In computability theory the Myhill isomorphism theorem, named after John Myhill, provides a characterization for two numberings to induce the same notion of computability on a set.
Theorem
Definitions
Sets A and B of natural numbers are said to be recursively isomorphic if there is a total computable bijective function f on the natural numbers such that for any , .
A set A of natural numbers is said to be one-one reducible to a set B if there is a total computable injective function f on the natural numbers such that and .
Stetement
Myhill's isomorphism theorem states that two sets A and B of natural numbers are recursively isomorphic if and only if A is one-reducible to B and B is one-reducible to A.
Corollaries
Two total numberings are one-equivalent if and only if they are recursively isomorphic.
Discussion
The theorem implies that given two injective reductions in opposing directions, there is a computable bijection on the naturals that puts the sets in question in bijective correspondence. This is reminiscent of the Schröder–Bernstein theorem about general sets, and Myhill's theorem has been called a constructive version of it. Their proofs are however different. The proof of Schröder-Bernstein uses the inverses of the two injections, which is impossible in the setting of the Myhill theorem since these inverses might not be recursive. The proof of the Myhill theorem, on the other hand, defines the bijection inductively, which is impossible in the setting of Schröder-Bernstein unless one uses the Axiom of Choice (which is not necessary for the proof of the Myhill theorem).
See also
Berman–Hartmanis conjecture, an analogous statement in computational complexity theory
References
. . Soare, Robert I. (1987), Recursively enumerable sets and degrees : a study of computable functions and computably generated sets, Perspectives in Mathematical Logic, Berlin Heidelberg : Springer-Verlag, ISBN 978-3-540-66681-3, MR 0882921