Slide trí tuệ nhân tạo propositional logic

❑Syntax ❑Semantics ❑A simple knowledge base ❑Logical Inference Problem ❑CNF Form... Propositional logic: Syntax❑Propositional logic is the simplest logic – illustrates basic ideas ❑Const

Introduction to Artificial Intelligence

Chapter 3: Knowledge Representation and Reasoning

(2) Propositional Logic

Nguyễn Hải Minh, Ph.D nhminh@fit.hcmus.edu.vn

❑Syntax

❑Semantics

❑A simple knowledge base

❑Logical Inference Problem

❑CNF Form

Propositional logic: Syntax

❑Propositional logic is the simplest logic – illustrates basic ideas

❑Constants: TRUE or FALSE

❑Symbols to stand for propositions (sentences): P, Q, R, P 1 ,

o IMPLIES  Implication (if then)

o Iff  Equivalence, biconditional (if and only if)

❑Literal: an atomic sentence (P) or negated atomic sentence (  P)

Backus-Naur Form (BNF) Grammar

BNF – a formal grammar of propositional logic

Propositional logic: Semantics

❑Each model specifies true/false for each proposition symbol

o E.g P 1,2 P 2,2 P 3,1

false true false

❑With these symbols, 8 possible models can be enumerated

automatically.

❑Rules for evaluating truth with respect to a model m:

o  S is true iff S is false

o S 1  S 2 is true iff S 1 is true and S 2 is true

o S 1  S 2 is true iff S 1 is true or S 2 is true

o S 1  S 2 is true iff S 1 is false or S 2 is true

o i.e., is false iff S 1 is true and S 2 is false

o S 1  S 2 is true iff S 1  S 2 is true and S 2  S 1 is true

❑ Simple recursive process evaluates an arbitrary sentence,

o e.g.,  P 1,2  (P 2,2  P 3,1 ) = true  (true  false) = true  true = true

Truth tables for connectives

A simple knowledge base:

Wumpus world

❑Symbols for each position [𝑖, 𝑗]

o 𝑃 𝑖, 𝑗 is true if there is a pit in [𝑖, 𝑗]

o 𝑊 𝑖, 𝑗 is true if there is a wumpus in [𝑖, 𝑗]

o 𝐵 𝑖, 𝑗 is true if there is a breeze in [𝑖, 𝑗]

o 𝑆 𝑖, 𝑗 is true if there is a stench in [𝑖, 𝑗]

❑Sentences in Wumpus world’s KB:

Logical Inference Problem

❑In other words:

o In all interpretations in which sentences in KB are true, is α also true?

Solving the Logical Inference Problem

Truth table for the KB of Wumpus World

A truth table constructed for the KB of Wumpus World

No pit in [1,2]

Inference by enumeration

❑ Depth-first enumeration of all models is sound and complete

❑For n symbols, time complexity is O(2 n ), space complexity is O(n)

2 Inference Rules Approach

❑Other name:

o Theorem proving

❑Applying rules of inference directly to the

sentences in KB to construct a proof of the

desired sentence without consulting models

→ Efficient than model checking if the number of

models is large but length of proof is short

❑New concepts:

o Logical equivalence

o Validity

o Satisfiability

Logical equivalence

❑Two sentences are logically equivalent iff true in same models: α ≡ ß iff α╞ β and β╞ α

Validity and satisfiability

❑A sentence is valid if it is true in all models,

o e.g., True, A  A, A  A, (A  (A  B))  B

❑Validity is connected to inference via the Deduction

Theorem:

o KB ╞ α if and only if (KB  α) is valid

❑A sentence is satisfiable if it is true in some model

o e.g., A  B, C

❑A sentence is unsatisfiable if it is true in no models

o e.g., A  A

❑Satisfiability is connected to inference via the following:

o KB ╞ α if and only if (KB  α) is unsatisfiable

Validity and satisfiability

❑Validity and satisfiability are connected:

o α is valid iff ¬α is unsatisfiable;

o α is satisfiable iff ¬α is not valid

❑Result:

o α |= β if and only if the sentence (α ∧ ¬β) is

unsatisfiable.

Apply Inference Rules to derive a

Proof

❑Proof:

o A chain of conclusions leads to the desired goal

❑Example sound rules of inference:

Inference Rules in Wumpus World

finding a proof can be more efficient because the proof can ignore irrelevant propositions, no matter how many of them there are.

Proof by Resolution Inference Rule

❑Problem of Proof by Inference Rules:

o If the rules are inadequate, then the goal is not reachable → the algorithm is not complete

❑Resolution Rule:

o A single inference rule

o Or: ¬α ⇒ β, β ⇒ γ |- ¬ α ⇒ γ

o Yields complete inference algorithm when

coupled with any complete search algorithm

α ∨ β, ¬ β V γ |- α ∨ γ

Soundness of Resolution Rule

We highlighted the cases when both premises are true

The resolution rule is sound because the conclusions are true in all cases (here 4) where the premises are true

Resolution in Wumpus World

Resolution in Wumpus World

Resolution in Wumpus World

Conjunctive Normal Form (CNF)

❑Resolution rule applies only to clauses

❑Convert the following sentences into CNF:

Resolution Algorithm

❑Proof by contradiction, i.e., show KB  α unsatisfiable

Resolution Algorithm – Example

❑Wumpus World:

o KB = (B 1,1  (P 1,2  P 2,1 ))  B 1,1

o α =  P 1,2

Problem of Inference Rules

❑Too many propositions to handle

o The statement “Do not go forward if the Wumpus is in front

of you” requires 16 squares x 4 orientations = 64

propositional rules

o It will take thousands of rules to build an agent

❑Change of the KB over time is difficult to represent

o Standard technique is to index facts with the time when

they’re true

o This means we have a separate KB for every time point

Next week

❑Chapter 3: Knowledge

Representation and Reasoning (cont.)