Effective categoricity of equivalence Structures

TL;DR

This paper characterizes the levels of computable categoricity for equivalence structures, providing exact results for certain levels and discussing open cases for others.

Contribution

It offers a precise classification of $ ext{Delta}^0_eta$ categoricity for equivalence structures at various levels, including new results for $eta=1, 2,$ and $eta eq 2$.

Findings

Exact characterization for $eta=1$ and $eta eq 2,3$

Extensive results and open cases for $eta=2$

Complete description of open cases in the classification

Abstract

An equivalence structure is a set with a single binary relation, satisfying sentences stating that the relation is an equivalence relation. A computable structure A is said to be $\Delta^0_\alpha$ categorical if for any computable structure B isomorphic to A there is a $\Delta^0_\alpha$ function witnessing that the two are isomorphic. The present paper gives an exact characterization of $\Delta^0_\alpha$ equivalence structures where $\alpha = 1$ or $\alpha \geq 3$. Extensive results for $\alpha = 2$ are also given, and open cases are exhaustively described.