Home13MoreReduction - CS 341 Automata Theory Elaine Rich...

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CS 341 Automata Theory Elaine Rich Homework 13 Due Friday, Dec. 8 at 11:59 pm 1) Let L 1 and L 2 be two languages such that L 1 L 2 . Prove that: a) If L 1 is not in D , then L 2 is not in D . b) If L 1 is not in SD , then L 2 is not in SD . Hint: These are not hard. You just need to state clearly the argument given in class. 2) For each of the following languages, state whether or not it is in D , SD / D ( SD but not D ), or neither. Prove your answer. Do not use Rice’s Theorem. If you claim that L is not in SD , first prove that it’s not in D (for practice), then prove that it’s not in SD . a) L = {< M > : L ( M ) contains at least two strings} b) L = {< M > : | L ( M )| > 0 and is prime} c) L = {< M > : M accepts the string < M , M > and does not accept the string < M >} d) L = {< M > : there exists a string w , | w | < 100, such that M accepts w } e) L = {< M > : M does not accept any string that ends with 0} f) L = {< M > : there are at least two strings w and x such that M halts on w and x in some number of steps s and s < 1000 and s is prime} 3) Let L 1 , L 2 , …, L k be a collection of languages over some alphabet Σ such that: For all i j , L i L j = . L 1 L 2 L k = Σ *. 2200 i , L i is in SD . Prove that each of the languages L 1 through L k is in D .
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CS 341 Automata Theory Elaine Rich Homework 13 Answers 1) Let L 1 and L 2 be two languages such that L 1 L 2 . Prove that: a) If L 1 is not in D , then L 2 is not in D . Let R be the TM that reduces L 1 to L 2 . If L 2 is in D, then there exists a Turing machine M 2 that decides it. But then a composition of M 2 with R decides L 1. But, if L 1 is not in D, such a machine cannot exist. Contradiction. So L 2 must not be in D. b) If L 1 is not in SD , then L 2 is not in SD . Let R be the Turing machine that reduces L 1 to L 2 . If L 2 is in SD , then there exists a Turing machine M 2 that semidecides it. Then the composition of M 2 with R semidecides L 1 . But, if L 1 is not in SD , such a machine cannot exist. Contradiction. So L 2 must not be in SD . 2) For each of the following languages, state whether or not it is in D, SD/D (SD but not D), or neither. Prove your answer. Do not use Rice’s Theorem. If you claim that L is not in SD, first prove that it’s not in D (for practice), then prove that it’s not in SD. a)
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Home13MoreReduction - CS 341 Automata Theory Elaine Rich...

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