hw6sol - CS 181 Winter 2008 Formal Languages and Automata...

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CS 181 — Winter 2008 Formal Languages and Automata Theory Problem Set #6 Solutions Problem 6.1. (10 points) Show that a language for which there exists an enumerator that enumer- ates the language in increasing lexicographical order is a decidable language. [For this, give a high level description of a Turing machine that halts on all inputs and accepts exactly the language.] Solution Let L be a language and E an enumerator that lists the words of L in increasing lexicographical order (i.e. by length, breaking ties lexicographically). We will show how build a Turing machine M that decides L . Given a word w , we can check to see if w L by comparing w with each of the words enumerated by E in order. If w L then eventually we’ll find a match. However, if w / L , then we may continue forever. Yet we can take advantage of the fact that E enumerates the words in increasing length. In particular, we can check w against the enumerated words until the length of the enumerated words becomes larger than that of w . If this is the case, then we know w will never appear in E ’s list and so must not be in L . Then our Turing machine M decides L by enumerating words (using E ) one at a time and comparing against w . If w is found, then it accepts. If
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  • Spring '08
  • griebach
  • #, Halting problem, #w2, #wi

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