Automated reasoning with propositional and predicate logics

1. Automated reasoning with propositional and predicate logics Spring 2007,Juris Vīksna

2. Propositional logic

3. Propositional logic A formal definition of propositional logic formulas: [Adapted from M.Davis, E.Weyukerl]

4. Propositional logic - assignments [Adapted from M.Davis, E.Weyukerl]

5. Propositional logic How useful is propositional logic?

6. Satisfiability and tautologies In Latvian: satisfiable = nepretrunīga unsatisfiable = pretrunīga [Adapted from M.Davis, E.Weyukerl]

7. Some useful equivalences [Adapted from M.Davis, E.Weyukerl]

8. Normal forms - DNF

9. Normal forms - CNF

10. Logical consequence [Adapted from M.Davis, E.Weyukerl]

11. How it looks for CNFs/DNFs? [Adapted from M.Davis, E.Weyukerl]

12. Logical consequence [Adapted from M.Davis, E.Weyukerl] Deciding (1) is much simpler for DNFs Deciding (2) is much simpler for CNFs Unfortunately: When applying case (1) it is easy to obtain CNF, not DNF... When applying case (2) it is easy to obtain DNF, not CNF... (This also means that CNFDNF conversion in general requires an exponential time) Largely by following the tradition we will use case (2)

13. Conversion of formula to CNF (III) Use distributive laws to obtain CNF [Adapted from M.Davis, E.Weyukerl]

14. CNF - some more simplifications [Adapted from M.Davis, E.Weyukerl]

15. Thus, we currently have: [Adapted from M.Davis, E.Weyukerl]

16. Some useful notation [Adapted from M.Davis, E.Weyukerl]

17. Empty clauses and empty formulas [Adapted from M.Davis, E.Weyukerl]

18. Yet more of notation [Adapted from M.Davis, E.Weyukerl]

19. Davis-Putnam rules I [Adapted from M.Davis, E.Weyukerl]

20. Davis-Putnam rules I [Adapted from M.Davis, E.Weyukerl]

21. Davis-Putnam rules I [Adapted from M.Davis, E.Weyukerl]

22. Davis-Putnam rules II and III [Adapted from M.Davis, E.Weyukerl]

23. Davis-Putnam procedure Use rules II, III and I (in this order of preference) [Adapted from M.Davis, E.Weyukerl]

24. Complexity of Davis-Putnam procedure Each step decreases the number of literals by 1 (thus for n literals there will be up to n steps) Rules II and III do not increase the number of formulas to be checked Unfortunately, when only rule I applies, the number of formulas doubles In worst case the complexity might be (2n)

25. Davis-Putnam procedure - some improvements

26. Resolvent [Adapted from M.Davis, E.Weyukerl]

27. Notation again... [Adapted from M.Davis, E.Weyukerl]

28. Resolution method [Adapted from M.Davis, E.Weyukerl]

29. Resolution method [Adapted from M.Davis, E.Weyukerl]

30. Ground Resolution Theorem [Adapted from M.Davis, E.Weyukerl]

31. Ground Resolution Theorem [Adapted from M.Davis, E.Weyukerl]

32. Finite satisfiability This will be useful when we move up to predicate logic... [Adapted from M.Davis, E.Weyukerl]

33. Enumeration principle [Adapted from M.Davis, E.Weyukerl]

34. Finite satisfiability - some lemmas [Adapted from M.Davis, E.Weyukerl]

35. Finite satisfiability - some lemmas [Adapted from M.Davis, E.Weyukerl]

36. Finite satisfiability - some lemmas

37. Finite satisfiability - some lemmas [Adapted from M.Davis, E.Weyukerl]

38. Compactness theorem [Adapted from M.Davis, E.Weyukerl]

39. Predicate logic [Adapted from M.Davis, E.Weyukerl] Lets start with a formal definition:

40. Predicate logic - an alphabet [Adapted from M.Davis, E.Weyukerl]

41. Predicate logic - terms [Adapted from M.Davis, E.Weyukerl]

42. Predicate logic - formulas [Adapted from M.Davis, E.Weyukerl]

43. Free and bound occurrences [Adapted from M.Davis, E.Weyukerl]

44. Predicate logic - interpretations [Adapted from M.Davis, E.Weyukerl]

45. Interpretations - some notation [Adapted from M.Davis, E.Weyukerl]

46. Interpretations [Adapted from M.Davis, E.Weyukerl]

47. Interpretations [Adapted from M.Davis, E.Weyukerl]

48. Interpretations [Adapted from M.Davis, E.Weyukerl]

49. Interpretations [Adapted from M.Davis, E.Weyukerl]

50. Interpretations [Adapted from M.Davis, E.Weyukerl]