UTM theorem

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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.^[1] The universal function is an abstract version of the universal Turing machine, thus the name of the theorem.

Roger's equivalence theorem provides a characterization of the Gödel numbering of the computable functions in terms of the s_mn theorem and the UTM theorem.

The theorem states that a partial computable function u of two variables exists such that, for every computable function f of one variable, an e exists such that ${\displaystyle f(x)\simeq u(e,x)}$ for all x. This means that, for each x, either f(x) and u(e,x) are both defined and are equal, or are both undefined.^[2]

The theorem thus shows that, defining φ_e(x) as u(e, x), the sequence φ₁, φ₂, ... is an enumeration of the partial computable functions. The function ${\displaystyle u}$ in the statement of the theorem is called a universal function.

• Rogers, H. (1987) [1967]. The Theory of Recursive Functions and Effective Computability. First MIT press paperback edition. ISBN 0-262-68052-1.
• Soare, R. (1987). Recursively enumerable sets and degrees. Perspectives in Mathematical Logic. Springer-Verlag. ISBN 3-540-15299-7.

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