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 proves 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 $\backslash varphi$ be a Gödel numbering of the set of computable functions, then the function

$u\_\backslash varphi:\; \backslash subseteq\; \backslash mathbb\{N\}^2\; \backslash to\; \backslash mathbb\{N\}$

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

$u\_\backslash varphi$ is called the universal function for $\backslash varphi$

Theory of computation | Recursion theory