8 found
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  1.  13
    Linear Orders Realized by C.E. Equivalence Relations.Ekaterina Fokina, Bakhadyr Khoussainov, Pavel Semukhin & Daniel Turetsky - 2016 - Journal of Symbolic Logic 81 (2):463-482.
    LetEbe a computably enumerable equivalence relation on the setωof natural numbers. We say that the quotient set$\omega /E$realizesa linearly ordered set${\cal L}$if there exists a c.e. relation ⊴ respectingEsuch that the induced structure is isomorphic to${\cal L}$. Thus, one can consider the class of all linearly ordered sets that are realized by$\omega /E$; formally,${\cal K}\left = \left\{ {{\cal L}\,|\,{\rm{the}}\,{\rm{order}}\, - \,{\rm{type}}\,{\cal L}\,{\rm{is}}\,{\rm{realized}}\,{\rm{by}}\,E} \right\}$. In this paper we study the relationship between computability-theoretic properties ofEand algebraic properties of linearly ordered sets realized (...)
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  2.  12
    An Uncountably Categorical Theory Whose Only Computably Presentable Model Is Saturated.Denis R. Hirschfeldt, Bakhadyr Khoussainov & Pavel Semukhin - 2006 - Notre Dame Journal of Formal Logic 47 (1):63-71.
    We build an א₁-categorical but not א₀-categorical theory whose only computably presentable model is the saturated one. As a tool, we introduce a notion related to limitwise monotonic functions.
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  3.  16
    Finite Automata Presentable Abelian Groups.André Nies & Pavel Semukhin - 2010 - Annals of Pure and Applied Logic 161 (3):458-467.
    We give new examples of FA presentable torsion-free abelian groups. Namely, for every n2, we construct a rank n indecomposable torsion-free abelian group which has an FA presentation. We also construct an FA presentation of the group in which every nontrivial cyclic subgroup is not FA recognizable.
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  4.  67
    Automatic Models of First Order Theories.Pavel Semukhin & Frank Stephan - 2013 - Annals of Pure and Applied Logic 164 (9):837-854.
    Khoussainov and Nerode [14] posed various open questions on model-theoretic properties of automatic structures. In this work we answer some of these questions by showing the following results: There is an uncountably categorical but not countably categorical theory for which only the prime model is automatic; There are complete theories with exactly 3,4,5,…3,4,5,… countable models, respectively, and every countable model is automatic; There is a complete theory for which exactly 2 models have an automatic presentation; If LOGSPACE=PLOGSPACE=P then there is (...)
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  5. InlineEquation ID=" IEq4"> EquationSource Format=" TEX"> ImageObject Color=" BlackWhite" FileRef=" 153200613ArticleIEq4. Gif" Format=" GIF" Rendition=" HTML" Type=" Linedraw"/>-Presentations of Algebras. [REVIEW]Bakhadyr Khoussainov, Theodore Slaman & Pavel Semukhin - 2006 - Archive for Mathematical Logic 45 (6):769.
     
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  6.  28
    P 0 1 \pi^0_1 -Presentations of Algebras.Bakhadyr Khoussainov, Theodore Slaman & Pavel Semukhin - 2006 - Archive for Mathematical Logic 45 (6):769-781.
    In this paper we study the question as to which computable algebras are isomorphic to non-computable \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Pi_{1}^{0}$$\end{document}-algebras. We show that many known algebras such as the standard model of arithmetic, term algebras, fields, vector spaces and torsion-free abelian groups have non-computable\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Pi_{1}^{0}$$\end{document}-presentations. On the other hand, many of this structures fail to have non-computable \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma_{1}^{0}$$\end{document}-presentation.
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  7.  28
    $$\pi^0_1$$ -Presentations of Algebras.Bakhadyr Khoussainov, Theodore Slaman & Pavel Semukhin - 2006 - Archive for Mathematical Logic 45 (6):769-781.
    In this paper we study the question as to which computable algebras are isomorphic to non-computable $\Pi_{1}^{0}$ -algebras. We show that many known algebras such as the standard model of arithmetic, term algebras, fields, vector spaces and torsion-free abelian groups have non-computable $\Pi_{1}^{0}$ -presentations. On the other hand, many of this structures fail to have non-computable $\Sigma_{1}^{0}$ -presentation.
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  8.  23
    Prime Models of Finite Computable Dimension.Pavel Semukhin - 2009 - Journal of Symbolic Logic 74 (1):336-348.
    We study the following open question in computable model theory: does there exist a structure of computable dimension two which is the prime model of its first-order theory? We construct an example of such a structure by coding a certain family of c.e. sets with exactly two one-to-one computable enumerations into a directed graph. We also show that there are examples of such structures in the classes of undirected graphs, partial orders, lattices, and integral domains.
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