1. Courtesy Costas Busch - RPI1 Reducibility

2. Courtesy Costas Busch - RPI2 Problem is reduced to problem If we can solve problem then we can solve problem

3. Courtesy Costas Busch - RPI3 If is undecidable then is undecidable If is decidable then is decidable Problem is reduced to problem

4. Courtesy Costas Busch - RPI4 Example: the halting problem is reduced to the state-entry problem

5. Courtesy Costas Busch - RPI5 The state-entry problem Inputs: Turing Machine State Question: Does String enter state on input ?

6. Courtesy Costas Busch - RPI6 Theorem: The state-entry problem is undecidable Proof: Reduce the halting problem to the state-entry problem

7. Courtesy Costas Busch - RPI7 state-entry problem decider YES NO enters doesn’t enter Suppose we have a Decider for the state-entry algorithm:

8. Courtesy Costas Busch - RPI8 Halting problem decider YES NO halts on doesn’t halt on We want to build a decider for the halting problem:

9. Courtesy Costas Busch - RPI9 State-entry problem decider Halting problem decider YES NO YES NO We want to reduce the halting problem to the state-entry problem:

10. Courtesy Costas Busch - RPI10 Halting problem decider YES NO YES NO Convert Inputs ? We need to convert one problem instance to the other problem instance State-entry problem decider

11. Courtesy Costas Busch - RPI11 Convert to : Add new state From any halting state of add transitions to halting states Single halt state

12. Courtesy Costas Busch - RPI12 halts on input halts on state on input if and only if

13. Courtesy Costas Busch - RPI13 Generate Halting problem decider YES NO YES NO State-entry problem decider

14. Courtesy Costas Busch - RPI14 Since the halting problem is undecidable, the state-entry problem is undecidable END OF PROOF We reduced the halting problem to the state-entry problem

15. Courtesy Costas Busch - RPI15 Another example: the halting problem is reduced to the blank-tape halting problem

16. Courtesy Costas Busch - RPI16 The blank-tape halting problem Input:Turing Machine Question:Doeshalt when started with a blank tape?

17. Courtesy Costas Busch - RPI17 Theorem: Proof: Reduce the halting problem to the blank-tape halting problem The blank-tape halting problem is undecidable

18. Courtesy Costas Busch - RPI18 blank-tape halting problem decider YES NO halts on blank tape doesn’t halt on blank tape Suppose we have a decider for the blank-tape halting problem:

19. Courtesy Costas Busch - RPI19 halting problem decider YES NO halts on doesn’t halt on We want to build a decider for the halting problem:

20. Courtesy Costas Busch - RPI20 Blank-tape halting problem decider Halting problem decider YES NO YES NO We want to reduce the halting problem to the blank-tape halting problem:

21. Courtesy Costas Busch - RPI21 Blank-tape halting problem decider Halting problem decider YES NO YES NO We need to convert one problem instance to the other problem instance Convert Inputs ?

22. Courtesy Costas Busch - RPI22 Construct a new machine When started on blank tape, writes Then continues execution like then write step 1step2 if blank tapeexecute with input

23. Courtesy Costas Busch - RPI23 halts on input string halts when started with blank tape if and only if

24. Courtesy Costas Busch - RPI24 Generate blank-tape halting problem decider Halting problem decider YES NO YES NO

25. Courtesy Costas Busch - RPI25 Since the halting problem is undecidable, the blank-tape halting problem is undecidable END OF PROOF We reduced the halting problem to the blank-tape halting problem

26. Courtesy Costas Busch - RPI26 Does machine halt on input ? Summary of Undecidable Problems Halting Problem: Membership problem: Does machine accept string ?

27. Courtesy Costas Busch - RPI27 Does machine enter state on input ? Does machine halt when starting on blank tape? Blank-tape halting problem: State-entry Problem:

28. Courtesy Costas Busch - RPI28 Uncomputable Functions

29. Courtesy Costas Busch - RPI29 Uncomputable Functions A function is uncomputable if it cannot be computed for all of its domain Domain Values region

30. Courtesy Costas Busch - RPI30 An uncomputable function: maximum number of moves until any Turing machine with states halts when started with the blank tape

31. Courtesy Costas Busch - RPI31 Theorem:Function is uncomputable Proof: Then the blank-tape halting problem is decidable Assume for contradiction that is computable

32. Courtesy Costas Busch - RPI32 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

33. Courtesy Costas Busch - RPI33 Therefore, the blank-tape halting problem is decidable However, the blank-tape halting problem is undecidable Contradiction!!!

34. Courtesy Costas Busch - RPI34 Therefore, function in uncomputable END OF PROOF

35. Courtesy Costas Busch - RPI35 Undecidable Problems for Recursively Enumerable Languages

36. Courtesy Costas Busch - RPI36 is empty? is finite? contains two different strings of the same length? Take a recursively enumerable language Decision problems: All these problems are undecidable

37. Courtesy Costas Busch - RPI37 Theorem: For any recursively enumerable language it is undecidable to determine whether is empty Proof: We will reduce the membership problem to this problem

38. Courtesy Costas Busch - RPI38 empty language problem decider YES NO Suppose we have a decider for the empty language problem: Let be the TM with empty not empty

39. Courtesy Costas Busch - RPI39 membership problem decider YES NO accepts rejects We will build the decider for the membership problem:

40. Courtesy Costas Busch - RPI40 YES NO YES Membership problem decider empty language problem decider We want to reduce the membership problem to the empty language problem:

41. Courtesy Costas Busch - RPI41 YES NO YES Membership problem decider empty language problem decider We need to convert one problem instance to the other problem instance Convert inputs ?

42. Courtesy Costas Busch - RPI42 Construct machine : When enters a final state, compare with Accept only if executes the same as with On arbitrary input string

43. Courtesy Costas Busch - RPI43 is not empty if and only if

44. Courtesy Costas Busch - RPI44 construct YES NO YES Membership problem decider END OF PROOF empty language problem decider