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Rice ’ s Theorem

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Def: A property of the Turing-recognizable languages is simply a subset of all Turing- recognizable languages.

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A property is trivial if it is either empty or is all Turing-recognizable languages. Otherwise, it is nontrivial. Note that the empty property is different from the property { }.

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Thm: Every nontrivial property of the Turing- recognizable languages is undecidable.

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Pf: Let C be a nontrivial property. Assume , otherwise we can consider. Since C is nontrivial, there exists a non- empty language L C.

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Letbe a TM accepting (or recognizing) L. Given, construct the following TM : M w Accept Start x

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Thus, if we could decide or not,then we could decide. Therefore, C is undecidable.

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Eg. The following are undecidable: 1. 2. 3. 4.

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Eg. How about the following: 1. 2.

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