Modus ponens pw qw q y s y yw pw sw true

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Unformatted text preview: Canonical forms for resolution Conjunctive Normal Form (CNF) Implicative Normal Form (INF) P( w) ⇒ Q( w) True ⇒ P ( x) ∨ R ( x) Q( y ) ⇒ S ( y ) R( z ) ⇒ S ( z ) ¬P ( w) ∨ Q ( w) P( x) ∨ R ( x) ¬Q ( y ) ∨ S ( y ) ¬R ( z ) ∨ S ( z ) CS 460, Session 18 23 Inference in First-Order Logic • Resolution Proofs to show S(A) is entailed In a forward- or backward-chaining algorithm, just as Modus Ponens. P(w) ⇒ Q(w) Q( y) ⇒ S ( y) {y/w} P(w) ⇒ S(w) True ⇒ P ( x ) ∨ R ( x) {w/x} R( z ) ⇒ S ( z ) True ⇒ S ( x) ∨ R( x) {x/A,z/A} True ⇒ S ( A) CS 460, Session 18 24 Inference in First-Order Logic • Refutation proof to show S(A) is entailed P(w) ⇒ Q(w) Q( y) ⇒ S ( y) {y/w} P(w) ⇒ S(w) True ⇒ P ( x ) ∨ R ( x) {w/x} R( z ) ⇒ S ( z ) True ⇒ S ( x) ∨ R( x) {z/x} ( KB ∧ ¬P ⇒ False) ⇔ ( KB ⇒ P ) True ⇒ S ( A) S ( A) ⇒ False {x/A} True ⇒ False CS 460, Session 18 25 Example of Refutation Proof (in conjunctive normal form) (1) (2) (3) (4) Cats like fish Cats eat everything they like Josephine is a cat. Prove: Josephine eats fish. ¬cat (x) ∨ likes (x,fish) ¬cat (y) ∨ ¬likes (y,z) ∨ eats (y,z) cat (jo) eats (jo,fish) CS 460, Session 18 26 Backward Chaining Negation of goal wff: ¬ eats(jo, fish) ¬ eats(jo, fish) ¬ cat(y) ∨ ¬likes(y, z) ∨ eats(y, z) θ = {y/jo, z/fish} ¬ cat(jo) ∨ ¬likes(jo, fish) cat(jo) θ=∅ ¬ cat(x) ∨ likes(x, fish) ¬ likes(jo, fish) θ = {x/jo} ¬ cat(jo) cat(jo) ⊥ (contradiction) CS 460, Session 18 27 Forward chaining ¬cat (X) ∨ likes (X,fish) cat (jo) \ / ¬cat (Y) ∨ ¬likes (Y,Z) ∨ eats (Y,Z) likes (jo,fish) \ / ¬cat (jo) ∨ eats (jo,fish) cat (jo) \ / ¬ eats (jo,fish) eats (jo,fish) \ / CS 460, Session 18 28 Question: • When would you use forward chaining? What about backward chaining? • A: • FC: If expert needs to gather information before any inferencing • BC: If expert has a hypothetical solution CS 460, Session 18 29...
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