Bounded Reducibility for Computable Numberings

Abstract

The theory of numberings gives a fruitful approach to studying uniform computations for various families of mathematical objects. The algorithmic complexity of numberings is usually classified via the reducibility $\le$ between numberings. This reducibility gives rise to an upper semilattice of degrees, which is often called the Rogers semilattice. For a computable family S of c.e. sets, its Rogers semilattice R(S) contains the $\le$-degrees of computable numberings of S. Khutoretskii proved that R(S) is always either one-element, or infinite. Selivanov proved that an infinite R(S) cannot be a lattice.

We introduce a bounded version of reducibility between numberings, denoted by $\le _{bm}$. We show that Rogers semilattices $R_{bm}(S)$, induced by $\le _{bm}$, exhibit a striking difference from the classical case. We prove that the results of Khutoretskii and Selivanov cannot be extended to our setting: For any natural number $n\ge 2$, there is a finite family S of c.e. sets such that its semilattice $R_{bm}(S)$ has precisely $2^n-1$ elements. Furthermore, there is a computable family T of c.e. sets such that $R_{bm}(T)$ is an infinite lattice.

The work was supported by Nazarbayev University Faculty Development Competitive Research Grants N090118FD5342. The first author was partially supported by the grant of the President of the Russian Federation (No. MK-1214.2019.1). The third author was partially supported by the program of fundamental scientific researches of the SB RAS No. I.1.1, project No. 0314-2019-0002.