Week7 - CSE 3402: Intro to Artificial Intelligence...

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1 1 CSE3402 Winter 2012 Fahiem Bacchus & Yves Lesperance CSE 3402: Intro to Artificial Intelligence Inference in First-Order Logic Required Readings: 9.1, 9.2, and 9.5 Resolution Proofs. Part I: Convert to clausal form Part II: Dealing with variables (unification). Part III: Constructing Resolution Proofs. 2 CSE3402 Winter 2012 Fahiem Bacchus & Yves Lesperance Computing logical consequences We want procedures for computing logical consequences that can be implemented in our programs. This would allow us to reason with our knowledge Represent the knowledge as logical formulas Apply procedures for generating logical consequences These procedures are called proof procedures.
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2 3 CSE3402 Winter 2012 Fahiem Bacchus & Yves Lesperance Proof Procedures Interesting, proof procedures work by simply manipulating formulas. They do not know or care anything about interpretations. Nevertheless they respect the semantics of interpretations! We will develop a proof procedure for first- order logic called resolution. Resolution is the mechanism used by PROLOG 4 CSE3402 Winter 2012 Fahiem Bacchus & Yves Lesperance Properties of Proof Procedures Before presenting the details of resolution, we want to look at properties we would like to have in a (any) proof procedure. We write KB f to indicate that f can be proved from KB (the proof procedure used is implicit).
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3 5 CSE3402 Winter 2012 Fahiem Bacchus & Yves Lesperance Properties of Proof Procedures Soundness KB f KB f i.e all conclusions arrived at via the proof procedure are correct: they are logical consequences. Completeness KB f KB f i.e. every logical consequence can be generated by the proof procedure. Note proof procedures are computable, but they might have very high complexity in the worst case. So completeness is not necessarily achievable in practice. 6 CSE3402 Winter 2012 Fahiem Bacchus & Yves Lesperance Resolution Clausal form. Resolution works with formulas expressed in clausal form. A literal is an atomic formula or the negation of an atomic formula. dog(fido), ¬cat(fido) A clause is a disjunction of literals: ¬owns(fido,fred) ¬dog(fido) person(fred) We write (¬owns(fido,fred), ¬dog(fido), person(fred)) A clausal theory is a conjunction of clauses.
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4 7 CSE3402 Winter 2012 Fahiem Bacchus & Yves Lesperance Resolution Prolog Programs Prolog programs are clausal theories. However, each clause in a Prolog program is Horn. A horn clause contains at most one positive literal. The horn clause ¬q1 ¬q2 ¬qn p is equivalent to q1 q2 qn p and is written as the following rule in Prolog: p :- q1 , q2 ,… ,qn 8 CSE3402 Winter 2012 Fahiem Bacchus & Yves Lesperance Resolution Rule for Ground Clauses The resolution proof procedure consists of only one simple rule: From the two clauses (P, Q1, Q2, …, Qk)
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Week7 - CSE 3402: Intro to Artificial Intelligence...

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