KNOWLEDGE REPRESENTATION 최윤정
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Transcript of KNOWLEDGE REPRESENTATION 최윤정
KNOWLEDGE REPRESENTATION
최윤정
Knowledge Representation Methods
2
Declarative Methods --knowledge is knowing
WHAT Logical Approach
Predicate Calculus Nonstandard Logics Fuzzy Logic
Non-Logical Approach Semantic Net Frame (procedural features) Conceptual Dependency
Procedural Methods --knowledge is
knowing HOW
PLANNER, CONNIVER Rule-based systems
Semantic Net(1/3) (Quillian(1968) Psychological Model)
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Basic Constructs Node-Object, Concept Links-Relation
property inheritance-Property Inheritance is the main inference
mechanism!
Tweety Robin Bird
Wings Wings Wings
isa
isa has-part
Semantic Net(2/3)4
Example
Furniture Furniturestore
SeatChair
My-ChairPark
Person Leather Black
Sold-by
has-part
isa
isa
ownerisa cover
Color
Semantic Net(3/3)5
Internal Representation(LISP) My-Chair: ((ISA CHAIR)(COLOR BLACK) (OWNER PARK)(COVER LEATHER)) Chair: ((ISA FURNITURE)(HASPART SEAT))
(get `My-CHAIR `COLOR)= `BLACK
Action and Event(1/2)6
“John gave the book to Mary.”
Event
EV-1 BK-1 BookJohn
Mary
Give Past
isaobject isaagen
t
beneficiary
action
time
Action and Event(2/2)7
“John is taller than Bill.”
John Bill
John Bill
H1 Number H2
Is-taller
height
isa isa
height
greater-than
Reasoning with Semantic Nets(1/2)
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Spreading Activation
“What is the relation between John and Mary?”
John ? Mary
Reasoning with Semantic Nets(2/2)
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Matching
Fact Goal Net
Direct Match vs. Semantic Match What is Tweety?
Tweety Robin Bird
Tweety ?
isa isa
isa
isa
Problems of Semantic Net(1/2)10
1. Different people use different nets to represent the same thing.
John Mary
Marriage
Event
M1John Mary
married
isa
isa
femalemale
Problems of Semantic Net(2/2)11
2. Same Net interpreted differently by different person.
3. Quantification
Jack TomFather-of
Dealing with Exceptions(1/2)
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Fly
Bird
Ostrich
Henry
Exception node
CAN
ISA
ISA
Dealing with Exceptions(2/2)
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Inferential Distance(Touretzky)
Grey
Elephant
Royal Elephant
Circus Elephant
Clyde
isa
COLOR
ISA
ISA
ISA
Frame (1/2) -Minsky14
Slot-Filler Concept : Typical Expected Situation
[frame: Vehicle ISA: Object Slots: (Weight (a wt-measure)) (color (a color(default black))) (number-of-wheel (a integer))]
[frame: Trailer-Truck ISA: Vehicle Slots: (trailer-size (a length-measure)) (weight (default 8)) (number-of-wheel (default 18))]
[frame: Sedan ISA: Vehicle Slots: (number-of-
wheel 4)]
[frame: My-truck instance-of: Trailer-
Truck Slots: (trailer-size = 12) (color red)]
Frame(2/2)15
Object
Vehicle
Trailer-truck
My-Truck
weight=color= blackwheel=
weight=8trailer-size=wheel=18Color=black
Weight=8Trailer-size=12Wheel=18Color=red
ISA
ISA
ISA
Procedural Attachment(1/2)-Procedural knowledge is attached to slots
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If-Added: Triggered to fill in If-Needed: Triggered when filled in If-Modified: Triggered when changed
Employee: ISA: Person Sex: (M, F) Birthday: Date
Age: integer If-Needed: CALC-AGE
Skill: code If-Added: ADD-TO-SKILL-FILE
Procedural Attachment(2/2)17
Proc CALC-AGE; x:= get-current-year; y:= get-birth-year; age:= x-y end;
Proc ADD-TO-FILE; c:= get-code; open-file(skill); put-file(c, skill) end;
LOGICAL PRELIMINARIES18
LOGIC- ARTIFICIAL LANGUAGE TALKING ABOUT “TRUTH”
LOGIC AS LANGUAGE SYNTAX(GRAMMAR) -Symbol -WFF(Well Formed Formula) -Deductive Closure -Proof Theory SEMANTICS -Meaning -Model -Validity, Consistency -Model Theory
LOGIC AS A PROGRAMMING LANGUAGE SYNTAX SEMANTICS
LOGIC - LANGUAGE and its MEANING19
LOGIC LANGUAGE MODEL
PROPOSITIONALLOGIC
PREDICATECALCULUS
MODAL LOGIC
P, Q, P->Q(P->((-Q->R) ∨P))
Variables x,yFunctions f, gPredicates P,QQuantifiers ∀,∃
ᄆ P, ◇P
Truth Assignment
First Order Structure<D,C,F,P>
Kripke Structure<W, R, V>
Possible WorldSemantics
PROPOSITIONAL LOGIC (1/2)20
(P∧Q)R
P Q R P∧Q (P∧Q)->R F F FF F TF T F
F T T T F F T F TT T FT T T
F TF TF TF TF TF T
T [F]T T
Falsifying Model
PROPOSITIONAL LOGIC (2/2)21
((PQ)∧~Q)~P
VALID TRUE in Every Model(Tautology) INCONSISTENT FALSE in Every Model CONSISTENT TRUE in at least ONE Model
VALID INVALID INCONSISTENT CONSISTENT (Unsatisfiable) (Satisfiable)
P Q P->Q (P->Q)∧~Q ((P->Q) ∧~Q)->~PF FF TT FT T
T T T T F T F F T T F T
FORMAL SYSTEM22
Well Formed Formula Language
AXIOM + THEOREM ├ AINFERENCE RULES
VALID ╞ A
PROOF THEORY MODEL THEORY
THEOREM VALID SOUNDNESS (→) COMPLETENESS (←)
Types of Logical Reasoning23
Deduction Given A, AB infer B Induction Given A, B find the rule AB Abduction (Not logically valid!) Given AB, B infer A Refutation Proof
Proof by Cases24
Is there a Red Box right next to a Non-Red Box?
?
Refutation Proof25
A, AB want to prove B
Assume ¬B and find a contradiction
Most Common Method using Computer- Resolution, Tableau Method etc.
PROVING VALIDITY in PROPOSITIONAL LOGIC
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1. TRUTH TABLE 2. TABLEAU METHOD 3. SEQUENT CALCULUS 4. RESOLUTON
TABLEAU METHOD27
Refutation Method(Assuming FALSE and draw CONTRADICTION)
(( P Q) ∧ -Q ) -P
F T F
T T TT F CONTRADICTION
RESOLUTION (Robinson)28
A B, B C, A C?
-A ∨ B A-B ∨ C -C
-A ∨ C
C
Example 1 (1/2)29
“Head I win, Tail you lose.”Prove I win.H: headT: tail H WW: I win T LL: You lose(Hidden information)H ∨ TW LL W
Example 1 (2/2)30
-H∨W -T∨L H∨T -W∨L -L∨W -W
HW
~T
~L
Example: Lion Sleeps Tonight31
Lion always sleeps except when he is hunting.
Lion cannot sleep when he is hungry. When he is tired he cannot hunt. Lion is tired when he does not sleep. Prove Lion is not hungry.
Resolution Strategies (1/4)32
1. UNIT RESOLUTION (Wos) Not complete
UNIT PREFERENCE RESOLUTION Unit clause always reduces the size!
P ∨ Q P ∨ -Q -P ∨ Q
P
Q -Q
-P ∨ -Q
Resolution Strategies (2/4)33
2. INPUT RESOLUTION Not complete
INPUT CLAUSES HAVE MEANINGFUL INFORMATION
P
Q -Q
INPUT CLAUSES
P ∨ Q P ∨ -Q -P ∨ Q -P ∨ -Q
Resolution Strategies (3/4)34
3. LINEAR RESOLUTION(Loveland) Chain of Reasoning
Depth First COMPLETE
P
Q-P
P ∨ Q P ∨ -Q -P ∨ Q -P ∨ -Q
Resolution Strategies (4/4)35
4. LOCK RESOLUTION(BOYER) COMPLETE Index every literal (Lock) : consider smallest
P1 Q2 P3 -Q4 -P6 Q5
-P6
Q2 -Q4
-P8 -Q7
Predicate Calculus36
Variable : object x, y, z, .. Constant : a, b, c, tom, 1, 2, .. Function : f, g, h, father(tom),… Predicate : P, Q, R Quantifier : ∀, ∃
Well Formed Formula37
Term constant, variable, f(t1, .. tn): ti term Atom P(t1,..tn) Formula(wff) 1. atom 2. F∨G, -F, FG 3. (∀x)F, (∃x)F
Nested Quantifiers38
Describe each statement ∀x ∀y Love(x, y)
∀x ∃y Love(x, y)
∃x ∀y Love(x, y)
∃x ∃y Love(x, y)
∀x ∃y Love(y, x)
∃x ∀y Love(y, x)
Negation of these?
Symbolize the Statement39
Every rational number is a real number. There exists a number that is prime. For every number x, there exists a number y such that x<y. Not every real number is a rational number. Everybody has somebody who loves him. There is someone whom everybody loves. Mimi loves only those who is younger than her. Everyone who eats BigMac listens Jazz music.
Clausal Form Conversion40
1. Eliminate 2. Reduce the scope of ~ 3. Rename the variables 4. Move quantifiers to the left
(prenex normal form) 5. Eliminate ∃ : Skolemize 6. Eliminate ∀ 7. Conjunctive Normal Form
Reducing the Scope of ~41
~ (p ∧ q) = ~p ∨ ~q ~ (p ∨ q) = ~p ∧ ~q
~ Q1Q2..Qn P(x,y,..) = Q’1Q’2..Q’n ~P(x,y,..) where Q’ = ∃ if Q=∀ ∀ if Q= ∃
Prenex Normal Form42
Prenex normal form:
Q1Q2..Qn P(x,y,..) where Q i = ∀, ∃
∀x (P(x) ∨ ∀y (Q(y)))
= ∀x ∀y (P(x) ∨ Q(y))
∀x (P(x) ∨ ∃y (Q(x, y)))
= ∀x ∃y (P(x) ∨ Q(x, y))
Skolem Function43
Eliminating ∃’s
∃x P(x) P(sk1) : sk1 is skolem constant ∃x ∀y P(x,y) ∀y P(sk1, y) ∀y ∃x P(x,y) ∀y P(sk1(y),y) ∀x ∃y ∀z ∃w P(x,y,z,w) ∀x ∀z P(x,sk1(x),z,sk2(x,z))
Example : Clausal Form44
∀x((∀y P(x,y) ~∀y(Q(x,y) R(x,y)))
∀x( ~∀y P(x,y) ∨ ~∀y (~Q(x,y) ∨ R(x,y))) … (1)
∀x( ∃y ~P(x,y) ∨ ∃y (Q(x,y) ∧~R(x,y))) … (2)
∀x( ∃y ~P(x,y) ∨ ∃z (Q(x,z) ∧~R(x,z))) … (3)
∀x∃y∃z (~P(x,y) ∨((Q(x,z) ∧~R(x,z))) … (4)
∀x(~P(x,s1(x))∨((Q(x,s2(x))∧~R(x,s2(x)))) … (5)
~P(x,s1(x))∨((Q(x,s2(x))∧~R(x,s2(x))) … (6)
(~P(x,s1(x))∨Q(x,s2(x)))∧(~P(x,s1(x))∨~R(x,s2(x)))
Matching45
Rule : Mother(x, y) Like(x, y)
“Every Mother Like their Son” Fact
Like(Joe, Jack), Like(Kim, Mary)
Mother(Judy, Jack), Mother(Mary, Jay) Query
Like(Judy Jack)?
Like(Mary, ?)
Unification(2-way Matching)46
Find a substitution σ(unifier) which makes two clause equal
Essential step for Resolution of Predicate Calculus Usually unification tries to find a most general unifier
Most General Unifier (mgu)47
Substitution : σ C = Like(x, father(x)) σ = {jack/x} C • σ = Like(Jack, father(Jack)) C, D are unifiable iff there is σ s.t. C • σ = D • σ (σ is called unifier) Mgu least specific unifier Like(x, y), Like(Jack, y) σ1={Jack/x}, σ2={Jack/x, Mary/y}
Unification - Examples48
Like(x, y) Like(joe, father(joe))Like(jack, y) Like(x, father(x))Like(x, father(x)) Like(joe, y)Like(x, father(joe)) Like(jack, father(y))Like(x, father(x)) Like(jack, father(joe)) Like(x, father(x)) Like(father(y),z)Like(x, x) Like(father(z), z) ?
Factoring49
If 2 literals in a clause C have mgu σ then C • σ is called a factor of C If C = P(x) ∨ P(f(y)) ∨ ~Q(x) Then σ = {f(y)/x} P(f(y))∨P(f(y))∨~Q(f(y)) P(f(y))∨~Q(f(y)) : factor of C
Subsumption50
Clause C subsumes D iff
∀ C D
(A ∧ B) subsumes A
A subsumes A ∨ B Subsumed clause can be deleted C=P(x) D=P(a) delete P(a) Note: factoring – within a clause
subsumption – between two clauses
Example51
1. All KU students are handsome&pretty.
2. Kim only likes an intelligent girl.
3. Pretty girls do not read books.
4. Intelligent girls are either good reader or music lovers.
5. Kim likes Mimi who attends to KU.
* Prove that Mimi is a music lover
E(x): KU students P(x): Pretty
B(x): Book reader M(x): Music lover
Like(x, y): x likes y I(x): intelligent
Example - Skolemize52
Customer officials search everyone who entered the country who is not a VIP
Some of the drug dealers entered the country and they were only searched by drug dealers.
No drug dealer was a VIP Conclusion: Some of the officials were drug dealers.
E(x) : x entered the country V(x): VIP
C(x) : custom official D(x): drug dealer
S(x,y): x searched y
Types of Question - Resolution53
Type1: Yes/No Question “Is Mimi a Music Lover?” Type2: Short Answer “Who is a Music Lover?” Use special predicate: Ans(x) Type3: How to type Question
Type2 Question54
Every Pompeian died in 79. Marcus was a Pompeian. When was Marcus died?
~Died(marcus, x) ∨ Ans(x)
Type3:Monkey Banana Problem
55
P(x, y, z, s) : x: monkey y: banana z: chair s: state R(s) : monkey reachable to banana at s Functions walk(l1, l2, s) : at state s, monkey walk from l1 to l2 carry(l1, l2, s) : carry chair from l1 to l2 climb(s) : at s, monkey climb to the chair
Monkey – continue56
P(x,y,z,s) P(z,y,z, walk(x,z,s))P(x,y,x,s) P(y,y,y, carry(x,y,s))P(kit,kit,kit,s) R(climb(s))P(liv, kit, din, s1)R(x) Ans(x)-----Ans(climb(carry(din,kit,(walk(liv,din,s1)))))
Merit of using Logic as KR57
Flexible (Rich) Representation Natural Language > Logic Formula Note: flexibility is also a weak point of logic
Powerful Inference Mechanism Resolution, Graphical methods
Theoretical Background Solid
Weak Points of Logic a KR58
Too Rigid (Not Flexible) - no contradiction - no exception - no change
Complexity - NP-complete Horn-clause (restricted form)
Spin-off Products of Logic59
Prolog Language - Specification Language - Horn clause
Knowledge-base Systems - Rules & Inference Engine
New Breed of Logics
Real World vs Logic World60
Dynamic - Static
- keeps on changing Incomplete Knowledge
- implicit vs explicit, Default Rules, Closed World Assumption
Belief vs Truth Non-monotonic vs Monotonic Uncertainty – Statistical Reasoning
TMS (1/2)61
Truth Maintenance System - Doyle
Intended to Model Belief Changes Information is linked together by its justifications Dependency-directed backtracking Basic Data Structure
Node: belief Justification: reason to believe http://www.aistudy.com/problem/exercise/%
EC%A7%84%EB%A6%AC%EA%B0%92%20%EC%9C%A0%EC%A7%80%20%EC%8B%9C%EC%8A%A4%ED%85%9C.htm
http://www.aistudy.com/ai/logic_rich.htm
TMS (2/2)62
2 states of node IN – current belief OUT – not believed (cf. believed to be not true)
A node is assigned a justification set A node is IN iff there is at least one valid
justification A node is OUT iff there is no valid justification
SL justification (1/3)63
(SL (list of IN-nodes)(list of OUT-nodes)) SL-justification is valid if all the nodes in the IN-node list
are currently IN, and those in the OUT-node list are OUT.
Statement-1: (SL (x)(y)) Meaning:
If x is believed and y is not believed, the statement-1 is believed.
SL justification (2/3)64
Special nodes: Premise – nodes with (SL ()()) always IN Assumption – nodes with nonempty OUT-list which is
currently IN. (Default Rule) eg. 1. There is other schedule 2. I will be at the party. (SL ()(1))
“Unless there is other schedule, I will be at a party.”
SL justification (3/3)65
1. X : (SL (2)()) : If Y Then X (normal rule) 2. Y
1. X : (SL ()(2)) : X unless Y (strong default rule – CWA)
2. Y 1. X: (SL (2)(3)) : If Y Then X Unless Z (default
rule) 2. Y 3. Z
Example 1- Dream66
If I win Lotto, I’ll be Rich unless it is a Dream.
1. Rich : (SL (2)(3)) 2. Lotto Win! 3. Dream I win Lotto!!!
1. Rich : (SL (2)(3)) 2. Lotto Win! ------------- (SL ()()) 3. Dream I pinched myself, and woke up.. 1. Rich : (SL (2)(3)) 2. Lotto Win! (SL()()) 3. Dream -----------------(SL ()())
Example 267
1. It is winter OUT 2. It is cold (SL(1)(3)) OUT 3. It is warm (SL(4)(2)) IN 4. It is summer (SL()(1)) IN
It is winter. 1. It is winter (SL()()) IN 2. It is cold (SL(1)(3)) ? 3. It is warm (SL(4)(2)) ? 4. It is summer (SL()(1)) ?
It is warm outside. 1. It is winter (SL()()) IN 2. It is cold (SL(1)(3)) ? 3. It is warm (SL(4)(2)) (SL()()) IN 4. It is summer (SL()(1)) ?
Example 3 (1/5)68
This is how Mimi likes to see as her marriage partner. Not OK unless she really likes him. She likes a rich man as long as he doesn’t have a problem. She likes a man if he is healthy and kind as long as he does not
have a problem and is not the eldest son. A man is problematic if he is older than 35 unless he is
exceptional. Married man is problematic Love is an exception.
Example 3(2/5)69
Nodes: 1. Not OK (SL()(2)) OUT IN 2. She likes him (SL(3)(4)) (SL(5,6)(4,7)) OUT 3. He is rich OUT 4. He has a problem (SL(8)(9)) (SL(10)()) OUT 5. He is healthy OUT 6. Kind OUT 7. The eldest son OUT 8. Older than 35 OUT 9. Exception (SL(11)()) OUT10. Married OUT11. Love OUT
Example 3(3/5)70
He looks healthy and kind 1. Not OK (SL()(2)) IN -- OUT 2. She likes him (SL(3)(4)) (SL(5,6)(4,7)) OUT -- IN 3. He is rich OUT 4. He has a problem (SL(8)(9)) (SL(10)()) OUT 5. He is healthy(SL()())IN 6. Kind (SL()())IN 7. The eldest son OUT 8. Older than 35 OUT 9. Exception (SL(11)()) OUT10. Married OUT11. Love OUT
Currnet belief:He is healthy and kindShe likes him --- OK
Example 3(4/5)71
His age is 38! 1. Not OK (SL()(2)) OUT -- IN 2. She likes him (SL(3)(4)) (SL(5,6)(4,7)) IN -- OUT 3. He is rich OUT 4. He has a problem (SL(8)(9)) (SL(10)()) OUT -- IN 5. He is healthy IN 6. Kind IN 7. The eldest son OUT 8. Older than 35 (SL()())IN 9. Exception (SL(11)()) OUT10. Married OUT11. Love OUT
Example 3(5/5)72
Mimi finds herself that she is in love with him. 1. Not OK (SL ()(2)) IN -- OUT 2. She likes him (SL(3)(4)) (SL(5,6)(4,7)) OUT -- IN 3. He is rich OUT 4. He has a problem (SL(8)(9)) (SL(10)()) IN -- OUT 5. He is healthy IN 6. Kind IN 7. The eldest son OUT 8. Older than 35 (SL()())IN 9. Exception (SL(11)()) OUT -- IN10. Married OUT11. Love (SL()())IN
-- So they married, and happily there
after …
Cyc : KB for Commonsense73
Lenat,D (MCC) Limitation of Logic
Predicate Symbol – No Semantics
Need More Background Knowledge Build a Huge Knowledgebase to cover human everyday
commonsense Enough to Understand –
Newspaper Article or Encyclopedia
Cyc: Structure74
Cyc KB – Knowledgebase CycL – Representation Language Environment (UE, MUE) Interface Editing/BrowsingUE: Spread Sheet TypeMUE: Museum Type (Graphic)(Note: Cyc Needs Lot of Update/Expansion)
CycL : Cyc Language75
CycL is Frame-Based Slot-Value Texas capital: (Austin) residents: (Fred Tom Park) stateOf: (UnitedStatesOfAmerica)
CycL (2)76
Constraint Language (on Top of Frame) Predicate Calculus Type (#%ForAll x #%Number (#%LogImplication (#%GreaterThan x 1) (#%GreaterThan (#%NumOfDiv x) 1)))
Constraint Language77
First Order Logic ‘All of Fred’s Friends are artists’(#%ForAll x(#%Fred #%friends) (#%allInstanceOF x #%Artist))
‘Some of Fred’s Friends are artists’(#%ThereExists x(#%Fred #%friends) (#%allInstanceOF x #%Artist))
Frame Types of Cyc78
Normal Texas, Fred, Red, Walking.. Etc. SlotUnit Frames to Define Slots SeeUnit Meta-level Info for certain slot of a unit SlotEntryDetail SeeUnit for a member of slot entry (eg. Park of resident slot of Texas frame)
Slot Unit79
Slot Frame is a Frame about a Slot Define, Constraints, Interrelationships among SlotsResidents instanceOf: (Slot) inverse: (residentOf) entryIsA: (Person) …..
SeeUnit80
Metalevel Information for particular slot for particular unit (footnote)
Texas capital: (Austin) *residents: (Tom Jack Park..) … SeeUnitFor-residents.Texas instanceOf: (SeeUnit) modifiesUnit: (Texas) *rateOfChange: ..
SlotEntry-Details81
Similar to SeeUnit except it talks about single entry of a slot
Texas capital: (Austin) residents: (Tom Jack *Park..) … SeeUnitFor-Park∈residents.Texas instanceOf: (SlotEntryDetailTypeofSeeUnit) modifiesUnit: (Texas) modifiesSlot: (residents) modifiesEntry: (Park) …
Inference in CycL82
What Does Cyc “Do”? More than 20 Special Purpose Inference
Schemes Inheritance Automatic Classification Constraint Maintenance TMS Guessing by Closed World Assumption Analogy Reasoning
Meta-Level Inference83
Inference Schemes are Divided into Several Levels
Simple and Fast Schemes are Used before more Slow and Complex Ones
Level1: Simply Access the Data Structure Level2: Inheritance Level3: Subsumption, Classification Level4: Constraint …. Level n : Analogy, Guess, etc.
Cyc Review84
First Attempt for Global Ontology Frame-based Mixture of Inference Reference: ‘Building Large Knowledge-Based Systems’ by
Lenat & Guha