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decid-2 - We mentioned earlier that another term that is...

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We mentioned earlier that another term that is used for decidable is recursive , and another term that is used for semi-decidable is recursively enumerable. We now explain the enumerable part of recursively enumerable. Definition 2.4. Let L be an infinite language on an alphabet Σ. An enumeration algorithm for L is a a computer program, M , (written in your favorite language) which runs on an idealized computer and does the following: M takes no input. M never halts. As it runs M is progressively outputting a list of strings on Σ. For all strings w Σ * , w L iff w is one of the strings which is eventually output from M . Definition 2.5. A language L is algorithmically enumerable iff there exists an enumeration algo- rithm for L . Lemma 2.6. Let L be an infinite language. Then L is semi-decidable iff L is algorithmically enumerable. Proof. First assume that L is algorithmically enumerable. Let M be an enumeration algorithm for L . We will now describe an algorithm M 1 which is a semi-acceptor for L : (1) input(w); (2) start running M ; (3) if M ever outputs w then accept an halt; If w L then M will eventually output w and so M 1 will eventually accept w and halt. If w / L then M will never output w and so M 1 will never halt. Thus M 1 is a semi-acceptor for L .
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