51+PS9 - Handout #51 November 17, 2010 CS103 Robert Plummer...

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Handout #51 CS103 November 17, 2010 Robert Plummer Problem Set #9—Due Friday , December 3 in class No late days may be used for this assignment 1. Suppose that B is a decidable language, that w 1 B, and that w 2 B. Prove that a language A is decidable if and only if A m B. The parts of your proof will thus be: (a) Prove that if A m B, then A is decidable. (b) Prove that if A is decidable, then A m B. In this part of the proof, you need to specify a mapping. Do this in the style of Example 5.24; that is, specify a Turing machine that does the mapping, in Sipser's style: "F = "On input . ..". Then you must argue why your mapping is correct; that is, why it meets the requirements necessary for mapping reductions. 2. We have seen that some languages are neither Turing recognizable nor co-Turing recognizable. In this problem, we consider another such language, defined as follows: L = { w | either w = 0x for some x A TM , or w = 1y for some y A TM } Show that neither L nor L is Turing recognizable. 3. Prove or disprove: there exists an undecidable language A such that A
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51+PS9 - Handout #51 November 17, 2010 CS103 Robert Plummer...

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