39.pdf - CMPSCI 250 Introduction to Computation Lecture#39...

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CMPSCI 250: Introduction to Computation Lecture #39: The Halting Problem and Unsolvability David Mix Barrington 11 December 2017
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Halting and Unsolvability Simulating an NDTM With a DTM Proving Something to be Impossible Representing TM’s By Strings The Universal Turing Machine The Barber of Seville Language Undecidable, Non-Recognizable Languages Getting More Undecidable Languages Turing Complete Languages
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Simulating NDTM’s with DTM’s To simulate an NDTM with a DTM, we first build a DTM with three tapes. The first tape will store the input string w and will never change. The second will be a work tape to exactly simulate a particular computation of the NDTM. The third tape will hold a choice sequence , which is a string of symbols telling the NDTM which of its options to take on each of its moves.
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Simulating NDTM’s with DTM’s The input w is in the language of the NDTM N if and only if there exists a choice sequence that causes N to halt, starting from i w. So our simulation tests all possible choice sequences, starting with the one of length 0, then all the ones of length 1, then length 2, and so forth. For each choice sequence, the DTM clears its work tape, copies w onto it, then runs N using the sequence. If the simulated N ever halts, the DTM accepts w.
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Simulating NDTM’s with DTM’s So if w L(N), the DTM will eventually reach a good choice sequence and will accept w. If w L(N), the DTM will run forever because it will keep trying longer and longer choice sequences. Thus the DTM accepts exactly those strings in L(N), and so simulates N. If the DTM accepts, it takes exponentially longer to do so than N did on that input.
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Proving Things to Be Impossible When a problem can be solved with a particular set of resources, we can prove this to be the case by showing how to do it. But what about when a problem can’t be solved with those resources? We can’t just show algorithms that don’t work, because these don’t rule out the existence of other algorithms that do.
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Proving Things to Be Impossible We have one example in this course -- if a language cannot be decided by a DFA, the Myhill-Nerode Theorem can be used to prove it. This also shows that the language has no regular expression. Gödel proved in 1931 that there is a true statement of number theory that can’t be proved (or a false statement that can be proved). The idea is that the statement can be interpreted as “I am not provable”.
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Clicker Question #1 Suppose that Statement n means “there is no proof of Statement n in the system”. Which one of these statements could not be true?
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  • Fall '09
  • Turing, Turing Complete Languages, Seville Language

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