this paper we are interested in those structures in which the basic computations can be performed by Turing machines.

Effective model theory studies model theoretic notions with an eye towards issues of computability and effectiveness. We consider two possible starting points. If the basic objects are taken to be theories, then the appropriate effective version investigates decidable theories (the set of theorems is computable) and decidable structures (ones with decidable theories). If the objects of initial interest are typical mathematical structures, then the starting point is computable structures. We present an introduction to both of these aspects of effective model theory organized roughly around the themes of the number and types of models of theories with particular attention to categoricity (as either a hypothesis or a conclusion) and the analysis of various computability issues in families of models.

iii In memory of my father. iv

We investigate effective categoricity of computable equivalence structures A. We show that A is computably categorical if and only if A has only finitely many finite equivalence classes, or A has only finitely many infinite classes, bounded character, and at most one finite k such that there are infinitely many classes of size k. We also prove that all computably categorical structures are relatively computably categorical, that is, have computably enumerable Scott families of existential formulas. Since all computable equivalence structures are relatively ∆ 0 3 categorical, we further investigate when they are ∆ 0 2 categorical. We also obtain results on the index sets of computable equivalence structures. ∗ The authors would like to thank the anonymous referee for his comments and suggestions. † Calvert was partially supported by the NSF grants DMS-9970452, DMS-0139626, and DMS-0353748, Harizanov by the NSF grant DMS-0502499, and the last three authors by the NSF binational grant DMS-0075899. Harizanov and Morozov also gratefully acknowledge the

We investigate effective categoricity of computable Abelian p-groups A. We prove that all computably categorical Abelian p-groups are relatively computably categorical, that is, have computably enumerable Scott families of existential formulas. We investigate which computable Abelian p-groups are ∆ 0 2 categorical and relatively ∆ 0 2 categorical. 1

Abstract. We study the notion of computable categoricity of computable structures, comparing it especially to the notion of relative computable categoricity and its relativizations. We show that every 1-decidable computably categorical structure is relatively ∆0 2-categorical. We study the complexity of various index sets associated with computable categoricity and relative computable categoricity. We also introduce and study a variation of relative computable categoricity, comparing it to both computable categoricity and relative computable categoricity and its relativizations. 1.