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Unformatted text preview: Solutions for Problem Set 7 CS 373: Theory of Computation Assigned: October 19, 2010 Due on: October 26, 2010 at 10am Homework Problems Problem 1 . [Category: Proof] Solve problem 3.19. Hint: Use the result of problem 3.18, which was solved in discussion 9. Solution: Let L be an infinite recursively enumerable language. Since L is recursively enumerable, L has an enumerator M such that E ( M ) = L . Problem 3.18 (and the discussion section problem) provide a characterization of decidable languages in terms of enumeration: L 1 is decidable iff there is an enumerator M 1 that enumerates the strings of L 1 in lexicographic order. We will use this result to identify an infinite, decidable subset of L . Consider the following enumerator M 1 last-string = Run M Whenever M outputs a string (say) w if ((last-string = ) or ( w > last-string) then output w last-string = w In the above algorithm, the check w > last-string means that w is after last-string in the lexicographic ordering. Observe that, by construction, any string output by M...
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