Finitary reducibility on equivalence relations

TL;DR

This paper introduces finitary computable reducibility for equivalence relations, demonstrating it differs from traditional reducibility and allows for higher complexity classifications, thus refining the understanding of equivalence relation hierarchies.

Contribution

It defines a new, weaker form of reducibility, shows it can classify higher complexity equivalence relations, and establishes that the hierarchy of finitary reducibilities does not collapse.

Findings

Existence of $oldsymbol{oldsymbol{oldsymbol{oldsymbol{ ext{Pi}}_{n+2}}}}$-complete relations under finitary reducibility

Hierarchy of finitary reducibilities is non-collapsing

New results on complexity of natural equivalence relations in the arithmetical hierarchy

Abstract

We introduce the notion of finitary computable reducibility on equivalence relations on the natural numbers. This is a weakening of the usual notion of computable reducibility, and we show it to be distinct in several ways. In particular, whereas no equivalence relation can be $\Pi_{n+2}$-complete under computable reducibility, we show that, for every $n$, there does exist a natural equivalence relation which is $\Pi_{n+2}$-complete under finitary reducibility. We also show that our hierarchy of finitary reducibilities does not collapse, and illustrate how it sharpens certain known results. Along the way, we present several new results which use computable reducibility to establish the complexity of various naturally defined equivalence relations in the arithmetical hierarchy.