Lecture 15 Logic Intro and PDCL - cs.ubc.caconati/322/322-2017W1/slides... · Technique. Variable....

Computer Science CPSC 322

Lecture 15

Logic Intro and PDCL

1

Announcements

• Marked midterms will be available on Thursday (with solutions)

• Assignment 3 will be posted on Th.• Due Wed. Nov 15• 2 late days allowed

• Marks for assignment 3 will likely be available on Th

Lecture Overview

• Intro to Logic

• Propositional Definite Clauses: • Syntax

• Semantics

• Proof procedures (time permitting)

Where Are We?Environment

Problem Type

Query

Planning

Deterministic Stochastic

Constraint Satisfaction Search

Arc Consistency

Search

Search

Logics

STRIPS

Vars + Constraints

Value Iteration

VariableElimination

Belief Nets

Decision Nets

Markov Processes

Static

Sequential

RepresentationReasoningTechnique

VariableElimination

First Part of the Course 4

Where Are We?Environment

Problem Type

Query

Planning

Deterministic Stochastic

Constraint Satisfaction Search

Arc Consistency

Search

Search

Logics

STRIPS

Vars + Constraints

Value Iteration

VariableElimination

Belief Nets

Decision Nets

Markov Processes

Static

Sequential

RepresentationReasoningTechnique

VariableElimination

Back to static problems, but with

richer representation 5

Logics in AI: Similar slide to the one for planning

Propositional Logics

First-Order Logics

Propositional Definite Clause Logics

Semantics and Proof Theory

Satisfiability Testing (SAT)

Description Logics

Cognitive Architectures

Video Games

Ontologies

Semantic Web

Information Extraction

Summarization

Production Systems

Tutoring Systems

Hardware Verification

Product Configuration

Software Verification

Applications

6

Logics in AI: Similar slide to the one for planning

Propositional Logics

First-Order Logics

Propositional Definite Clause Logics

Semantics and Proof Theory

Description Logics

Cognitive Architectures

Video Games

Hardware Verification

Product ConfigurationOntologies

Semantic Web

Information Extraction

Summarization

Production Systems

Tutoring Systems

Software Verification

You will know

You will know a little

Applications

Satisfiability Testing (SAT)

7

What you already know about logic...• From programming: Some logical operators

• If ((amount > 0) && (amount < 1000)) || !(age < 30) • ...

Logic is the language of Mathematics. To define formal structures (e.g., sets, graphs) and to prove statements about those

You know what they mean in a “procedural” way

We use logic as a Representation and Reasoning Systemthat can be used to formalize a domain and to reason about it

8

Logic: a framework for representation & reasoning

• When we represent a domain about which we have only partial (but certain) information, we need to represent….

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Logic: a framework for representation & reasoning

• When we represent a domain about which we have only partial (but certain) information, we need to represent….• Objects, properties, sets, groups, actions, events, time, space, …

• All these can be represented as • Objects• Relationships between objects

• Logic is the language to express knowledge about the world this way• http://en.wikipedia.org/wiki/John_McCarthy (1927 - 2011)

Logic and AI “The Advice Taker”Coined “Artificial Intelligence”. Dartmouth W’shop (1956)

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http://en.wikipedia.org/wiki/John_McCarthy

Why Logics?

• “Natural” way to express knowledge about the world

e.g. “Every 101 student will pass the course”Course (c1)Name-of (c1, 101)

• It is easy to incrementally add knowledge • It is easy to check and debug knowledge• Provides language for asking complex queries• Well understood formal properties

)1,(_)1,(&)()( czpasswillczregisteredzstudentz →∀

11

Logic: A general framework for reasoning

General problem: Query answering• tell the computer how the world works• tell the computer some facts about the world• ask a yes/no question about whether other facts must be true

Solving it with Logic1. Begin with a task domain.2. Distinguish those things you want to talk about (the ontology)3. Choose symbols in the computer to denote elements of your ontology4. Tell the system knowledge about the domain

12

Example: Electrical Circuit

/ up

/down

13

/ up

/down

14

Logic: A general framework for reasoning

General problem: Query answering• tell the computer how the world works• tell the computer some facts about the world• ask a yes/no question about whether other facts must be true

Solving it with Logic1. Begin with a task domain.2. Distinguish those things you want to talk about (the ontology)3. Choose symbols in the computer to denote elements of your ontology4. Tell the system knowledge about the domain5. Ask the system whether new statements about the domain are true or

false

15

/ up

/down

16

live_w4? lit_l2?

To Define a Logic We Need

• Syntax: specifies the symbols used, and how they can be combined to form legal sentences • Knowledge base is a set of sentences in the

language

• Semantics: specifies the meaning of symbols and sentences

• Reasoning theory or proof procedure: a specification of how an answer can be produced.• Sound: only generates correct answers with

respect to the semantics• Complete: Guaranteed to find an answer if it exists 17

Propositional Definite ClausesWe will start with a simple logic

• Primitive elements are propositions: Boolean variables that can be {true, false}

Two kinds of statements:• that a proposition is true• that a proposition is true if one or more other propositions are true

Why only propositions?• We can exploit the Boolean nature for efficient reasoning• Starting point for more complex logics

We need to specify: syntax, semantics, proof procedure

18

Lecture Overview

• Intro to Logic


• Semantics

• Proof Procedures



language




Propositional Definite Clauses: SyntaxDefinition (atom)An atom is a symbol starting with a lower case letter

Definition (body)A body is an atom or is of the form b1 ∧ b2 where b1 and b2 are bodies.

Definition (definite clause)A definite clause is - an atom or - a rule of the form h ← b where h is an atom (“head”) and b is a body. (Read this as “h if b”.)

Definition (KB)A knowledge base (KB) is a set of definite clauses

Examples: p1;live_l1

Examples: p1 ∧ p2; ok_w1 ∧ live_w0

Examples: p1 ← p2;live_w0 ← live_w1 ∧ up_s2

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atoms

rules

definiteclauses, KB

22

PDCL Syntax: more examples

a) Sunny_today

b) sunny_today ∨ cloudy_today

c) vdjhsaekwrq

d) high_pressure_system ← sunny-today

e) sunny_today ← high_pressure_system ∧ summer

f) sunny_today ← high_pressure-system ∧ ¬ winter

g) ai_is_fun ← f(time_spent, material_learned)

h) summer ← sunny_today ∧ high_pressure_system

How many of the clauses below are legal PDCL clauses?

Definition (definite clause)A definite clause is - an atom or - a rule of the form h ← b where h is an atom (‘head’) and b is a body.

(Read this as ‘h if b.’)

23

PDCL Syntax: more examples

a) Sunny_today


c) vdjhsaekwrq






How many of the clauses below are legal PDCL clauses?

Definition (definite clause)A definite clause is - an atom or - a rule of the form h ← b where h is an atom (‘head’) and b is a body.

(Read this as ‘h if b.’)

24

A. 3

B. 4

C. 5

D. 6

PDC Syntax: more examples

a) Sunny_today


c) vdjhsaekwrq






Do any of these statements mean anything? Syntax doesn't answer this question!

Legal PDC clause Not a legal PDC clause

25

B. 4



language




Lecture Overview

• Intro to Logic


• Semantics


Propositional Definite Clauses: Semantics

• Semantics allows one to relate the symbols in the logic to the domain to be modeled.

• If our domain has 8 atoms, how many interpretations are there?

Definition (interpretation)An interpretation I assigns a truth value to each atom.

B. 8*2 C. 82A. 8+2 D. 28

28



• If our domain has 8 atoms, how many interpretations are there?• 2 values for each atom, so 28 combinations• Similar to possible worlds in CSPs


29



We can use the interpretation to determine the truth value of clauses


Definition (truth values of statements)• A body b1 ∧ b2 is true in I if and only if b1 is true in I

and b2 is true in I.• A rule h ← b is false in I if and only if b is true in I

and h is false in I.

30

h b h ← bI1 F F TI2 F T FI3 T F TI4 T T T

PDC Semantics: Example

a1 a2 a1 ∧ a2

I1 F F FI2 F T FI3 T F FI4 T T T

Truth values under different interpretationsF=false, T=true

31


PDC Semantics: ExampleTruth values under different interpretations

F=false, T=true

h ← b is false only whenb is true and h is false

h a1 a2 h ← a1 ∧ a2

I1 F F FI2 F F TI3 F T FI4 F T TI5 T F FI6 T F TI7 T T FI8 T T T

33


PDC Semantics: Example for truth values

Truth values under different interpretationsF=false, T=true

h ← a1 ∧ a2

Body of the clause is a1 ∧ a2

Body is true only if both a1 and a2 are true in I

h a1 a2 h ← a1 ∧ a2

I1 F F F TI2 F F T TI3 F T F TI4 F T T FI5 T F F TI6 T F T TI7 T T F TI8 T T T T

34


Semantics allows you to relate the symbols in the logic to the domain you're trying to model.

We can use the interpretation to determine the truth value of clauses


Definition (truth values of statements)• A body b1 ∧ b2 is true in I if and only if b1 is true in

I and b2 is true in I.• A rule h ← b is false in I if and only if b is true in I

and h is false in I.

• A knowledge base KB is true in I if and only if every clause in KB is true in I. 35


Definition (model)A model of a knowledge base KB is an interpretation in which KB is true.

Similar to CSPs: a model of a set of clauses is an interpretation that makes all of the clauses true


Definition (truth values of statements)• A body b1 ∧ b2 is true in I if and only if b1 is true in

I and b2 is true in I.• A rule h ← b is false in I if and only if b is true in I

and h is false in I.• A knowledge base KB is true in I if and only if

every clause in KB is true in I.

36

PDC Semantics: Knowledge Base (KB)

p q r s

I1 false true true false

prs ← q ∧ p

rs ← pq ← p ∧ s

rqs ← q

A. KB1 C. KB2B. KB3

Which of the three KB above is True in I1

• A knowledge base KB is true in I if and only if every clause in KB is true in I.

37

PDC Semantics: Knowledge Base (KB)

p q r s

I1 false true true false

rs ← pq ← p ∧ s

C. KB2

Which of the three KB above is True in I1

• A knowledge base KB is true in I if and only if every clause in KB is true in I.

38

PDC Semantics: Example for models

p ← qKB = s

s ← r

Definition (model)A model of a knowledge base KB is an interpretation in which every clause in KB is true.

p q r s

I1 T T T T

I2 F F F T

I3 T T F F

I4 T T T F

I5 T T F T

Which of the interpretations below are models of KB?

B. I1 , I2

C. I1, I2, I5

D. All of them

A. I1

39


p q r s p ← q s s ← r KB

I1 T T T T

I2 F F F T

I3 T T F F

I4 T T T F

I5 T T F T


p ← qKB = s

s ← r

Which of the interpretations below are models of KB?All interpretations where KB is true

40



p ← qKB = s

s ← r

Which of the interpretations below are models of KB?All interpretations where KB is true: I1, I2, and I5

41

p q r s p ← q s s ← r KB

I1 T T T T T T T T

I2 F F F T T T T T

I3 T T F F T F T F

I4 T T T F T F T F

I5 T T F T T T T T

C. I1, I2, I5

OK but….…. Who cares? Where are we going with this?

Remember what we want to do with Logic

1) Tell the system knowledge about a task domain.• This is your KB• which expresses true statements about the world

2) Ask the system whether new statements about the domain are true or false.• We want the system responses to be

– Sound: only generates correct answers with respect to the semantics

– Complete: Guaranteed to find an answer if it exists

42

For Instance…

1) Tell the system knowledge about a task domain.

2) Ask the system whether new statements about the domain are true or false

p? r? s?

←

←=

..

.

srq

qpKB

43

Or, More Interestingly

1) Tell the system knowledge about a task domain.

2) Ask the system whether new statements about the domain are true or false

• live_w4? lit_l2? 44

To Obtain This We Need One More Definition

45

To Obtain This We Need One More Definition

Definition (logical consequence)If KB is a set of clauses and G is a conjunction of atoms,

G is a logical consequence of KB, written KB ⊧ G, if G is true in every model of KB.

• we also say that G logically follows from KB, or that KBentails G.

• In other words, KB ⊧ G if there is no interpretation in which KB is true and G is false.• when KB is TRUE, then G must be TRUE

• We want a reasoning procedure that can find all and only the logical consequences of a knowledge base

46

∧←

=.

.q

rqpKB

Example of Logic Entailment

How many models are there?

A. 1 B. 2 C. 3 D. 4

E. 5

47

∧←

=.

.q

rqpKB

Interpretationsr q pT T TT T FT F TT F FF T TF T FF F TF F F


How many models are there?

A. 1 B. 2 C. 3 D. 4

E. 5

48

∧←

=.

.q

rqpKB


Models

Which atoms are logically entailed?


• We want a reasoning procedure that can find all and only the logical consequences of a knowledge base 49

C. 3

∧←

=.

.q

rqpKB


Models

Which atoms are logically entailed?q


• We want a reasoning procedure that can find all and only the logical consequences of a knowledge base 50

C. 3

Example: Logical Consequences

←

←=

..

.

srq

qpKB

p q r sI1 true true true trueI2 true true true falseI3 true true false falseI4 true true false trueI5 false true true trueI6 false true true falseI7 false true false falseI8 false true false trueI9 true false true trueI10 true false true falseI11 true false false falseI12 true false false false….. …… ….. …… …….

Which of the following is true?

A. KB ⊧ q and KB ⊧ r

B. KB ⊧ q, and KB ⊧ s

C. KB ⊧ q, and KB ⊧ p

D. KB ⊧ r, KB ⊧ s,

E. None of the above51


←

←=

..

.

srq

qpKB



•KB ⊧ q,

•KB ⊧ p,

•KB ⊧ s,

•KB ⊧ r52


←

←=

..

.

srq

qpKB



•KB ⊧ q,

•KB ⊧ p,

•KB ⊧ s,

•KB ⊧ r53

models


←

←=

..

.

srq

qpKB



•KB ⊧ q,

•KB ⊧ p,

•KB ⊧ s,

•KB ⊧ r

T

T

F

F

54

models

C. KB ⊧ q, and KB ⊧ p

User’s View of Semantics

• Choose a task domain: intended interpretation.

• For each proposition you want to represent, associate a proposition symbol in the language.

• Tell the system clauses that are true in the intended interpretation: axiomatize the domain.

• Ask questions about the intended interpretation.

• If KB |= g , then g must be true in the intended interpretation.

55

Computer’s View of Semantics

• The computer doesn’t have access to the intended interpretation.

• All it knows is the knowledge base.

• The computer can determine if a formula is a logical consequence of KB.

• If KB |= g then g must be true in the intended interpretation.

• Otherwise, there is a model of KB in which g is false. This could be the intended interpretation.

The computer wouldn't know! 56


• Otherwise, there is a model of KB in which g is false. This could be the intended interpretation.The computer wouldn't know

←

←=

..

.

srq

qpKB

p q r sI1 true true true trueI2 true true true false

57


• Otherwise, there is a model of KB in which g is false. This could be the intended interpretation.The computer wouldn't know

←

←=

..

.

srq

qpKB

p q r sI1 true true true trueI2 true true true false

58

I1 and I2above are both models for KB, each could be the intended interpretation. The computer cannot know, thus it cannot say anything about the truth value of s

Learning Goals for Logic Up To Here

• PDCL syntax & semantics- Verify whether a logical statement belongs to the

language of propositional definite clauses- Verify whether an interpretation is a model of a PDCL KB.

‾ Verify when a conjunction of atoms is a logical consequence of a knowledge base

Next: Proof Procedures (5.2.2)

59

Lecture Overview

• Intro to Logic


• Semantics




language


• Reasoning theory or proof procedure: a specification of how an answer can be produced (sound and complete)• Bottom-up and Top-Down Proof Procedure for Finding

Logical Consequence

Slide 61

Proof Procedures• A proof procedure is a mechanically derivable

demonstration that a formula logically follows from a knowledge base.

• Given a proof procedure P, KB ⊦ P g means g can be derived from knowledge base KB with the proof procedure.

• If I tell you I have a proof procedure for PDCL• What do I need to show you in order for you to trust my

procedure?

That is sound and complete

Slide 62

Soundness and Completeness

• Completeness of proof procedure P: need to prove that

If g is true in all models of KB (KB ⊧ g)then g is derived by the procedure (KB ⊦P g)

Definition (completeness)A proof procedure P is complete if KB ⊧ g implies KB ⊦P g.

complete: every atom that logically follows from KB is derived by P

• Soundness of proof procedure P: need to prove that

Definition (soundness)A proof procedure P is sound if KB ⊦P g implies KB ⊧ g.

If g can be derived by the procedure (KB ⊦P g)then g is true in all models of KB (KB ⊧ g)

sound: every atom derived by P follows logically from KB (i.e. is true in every model)

Slide 63

Simple Proof Procedure

problem with this approach?

Simple proof procedure S• Enumerate all interpretations• For each interpretation I, check whether it is a model of KB

i.e., check whether all clauses in KB are true in I

• KB ⊦S g if g holds in all such models

64



• If there are n propositions in the KB, must check all the interpretations!




65



• If there are n propositions in the KB, must check all the 2n interpretations!

Goal of proof theory • find sound and complete proof procedures that allow

us to prove that a logical formula follows from a KB avoiding to do the above




66

Bottom-up proof procedure

• One rule of derivation, a generalized form of modus ponens:

• If “h← b1 ∧ … ∧ bm" is a clause in the knowledge base, andeach bi has been derived,

then h can be derived.

• This rule also covers the case when m = 0.

67

Bottom-up (BU) proof procedureC :={};repeat

select clause “h ← b1 ∧ … ∧ bm” in KB such that bi ∈ C for all i, and h ∉ C;

C := C ∪ { h }until no more clauses can be selected.

KB ⊦BU G if G ⊆ C at the end of this procedure

The C at the end of BU procedure is a fixed point:• Further applications of the rule of derivation will not

change C!68

C := {};repeat

select clause h ← b1 ∧… ∧ bm in KB such that bi ∈ C for all i, and h ∉ C;

C := C ∪ {h}until no more clauses can be selected.

Bottom-up proof procedure: example

a ← b ∧ ca ← e ∧ fb ← f ∧ kc ← ed ← ke.f ← j ∧ ef ← cj ← c

{}

69

C := {};repeat

select clause h ← b1 ∧… ∧ bm in KB such that bi ∈ C for all i, and h ∉ C;

C := C ∪ {h}until no more clauses can be selected.

Bottom-up proof procedure: example

a ← b ∧ ca ← e ∧ fb ← f ∧ kc ← ed ← ke.f ← j ∧ ef ← cj ← c

{}{e}{c,e}{c,e,f}{c,e,f,j}{a,c,e,f,j}

Done.70

Lecture 15 Logic Intro and PDCL - cs.ubc.caconati/322/322-2017W1/slides... · Technique. Variable....

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