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  1. Barbara F. Csima, Valentina S. Harizanov, Russell Miller & Antonio Montalbán (2011). Computability of Fraïssé Limits. Journal of Symbolic Logic 76 (1):66 - 93.
    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 definable by (...)
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  2. Malgorzata A. Dabkowska, Mieczyslaw K. Dabkowski, Valentina S. Harizanov & Amir A. Togha (2010). Spaces of Orders and Their Turing Degree Spectra. Annals of Pure and Applied Logic 161 (9):1134-1143.
    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 [28] (...)
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  3. Wesley Calvert, Douglas Cenzer, Valentina S. Harizanov & Andrei Morozov (2009). Effective Categoricity of Abelian P-Groups. Annals of Pure and Applied Logic 159 (1):187-197.
    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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  4. John Chisholm, Ekaterina B. Fokina, Sergey S. Goncharov, Valentina S. Harizanov, Julia F. Knight & Sara Quinn (2009). Intrinsic Bounds on Complexity and Definability at Limit Levels. Journal of Symbolic Logic 74 (3):1047-1060.
    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 $ (...)
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  5. Valentina S. Harizanov, Carl G. Jockusch Jr & Julia F. Knight (2009). Chains and Antichains in Partial Orderings. Archive for Mathematical Logic 48 (1):39-53.
    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 (...)
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  6. Jennifer Chubb, Valentina S. Harizanov, Andrei S. Morozov, Sarah Pingrey & Eric Ufferman (2008). Partial Automorphism Semigroups. Annals of Pure and Applied Logic 156 (2):245-258.
    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 computable structure, up to computable isomorphism, from the isomorphism type of its inverse semigroup of computable (...)
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  7. John Chisholm, Jennifer Chubb, Valentina S. Harizanov, Denis R. Hirschfeldt, Carl G. Jockusch, Timothy McNicholl & Sarah Pingrey (2007). $\Pi _{1}^{0}$ Classes and Strong Degree Spectra of Relations. Journal of Symbolic Logic 72 (3):1003 - 1018.
    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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  8. Barbara F. Csima, Valentina S. Harizanov, Denis R. Hirschfeldt & Robert I. Soare (2007). Bounding Homogeneous Models. Journal of Symbolic Logic 72 (1):305 - 323.
    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 (...)
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  9. Valentina S. Harizanov & Russel G. Miller (2007). Spectra of Structures and Relations. Journal of Symbolic Logic 72 (1):324 - 348.
    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 (...)
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  10. Rumen D. Dimitrov, Valentina S. Harizanov & Andrei S. Morozov (2005). Dependence Relations in Computably Rigid Computable Vector Spaces. Annals of Pure and Applied Logic 132 (1):97-108.
    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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  11. Sergey S. Goncharov, Valentina S. Harizanov, Julia F. Knight & Richard A. Shore (2004). Π11 Relations and Paths Through. Journal of Symbolic Logic 69 (2):585-611.
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  12. Sergey S. Goncharov, Valentina S. Harizanov, Julia F. Knight & Richard A. Shore (2004). Π 1 1 Relations and Paths Through. Journal of Symbolic Logic 69 (2):585-611.
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  13. Sergei S. Goncharov, Valentina S. Harizanov, Julia F. Knight & Charles F. D. McCoy (2003). Simple and Immune Relations on Countable Structures. Archive for Mathematical Logic 42 (3):279-291.
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  14. Valentina S. Harizanov (2003). Turing Degrees of Hypersimple Relations on Computable Structures. Annals of Pure and Applied Logic 121 (2-3):209-226.
    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 (...)
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  15. Valentina S. Harizanov (2002). Computability-Theoretic Complexity of Countable Structures. Bulletin of Symbolic Logic 8 (4):457-477.
  16. Valentina S. Harizanov, Julia F. Knight & Andrei S. Morozov (2002). Sequences of N-Diagrams. Journal of Symbolic Logic 67 (3):1227-1247.
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  17. Valentina S. Harizanov (1998). Turing Degrees of Certain Isomorphic Images of Computable Relations. Annals of Pure and Applied Logic 93 (1-3):103-113.
    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 (...)
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  18. Valentina S. Harizanov (1996). Effectively and Noneffectively Nowhere Simple Sets. Mathematical Logic Quarterly 42 (1):241-248.
    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 (...)
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  19. Valentina S. Harizanov (1993). The Possible Turing Degree of the Nonzero Member in a Two Element Degree Spectrum. Annals of Pure and Applied Logic 60 (1):1-30.
    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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  20. Valentina S. Harizanov (1991). Some Effects of Ash–Nerode and Other Decidability Conditions on Degree Spectra. Annals of Pure and Applied Logic 55 (1):51-65.
    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 (...)
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  21. Valentina S. Harizanov (1991). Uncountable Degree Spectra. Annals of Pure and Applied Logic 54 (3):255-263.
    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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