Turing-Enumerable (Part II)

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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
Homework for Monday, Nov. 29. Construct the two
Turing machines Such that ? gets
enumerated ?a?b?aa?ab?ba?bb?
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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