cs440-lec22-FOL-resolution - Artificial Intelligence Lec....

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Unformatted text preview: Artificial Intelligence Lec. 21-22: First-Order Inference UIUC CS 440 / ECE 448 Professor: Eyal Amir Spring Semester 2008 Now Until now: First-order logic basics Propositional Resolution Now: Resolution refutation for FOL Resolution Theorem Proving Given: KB a set of first-order sentences Query Q a logical sentence Calling procedure: 1. Add Q to KB 2. Convert KB into clausal form 3. Run theorem prover . If we prove contradiction, return T. Clausal Form Every FOL formula is consistency- equivalent to conjunction of F.O. clauses. First-order clause Only universal quantifiers (which are implicit) Disjunction of literals (atoms or their negation) ))) , ( ) ( ( ) ( ( w m loves m man m w woman w 5 2200 )) ), ( ( ) ( )) ( ( ) ( v v SKm loves v woman w SKm man w woman First-Order Resolution Resolution inference rule: C1: P(t1,,tk) C1(t1,,tk) C2: P(s1,,sk) C2(s1,,sk) mgu(<t1,,tk>,<s1,,sk>) = {r1,,rn}-------------------------------------------- C3: (C1 C2) {r1,,rn} First-Order Resolution Resolution algorithm (saturation): 1. While there are unresolved C1,C2: (1) Select C1, C2 in KB (2) If C1, C2 are resolvable, resolve them into a new clause C3 (3) Add C3 to KB (4) If C3={ } return T. 1. STOP C1: P(t1,,tk) C1(t1,,tk) C2: P(s1,,sk) C2(s1,,sk) mgu(<t1,,tk>,<s1,,sk>) = {r1,,rn}-------------------------------------------- C3: (C1 C2) {r1,,rn} Resolution in Action ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( A a B f A c B f C e r c w f C e z a B f y c A a s f s a x c x a On board Negated Query KB C1: P(t1,,tk) C1(t1,,tk) C2: P(s1,,sk) C2(s1,,sk) mgu(<t1,,tk>,<s1,,sk>) = {r1,,rn}-------------------------------------------- C3: (C1 C2) {r1,,rn} First-Order Resolution Resolution algorithm (saturation): 1. While there are unresolved C1,C2: (1) Select C1, C2 in KB (1) If C1, C2 are resolvable, resolve them into a new clause C3 (1) Add C3 to KB (2) If C3={ } return T. 1. STOP C1: P(t1,,tk) C1(t1,,tk) C2: P(s1,,sk) C2(s1,,sk) mgu(<t1,,tk>,<s1,,sk>) = {r1,,rn}-------------------------------------------- C3: (C1 C2) {r1,,rn} First-Order Resolution Rule (2) If C1, C2 are resolvable, resolve them into a new clause C3 If C1,C2 have two literals l1,l2 with same predicates (P) and opposite polarity, and If l1= P(t1,,tk), l2= P(s1,,sk), unifiable with mgu (most general unifier) {r1,,rn}, then C1: P(t1,,tk) C1(t1,,tk) C2: P(s1,,sk) C2(s1,,sk) mgu(<t1,,tk>,<s1,,sk>) = {r1,,rn}-------------------------------------------- C3: (C1 C2) {r1,,rn} Unification P(t1,,tk), P(s1,,sk), unifiable with mgu (most general unifier)...
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This note was uploaded on 05/04/2008 for the course CS 440 taught by Professor Eyalamir during the Spring '08 term at University of Illinois at Urbana–Champaign.

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cs440-lec22-FOL-resolution - Artificial Intelligence Lec....

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