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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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##### 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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##### 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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##### 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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##### 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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##### 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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##### 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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##### Some effects of Ash–Nerode and other decidability conditions on degree spectra Annals 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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##### 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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##### 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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##### 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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##### Turing degrees of hypersimple relations on computable structures Annals 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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##### 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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##### $\Pi _{1}^{0}$ 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