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Unformatted text preview: Artificial Intelligence Lec. 2122: FirstOrder Inference UIUC CS 440 / ECE 448 Professor: Eyal Amir Spring Semester 2008 Now Until now: Firstorder logic basics Propositional Resolution Now: Resolution refutation for FOL Resolution Theorem Proving Given: KB a set of firstorder 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. Firstorder 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 FirstOrder 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} FirstOrder 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} FirstOrder 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} FirstOrder 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.
 Spring '08
 EyalAmir
 Artificial Intelligence

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