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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
Comparing classes of finite structures
 Algebra and Logic
"... In many branches of mathematics, there is work classifying a collection of objects, up to isomorphism or other important equivalence, in terms of nice invariants. In descriptive set theory, there is a body of work using a notion of ..."
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Cited by 5 (3 self)
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In many branches of mathematics, there is work classifying a collection of objects, up to isomorphism or other important equivalence, in terms of nice invariants. In descriptive set theory, there is a body of work using a notion of
Enumerations in computable structure theory
 Ann. Pure Appl. Logic
"... Goncharov, Harizanov, Knight, Miller, and Solomon gratefully acknowledge NSF support under binational ..."
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Cited by 4 (1 self)
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Goncharov, Harizanov, Knight, Miller, and Solomon gratefully acknowledge NSF support under binational
On the complexity of the successivity relation in computable linear orderings, in preparation
"... Abstract. In this paper, we solve a longstanding open question (see, e.g., Downey [6, §7] and Downey and Moses [11]), about the spectrum of the successivity relation on a computable linear ordering. We show that if a computable linear ordering L has infinitely many successivities, then the spectrum ..."
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Cited by 3 (2 self)
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Abstract. In this paper, we solve a longstanding open question (see, e.g., Downey [6, §7] and Downey and Moses [11]), about the spectrum of the successivity relation on a computable linear ordering. We show that if a computable linear ordering L has infinitely many successivities, then the spectrum of the successivity relation is closed upwards in the computably enumerable Turing degrees. To do this, we use a new method of constructing ∆ 0 3isomorphisms, which has already found other applications such as Downey, Kastermans and Lempp [9] and is of independent interest. It would seem to promise many further applications.
Computable Structures: Presentations Matter
 IN PROCEEDINGS OF THE INTL. CONG. LMPS
, 1999
"... The computability properties of a relation R not included in the language of a computable structure A can vary from one computable presentation to another. We describe some classic results giving conditions on A or R that restrict the possible variations in the computable dimension of A (i.e. the ..."
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Cited by 1 (1 self)
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The computability properties of a relation R not included in the language of a computable structure A can vary from one computable presentation to another. We describe some classic results giving conditions on A or R that restrict the possible variations in the computable dimension of A (i.e. the number of isomorphic copies of A up to computable isomorphism) and the computational complexity of R. For example, what conditions guarantee that A is computably categorical (i.e. of dimension 1) or that R is intrinsically computable (i.e. computable in every presentation). In the absence of such conditions, we discuss the possible computable dimensions of A and variations (in terms of Turing degree) of R in different presentations (the degree spectrum of R). In particular, various classic theorems and more recent ones of the author, B. Khoussainov, D. Hirschfeldt and others about the possible degree spectra of computable relations on computable structures and the connections with ...
Π 0 1 CLASSES AND STRONG DEGREE SPECTRA OF RELATIONS
"... Abstract. We study the weak truthtable and truthtable degrees of the images of subsets of computable structures under isomorphisms between computable structures. In particular, we show that there is a low c.e. set that is not weak truthtable reducible to any initial segment of any scattered compu ..."
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Cited by 1 (1 self)
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Abstract. We study the weak truthtable and truthtable degrees of the images of subsets of computable structures under isomorphisms between computable structures. In particular, we show that there is a low c.e. set that is not weak truthtable reducible to any initial segment of any scattered computable linear order. Countable Π 0 1 subsets of 2 ω and Kolmogorov complexity play a major role in the proof.
Computably Enumerable Vector Spaces, Dependence Relations, and Turing Degrees By
, 2002
"... I would like to thank my advisor, Valentina Harizanov, for all her support and help. She worked with me and encouraged my research and studies throughout the Ph.D. program. She has been my mentor, collaborator, and a good friend. IamgratefultoRodDowney,AliEnayat,AndreiMorozov,MichaelMoses, and Frank ..."
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Cited by 1 (1 self)
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I would like to thank my advisor, Valentina Harizanov, for all her support and help. She worked with me and encouraged my research and studies throughout the Ph.D. program. She has been my mentor, collaborator, and a good friend. IamgratefultoRodDowney,AliEnayat,AndreiMorozov,MichaelMoses, and Frank Stephan for the useful mathematical discussions. I am also indebted to the other committee members, Michele Friend and Irving Glick, for their suggestions on the final form of my thesis. I would like to thank the Faculty members and graduate students from the Department of Mathematics for their various support and for organizing various stimulating seminars and discussions. I would like to thank my wife Anna for supporting my family and being patient during my long days and nights these years. We present some structural theorems on the lattice L(V∞) of computably enumerable vector spaces and its factorlattices. The reader who is not familiar with