62 PS9 Solutions

# 62 PS9 Solutions - Handout#62 June 2 2010 CS103 Robert...

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Handout #62 CS103 June 2, 2010 Robert Plummer Problem Set #9 Solutions All problems are from Sipser. 1. Problem 5.24, p. 212. To prove that J is not Turing-recognizable, we show that A TM m J. The reduction maps any string y to the string 1y. Then y A TM 1y J. To prove that J is not Turing-recognizable, we show that A TM m J. (Note that A TM m J A TM m J .) The reduction maps any string x to the string 0x. Then x A TM 0x J. 2. Problem 5.25, p. 212. Hint: use Problem 5.24. Use the language J from Problem 5.24. That problem showed that J is undecidable. Here we show that J m J . The reduction maps to itself and for other strings x, 0s if x = 1s for some string s * 1s if x = 0s for some string s * 3. Problem 5.33, p. 213. REMOVED 4. Exercise 7.7, p. 295. NP is closed under union. For any two NP languages L 1 and L 2 , let M 1 and M 2 be the NTMs that decide them in polynomial time. We construct an NTM M' that decides L 1 L 2 in polynomial time: M' = "On input <w>: 1. Run M 1 on w. If it accepts, accept. 2. Run M 2 on w. If it accepts, accept. Otherwise, reject." In both stages 1 and 2, M' uses its nondeterminism when the machines being run make nondeterministic steps. M' accepts w if and only if either M

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62 PS9 Solutions - Handout#62 June 2 2010 CS103 Robert...

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