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13
Degree spectra and computable dimension in algebraic structures
 Annals of Pure and Applied Logic 115 (2002
, 2002
"... \Lambda \Lambda ..."
Degree spectra of prime models
 J. Symbolic Logic
, 2004
"... 2.1 Notation from model theory................... 4 2.2 F ..."
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2.1 Notation from model theory................... 4 2.2 F
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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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.
Questions in Computable Algebra and Combinatorics
, 1999
"... this article, we will focus on two areas of computable mathematics, namely computable algebra and combinatorics. The goal of this article is to present a number of open questions in both computable algebra and computable combinatorics and to give the reader a sense of the research activity in these ..."
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this article, we will focus on two areas of computable mathematics, namely computable algebra and combinatorics. The goal of this article is to present a number of open questions in both computable algebra and computable combinatorics and to give the reader a sense of the research activity in these elds. Our philosophy is to try to highlight questions, whose solutions we feel will either give insight into algebra or combinatorics, or will require new technology in the computabilitytheoretical techniques needed. A good historical example of the rst phenomenom is the word problem for nitely presented groups which needed the development of a great deal of group theoretical machinery for its solution by Novikov [110] and Boone [10]. A good example of the latter phenomenon is the recent solution by Coles, Downey and Slaman [17] of the question of whether all rank one torsion free 1991 Mathematics Subject Classi cation. Primary 03D45; Secondary 03D25
Computable trees, prime models, and relative decidability
 PROC. AMER. MATH. SOC
, 2005
"... We show that for every computable tree T with no dead ends and all paths computable, and every D>T ∅, there is a Dcomputable listing of the isolated paths of T. It follows that for every complete decidable theory T such that all the types of T are computable and every D>T ∅, there is a Ddecidable ..."
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We show that for every computable tree T with no dead ends and all paths computable, and every D>T ∅, there is a Dcomputable listing of the isolated paths of T. It follows that for every complete decidable theory T such that all the types of T are computable and every D>T ∅, there is a Ddecidable prime model of T. This result extends a theorem of Csima and yields a stronger version of the theorem, due independently to Slaman and Wehner, that there is a structure with presentations of every nonzero degree but no computable presentation.
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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Goncharov, Harizanov, Knight, Miller, and Solomon gratefully acknowledge NSF support under binational
Computability, Definability and Algebraic Structures
, 1999
"... In a later section, we will look at a result of Coles, Downey and Slaman [16] of pure computability theory. The result is that, for any set X, the set ..."
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In a later section, we will look at a result of Coles, Downey and Slaman [16] of pure computability theory. The result is that, for any set X, the set
On mass problems of presentability
 Li (Eds.): TAMC2006. LNCS 3959
, 2006
"... Abstract. We consider the notion of mass problem of presentability for countable structures, and study the relationship between Medvedev and Muchnik reducibilities on such problems and possible ways of syntactically characterizing these reducibilities. Also, we consider the notions of strong and wea ..."
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Abstract. We consider the notion of mass problem of presentability for countable structures, and study the relationship between Medvedev and Muchnik reducibilities on such problems and possible ways of syntactically characterizing these reducibilities. Also, we consider the notions of strong and weak presentability dimension and characterize classes of structures with presentability dimensions 1. 1 Basic notions and facts The main problem we consider in this paper is the relationship between presentations of countable structures on natural numbers and on admissible sets. Most of notations and terminology we use here are standard and corresponds to [4, 1, 13]. We denote the domains of a structures M, N,... by M, N..... For any arbitrary structure M the hereditary finite superstructure HF(M), which is the least admissible set containing the domain of M as a subset, enables us to study effective (computable) properties of M by means of computability theory for admissible sets. The exact definition is as follows: the hereditary finite
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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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 ...