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Decidability

Decidability. continued. Undecidable Problems. Halting Problem:. Does machine halt on input ?. State-entry Problem:. Does machine enter state halt on input ?. Blank-tape halting problem:. Does machine halt when starting on blank tape?.

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Decidability

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  1. Decidability continued

  2. Undecidable Problems Halting Problem: Does machine halt on input ? State-entry Problem: Does machine enter state halt on input ?

  3. Blank-tape halting problem: Does machine halt when starting on blank tape? Membership problem: Is a string member of a recursively enumerable language ?

  4. Uncomputable Functions Values region Domain A function is uncomputable if it cannot be computed for all the domain

  5. Function : maximum number of moves until any Turing machine with states halts when started with the blank tape

  6. Theorem: Function is uncomputable Proof: If was computable then the blank-tape halting problem would be decidable

  7. Algorithm 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

  8. Rice’s Theorem

  9. Non-trivial property of recursively enumerable languages: any property possessed by some (not all) recursively enumerable languages

  10. Some non-trivial properties of recursively enumerable languages: • is empty • is finite • contains two different strings • of the same length

  11. Rice’s Theorem: Any non-trivial property of a recursively enumerable language is undecidable

  12. Theorem: For a recursively enumerable language it is undecidable to determine whether is empty Proof: We will reduce the membership problem to this problem

  13. Membership problem: Inputs: machine and string Question: ?

  14. Construct machine : When enters a final state, compare input with Observations: if and only if is empty

  15. Algorithm for membership problem: Inputs: machine and string 1. Construct 2. Determine if is empty Yes: then No: then

  16. Membership machine yes no Check if construct no is empty yes

  17. Theorem: For a recursively enumerable language it is undecidable to determine whether is finite Proof: We will reduce the halting problem to this problem

  18. Halting problem: Inputs: machine and string Question: does halt on input ?

  19. Construct machine : When enters a halt state, accept any input Initially, simulates on input (virtual input)

  20. Observations: If is finite then halts on if and only if is infinite

  21. Algorithm for halting problem: Inputs: machine and string 1. Construct 2. Determine if is finite Yes: then doesn’t halt on No: then halts on

  22. Machine for halting problem yes no Check if construct no is finite yes

  23. Theorem: For a recursively enumerable language it is undecidable to determine whether contains two different string of same length Proof: We will reduce the halting problem to this problem

  24. Halting problem: Inputs: machine and string Question: does halt on input ?

  25. Construct machine : When enters a halt state, accept symbols or Initially, simulates on input (virtual input)

  26. Observation: halts on if and only if accepts and (strings of equal length)

  27. Algorithm for halting problem: Inputs: machine and string 1. Construct 2. Determine if accepts strings of equal length Yes: then halts on No: then doesn’t halt on

  28. Machine for halting problem yes yes Check if construct no Has equal length strings no

  29. The Post Correspondence Problem

  30. Some undecidable problems for context-free languages: • Is context-free grammar ambiguous? • Is ?

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

  32. The Post Correspondence Problem Input: Two sequences of strings

  33. There is a Post Correspondence Solution if there is a sequence such that: PC-solution

  34. Example: PC-solution:

  35. Example: There is no solution Because total length of strings from is smaller than total length of strings from

  36. The Modified Post Correspondence Problem Inputs: MPC-solution:

  37. Example: MPC-solution:

  38. We will show that: The Modified Post Correspondence Problem is undecidable In other words: There is not MPC-solution for any pair

  39. Proof Technique: We will reduce the membership problem to the MPC problem

  40. Membership problem: Does Turing machine accept string Equivalent Problem: Does unrestricted Grammar generate string ?

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