A note on decidable categoricity and index sets

Abstract

A structure S is decidably categorical if S has a decidable copy, and for any decidable copies A and B of S, there is a computable isomorphism from A onto B. Goncharov and Marchuk proved that the index set of decidably categorical graphs is $\Sigma^0_{\omega+2}$ complete. In this paper, we isolate two familiar classes of structures $K$ such that the index set for decidably categorical members of K has a relatively low complexity in the arithmetical hierarchy. We prove that the index set of decidably categorical real closed fields is $\Sigma^0_3$ complete. We obtain a complete characterization of decidably categorical equivalence structures. We prove that decidably presentable equivalence structures have a $\Sigma^0_4$ complete index set. A similar result is obtained for decidably categorical equivalence structures.