1 Resolution in First Order Logic CS 171/271 (Chapter 9, continued) Some text and images in these slides were drawn from Russel & Norvig’s published material

2 PL Resolution Revisited Recall PL Resolution algorithm: Convert (KB   ) to CNF Repeatedly get pairs of clauses and eliminate complementary literals If an empty clause results, KB ╞  Resolution applies to FOL, but we need to refine definitions of: CNF (for quantified sentences) Resolution inference rule / complimentary literals

3 CNF Conversion in FOL 1. Eliminate biconditionals and implications 2. Move  inwards (De Morgan’s) For quantifiers:  x P   x  P,  x P   x  P 3. Standardize variables Eliminate possible name clashes 4. Skolemize (Apply EI to existential sentences) Introduce Skolem constants or functions 5. Drop universal quantifiers 6. Distribute  over 

4 Example Everyone who loves all animals is loved by someone:  x [  y Animal(y)  Loves(x,y)]  [  y Loves(y,x)] 1. Eliminate biconditionals and implications  x [  y  Animal(y)  Loves(x,y)]  [  y Loves(y,x)] 2. Move  inwards:  x [  y  (  Animal(y)  Loves(x,y))]  [  y Loves(y,x)]  x [  y  Animal(y)   Loves(x,y)]  [  y Loves(y,x)]  x [  y Animal(y)   Loves(x,y)]  [  y Loves(y,x)] 3. Standardize variables: each quantifier should use a different one  x [  y Animal(y)   Loves(x,y)]  [  z Loves(z,x)]

5 Example, continued 4. Skolemize: a more general form of existential instantiation. Each existential variable is replaced by a Skolem function of the enclosing universally quantified variables:  x [Animal(F(x))   Loves(x,F(x))]  Loves(G(x),x) 5. Drop universal quantifiers: [Animal(F(x))   Loves(x,F(x))]  Loves(G(x),x) 6. Distribute  over  : [Animal(F(x))  Loves(G(x),x)]  [  Loves(x,F(x))  Loves(G(x),x)]

6 Resolution Lifted version of resolution inference rule: l 1  ···  l k, m 1  ···  m n (l 1  ···  l i-1  l i+1  ···  l k  m 1  ···  m j-1  m j+1  ···  m n )θ where Unify (l i,  m j ) = θ. The two clauses are assumed to be standardized apart so that they share no variables For example,  Rich(x)  Unhappy(x), Rich(Ken) Unhappy(Ken) with θ = {x/Ken}

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8 Making Resolution More Efficient Can favor particular clauses in the KB Unit preference (unit clauses) Sets of support Input resolution (e.g. “single spine”) Subsumption Reduces size of KB by eliminating redundant sentences