| Rice's Theorem |
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Information AboutRice's Theorem |
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Another way of stating this problem that is more useful in ''S''. Then the problem of deciding whether the language of a given Turing Machine is in ''S'' is Undecidable , provided that there exists a Turing machine that recognizes a language in ''S'' and a Turing machine that recognizes a language not in ''S''. Effectively this means that there is no machine that can always correctly decide whether the language of a given Turing machine has a particular nontrivial property. Special cases include the undecidability of whether a Turing machine accepts a particular string, whether a Turing machine recognizes a particular recognizable language, and whether the language recognized by a Turing machine could be recognized by a nontrivial simpler machine, such as a Finite Automaton . It is important to note that Rice's theorem does not say anything about properties of ''machines'', only of ''functions'' and ''languages''. For example, asking whether a machine runs for more than 100 steps on some input, whether it has more than 5 states, or whether it ever moves its tape head to the left are all nontrivial but decidable properties. They are not properties of the language because it is possible to find two machines recognizing exactly the same language, one which has the property and one which does not. Using . As an example, consider the following variant of the Halting Problem : Take the property a partial function F has if F is defined for argument 1. It is obviously non-trivial, since there are partial functions that are defined for 1 and others that are undefined at 1. The ''1-halting problem'' is the problem of deciding of any algorithm whether it defines a function with this property, i.e., whether the algorithm halts on input 1. By Rice's theorem, the 1-halting problem is undecidable. FORMAL STATEMENT Given a Gödel Numbering : of the Computable Function s then for any property : is recursive if and only if or . EXAMPLES |
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