UTM theorem

In computability theory, the Template:Abbr 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.Template:Sfn 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.

Theorem

The theorem states that a partial computable function u of two variables exists such that, for every computable function f of one variable, a number e exists such that $f(x)\simeq u(e,x)$ 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.Template:Sfn

The theorem thus shows that, defining φ_e(x) as u(e, x), the sequence φ₁, φ₂, ... is an enumeration (numbering) of the partial computable functions. In fact, u can be chosen so this is the standard numbering. The function $u$ in the statement of the theorem is called a universal function.

References

• Template:Cite book
• Template:Cite book