hd3_3 - SELECTED SOLUTIONS 163 3.18 Prove that there exists...

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SELECTED SOLUTIONS 163 3.18 Prove that there exists an infinite decidable subset of A TM . 3.19 Show that a language is decidable iff some enumerator enumerates the language in lexicographic order. 3.20 Show that single-tape TM s that cannot write on the portion of the tape containing the input string recognize only regular languages. 3.21 Let c 1 x n + c 2 x n - 1 + · · · + c n x + c n +1 be a polynomial with a root at x = x 0 . Let c max be the largest absolute value of a c i . Show that | x 0 | < ( n + 1) c max | c 1 | . A 3.22 Let A be the language containing only the single string s , where s = 0 if life never will be found on Mars. 1 if life will be found on Mars someday . Is A decidable? Why or why not? For the purposes of this problem, assume that the question of whether life will be found on Mars has an unambiguous Y ES or N O answer. SELECTED SOLUTIONS 3.1 (b) q 1 00 , q 2 0 , x q 3 , q 5 x , q 5 x , q 2 x , x q 2 , x q accept 3.2 (a) q 1 11 , q 3 1 , 1 q 3 , 1 q reject . 3.3 We prove both directions of the “iff.” First, if a language L is decidable, it can be decided by a deterministic Turing machine, and that is automatically a nondeter- ministic Turing machine. Second, if a language L is decided by a nondeterministic TM N , we construct a deterministic TM D 2 that decides L . Machine D 2 runs the same algorithm that appears in the TM D described in the proof of Theorem 3.3, with an additional
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