122983141983155983151983148983157983156983145983151983150

7 Convert to CNF

[notroman(X) v notknow(X Marcus)] v

[hate(X Ceasar) v

(nothate(YZ) v thinkcrazy(XY))]=

notroman(X) v notknow(X Marcus) v

hate(X Ceasar) v nothate(YZ) v thinkcrazy(XY)

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 5880

Artificial Intelligence Predicate Logic Slide 58

983122983141983155983151983148983157983156983145983151983150

8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 5980

Artificial Intelligence Predicate Logic Slide 59

983122983141983155983151983148983157983156983145983151983150

9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 60

983122983141983155983151983148983157983156983145983151983150

Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 61

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 62

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 63

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 64

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6580

Artificial Intelligence Predicate Logic Slide 65

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 66

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6780

Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 69

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 71

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 72

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Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7380

Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 74

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 75

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 76

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Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7780

Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

892019 First Order(Predicate)Logic

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892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 5880

Artificial Intelligence Predicate Logic Slide 58

983122983141983155983151983148983157983156983145983151983150

8 Separate clauses

If having the clause form

(a v notb) ^ (nota v c v d) ^ (a v notc v e)

Then

1 (a v notb)

2 (nota v c v d)

3 (a v notc v e)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 59

983122983141983155983151983148983157983156983145983151983150

9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 60

983122983141983155983151983148983157983156983145983151983150

Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 61

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 62

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θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 63

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Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6480

Artificial Intelligence Predicate Logic Slide 64

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6580

Artificial Intelligence Predicate Logic Slide 65

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 66

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Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6780

Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 69

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Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 71

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 72

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 74

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 75

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

892019 First Order(Predicate)Logic

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Artificial Intelligence Predicate Logic Slide 76

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7780

Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7880

Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7980

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 8080

Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 5980

Artificial Intelligence Predicate Logic Slide 59

983122983141983155983151983148983157983156983145983151983150

9 Add every qualifier to clause

That is

(forallX P(X) ^ Q(X) )=

forallX P(X) ^ forallX Q(X)

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6080

Artificial Intelligence Predicate Logic Slide 60

983122983141983155983151983148983157983156983145983151983150

Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6180

Artificial Intelligence Predicate Logic Slide 61

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6280

Artificial Intelligence Predicate Logic Slide 62

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6380

Artificial Intelligence Predicate Logic Slide 63

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6480

Artificial Intelligence Predicate Logic Slide 64

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6580

Artificial Intelligence Predicate Logic Slide 65

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6680

Artificial Intelligence Predicate Logic Slide 66

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6780

Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6880

Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6980

Artificial Intelligence Predicate Logic Slide 69

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7080

Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7180

Artificial Intelligence Predicate Logic Slide 71

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7280

Artificial Intelligence Predicate Logic Slide 72

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7380

Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7480

Artificial Intelligence Predicate Logic Slide 74

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7580

Artificial Intelligence Predicate Logic Slide 75

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7680

Artificial Intelligence Predicate Logic Slide 76

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7780

Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7880

Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7980

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 8080

Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6080

Artificial Intelligence Predicate Logic Slide 60

983122983141983155983151983148983157983156983145983151983150

Exercises

Convert to clause form for sentences in previous

example

Convert to clause form for

1 forallX A(X) v existX B(X) forallXC(X) ^ existX D(X)

2 forallX (p(X) v q(X)) forallX p(X) v forallX q(X)

3 existX p(X) ^ existX q(X)existX(p(X) ^ q(X))4 forallX existY p(XY) existY forallX p(XY)

5 forallX (p(X f(X)) p(XY))

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6180

Artificial Intelligence Predicate Logic Slide 61

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6280

Artificial Intelligence Predicate Logic Slide 62

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6380

Artificial Intelligence Predicate Logic Slide 63

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6480

Artificial Intelligence Predicate Logic Slide 64

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6580

Artificial Intelligence Predicate Logic Slide 65

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6680

Artificial Intelligence Predicate Logic Slide 66

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6780

Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6880

Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6980

Artificial Intelligence Predicate Logic Slide 69

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7080

Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7180

Artificial Intelligence Predicate Logic Slide 71

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7280

Artificial Intelligence Predicate Logic Slide 72

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7380

Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7480

Artificial Intelligence Predicate Logic Slide 74

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7580

Artificial Intelligence Predicate Logic Slide 75

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7680

Artificial Intelligence Predicate Logic Slide 76

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7780

Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7880

Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7980

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 8080

Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6180

Artificial Intelligence Predicate Logic Slide 61

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

X Variable t term

xt value of x will be t OR x is replaced by t

A substitution = x1 t1 x2 t2 xn tn

Empty substitution ε

E Predicate expression

θ A substitution

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6280

Artificial Intelligence Predicate Logic Slide 62

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6380

Artificial Intelligence Predicate Logic Slide 63

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6480

Artificial Intelligence Predicate Logic Slide 64

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6580

Artificial Intelligence Predicate Logic Slide 65

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6680

Artificial Intelligence Predicate Logic Slide 66

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6780

Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6880

Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6980

Artificial Intelligence Predicate Logic Slide 69

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7080

Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7180

Artificial Intelligence Predicate Logic Slide 71

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7280

Artificial Intelligence Predicate Logic Slide 72

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7380

Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7480

Artificial Intelligence Predicate Logic Slide 74

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7580

Artificial Intelligence Predicate Logic Slide 75

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7680

Artificial Intelligence Predicate Logic Slide 76

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7780

Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7880

Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7980

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 8080

Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6280

Artificial Intelligence Predicate Logic Slide 62

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

θ A substitution

Eθ Apply θ to E

ExampleE = p(XYf(X))

θ = Xa yf(b)Then Eθ = p(a f(b) f(a))

Union of θ and δ

E is a expression thenE(θδ) = (Eθ)δ

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6380

Artificial Intelligence Predicate Logic Slide 63

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6480

Artificial Intelligence Predicate Logic Slide 64

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6580

Artificial Intelligence Predicate Logic Slide 65

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6680

Artificial Intelligence Predicate Logic Slide 66

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6780

Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6880

Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6980

Artificial Intelligence Predicate Logic Slide 69

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7080

Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7180

Artificial Intelligence Predicate Logic Slide 71

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7280

Artificial Intelligence Predicate Logic Slide 72

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7380

Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7480

Artificial Intelligence Predicate Logic Slide 74

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7580

Artificial Intelligence Predicate Logic Slide 75

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7680

Artificial Intelligence Predicate Logic Slide 76

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7780

Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7880

Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7980

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 8080

Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6380

Artificial Intelligence Predicate Logic Slide 63

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ = x1 t1 x2 t2 xn tn

δ = y1 s1 y2 s2 yn sn

θδ is calculated as follows

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6480

Artificial Intelligence Predicate Logic Slide 64

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6580

Artificial Intelligence Predicate Logic Slide 65

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6680

Artificial Intelligence Predicate Logic Slide 66

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6780

Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6880

Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6980

Artificial Intelligence Predicate Logic Slide 69

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7080

Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7180

Artificial Intelligence Predicate Logic Slide 71

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7280

Artificial Intelligence Predicate Logic Slide 72

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7380

Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7480

Artificial Intelligence Predicate Logic Slide 74

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7580

Artificial Intelligence Predicate Logic Slide 75

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7680

Artificial Intelligence Predicate Logic Slide 76

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7780

Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7880

Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7980

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 8080

Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6480

Artificial Intelligence Predicate Logic Slide 64

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θδ is calculated as follows

1 Apply δ to denominator of θx1 t1δ x2 t2δ xn tnδ

2 Remove all the form xi xi from 1

3 Remove from δ the form yi si

if yi isin x1x2 xn

4 θδ = union( line 2 and 3)

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6580

Artificial Intelligence Predicate Logic Slide 65

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6680

Artificial Intelligence Predicate Logic Slide 66

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6780

Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6880

Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6980

Artificial Intelligence Predicate Logic Slide 69

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7080

Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7180

Artificial Intelligence Predicate Logic Slide 71

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7280

Artificial Intelligence Predicate Logic Slide 72

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7380

Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7480

Artificial Intelligence Predicate Logic Slide 74

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7580

Artificial Intelligence Predicate Logic Slide 75

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7680

Artificial Intelligence Predicate Logic Slide 76

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7780

Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7880

Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7980

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 8080

Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6580

Artificial Intelligence Predicate Logic Slide 65

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Union of θ and δ

θ δ substitution

ε empty substitution

E Expression

Properties

1 E(θδ) = (Eθ)δ

2 Eε = E

3 θε = εθ = θ

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6680

Artificial Intelligence Predicate Logic Slide 66

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6780

Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6880

Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6980

Artificial Intelligence Predicate Logic Slide 69

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7080

Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7180

Artificial Intelligence Predicate Logic Slide 71

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7280

Artificial Intelligence Predicate Logic Slide 72

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7380

Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7480

Artificial Intelligence Predicate Logic Slide 74

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7580

Artificial Intelligence Predicate Logic Slide 75

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7680

Artificial Intelligence Predicate Logic Slide 76

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7780

Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7880

Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7980

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 8080

Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6680

Artificial Intelligence Predicate Logic Slide 66

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification

If θ The most general unifier of S containingexpression then Sθ is a set having singleelement

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6780

Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6880

Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6980

Artificial Intelligence Predicate Logic Slide 69

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7080

Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7180

Artificial Intelligence Predicate Logic Slide 71

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7280

Artificial Intelligence Predicate Logic Slide 72

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7380

Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7480

Artificial Intelligence Predicate Logic Slide 74

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7580

Artificial Intelligence Predicate Logic Slide 75

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7680

Artificial Intelligence Predicate Logic Slide 76

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7780

Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7880

Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7980

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 8080

Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6780

Artificial Intelligence Predicate Logic Slide 67

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

mgu most general unifier

S set of expressions

θ mgu of S if 983111983145983158983141983150 983137983150983161 983157983150983145983142983145983141983154 δ 983151983142 983123 983156983144983141983150

exist α 983155983157983139983144 983156983144983137983156983098

δ 983101 θα

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6880

Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6980

Artificial Intelligence Predicate Logic Slide 69

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7080

Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7180

Artificial Intelligence Predicate Logic Slide 71

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7280

Artificial Intelligence Predicate Logic Slide 72

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7380

Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7480

Artificial Intelligence Predicate Logic Slide 74

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7580

Artificial Intelligence Predicate Logic Slide 75

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7680

Artificial Intelligence Predicate Logic Slide 76

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7780

Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7880

Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7980

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 8080

Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6880

Artificial Intelligence Predicate Logic Slide 68

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

S Set of expressions

Find the common longest substring from the

begining of each expression in S

Dis-set = set of right next term in all the

expressions

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6980

Artificial Intelligence Predicate Logic Slide 69

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7080

Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7180

Artificial Intelligence Predicate Logic Slide 71

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7280

Artificial Intelligence Predicate Logic Slide 72

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7380

Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7480

Artificial Intelligence Predicate Logic Slide 74

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7580

Artificial Intelligence Predicate Logic Slide 75

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7680

Artificial Intelligence Predicate Logic Slide 76

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7780

Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7880

Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7980

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 8080

Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 6980

Artificial Intelligence Predicate Logic Slide 69

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Dis-set

Example

S = p(X f(X) y) p(XYZ) p(X f(a) b)

Common longest substring = ldquop(Xrdquo

Dis-set = f(X) Y f(a)

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7080

Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7180

Artificial Intelligence Predicate Logic Slide 71

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7280

Artificial Intelligence Predicate Logic Slide 72

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7380

Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7480

Artificial Intelligence Predicate Logic Slide 74

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7580

Artificial Intelligence Predicate Logic Slide 75

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7680

Artificial Intelligence Predicate Logic Slide 76

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7780

Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7880

Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7980

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 8080

Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7080

Artificial Intelligence Predicate Logic Slide 70

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Unification algorithm

Input S = atoms Output mgu(S) or not unifiable(S)

1 Set K =0 and θ0 = ε Goto 22 Compute Sθk

If it is a set having single element stop mgu = θk Else D

k

= dis-set(Sθk

) Goto 3

3 If Dk contains variable V and a term tand V is not in term tThen compute θk+1 = θk Vt set K = K + 1 Goto 2

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7180

Artificial Intelligence Predicate Logic Slide 71

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7280

Artificial Intelligence Predicate Logic Slide 72

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7380

Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7480

Artificial Intelligence Predicate Logic Slide 74

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7580

Artificial Intelligence Predicate Logic Slide 75

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7680

Artificial Intelligence Predicate Logic Slide 76

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7780

Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7880

Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7980

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 8080

Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7180

Artificial Intelligence Predicate Logic Slide 71

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Clausersquos properties

1 There is no common variable between twoclauses

2 There exist one or many atoms L1 L2 Lk in a

clause one or many literals notM1 notM2 notMn inanother clause so that there is a mgu for set L1L2 Lk M1 M2 Mn

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7280

Artificial Intelligence Predicate Logic Slide 72

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7380

Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7480

Artificial Intelligence Predicate Logic Slide 74

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7580

Artificial Intelligence Predicate Logic Slide 75

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7680

Artificial Intelligence Predicate Logic Slide 76

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7780

Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7880

Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7980

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 8080

Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7280

Artificial Intelligence Predicate Logic Slide 72

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

N)-(DvN)-(C

D vMvv

C vLvv

n1

k 1

notthere4

notnot

θ θ

M

L

N = Liθ

= M jθ

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7380

Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7480

Artificial Intelligence Predicate Logic Slide 74

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7580

Artificial Intelligence Predicate Logic Slide 75

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7680

Artificial Intelligence Predicate Logic Slide 76

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7780

Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7880

Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7980

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 8080

Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7380

Artificial Intelligence Predicate Logic Slide 73

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

1 Verify the difference of variable name betweentwo clauses

2 Find mgu θ for L1 L2 Lk M1 M2 Mn

3 Apply θ into C vagrave D4 Compute N = L1θ

5 Remove N from Cθ

6 Remove notN from Dθ

7 Compute union of 5 and 6 and result resovant

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7480

Artificial Intelligence Predicate Logic Slide 74

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7580

Artificial Intelligence Predicate Logic Slide 75

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7680

Artificial Intelligence Predicate Logic Slide 76

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7780

Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7880

Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7980

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 8080

Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7480

Artificial Intelligence Predicate Logic Slide 74

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Resolution rule

Properties of resolvants

E F are two clauses

G is their resolvant

EF is unsatisfiable if and only if

EFG is unsatisfiable

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7580

Artificial Intelligence Predicate Logic Slide 75

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7680

Artificial Intelligence Predicate Logic Slide 76

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7780

Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7880

Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7980

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 8080

Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7580

Artificial Intelligence Predicate Logic Slide 75

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Proof with resolution rule

Set of expression F Prove P

Procedure

1 Convert F into clauses

2 Take notP convert notP into clauses Add to resultof Step 1

3 Apply resolution rule to produce empty set iefind a contradiction

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7680

Artificial Intelligence Predicate Logic Slide 76

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7780

Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7880

Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7980

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 8080

Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7680

Artificial Intelligence Predicate Logic Slide 76

983123983157983138983155983156983145983156983157983156983145983151983150 983137983150983140 983125983150983145983142983145983139983137983156983145983151983150

Example

Tryassassinate(Marcus Caesar)8

notman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v loyalto(X4 Y1)7

Loyalto(X3 f1(X3))6

notRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

Ruler(Caesar)4notPompiean(X1) v Roman(X1)3

Pompiean(Marcus)2

man(Marcus)1

ClausesSTT

Prove ldquoMarcus hate Caesarrdquo or

hate(Marcus Ceasar)

Proof

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7780

Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7880

Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7980

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 8080

Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7780

Artificial Intelligence Predicate Logic Slide 77

310 X1 = MarcusnotPompiean(Marcus) v loyalto(Marcus Caesar)11

Pman(Marcus)1

PPompiean(Marcus)2

PnotPompiean(X1) v Roman(X1)3

PRuler(Caesar)4

PnotRoman(X2) v loyalto(X2 Caesar) v hate(X2 Caesar)5

PLoyalto(X3 f1(X3))6Pnotman(X4) v notruler(Y1) v nottryassassinate(X4 Y1) v

notloyalto(X4 Y1)

7

PTryassassinate(Marcus Caesar)8

Pnothate(Marcua Ceasar)959 X2= MarcusnotRoman(Marcus) v loyalto(Marcus Caesar)10

NoteClauses

Proof

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7880

Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7980

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 8080

Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7880

Artificial Intelligence Predicate Logic Slide 78

15816

144nottryassassinate(Marcus Ceasar)15

131notruler(Ceasar) v nottryassassinate(Marcus

Ceasar)

14

127X4=Marcus y1=Ceasarnotman(Marcus) v notruler(Ceasar) v

nottryassassinate(Marcus Ceasar)

13

211loyalto(Marcus Ceasar)12

Ghi chuacuteClausesSTT

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7980

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 8080

Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 7980

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 8080

Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH

892019 First Order(Predicate)Logic

httpslidepdfcomreaderfullfirst-orderpredicatelogic 8080

Artificial Intelligence Predicate Logic Slide 80

983109983160983141983154983139983145983155983141983155

You know

All people living in PMH are rich

Rich people have much money

People who work only in government sector (or

called government staff) do not have muchmoney

An is a government staff Prove

An does not live in PMH