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30San Antonio Convention Center San Antonio, Texas January 14–15, 2006Bulletin of Symbolic Logic 12 (4). 2006.
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26Turing degrees of hypersimple relations on computable structuresAnnals of Pure and Applied Logic 121 (2-3): 209-226. 2003.Let be an infinite computable structure, and let R be an additional computable relation on its domain A. The syntactic notion of formal hypersimplicity of R on , first introduced and studied by Hird, is analogous to the computability-theoretic notion of hypersimplicity of R on A, given the definability of certain effective sequences of relations on A. Assuming that R is formally hypersimple on , we give general sufficient conditions for the existence of a computable isomorphic copy of on whose d…Read more
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59Computability of fraïssé limitsJournal of Symbolic Logic 76 (1). 2011.Fraïssé studied countable structures S through analysis of the age of S i.e., the set of all finitely generated substructures of S. We investigate the effectiveness of his analysis, considering effectively presented lists of finitely generated structures and asking when such a list is the age of a computable structure. We focus particularly on the Fraïssé limit. We also show that degree spectra of relations on a sufficiently nice Fraïssé limit are always upward closed unless the relation is defi…Read more
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25Effective categoricity of Abelian p -groupsAnnals of Pure and Applied Logic 159 (1-2): 187-197. 2009.We investigate effective categoricity of computable Abelian p-groups . 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 categorical and relatively categorical
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17Regular relations and the quantifier “there exist uncountably many”Mathematical Logic Quarterly 29 (3): 151-161. 1983.
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55Computability-theoretic complexity of countable structuresBulletin of Symbolic Logic 8 (4): 457-477. 2002.Computable model theory, also called effective or recursive model theory, studies algorithmic properties of mathematical structures, their relations, and isomorphisms. These properties can be described syntactically or semantically. One of the major tasks of computable model theory is to obtain, whenever possible, computability-theoretic versions of various classical model-theoretic notions and results. For example, in the 1950's, Fröhlich and Shepherdson realized that the concept of a computabl…Read more
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54New Orleans Marriott and Sheraton New Orleans New Orleans, Louisiana January 7–8, 2007Bulletin of Symbolic Logic 13 (3). 2007.
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64Intrinsic bounds on complexity and definability at limit levelsJournal of Symbolic Logic 74 (3): 1047-1060. 2009.We show that for every computable limit ordinal α, there is a computable structure A that is $\Delta _\alpha ^0 $ categorical, but not relatively $\Delta _\alpha ^0 $ categorical (equivalently. it does not have a formally $\Sigma _\alpha ^0 $ Scott family). We also show that for every computable limit ordinal a, there is a computable structure A with an additional relation R that is intrinsically $\Sigma _\alpha ^0 $ on A. but not relatively intrinsically $\Sigma _\alpha ^0 $ on A (equivalently,…Read more
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24The possible turing degree of the nonzero member in a two element degree spectrumAnnals of Pure and Applied Logic 60 (1): 1-30. 1993.We construct a recursive model , a recursive subset R of its domain, and a Turing degree x 0 satisfying the following condition. The nonrecursive images of R under all isomorphisms from to other recursive models are of Turing degree x and cannot be recursively enumerable
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20Review: C. J. Ash, J. Knight, Computable Structures and the Hyperarithmetical Hierarchy (review)Bulletin of Symbolic Logic 7 (3): 383-385. 2001.
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91Simple and immune relations on countable structuresArchive for Mathematical Logic 42 (3): 279-291. 2003.Let ???? be a computable structure and let R be a new relation on its domain. We establish a necessary and sufficient condition for the existence of a copy ℬ of ???? in which the image of R (¬R, resp.) is simple (immune, resp.) relative to ℬ. We also establish, under certain effectiveness conditions on ???? and R, a necessary and sufficient condition for the existence of a computable copy ℬ of ???? in which the image of R (¬R, resp.) is simple (immune, resp.).
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31Spaces of orders and their Turing degree spectraAnnals of Pure and Applied Logic 161 (9): 1134-1143. 2010.We investigate computability theoretic and topological properties of spaces of orders on computable orderable groups. A left order on a group G is a linear order of the domain of G, which is left-invariant under the group operation. Right orders and bi-orders are defined similarly. In particular, we study groups for which the spaces of left orders are homeomorphic to the Cantor set, and their Turing degree spectra contain certain upper cones of degrees. Our approach unifies and extends Sikora’s …Read more
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41Effective categoricity of equivalence structuresAnnals of Pure and Applied Logic 141 (1): 61-78. 2006.We investigate effective categoricity of computable equivalence structures . We show that is computably categorical if and only if has only finitely many finite equivalence classes, or 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. Sin…Read more
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36Effectively and Noneffectively Nowhere Simple SetsMathematical Logic Quarterly 42 (1): 241-248. 1996.R. Shore proved that every recursively enumerable set can be split into two nowhere simple sets. Splitting theorems play an important role in recursion theory since they provide information about the lattice ϵ of all r. e. sets. Nowhere simple sets were further studied by D. Miller and J. Remmel, and we generalize some of their results. We characterize r. e. sets which can be split into two effectively nowhere simple sets, and r. e. sets which can be split into two r. e. non-nowhere simple sets.…Read more
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68Enumerations in computable structure theoryAnnals of Pure and Applied Logic 136 (3): 219-246. 2005.We exploit properties of certain directed graphs, obtained from the families of sets with special effective enumeration properties, to generalize several results in computable model theory to higher levels of the hyperarithmetical hierarchy. Families of sets with such enumeration features were previously built by Selivanov, Goncharov, and Wehner. For a computable successor ordinal α, we transform a countable directed graph into a structure such that has a isomorphic copy if and only if has a com…Read more
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45Degree spectra of the successor relation of computable linear orderingsArchive for Mathematical Logic 48 (1): 7-13. 2009.We establish that for every computably enumerable (c.e.) Turing degree b the upper cone of c.e. Turing degrees determined by b is the degree spectrum of the successor relation of some computable linear ordering. This follows from our main result, that for a large class of linear orderings the degree spectrum of the successor relation is closed upward in the c.e. Turing degrees
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George Washington UniversityRegular Faculty
Areas of Interest
Logic and Philosophy of Logic |
General Philosophy of Science |