Computable isomorphism

In computability theory two sets $A,B$ of natural numbers are computably isomorphic or recursively isomorphic if there exists a total computable and bijective function $f\colon \mathbb {N} \to \mathbb {N}$ such that the image of $f$ restricted to $A\subseteq \mathbb {N}$ equals $B\subseteq \mathbb {N}$, i.e. $f(A)=B$.

Further, two numberings $\nu$ and $\mu$ (of the same set of objects) are called computably isomorphic if there exists a computable bijection $f$ so that $\nu =\mu \circ f$. Computably isomorphic numberings induce the same notion of computability on a set.

Theorems

By the Myhill isomorphism theorem, the relation of computably isomorphic coincides with the relation of mutual one-one reducibility.

References

• Rogers, Hartley Jr. (1987), Theory of recursive functions and effective computability (2nd ed.), Cambridge, MA: MIT Press, ISBN 0-262-68052-1, MR 0886890.