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Turing-Enumerable (Part II)
Héctor Muñoz-Avila
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Enumerability and Turing Machines
Definition: A language L is Turing-enumerable if there is a Turing machine that enumerates all words in L in its tape (and will run forever if L is infinite): w1 w2 w3… Theorem 1. If a language L is decidable then L is Turing-enumerable Theorem 2. If a language L is Turing-enumerable then L is semi-decidable
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* is Enumerable Lemma. If is finite then * is enumerable 2
2 1 a1 a2 1 2 2 Increase word in second tape copy word into first tape Homework for Monday, Nov Construct the two Turing machines Such that * gets enumerated: abaaabbabb…
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Decidable Implies Enumerable
Let L be a language over the alphabet If L is decidable there is a Turing Machine that decides M Let * be the Turing machine that enumerates * Construct a 3-tape Turing machine that enumerates L as follows: M* will output words on the 2nd tape Every time M* outputs a word w, it is copied in the 3rd tape, where M checks if w is in L If w is in L, then w is appended to the end of the 1st tape
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Enumerable Implies Semi-decidable
Let L be a Turing-enumerable language Thus, there is a Turing machine M that enumerates words in L We can construct a 3-tape Turing machine ML that semi-decides if a word w is in L as follows: Put w in the first tape Run M enumerating all elements in the 2nd tape: Every time a new word w’ is added to the second tape, we copy it to the 3rd tape If w = w’ then halt otherwise continue with Step 1
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