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Computable Isomorphisms, Degree Spectra of Relations, and Scott Families
 Ann. Pure Appl. Logic
, 1998
"... this paper we are interested in those structures in which the basic computations can be performed by Turing machines. ..."
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Cited by 26 (12 self)
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this paper we are interested in those structures in which the basic computations can be performed by Turing machines.
Effective model theory: the number of models and their complexity
 MODELS AND COMPUTABILITY
, 1999
"... 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 i ..."
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Cited by 18 (6 self)
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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.
Effective Categoricity of Equivalence Structures
 Annals of Pure and Applied Logic 141 (2006
, 2005
"... 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 inf ..."
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Cited by 13 (9 self)
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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 DMS9970452, DMS0139626, and DMS0353748, Harizanov by the NSF grant DMS0502499, and the last three authors by the NSF binational grant DMS0075899. Harizanov and Morozov also gratefully acknowledge the
Computable categoricity of trees of finite height
 Journal of Symbolic Logic
"... We characterize the structure of computably categorical trees of finite height, and prove that our criterion is both necessary and sufficient. Intuitively, the characterization is easiest to express in terms of isomorphisms of (possibly infinite) trees, but in fact it is equivalent to a Σ0 3conditi ..."
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Cited by 7 (1 self)
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We characterize the structure of computably categorical trees of finite height, and prove that our criterion is both necessary and sufficient. Intuitively, the characterization is easiest to express in terms of isomorphisms of (possibly infinite) trees, but in fact it is equivalent to a Σ0 3condition. We show that all trees which are not computably categorical have computable dimension ω. Finally, we prove that for every n ≥ 1 in ω, there exists a computable tree of finite height which is ∆0 n+1categorical but not ∆0 ncategorical.
Effective Categoricity of Abelian pGroups
, 2007
"... We investigate effective categoricity of computable Abelian pgroups A. We prove that all computably categorical Abelian pgroups are relatively computably categorical, that is, have computably enumerable Scott families of existential formulas. We investigate which computable Abelian pgroups are ∆ ..."
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We investigate effective categoricity of computable Abelian pgroups A. We prove that all computably categorical Abelian pgroups are relatively computably categorical, that is, have computably enumerable Scott families of existential formulas. We investigate which computable Abelian pgroups are ∆ 0 2 categorical and relatively ∆ 0 2 categorical. 1
COMPUTABLE CATEGORICITY VERSUS RELATIVE COMPUTABLE CATEGORICITY
"... 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 1decidable computably categorical structure is relatively ∆0 2categorical. We study the complexit ..."
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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 1decidable computably categorical structure is relatively ∆0 2categorical. 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.