1. PAI Recitation 3 – Logic Yuxin Chen 12-10-2012

2. Inference Truth tables Modus ponens Resolution …

3. Resolution Transform KB into CNF

4. Resolution: Example H: Heads up T: Tails up IW: I win UL : You lose Heads, I win Tails, you lose Either heads or tails I win or you lose I don’t win H => IW T => UL H ¬T IW UL ¬IW

5. Resolution Why can’t we just use resolution to derive the sentence we are interested in (instead of adding its negation and derive false)? Use resolution to derive from KB?

6. Resolution Why can’t we just use modus ponens instead of resolution to derive the sentence we are interested in? Modus ponens Example: P implies Q; Not P implies Q. Show that Q is ture.

7. Resolution Resolution is sound and complete Recall:

8. First order logic A superset of propositional logic Don’t treat propositions as “atomic” entities. Sentences refer to objects, their relationships, and their properties. TypesExample Objects Bob, October 12 Relations/predicatesMotherOf, ColorOf FunctionsIsMotherOf, IsBlue Connectives ∧∨ ¬ ⇔ ⇒ Quantifiers Universal: ∀ x:(isMan(x)) ⇒ isMortal(x)) Existential: ∃ y: ( isFather (y, Fred) )

9. Example: Convert to First-Order Logic ∀ x isSibling (x, Bob) ⇒ isKind(x) ∀ x isSibling (x, Bob) ∧ isKind(x) ∃ x At(x, ETH) ⇒ isSmart(x) ∃ x At(x, ETH) ∧ isSmart(x)

10. Example: Convert to English There is only one correct answer (do not use ∃ !) First cousin (child of your parent’s siblings) All grad students watch the same TV shows Any person can fool some of the people all of the time, all of the people some of the time but not all of the people all of the time

11. Example: Convert to English and FOL

12. Example (Cont..) In the sentences below, a symbol is missing. What symbol is it most likely to be? – ∀ x Something(x) ??? SomethingElse(x) – ∃ x Something(x) ??? SomethingElse(x)

13. Example (Cont..) Is this correctly written? – Peter has at least two children. – ∃ x,y ParentOf(Peter, x) ∧ ParentOf(Peter, y)

14. Resolution in FOL: Example Jack owns a dog. Every dog owner is an animal lover, and no animal lover kills an animal. Either Jack or Curiosity killed the cat, who is named Tuna. Did Curiosity kill the cat?

15. Resolution in ROL: Conjunctive Normal Form Jack owns a dog. Every dog owner is an animal lover. No animal lover kills an animal. Either Jack or Curiosity killed the cat, who is named Tuna. Did Curiosity kill the cat?

16. Resolution in ROL

17. Summary