AI=Knowledge Representation & Reasoning

1. AI=Knowledge Representation & Reasoning • Syntax • Semantics • Inference Procedure • Algorithm • Sound? • Complete? • Complexity

2. Some KR Languages • Propositional Logic • Predicate Calculus • Frame Systems • Rules with Certainty Factors • Bayesian Belief Networks • Influence Diagrams • Semantic Networks • Concept Description Languages • Nonmonotonic Logic

3. Propositional Logic • Syntax • Atomic sentences: P, Q, … • Connectives:  , , ,  • Semantics • Truth Tables • Inference • Modus Ponens • Resolution • Soundness and completeness • Complexity issues.

4. Inference Sentences Sentences Representation Semantics Semantics World Facts Facts Semantics • Syntax: a description of the legal arrangements of symbols (Def “sentences”) • Semantics: what the arrangement of symbols means in the world

5. Propsitional Logic: Syntax • Atoms • Literals • Sentences • Any literal is a sentence • If S1 and S2 are sentences, then • Then (S1  S2) is a sentence • Then (S1  S2) is a sentence • Then (S1  S2) is a sentence • Then S1 is a sentence

6. Q Q T T F F T T T F T T P P F F F F T F Propositional Logic: SEMANTICS • An interpretation is an assignment to each variable either True or False. • Assignments to compound sentences are defined by the standard truth tables: • A propositional knowledge base says which sentences must be true in the world. Q T T F P F F T P  Q P  Q  P

7. Example Knowledge Base • (Smoke  fire) <=> Alarm • Alarm

8. More Definitions • valid = tautology = always true • satisfiable = sometimes true • unsatisfiable = never true 1) smoke  fire 2) smoke  smoke 3) smoke  fire fire 4) (smoke  fire)  (smoke fire)

9. Making Inferences • A knowledge base gives us partial information about the world: it constrains the world to a set of possible truth assignments. • By inference, we decide what else holds in all of the truth assignments allowed by the knowledge base. • Inference question: does KB = S ?

10. Proof Procedures • To decide whether KB = S, we can try to look for a proof of S from KB. • A proof procedure is some algorithm that we apply to a KB to produce its logical consequences. • A proof uses: • the knowledge base, • axiom schemas • inference rules.

11. Soundness and Completeness • KB |- S: S is provable from KB. • A proof procedure is sound if: • If KB |- S, then KB |= S. • That is, the procedure produces only correct consequences. • A proof procedure is complete if: • If KB |= S, then KB |- S. • That is, the procedure produces all the consequences. • Ideally, the procedure should be sound and complete. (Ideals are nice in theory).

12. Modus Ponens • From A and A  B, infer B. • Modus ponens with a few axiom schemas is sound and complete: • A  (B  A) • A  (B  C)  ((A B)  (A  C)) • ( A   B)  (B  A) • More in the book.

13. Normal Forms • CNF = Conjunctive Normal Form • Conjunction of disjuncts (each disjunct = “clause”) (P  Q)  R (P  Q)  R (P  Q)  R P Q  R (P Q)  R (P  R)  (Q  R)

14. A  B  C, C D  E A  B  D  E Resolution • Refutation Complete • Given an unsatisfiable KB in CNF, • Resolution will eventually deduce the empty clause • Proof by Contradiction • To show = Q • Show  {Q} is unsatisfiable!

15. Resolution Example prove P (A B C) (B) (B D) (C A D) (D P Q) (Q)

16. Computational Complexity • Determining satisfiability is NP-complete. • Even when all clauses have at most 3 literals. • Hence, also validity and entailment testing are NP-complete • If all clauses have at most 2 literals, it is polynomial. • But if the KB is in DNF, satisfiability is polynomial. • What does this tell us about transforming a CNF into a DNF knowledge base?

17. Horn Clauses • If every sentence in KB is of the form: At most one positive literal A  B  C  ...  F  Z equivalently A  B  C  ... F  Z • Then Modus Ponens is • Polynomial time, and • Complete! Clause means a big disjunction