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Undecidable problems for Recursively enumerable languages

Undecidable problems for Recursively enumerable languages. continued…. Take a recursively enumerable language. Decision problems:. is empty?. is finite?. contains two different strings of the same length?. All these problems are undecidable. Theorem:.

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Undecidable problems for Recursively enumerable languages

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  1. Undecidable problemsfor Recursively enumerable languages continued… COMP 335

  2. Take a recursively enumerable language Decision problems: • is empty? • is finite? • contains two different strings • of the same length? All these problems are undecidable COMP 335

  3. Theorem: For a recursively enumerable language it is undecidable to determine whether is finite Proof: We will reduce the halting problem to this problem COMP 335

  4. Let be the TM with Suppose we have a decider for the finite language problem: YES finite finite language problem decider NO not finite COMP 335

  5. We will build a decider for the halting problem: YES halts on Halting problem decider doesn’t halt on NO COMP 335

  6. We want to reduce the halting problem to the finite language problem Halting problem decider NO YES finite language problem decider YES NO COMP 335

  7. We need to convert one problem instance to the other problem instance Halting problem decider NO YES convert input ? finite language problem decider YES NO COMP 335

  8. Construct machine : On arbitrary input string Initially, simulates on input If enters a halt state, accept ( inifinite language) Otherwise, reject ( finite language) COMP 335

  9. halts on if and only if is infinite COMP 335

  10. halting problemdecider NO YES finite language problem decider construct YES NO COMP 335

  11. Take a recursively enumerable language Decision problems: • is empty? • is finite? • contains two different strings • of the same length? All these problems are undecidable COMP 335

  12. 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 COMP 335

  13. Let be the TM with Suppose we have the decider for the two-strings problem: YES contains Two-strings problem decider Doesn’t contain NO two equal length strings COMP 335

  14. We will build a decider for the halting problem: YES halts on Halting problem decider doesn’t halt on NO COMP 335

  15. We want to reduce the halting problem to the empty language problem Halting problem decider YES YES Two-strings problem decider NO NO COMP 335

  16. We need to convert one problem instance to the other problem instance Halting problem decider YES YES Two-strings problem decider convert inputs ? NO NO COMP 335

  17. Construct machine : On arbitrary input string Initially, simulate on input When enters a halt state, accept if or (two equal length strings ) Otherwise, reject ( ) COMP 335

  18. halts on if and only if accepts two equal length strings accepts and COMP 335

  19. Halting problem decider YES YES Two-strings problem decider construct NO NO COMP 335

  20. Rice’s Theorem COMP 335

  21. Definition: Non-trivial properties of recursively enumerable languages: any property possessed by some (not all) recursively enumerable languages COMP 335

  22. Some non-trivial properties of recursively enumerable languages: • is empty • is finite • contains two different strings • of the same length COMP 335

  23. Rice’s Theorem: Any non-trivial property of a recursively enumerable language is undecidable COMP 335

  24. The Post Correspondence Problem COMP 335

  25. Some undecidable problems for context-free languages: • Is ? are context-free grammars • Is context-free grammar ambiguous? COMP 335

  26. We need a tool to prove that the previous problems for context-free languages are undecidable: The Post Correspondence Problem COMP 335

  27. The Post Correspondence Problem Input: Two sequences of strings COMP 335

  28. There is a Post Correspondence Solution if there is a sequence such that: PC-solution: Indices may be repeated or omitted COMP 335

  29. Example: PC-solution: COMP 335

  30. Example: There is no solution Because total length of strings from is smaller than total length of strings from COMP 335

  31. We will show: 1. The MPC problem is undecidable (by reducing the membership to MPC) 2. The PC problem is undecidable (by reducing MPC to PC) COMP 335

  32. Theorem: The PC problem is undecidable Proof: We will reduce the MPC problem to the PC problem COMP 335

  33. Some undecidable problems for context-free languages: • Is ? are context-free grammars • Is context-free grammar • ambiguous? We reduce the PC problem to these problems COMP 335

  34. Theorem: Let be context-free grammars. It is undecidable to determine if Rdeduce the PC problem to this problem Proof: COMP 335

  35. Suppose we have a decider for the empty-intersection problem Context-free grammars Empty- interection problem decider YES NO COMP 335

  36. For a context-free grammar , Theorem: it is undecidable to determine if G is ambiguous Reduce the PC problem to this problem Proof: COMP 335

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