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13 Slides--Resolution and Satisfiability

# 13 Slides--Resolution and Satisfiability - CS103 HO#13...

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CS103 HO#13 Slides--Resolution and Satisfiability April 12, 2010 1 Premises: A ¬ B Prove: A C B C A A C ¬B B A C Contradiction Elimination C A C A C A C Resolution Premises: A ¬ B Prove: A C B C A A C ¬B C Disjunctive syllogism A C A C Resolution Dodec(e) Large(e) ¬Dodec(e) Dodec(f) Small(e) ¬ Dodec(e) Dodec(f) Dodec(f) Dodec(f) Small(e) Dodec(f) Dodec(f) Informal proof Suppose e is a Large Dodec, and ¬Dodec(e) Dodec(f) Small(e). Show that f is a Dodec. PROOF: Since it is given that e is a Large Dodec, we know that ¬Dodec(e) is false and Small(e) is false. Thus f must be a Dodec for the third premise to be true. Proving a Biconditional ... P Q P Q Q P Q P P Q ¬ (P Q) P P ∨¬ P P ¬ (P Q) P ¬ P ¬ (P Q) P ¬ (P Q) P Law of the Excluded Middle

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CS103 HO#13 Slides--Resolution and Satisfiability April 12, 2010 2 P ∨¬ P P ¬ (P Q) P ¬ P P Q P ¬ (P Q) ¬ (P Q) P ¬ (P Q) P Law of the Excluded Middle P ∨¬ P P ¬ (P Q) P ¬ P ¬ P ∨¬ Q ¬ (P Q) ¬ (P Q) P ¬ (P Q) P Law of the Excluded Middle Prove that ¬ (P Q) P is a tautology. PROOF by cases on the Law of Excluded Middle, P ∨¬ P. If P is true, then ¬ (P Q) P follows by the Addition rule. If P is false, then ¬ P ∨¬ Q follows by Addition, we have ¬ (P Q) by De Morgan's Law, and the desired conclusion by Addition. Thus ¬ (P Q) P is always true, i.e., it is a tautology. Rapid-fire Examples 1) Given: (A v B) (A v C) Prove: A 2) Given: (A v B) (A C) Prove: A 3) Given: (A B) v (A C) Prove: A (B C) 4) Given: A v B C v D ¬C ¬B Prove: A D 5) Given: (A B) C ¬A Prove: C Resolution P Q ¬Q R P R Resolution is a valid rule of inference, which we can use as a proof procedure. To apply it, we convert the formulas to conjunctive normal form . Conjunctive Normal Form
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13 Slides--Resolution and Satisfiability - CS103 HO#13...

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