1. Fall 2004COMP 3351 Undecidable problems for Recursively enumerable languages continued…

2. Fall 2004COMP 3352 is empty? is finite? contains two different strings of the same length? Take a recursively enumerable language Decision problems: All these problems are undecidable

3. Fall 2004COMP 3353 Theorem: For a recursively enumerable language it is undecidable to determine whether is finite Proof: We will reduce the halting problem to this problem

4. Fall 2004COMP 3354 finite language problem decider YES NO Suppose we have a decider for the finite language problem: Let be the TM with finite not finite

5. Fall 2004COMP 3355 Halting problem decider YES NO halts on We will build a decider for the halting problem: doesn’t halt on

6. Fall 2004COMP 3356 YES NO YES Halting problem decider finite language problem decider We want to reduce the halting problem to the finite language problem

7. Fall 2004COMP 3357 YES NO YES Halting problem decider finite language problem decider We need to convert one problem instance to the other problem instance convert input ?

8. Fall 2004COMP 3358 Construct machine : If enters a halt state, accept ( inifinite language) Initially, simulates on input Otherwise, reject ( finite language) On arbitrary input string

9. Fall 2004COMP 3359 halts on is infinite if and only if

10. Fall 2004COMP 33510 construct YES NO YES halting problem decider finite language problem decider

11. Fall 2004COMP 33511 is empty? is finite? contains two different strings of the same length? Take a recursively enumerable language Decision problems: All these problems are undecidable

12. Fall 2004COMP 33512 Theorem: For a recursively enumerable language it is undecidable to determine whether contains two different strings of same length Proof: We will reduce the halting problem to this problem

13. Fall 2004COMP 33513 Two-strings problem decider YES NO Suppose we have the decider for the two-strings problem: Let be the TM with contains Doesn’t contain two equal length strings

14. Fall 2004COMP 33514 Halting problem decider YES NO halts on We will build a decider for the halting problem: doesn’t halt on

15. Fall 2004COMP 33515 YES NO YES NO Halting problem decider Two-strings problem decider We want to reduce the halting problem to the empty language problem

16. Fall 2004COMP 33516 YES NO YES NO Halting problem decider Two-strings problem decider We need to convert one problem instance to the other problem instance convert inputs ?

17. Fall 2004COMP 33517 Construct machine : When enters a halt state, accept if or Initially, simulate on input (two equal length strings ) On arbitrary input string Otherwise, reject ( )

18. Fall 2004COMP 33518 halts on if and only if accepts two equal length strings accepts and

19. Fall 2004COMP 33519 construct YES NO YES NO Halting problem decider Two-strings problem decider

20. Fall 2004COMP 33520 Rice’s Theorem

21. Fall 2004COMP 33521 Non-trivial properties of recursively enumerable languages: any property possessed by some (not all) recursively enumerable languages Definition:

22. Fall 2004COMP 33522 Some non-trivial properties of recursively enumerable languages: is empty is finite contains two different strings of the same length

23. Fall 2004COMP 33523 Rice’s Theorem: Any non-trivial property of a recursively enumerable language is undecidable

24. Fall 2004COMP 33524 The Post Correspondence Problem

25. Fall 2004COMP 33525 Some undecidable problems for context-free languages: Is context-free grammar ambiguous? Is ? are context-free grammars

26. Fall 2004COMP 33526 We need a tool to prove that the previous problems for context-free languages are undecidable: The Post Correspondence Problem

27. Fall 2004COMP 33527 The Post Correspondence Problem Input: Two sequences of strings

28. Fall 2004COMP 33528 There is a Post Correspondence Solution if there is a sequence such that: PC-solution: Indices may be repeated or omitted

29. Fall 2004COMP 33529 Example: PC-solution:

30. Fall 2004COMP 33530 Example: There is no solution Because total length of strings from is smaller than total length of strings from

31. Fall 2004COMP 33531 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:

32. Fall 2004COMP 33532 Theorem: The PC problem is undecidable Proof: We will reduce the MPC problem to the PC problem

33. Fall 2004COMP 33533 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

34. Fall 2004COMP 33534 Theorem: Proof: Let be context-free grammars. It is undecidable to determine if Rdeduce the PC problem to this problem

35. Fall 2004COMP 33535 Suppose we have a decider for the empty-intersection problem Empty- interection problem decider YES NO Context-free grammars

36. Fall 2004COMP 33536 For a context-free grammar, Theorem: it is undecidable to determine if G is ambiguous Proof: Reduce the PC problem to this problem