Every computable set is generically reducible to every computable set that does not have density 0 or 1

TL;DR

This paper proves that all non-trivially dense computable sets are mutually generically reducible, and every computable set can be generically reduced to such sets, offering a complete classification of computable sets under this reducibility.

Contribution

It improves previous results by showing all non-trivially dense computable sets are equivalent under generic reducibility and that every computable set reduces to these, completing the classification.

Findings

All non-trivially dense computable sets are generically equivalent.

Every computable set reduces to any non-trivially dense computable set.

Provides a complete classification of computable sets under generic reducibility.

Abstract

The notion of generic reducibility was introduced by A.Rybalov in his CiE 2018 paper: a set A is generically reducible to set B if there exists a total computable function f that m-reduces A to B such that the f-preimage of every set that has density 0 has density 0. It may be considered as the ``generic version'' of the notion of m-reducibility. In this note we improve one of his results and show that every two computable sets that do not have density 0 or 1 are equivalent with respect to generic reducibility, and that every computable set is reducible to every computable set that does not have density 0 or 1, thus providing a complete classification of computable sets with respect to generic reducibility.