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| {{lowercase|title=utm theorem}}
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| In [[computability theory]] the '''utm theorem''', or '''[[Universal Turing machine]] theorem''', is a basic result about [[Gödel numbering]]s of the set of [[computable function]]s. It affirms the existence of a computable '''universal function''' which is capable of calculating any other computable function. The universal function is an abstract version of the [[universal turing machine]], thus the name of the theorem.
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| [[Roger's equivalence theorem|Rogers equivalence theorem]] provides a characterization of the Gödel numbering of the computable functions in terms of the [[smn theorem]] and the utm theorem.
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| == utm theorem ==
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| Let <math>\varphi_1, \varphi_2, \varphi_3, ...</math> be an enumeration of Gödel numbers of computable functions. Then the partial function
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| :<math>u: \mathbb{N}^2 \to \mathbb{N}</math>
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| defined as
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| :<math>u(i,x) := \varphi_i(x) \qquad i,x \in \mathbb{N}</math>
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| is computable.
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| <math>u</math> is called the '''universal function'''.
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| == References ==
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| *{{cite book | author = Rogers, H. | title = The Theory of Recursive Functions and Effective Computability | publisher = First MIT press paperback edition | year = 1987 | origyear = 1967 | isbn = 0-262-68052-1 }}
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| *{{cite book | author = Soare, R.| title = Recursively enumerable sets and degrees | series = Perspectives in Mathematical Logic | publisher = Springer-Verlag | year = 1987 | isbn = 3-540-15299-7 }}
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| [[Category:Theory of computation]]
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| [[Category:Computability theory]]
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Revision as of 11:02, 19 February 2014
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