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##### Chains and antichains in partial orderings with Carl G. Jockusch and Julia F. Knight Archive 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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Let
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##### Intrinsic bounds on complexity and definability at limit levels with John Chisholm, Ekaterina B. Fokina, Sergey S. Goncharov, Julia F. Knight, and Sara Quinn Journal 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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##### Bounding Homogeneous Models with Barbara F. Csima, Denis R. Hirschfeldt, and Robert I. Soare Journal 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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##### Isomorphism relations on computable structures with Ekaterina B. Fokina, Sy-David Friedman, Julia F. Knight, Charles Mccoy, and Antonio Montalbán Journal 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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##### Degree spectra of the successor relation of computable linear orderings with Jennifer Chubb and Andrey Frolov Archive 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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##### Spaces of orders and their Turing degree spectra with Malgorzata A. Dabkowska, Mieczyslaw K. Dabkowski, and Amir A. Togha Annals 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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##### Turing degrees of certain isomorphic images of computable relations Annals of Pure and Applied Logic 93 (1-3): 103-113. 1998.
A model is computable if its domain is a computable set and its relations and functions are uniformly computable. Let be a computable model and let R be an extra relation on the domain of . That is, R is not named in the language of . We define to be the set of Turing degrees of the images f under all isomorphisms f from to computable models. We investigate conditions on and R which are sufficient and necessary for to contain every Turing degree. These conditions imply that if every Turing degre…Read more
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##### Spectra of Structures and Relations with Russel G. Miller Journal of Symbolic Logic 72 (1). 2007.
We consider embeddings of structures which preserve spectra: if g: M → S with S computable, then M should have the same Turing degree spectrum (as a structure) that g(M) has (as a relation on S). We show that the computable dense linear order L is universal for all countable linear orders under this notion of embedding, and we establish a similar result for the computable random graph G. Such structures are said to be spectrally universal. We use our results to answer a question of Goncharov, an…Read more
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##### Effectively and Noneffectively Nowhere Simple Sets Mathematical 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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##### Σ 1 0 and Π 1 0 equivalence structures with Douglas Cenzer and Jeffrey B. Remmel Annals of Pure and Applied Logic 162 (7): 490-503. 2011.
We study computability theoretic properties of and equivalence structures and how they differ from computable equivalence structures or equivalence structures that belong to the Ershov difference hierarchy. Our investigation includes the complexity of isomorphisms between equivalence structures and between equivalence structures
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##### Effective categoricity of equivalence structures with Wesley Calvert, Douglas Cenzer, and Andrei Morozov Annals 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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##### Computability of fraïssé limits with Barbara F. Csima, Russell Miller, and Antonio Montalbán Journal 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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##### Enumerations in computable structure theory with Sergey Goncharov, Julia Knight, Charles McCoy, Russell Miller, and Reed Solomon Annals 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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##### The possible turing degree of the nonzero member in a two element degree spectrum Annals 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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##### Uncountable degree spectra Annals of Pure and Applied Logic 54 (3): 255-263. 1991.
We consider a recursive model and an additional recursive relation R on its domain, such that there are uncountably many different images of R under isomorphisms from to some recursive model isomorphic to . We study properties of the set of Turing degrees of all these isomorphic images of R on the domain of
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##### Partial automorphism semigroups with Jennifer Chubb, Andrei S. Morozov, Sarah Pingrey, and Eric Ufferman Annals 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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##### $\Pi _{1}^{0}$ Classes and Strong Degree Spectra of Relations with John Chisholm, Jennifer Chubb, Denis R. Hirschfeldt, Carl G. Jockusch, Timothy McNicholl, and Sarah Pingrey Journal 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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##### Effective categoricity of Abelian p -groups with Wesley Calvert, Douglas Cenzer, and Andrei Morozov Annals 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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##### Dependence relations in computably rigid computable vector spaces with Rumen D. Dimitrov and Andrei S. Morozov Annals of Pure and Applied Logic 132 (1): 97-108. 2005.
We construct a computable vector space with the trivial computable automorphism group, but with the dependence relations as complicated as possible, measured by their Turing degrees. As a corollary, we answer a question asked by A.S. Morozov in [Rigid constructive modules, Algebra and Logic, 28 570–583 ; 379–387 ]
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