Undecidability in Computer Science

1. Reducibility Costas Busch - RPI

2. Problem is reduced to problem If we can solve problem then we can solve problem Costas Busch - RPI

3. Problem is reduced to problem If is decidable then is decidable If is undecidable then is undecidable Costas Busch - RPI

4. Example: the halting problem is reduced to the state-entry problem Costas Busch - RPI

5. The state-entry problem Inputs: • Turing Machine • State • String Question: Does enter state on input ? Costas Busch - RPI

6. Theorem: The state-entry problem is undecidable Proof: Reduce the halting problem to the state-entry problem Costas Busch - RPI

7. Suppose we have a Decider for the state-entry algorithm: YES enters state-entry problem decider NO doesn’t enter Costas Busch - RPI

8. We want to build a decider for the halting problem: YES halts on Halting problem decider NO doesn’t halt on Costas Busch - RPI

9. We want to reduce the halting problem to the state-entry problem: Halting problem decider YES YES State-entry problem decider NO NO Costas Busch - RPI

10. We need to convert one problem instance to the other problem instance Halting problem decider YES Convert Inputs ? YES State-entry problem decider NO NO Costas Busch - RPI

11. Convert to : • Add new state • From any halting state of add transitions to Single halt state halting states Costas Busch - RPI

12. halts on input if and only if halts on state on input Costas Busch - RPI

13. Halting problem decider YES YES State-entry problem decider Generate NO NO Costas Busch - RPI

14. We reduced the halting problem to the state-entry problem Since the halting problem is undecidable, the state-entry problem is undecidable END OF PROOF Costas Busch - RPI

15. Another example: the halting problem is reduced to the blank-tape halting problem Costas Busch - RPI

16. The blank-tape halting problem Input: Turing Machine Question: Does halt when started with a blank tape? Costas Busch - RPI

17. Theorem: The blank-tape halting problem is undecidable Proof: Reduce the halting problem to the blank-tape halting problem Costas Busch - RPI

18. Suppose we have a decider for the blank-tape halting problem: halts on blank tape YES blank-tape halting problem decider NO doesn’t halt on blank tape Costas Busch - RPI

19. We want to build a decider for the halting problem: YES halts on halting problem decider NO doesn’t halt on Costas Busch - RPI

20. We want to reduce the halting problem to the blank-tape halting problem: Halting problem decider YES YES Blank-tape halting problem decider NO NO Costas Busch - RPI

21. We need to convert one problem instance to the other problem instance Halting problem decider YES YES Convert Inputs ? Blank-tape halting problem decider NO NO Costas Busch - RPI

22. Construct a new machine • When started on blank tape, writes • Then continues execution like step 1 step2 if blank tape execute then write with input Costas Busch - RPI

23. halts on input string if and only if halts when started with blank tape Costas Busch - RPI

24. Halting problem decider YES YES blank-tape halting problem decider Generate NO NO Costas Busch - RPI

25. We reduced the halting problem to the blank-tape halting problem Since the halting problem is undecidable, the blank-tape halting problem is undecidable END OF PROOF Costas Busch - RPI

26. Summary of Undecidable Problems Halting Problem: Does machine halt on input ? Membership problem: Does machine accept string ? Costas Busch - RPI

27. Blank-tape halting problem: Does machine halt when starting on blank tape? State-entry Problem: Does machine enter state on input ? Costas Busch - RPI

28. Uncomputable Functions Costas Busch - RPI

29. Uncomputable Functions Values region Domain A function is uncomputable if it cannot be computed for all of its domain Costas Busch - RPI

30. An uncomputable function: maximum number of moves until any Turing machine with states halts when started with the blank tape Costas Busch - RPI

31. Theorem: Function is uncomputable Proof: Assume for contradiction that is computable Then the blank-tape halting problem is decidable Costas Busch - RPI

32. Decider for blank-tape halting problem: Input: machine 1. Count states of : 2. Compute 3. Simulate for steps starting with empty tape If halts then return YES otherwise return NO Costas Busch - RPI

33. Therefore, the blank-tape halting problem is decidable However, the blank-tape halting problem is undecidable Contradiction!!! Costas Busch - RPI

34. Therefore, function in uncomputable END OF PROOF Costas Busch - RPI

35. Undecidable Problems for Recursively Enumerable Languages Costas Busch - RPI

36. Take a recursively enumerable language Decision problems: • is empty? • is finite? • contains two different strings • of the same length? All these problems are undecidable Costas Busch - RPI

37. Theorem: For any recursively enumerable language it is undecidable to determine whether is empty Proof: We will reduce the membership problem to this problem Costas Busch - RPI

38. Let be the TM with Suppose we have a decider for the empty language problem: YES empty empty language problem decider NO not empty Costas Busch - RPI

39. We will build the decider for the membership problem: YES accepts membership problem decider NO rejects Costas Busch - RPI

40. We want to reduce the membership problem to the empty language problem: Membership problem decider NO YES empty language problem decider YES NO Costas Busch - RPI

41. We need to convert one problem instance to the other problem instance Membership problem decider NO YES Convert inputs ? empty language problem decider YES NO Costas Busch - RPI

42. Construct machine : On arbitrary input string executes the same as with When enters a final state, compare with Accept only if Costas Busch - RPI

43. if and only if is not empty Costas Busch - RPI

44. Membership problem decider NO YES empty language problem decider construct YES NO END OF PROOF Costas Busch - RPI