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22Partial automorphism semigroupsAnnals of Pure and Applied Logic 156 (2): 245-258. 2008.We study the relationship between algebraic structures and their inverse semigroups of partial automorphisms. We consider a variety of classes of natural structures including equivalence structures, orderings, Boolean algebras, and relatively complemented distributive lattices. For certain subsemigroups of these inverse semigroups, isomorphism of the subsemigroups yields isomorphism of the underlying structures. We also prove that for some classes of computable structures, we can reconstruct a c…Read more
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47Bounding Homogeneous ModelsJournal of Symbolic Logic 72 (1). 2007.A Turing degree d is homogeneous bounding if every complete decidable (CD) theory has a d-decidable homogeneous model A, i.e., the elementary diagram De (A) has degree d. It follows from results of Macintyre and Marker that every PA degree (i.e., every degree of a complete extension of Peano Arithmetic) is homogeneous bounding. We prove that in fact a degree is homogeneous bounding if and only if it is a PA degree. We do this by showing that there is a single CD theory T such that every homogene…Read more
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58$\Pi _{1}^{0}$ Classes and Strong Degree Spectra of RelationsJournal of Symbolic Logic 72 (3). 2007.We study the weak truth-table and truth-table 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 truth-table reducible to any initial segment of any scattered computable linear ordering. Countable $\Pi _{1}^{0}$ subsets of 2ω and Kolmogorov complexity play a major role in the proof
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16Some effects of Ash–Nerode and other decidability conditions on degree spectraAnnals of Pure and Applied Logic 55 (1): 51-65. 1991.With every new recursive relation R on a recursive model , we consider the images of R under all isomorphisms from to other recursive models. We call the set of Turing degrees of these images the degree spectrum of R on , and say that R is intrinsically r.e. if all the images are r.e. C. Ash and A. Nerode introduce an extra decidability condition on , expressed in terms of R. Assuming this decidability condition, they prove that R is intrinsically r.e. if and only if a natural recursive-syntacti…Read more
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46Chains and antichains in partial orderingsArchive for Mathematical Logic 48 (1): 39-53. 2009.We study the complexity of infinite chains and antichains in computable partial orderings. We show that there is a computable partial ordering which has an infinite chain but none that is ${\Sigma _{1}^{1}}$ or ${\Pi _{1}^{1}}$ , and also obtain the analogous result for antichains. On the other hand, we show that every computable partial ordering which has an infinite chain must have an infinite chain that is the difference of two ${\Pi _{1}^{1}}$ sets. Our main result is that there is a computa…Read more
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67Isomorphism relations on computable structuresJournal of Symbolic Logic 77 (1): 122-132. 2012.We study the complexity of the isomorphism relation on classes of computable structures. We use the notion of FF-reducibility introduced in [9] to show completeness of the isomorphism relation on many familiar classes in the context of all ${\mathrm{\Sigma }}_{1}^{1}$ equivalence relations on hyperarithmetical subsets of ω
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29San 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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58Computability 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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22Effective 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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63Intrinsic 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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19The 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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18Review: 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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90Simple 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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George Washington UniversityRegular Faculty
Areas of Interest
Logic and Philosophy of Logic |
General Philosophy of Science |