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20 - 20 Undecidable problems Reduction is the primary...

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20. Undecidable problems Reduction is the primary method for proving that a problem is computationally undecidable. Reducing a problem A to problem B means a solution for problem B can be used to solve problem A. Algorithm A Algorithm B To prove that a problem B is undecidable, we first assume on the contrary that B is decidable and show that, by making use of an algorithm for B (a black box), we would then be able to design an algorithm to solve another problem A that is already known to be undecidable, thus obtaining a contradiction. Your task is to design an algorithm for A : based on the input to problem A , decide what should be the input to the algorithm for B , and make use of the output of algorithm for B to decide what should be the output to problem A . 1
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Theorem 1 The following problems are all undecidable: 1. Given a Turing Machine M , does M halt on the empty tape? (i.e., e L ( M ) ?) 2. Given a Turing Machine M , is there any string at all upon which M halts? (i.e., L ( M ) = ?) 3. Given a Turing Machine M , does M halt on every input string? (i.e., L ( M ) = Σ ?) 4. Given two Turing Machines M 1 and M 2 , do they halt on exactly the same input strings? (i.e., L ( M 1 ) = L ( M 2 ) ?) 5. Given a Turing Machine M , is L ( M ) regular? Try this yourself. Will discuss this in next lecture. 2
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Theorem 2 K 1 = { M ” : Turing machine M halts on an empty tape } is not recursive.
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