13 Slides--Resolution and Satisfiability

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

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

This document was uploaded on 02/08/2011.

Page1 / 6

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online