hw11 - Computer Science 340 Reasoning about Computation...

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Computer Science 340 Reasoning about Computation Homework 11 Due on Friday, December 14, 2007 For this homework, read Section 5.3 on Mapping Reducibility in the notes distributed in class. Problem 1 Recall that E TM = { M | M is a TM and L ( M ) = ∅} . Prove that E TM is Turing-recognizable. Problem 2 Suppose that language A is mapping reducible to language B . (See definition 5.15 for an explanation of what this means.) Show that if A is not Turing-recognizable then B is not Turing-recognizable. Conclude that there is no mapping reduction from A TM to E TM . Recall that A TM = { M, w | M is a TM and M accepts on input w } . Problem 3 Let J = { w | w = 0 x for some x A
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Unformatted text preview: A TM or w = 1 y for some y ∈ A TM } . In other words, J is the union of all strings in A TM with a 0 appended in front of them, and all strings in A TM with a 1 appended in front of them. Show that neither J nor J is Turing-recognizable. Problem 4 Show that the Post Correspondence Problem (PCP) over a binary alphabet (i.e. over the alphabet Σ = { , 1 } ) is undecidable. Hint: Give a reduction from PCP over an arbitrary alphabet to PCP over a binary alphabet. You may use the fact that PCP over an arbitrary alphabet is undecidable....
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  • Fall '07
  • CharikarandChazelle
  • Computer Science, Logic, Computational complexity theory, Turing reduction, Many-one reduction, Log-space reduction

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