Myhill isomorphism 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.

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

Myhill isomorphism theorem

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 , .[1]

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 .

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, and has been called a constructive version of it.[2] 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

References
  1. P. Odifreddi, Classical Recursion Theory: The theory of functions and sets of natural numbers (pp.324--325). Studies in Logic and the Foundations of Mathematics, vol. 125 (1989), Elsevier 0-444-87295-7.
  2. P. Odifreddi, Classical Recursion Theory: The theory of functions and sets of natural numbers (p.320). Studies in Logic and the Foundations of Mathematics, vol. 125 (1989), Elsevier 0-444-87295-7.
  • 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
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.