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Revision as of 12:46, 1 January 2014

Template:Lowercase In computability theory the utm theorem, or Universal Turing machine theorem, is a basic result about Gödel numberings of the set of computable functions. 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.

Rogers equivalence theorem provides a characterization of the Gödel numbering of the computable functions in terms of the smn theorem and the utm theorem.

utm theorem

Let $\varphi _{1},\varphi _{2},\varphi _{3},...$ be an enumeration of Gödel numbers of computable functions. Then the partial function

u:\mathbb {N} ^{2}\to \mathbb {N}

defined as

u(i,x):=\varphi _{i}(x)\qquad i,x\in \mathbb {N}

is computable.

$u$ is called the universal function.

References

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+{{lowercase|title=utm theorem}}
+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.
+[[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.
+== utm theorem ==
+Let <math>\varphi_1, \varphi_2, \varphi_3, ...</math> be an enumeration of Gödel numbers of computable functions. Then the partial function
+:<math>u: \mathbb{N}^2 \to \mathbb{N}</math>
+defined as
+:<math>u(i,x) := \varphi_i(x) \qquad i,x \in \mathbb{N}</math>
+is computable.
+<math>u</math> is called the '''universal function'''.
+== References ==
+*{{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 }}
+*{{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 }}
+[[Category:Theory of computation]]
+[[Category:Computability theory]]

Fresnel diffraction: Difference between revisions

Revision as of 12:46, 1 January 2014

utm theorem

References

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