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Inference in First-Order Logic: Proofs, Unification, and Resolution Techniques

This content covers the core aspects of first-order logic inference, extending propositional logic to handle quantifiers. Key topics include proofs, unification, generalized modus ponens (GMP), forward and backward chaining, and completeness as demonstrated by Gödel’s theorem. The content elaborates on new inference rules for first-order logic: Universal Elimination (UE), Existential Elimination (EE), and Existential Introduction (EI). It also discusses the resolution procedure and its completeness along with logic programming concepts taught in CS 561 Sessions 16-18.

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Inference in First-Order Logic: Proofs, Unification, and Resolution Techniques

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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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