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cs440-lec22-FOL-resolution

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

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Artificial Intelligence Lec. 21-22: First-Order Inference UIUC CS 440 / ECE 448 Professor: Eyal Amir Spring Semester 2008

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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.

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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}

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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}

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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}

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Unification …P(t1,…,tk), ¬ P(s1,…,sk), unifiable with mgu (most general unifier) σ= {r1,…rk} Substitution : replace vars. by terms Term: constant, variable, or a function of terms Composition of substitutions {x/g(w,v)} {w/A,v/f(B,z)} = {x/g(A,f(B,z),w/A,v/f(B,z)} (P( x ) v Q(f( x )) v P(g(B, x )) v P(f( y )))
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cs440-lec22-FOL-resolution - Artificial Intelligence Lec...

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