UTM theorem explained

In computability theory, the 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.

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, an 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.

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

u in the statement of the theorem is called a universal function.

References

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