1. Costas Busch - LSU1 The Post Correspondence Problem

2. Costas Busch - LSU2 Some undecidable problems for context-free languages: Is context-free grammar ambiguous? Is ? are context-free grammars

3. Costas Busch - LSU3 We need a tool to prove that the previous problems for context-free languages are undecidable: The Post Correspondence Problem

4. Costas Busch - LSU4 The Post Correspondence Problem Input: Two sets of strings

5. Costas Busch - LSU5 There is a Post Correspondence Solution if there is a sequence such that: PC-solution: Indices may be repeated or omitted

6. Costas Busch - LSU6 Example: PC-solution:

7. Costas Busch - LSU7 Example: There is no solution Because total length of strings from is smaller than total length of strings from

8. Costas Busch - LSU8 The Modified Post Correspondence Problem Inputs: MPC-solution:

9. Costas Busch - LSU9 Example: MPC-solution:

10. Costas Busch - LSU10 1. The MPC problem is undecidable 2. The PC problem is undecidable (by reducing MPC to PC) (by reducing the membership to MPC) We will show:

11. Costas Busch - LSU11 Theorem: The MPC problem is undecidable Proof: We will reduce the membership problem to the MPC problem

12. Costas Busch - LSU12 Membership problem Input: Turing machine string Question: Undecidable

13. Costas Busch - LSU13 Membership problem Input: unrestricted grammar string Question: Undecidable

14. Costas Busch - LSU14 Suppose we have a decider for the MPC problem MPC solution? YES NO String Sequences MPC problem decider

15. Costas Busch - LSU15 We will build a decider for the membership problem YES NO Membership problem decider

16. Costas Busch - LSU16 MPC problem decider Membership problem decider The reduction of the membership problem to the MPC problem: yes no

17. Costas Busch - LSU17 MPC problem decider Membership problem decider We need to convert the input instance of one problem to the other yes no Reduction?

18. Costas Busch - LSU18 Convert grammar and string Reduction: to sets of strings and Such that: generates There is an MPC solution for

19. Costas Busch - LSU19 : special symbol For every symbol Grammar : start variable For every variable

20. Costas Busch - LSU20 Grammar For every production : special symbol string

21. Costas Busch - LSU21 Example: Grammar : String

22. Costas Busch - LSU22

23. Costas Busch - LSU23

24. Costas Busch - LSU24 Grammar :

25. Costas Busch - LSU25 Derivation:

26. Costas Busch - LSU26 Derivation:

27. Costas Busch - LSU27 Derivation:

28. Costas Busch - LSU28 Derivation:

29. Costas Busch - LSU29 Derivation:

30. Costas Busch - LSU30 has an MPC-solution if and only if

31. Costas Busch - LSU31 MPC problem decider Membership problem decider Construct yes no

32. Costas Busch - LSU32 Since the membership problem is undecidable, The MPC problem is undecidable END OF PROOF

33. Costas Busch - LSU33 Theorem: The PC problem is undecidable Proof: We will reduce the MPC problem to the PC problem

34. Costas Busch - LSU34 Suppose we have a decider for the PC problem PC solution? YES NO String Sequences PC problem decider

35. Costas Busch - LSU35 We will build a decider for the MPC problem MPC solution? YES NO String Sequences MPC problem decider

36. Costas Busch - LSU36 PC problem decider MPC problem decider The reduction of the MPC problem to the PC problem: yes no

37. Costas Busch - LSU37 PC problem decider MPC problem decider yes no Reduction? We need to convert the input instance of one problem to the other

38. Costas Busch - LSU38 : input to the MPC problem : input to the PC problem Translated to

39. Costas Busch - LSU39 For each replace

40. Costas Busch - LSU40 PC-solution Has to start with These strings

41. Costas Busch - LSU41 PC-solution MPC-solution

42. Costas Busch - LSU42 has a PC solution has an MPC solution if and only if

43. Costas Busch - LSU43 PC problem decider MPC problem decider yes no Construct

44. Costas Busch - LSU44 Since the MPC problem is undecidable, The PC problem is undecidable END OF PROOF

45. Costas Busch - LSU45 Some undecidable problems for context-free languages: Is context-free grammar ambiguous? Is ? are context-free grammars We reduce the PC problem to these problems

46. Costas Busch - LSU46 Theorem: Proof: Let be context-free grammars. It is undecidable to determine if Reduce the PC problem to this problem (intersection problem)

47. Costas Busch - LSU47 Suppose we have a decider for the intersection problem Empty- interection problem decider YES NO Context-free grammars

48. Costas Busch - LSU48 We will build a decider for the PC problem PC solution? YES NO String Sequences PC problem decider

49. Costas Busch - LSU49 PC problem decider The reduction of the PC problem to the empty-intersection problem: noyes no Intersection problem decider

50. Costas Busch - LSU50 PC problem decider noyes no Reduction? We need to convert the input instance of one problem to the other Intersection problem decider

51. Costas Busch - LSU51 Context-free grammar : Introduce new unique symbols: Context-free grammar :

52. Costas Busch - LSU52 if and only if has a PC solution

53. Costas Busch - LSU53 Because are unique There is a PC solution:

54. Costas Busch - LSU54 PC problem decider noyes no Intersection problem decider Construct Context-Free Grammars

55. Costas Busch - LSU55 Since PC is undecidable, the Intersection problem is undecidable END OF PROOF

56. Costas Busch - LSU56 For a context-free grammar, Theorem: it is undecidable to determine if G is ambiguous Proof: Reduce the PC problem to this problem

57. Costas Busch - LSU57 PC problem decider noyes no Ambiguous problem decider Construct Context-Free Grammar

58. Costas Busch - LSU58 start variable of

59. Costas Busch - LSU59 if and only if is ambiguous if and only if has a PC solution