The hierarchy of equivalence relations on the natural numbers under computable reducibility

TL;DR

This paper explores the hierarchy of computable reducibility among equivalence relations on natural numbers, comparing it with Borel reducibility and applying it to computably enumerable structures like graphs and groups.

Contribution

It extends the study of computable reducibility hierarchies, contrasting them with Borel hierarchies, and applies these concepts to classify computably enumerable structures.

Findings

Comparison of computable and Borel reducibility hierarchies

Application to equivalence relations on c.e. sets and structures

Identification of open questions in the field

Abstract

The notion of computable reducibility between equivalence relations on the natural numbers provides a natural computable analogue of Borel reducibility. We investigate the computable reducibility hierarchy, comparing and contrasting it with the Borel reducibility hierarchy from descriptive set theory. Meanwhile, the notion of computable reducibility appears well suited for an analysis of equivalence relations on the c.e.\ sets, and more specifically, on various classes of c.e.\ structures. This is a rich context with many natural examples, such as the isomorphism relation on c.e.\ graphs or on computably presented groups. Here, our exposition extends earlier work in the literature concerning the classification of computable structures. An abundance of open questions remains.