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Inference in First-Order Logic

Inference in First-Order Logic. Proofs Unification Generalized modus ponens Forward and backward chaining Completeness Resolution Logic programming. Inference in First-Order Logic. Proofs – extend propositional logic inference to deal with quantifiers Unification

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Inference in First-Order Logic

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  1. Inference in First-Order Logic • Proofs • Unification • Generalized modus ponens • Forward and backward chaining • Completeness • Resolution • Logic programming CS 561, Session 16-18

  2. Inference in First-Order Logic • Proofs – extend propositional logic inference to deal with quantifiers • Unification • Generalized modus ponens • Forward and backward chaining – inference rules and reasoning program • Completeness – Gödel’s theorem: for FOL, any sentence entailed by another set of sentences can be proved from that set • Resolution – inference procedure that is complete for any set of sentences • Logic programming CS 561, Session 16-18

  3. Remember:propositionallogic CS 561, Session 16-18

  4. Proofs CS 561, Session 16-18

  5. Proofs The three new inference rules for FOL (compared to propositional logic) are: • Universal Elimination (UE): for any sentence , variable x and ground term , x  {x/} • Existential Elimination (EE): for any sentence , variable x and constant symbol k not in KB, x  {x/k} • Existential Introduction (EI): for any sentence , variable x not in  and ground term g in ,  x {g/x} CS 561, Session 16-18

  6. Proofs The three new inference rules for FOL (compared to propositional logic) are: • Universal Elimination (UE): for any sentence , variable x and ground term , x  e.g., from x Likes(x, Candy) and {x/Joe} {x/} we can infer Likes(Joe, Candy) • Existential Elimination (EE): for any sentence , variable x and constant symbol k not in KB, x  e.g., from x Kill(x, Victim) we can infer {x/k} Kill(Murderer, Victim), if Murderer new symbol • Existential Introduction (EI): for any sentence , variable x not in  and ground term g in ,  e.g., from Likes(Joe, Candy) we can infer x {g/x} x Likes(x, Candy) CS 561, Session 16-18

  7. Example Proof CS 561, Session 16-18

  8. Example Proof CS 561, Session 16-18

  9. Example Proof CS 561, Session 16-18

  10. Example Proof CS 561, Session 16-18

  11. Search with primitive example rules CS 561, Session 16-18

  12. Unification CS 561, Session 16-18

  13. Unification CS 561, Session 16-18

  14. Generalized Modus Ponens (GMP) CS 561, Session 16-18

  15. Soundness of GMP CS 561, Session 16-18

  16. Properties of GMP • Why is GMP and efficient inference rule? - It takes bigger steps, combining several small inferences into one - It takes sensible steps: uses eliminations that are guaranteed to help (rather than random UEs) - It uses a precompilation step which converts the KB to canonical form (Horn sentences) Remember: sentence in Horn from is a conjunction of Horn clauses (clauses with at most one positive literal), e.g., (A  B)  (B  C  D), that is (B  A)  ((C  D)  B) CS 561, Session 16-18

  17. Horn form • We convert sentences to Horn form as they are entered into the KB • Using Existential Elimination and And Elimination • e.g., x Owns(Nono, x)  Missile(x) becomes Owns(Nono, M) Missile(M) (with M a new symbol that was not already in the KB) CS 561, Session 16-18

  18. Forward chaining CS 561, Session 16-18

  19. Forward chaining example CS 561, Session 16-18

  20. Backward chaining CS 561, Session 16-18

  21. Backward chaining example CS 561, Session 16-18

  22. Completeness in FOL CS 561, Session 16-18

  23. Historical note CS 561, Session 16-18

  24. Resolution CS 561, Session 16-18

  25. Resolution inference rule CS 561, Session 16-18

  26. Remember: normal forms “product of sums of simple variables or negated simple variables” “sum of products of simple variables or negated simple variables” CS 561, Session 16-18

  27. Conjunctive normal form CS 561, Session 16-18

  28. Skolemization CS 561, Session 16-18

  29. Resolution proof CS 561, Session 16-18

  30. Resolution proof CS 561, Session 16-18

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