Myhill isomorphism theorem

For the Goodman–Myhill theorem in constructive set theory, see Diaconescu's theorem.

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.

Myhill isomorphism theorem

Sets A and B of natural numbers are said to be recursively isomorphic if there is a total computable bijection f from the set of natural numbers to itself such that f(A) = B.^{[further explanation needed]}

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 ${\displaystyle f(A)\subseteq B}$  and ${\displaystyle f(\mathbb {N} \setminus A)\subseteq \mathbb {N} \setminus B}$ .

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.

The theorem is reminiscent of the Schroeder–Bernstein theorem. The proof is different, however. The proof of Schroeder–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 Schroeder–Bernstein unless one uses the Axiom of Choice (which is not necessary for the proof of the Myhill theorem).

A corollary of Myhill's theorem is that two total numberings are one-equivalent if and only if they are recursively isomorphic.

See also

• Berman–Hartmanis conjecture, an analogous statement in computational complexity theory

References

• Myhill, John (1955), "Creative sets", Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 1: 97–108, doi:10.1002/malq.19550010205, MR 0071379.
• Rogers, Hartley, Jr. (1987), Theory of recursive functions and effective computability (2nd ed.), Cambridge, MA: MIT Press, ISBN 0-262-68052-1, MR 0886890.
• 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