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Ekaterina B. Fokina [6]Ekaterina Fokina [5]
  1.  47
    Degrees of Categoricity of Computable Structures.Ekaterina B. Fokina, Iskander Kalimullin & Russell Miller - 2010 - Archive for Mathematical Logic 49 (1):51-67.
    Defining the degree of categoricity of a computable structure ${\mathcal{M}}$ to be the least degree d for which ${\mathcal{M}}$ is d-computably categorical, we investigate which Turing degrees can be realized as degrees of categoricity. We show that for all n, degrees d.c.e. in and above 0 (n) can be so realized, as can the degree 0 (ω).
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  2.  14
    Categoricity Spectra for Rigid Structures.Ekaterina Fokina, Andrey Frolov & Iskander Kalimullin - 2016 - Notre Dame Journal of Formal Logic 57 (1):45-57.
    For a computable structure $\mathcal {M}$, the categoricity spectrum is the set of all Turing degrees capable of computing isomorphisms among arbitrary computable copies of $\mathcal {M}$. If the spectrum has a least degree, this degree is called the degree of categoricity of $\mathcal {M}$. In this paper we investigate spectra of categoricity for computable rigid structures. In particular, we give examples of rigid structures without degrees of categoricity.
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  3.  43
    The Effective Theory of Borel Equivalence Relations.Ekaterina B. Fokina, Sy-David Friedman & Asger Törnquist - 2010 - Annals of Pure and Applied Logic 161 (7):837-850.
    The study of Borel equivalence relations under Borel reducibility has developed into an important area of descriptive set theory. The dichotomies of Silver [20] and Harrington, Kechris and Louveau [6] show that with respect to Borel reducibility, any Borel equivalence relation strictly above equality on ω is above equality on , the power set of ω, and any Borel equivalence relation strictly above equality on the reals is above equality modulo finite on . In this article we examine the effective (...)
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  4.  39
    Isomorphism Relations on Computable Structures.Ekaterina B. Fokina, Sy-David Friedman, Valentina Harizanov, Julia F. Knight, Charles Mccoy & Antonio Montalbán - 2012 - Journal of Symbolic Logic 77 (1):122-132.
    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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  5.  25
    On Σ1 1 Equivalence Relations Over the Natural Numbers.Ekaterina B. Fokina & Sy-David Friedman - 2012 - Mathematical Logic Quarterly 58 (1-2):113-124.
    We study the structure of Σ11 equivalence relations on hyperarithmetical subsets of ω under reducibilities given by hyperarithmetical or computable functions, called h-reducibility and FF-reducibility, respectively. We show that the structure is rich even when one fixes the number of properly equation imagei.e., Σ11 but not equation image equivalence classes. We also show the existence of incomparable Σ11 equivalence relations that are complete as subsets of ω × ω with respect to the corresponding reducibility on sets. We study complete Σ11 (...)
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  6.  20
    Degrees of Bi-Embeddable Categoricity of Equivalence Structures.Nikolay Bazhenov, Ekaterina Fokina, Dino Rossegger & Luca San Mauro - 2019 - Archive for Mathematical Logic 58 (5-6):543-563.
    We study the algorithmic complexity of embeddings between bi-embeddable equivalence structures. We define the notions of computable bi-embeddable categoricity, \ bi-embeddable categoricity, and degrees of bi-embeddable categoricity. These notions mirror the classical notions used to study the complexity of isomorphisms between structures. We show that the notions of \ bi-embeddable categoricity and relative \ bi-embeddable categoricity coincide for equivalence structures for \. We also prove that computable equivalence structures have degree of bi-embeddable categoricity \, or \. We furthermore obtain results (...)
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  7.  11
    Linear Orders Realized by C.E. Equivalence Relations.Ekaterina Fokina, Bakhadyr Khoussainov, Pavel Semukhin & Daniel Turetsky - 2016 - Journal of Symbolic Logic 81 (2):463-482.
    LetEbe a computably enumerable equivalence relation on the setωof natural numbers. We say that the quotient set$\omega /E$realizesa linearly ordered set${\cal L}$if there exists a c.e. relation ⊴ respectingEsuch that the induced structure is isomorphic to${\cal L}$. Thus, one can consider the class of all linearly ordered sets that are realized by$\omega /E$; formally,${\cal K}\left = \left\{ {{\cal L}\,|\,{\rm{the}}\,{\rm{order}}\, - \,{\rm{type}}\,{\cal L}\,{\rm{is}}\,{\rm{realized}}\,{\rm{by}}\,E} \right\}$. In this paper we study the relationship between computability-theoretic properties ofEand algebraic properties of linearly ordered sets realized (...)
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  8.  1
    Bi‐Embeddability Spectra and Bases of Spectra.Ekaterina Fokina, Dino Rossegger & Luca San Mauro - 2019 - Mathematical Logic Quarterly 65 (2):228-236.
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  9.  54
    Intrinsic Bounds on Complexity and Definability at Limit Levels.John Chisholm, Ekaterina B. Fokina, Sergey S. Goncharov, Valentina S. Harizanov, Julia F. Knight & Sara Quinn - 2009 - 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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  10.  11
    Computability-Theoretic Categoricity and Scott Families.Ekaterina Fokina, Valentina Harizanov & Daniel Turetsky - 2019 - Annals of Pure and Applied Logic 170 (6):699-717.
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  11.  9
    Index Sets for Some Classes of Structures.Ekaterina B. Fokina - 2009 - Annals of Pure and Applied Logic 157 (2-3):139-147.
    For a class K of structures, closed under isomorphism, the index set is the set I of all indices for computable members of K in a universal computable numbering of all computable structures for a fixed computable language. We study the complexity of the index set of class of structures with decidable theories. We first prove the result for the class of all structures in an arbitrary finite nontrivial language. After the complexity is found, we prove similar results for some (...)
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