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Busch Complexity Lectures: Reductions

This lecture series discusses reductions in computational complexity theory, demonstrating how problems can be reduced to other problems to facilitate their solution. The lecture covers the definition of reductions, the relationship between reducibility and decidability, and provides examples of reductions in various problem domains.

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Busch Complexity Lectures: Reductions

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  1. Busch Complexity Lectures:Reductions Prof. Busch - LSU

  2. Problem is reduced to problem If we can solve problem then we can solve problem Prof. Busch - LSU

  3. Definition: Language is reduced to language There is a computable function (reduction) such that: Prof. Busch - LSU

  4. Recall: Computable function : There is a deterministic Turing machine which for any string computes Prof. Busch - LSU

  5. Theorem: If: a: Language is reduced to b: Language is decidable Then: is decidable Proof: Basic idea: Build the decider for using the decider for Prof. Busch - LSU

  6. Decider for Reduction YES Input string YES accept accept compute Decider for (halt) (halt) NO NO reject reject (halt) (halt) END OF PROOF Prof. Busch - LSU

  7. Example: is reduced to: Prof. Busch - LSU

  8. We only need to construct: Turing Machine for reduction DFA Prof. Busch - LSU

  9. Let be the language of DFA Let be the language of DFA Turing Machine for reduction DFA construct DFA by combining and so that: Prof. Busch - LSU

  10. Prof. Busch - LSU

  11. Decider for Reduction Input string YES compute YES Decider NO NO Prof. Busch - LSU

  12. Theorem (version 1): If: a: Language is reduced to b: Language is undecidable Then: is undecidable (this is the negation of the previous theorem) Proof: Suppose is decidable Using the decider for build the decider for Contradiction! Prof. Busch - LSU

  13. If is decidable then we can build: Decider for Reduction YES Input string YES accept accept compute Decider for (halt) (halt) NO NO reject reject (halt) (halt) CONTRADICTION! END OF PROOF Prof. Busch - LSU

  14. Observation: In order to prove that some language is undecidable we only need to reduce a known undecidable language to Prof. Busch - LSU

  15. State-entry problem Input: • Turing Machine • State • String Question: Does enter state while processing input string ? Corresponding language: Prof. Busch - LSU

  16. Theorem: is undecidable (state-entry problem is unsolvable) Proof: Reduce (halting problem) to (state-entry problem) Prof. Busch - LSU

  17. Halting Problem Decider Decider for state-entry problem decider Reduction YES YES Compute Decider NO NO Given the reduction, if is decidable, then is decidable A contradiction! since is undecidable Prof. Busch - LSU

  18. We only need to build the reduction: Reduction Compute So that: Prof. Busch - LSU

  19. Construct from : special halt state halting states A transition for every unused tape symbol of Prof. Busch - LSU

  20. special halt state halting states halts halts on state Prof. Busch - LSU

  21. Therefore: halts on input halts on state on input Equivalently: END OF PROOF Prof. Busch - LSU

  22. Blank-tape halting problem Input: Turing Machine Question: Does halt when started with a blank tape? Corresponding language: Prof. Busch - LSU

  23. Theorem: is undecidable (blank-tape halting problem is unsolvable) Proof: Reduce (halting problem) to (blank-tape problem) Prof. Busch - LSU

  24. Halting Problem Decider Decider for blank-tape problem decider Reduction YES YES Compute Decider NO NO Given the reduction, If is decidable, then is decidable A contradiction! since is undecidable Prof. Busch - LSU

  25. We only need to build the reduction: Reduction Compute So that: Prof. Busch - LSU

  26. Construct from : Accept and halt no Tape is blank? yes Run Write on tape with input If halts then halt Prof. Busch - LSU

  27. Accept and halt no Tape is blank? yes Run Write on tape with input halts on input halts when started on blank tape Prof. Busch - LSU

  28. halts on input halts when started on blank tape Equivalently: END OF PROOF Prof. Busch - LSU

  29. Theorem (version 2): If: a: Language is reduced to b: Language is undecidable Then: is undecidable Proof: Suppose is decidable Then is decidable Using the decider for build the decider for Contradiction! Prof. Busch - LSU

  30. Suppose is decidable reject Decider for (halt) accept (halt) Prof. Busch - LSU

  31. Suppose is decidable Then is decidable (we have proven this in previous class) Decider for NO YES reject accept Decider for (halt) (halt) YES NO accept reject (halt) (halt) Prof. Busch - LSU

  32. If is decidable then we can build: Decider for Reduction YES Input string YES accept accept compute Decider for (halt) (halt) NO NO reject reject (halt) (halt) CONTRADICTION! Prof. Busch - LSU

  33. Alternatively: Decider for Reduction NO Input string YES reject accept compute Decider for (halt) (halt) YES NO accept reject (halt) (halt) CONTRADICTION! END OF PROOF Prof. Busch - LSU

  34. Observation: In order to prove that some language is undecidable we only need to reduce some known undecidable language to or to (theorem version 1) (theorem version 2) Prof. Busch - LSU

  35. Undecidable Problems for Turing Recognizable languages Let be a Turing-acceptable language • is empty? • is regular? • has size 2? All these are undecidable problems Prof. Busch - LSU

  36. Let be a Turing-acceptable language • is empty? • is regular? • has size 2? Prof. Busch - LSU

  37. Empty language problem Input: Turing Machine Question: Is empty? Corresponding language: Prof. Busch - LSU

  38. Theorem: is undecidable (empty-language problem is unsolvable) Proof: Reduce (membership problem) to (empty language problem) Prof. Busch - LSU

  39. membership problem decider Decider for empty problem decider Reduction YES YES Compute Decider NO NO Given the reduction, if is decidable, then is decidable A contradiction! since is undecidable Prof. Busch - LSU

  40. We only need to build the reduction: Reduction Compute So that: Prof. Busch - LSU

  41. Construct from : Tape of input string Turing Machine Accept yes yes • Write on tape, and accepts ? • Simulate on input Prof. Busch - LSU

  42. The only possible accepted string Louisiana Turing Machine Accept yes yes • Write on tape, and accepts ? • Simulate on input Prof. Busch - LSU Prof. Busch - LSU Prof. Busch - LSU 42 42

  43. accepts does not accept Turing Machine Accept yes yes • Write on tape, and accepts ? • Simulate on input Prof. Busch - LSU Prof. Busch - LSU Prof. Busch - LSU Prof. Busch - LSU 43 43 43

  44. Therefore: accepts Equivalently: END OF PROOF Prof. Busch - LSU

  45. Let be a Turing-acceptable language • is empty? • is regular? • has size 2? Prof. Busch - LSU

  46. Regular language problem Input: Turing Machine Question: Is a regular language? Corresponding language: Prof. Busch - LSU

  47. Theorem: is undecidable (regular language problem is unsolvable) Proof: Reduce (membership problem) to (regular language problem) Prof. Busch - LSU

  48. membership problem decider Decider for regular problem decider Reduction YES YES Compute Decider NO NO Given the reduction, If is decidable, then is decidable A contradiction! since is undecidable Prof. Busch - LSU

  49. We only need to build the reduction: Reduction Compute So that: Prof. Busch - LSU

  50. Construct from : Tape of input string Turing Machine Accept yes yes • Write on tape, and accepts ? • Simulate on input Prof. Busch - LSU Prof. Busch - LSU Prof. Busch - LSU Prof. Busch - LSU Prof. Busch - LSU 50 50 50 50

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