Switch to: Citations

Add references

You must login to add references.
  1. Degrees of Categoricity and the Hyperarithmetic Hierarchy.Barbara F. Csima, Johanna N. Y. Franklin & Richard A. Shore - 2013 - Notre Dame Journal of Formal Logic 54 (2):215-231.
    We study arithmetic and hyperarithmetic degrees of categoricity. We extend a result of E. Fokina, I. Kalimullin, and R. Miller to show that for every computable ordinal $\alpha$, $\mathbf{0}^{}$ is the degree of categoricity of some computable structure $\mathcal{A}$. We show additionally that for $\alpha$ a computable successor ordinal, every degree $2$-c.e. in and above $\mathbf{0}^{}$ is a degree of categoricity. We further prove that every degree of categoricity is hyperarithmetic and show that the index set of structures with degrees (...)
    Direct download (6 more)  
     
    Export citation  
     
    Bookmark   13 citations  
  • A Computably Categorical Structure Whose Expansion by a Constant has Infinite Computable Dimension.Denis R. Hirschfeldt, Bakhadyr Khoussainov & Richard A. Shore - 2003 - Journal of Symbolic Logic 68 (4):1199-1241.
    Cholak, Goncharov, Khoussainov, and Shore [1] showed that for each k > 0 there is a computably categorical structure whose expansion by a constant has computable dimension k. We show that the same is true with k replaced by ω. Our proof uses a version of Goncharov's method of left and right operations.
    Direct download (9 more)  
     
    Export citation  
     
    Bookmark   5 citations  
  • Degrees of Categoricity and Spectral Dimension.Nikolay A. Bazhenov, Iskander Sh Kalimullin & Mars M. Yamaleev - 2018 - Journal of Symbolic Logic 83 (1):103-116.
    A Turing degreedis the degree of categoricity of a computable structure${\cal S}$ifdis the least degree capable of computing isomorphisms among arbitrary computable copies of${\cal S}$. A degreedis the strong degree of categoricity of${\cal S}$ifdis the degree of categoricity of${\cal S}$, and there are computable copies${\cal A}$and${\cal B}$of${\cal S}$such that every isomorphism from${\cal A}$onto${\cal B}$computesd. In this paper, we build a c.e. degreedand a computable rigid structure${\cal M}$such thatdis the degree of categoricity of${\cal M}$, butdis not the strong degree of categoricity (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   3 citations  
  • Degrees of Categoricity on a Cone Via Η-Systems.Barbara F. Csima & Matthew Harrison-Trainor - 2017 - Journal of Symbolic Logic 82 (1):325-346.
    We investigate the complexity of isomorphisms of computable structures on cones in the Turing degrees. We show that, on a cone, every structure has a strong degree of categoricity, and that degree of categoricity is${\rm{\Delta }}_\alpha ^0 $-complete for someα. To prove this, we extend Montalbán’sη-system framework to deal with limit ordinals in a more general way. We also show that, for any fixed computable structure, there is an ordinalαand a cone in the Turing degrees such that the exact complexity (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   3 citations  
  • Degree Spectra of Relations on Computable Structures.Denis R. Hirschfeldt - 2000 - Bulletin of Symbolic Logic 6 (2):197-212.
    There has been increasing interest over the last few decades in the study of the effective content of Mathematics. One field whose effective content has been the subject of a large body of work, dating back at least to the early 1960s, is model theory. Several different notions of effectiveness of model-theoretic structures have been investigated. This communication is concerned withcomputablestructures, that is, structures with computable domains whose constants, functions, and relations are uniformly computable.In model theory, we identify isomorphic structures. (...)
    Direct download (10 more)  
     
    Export citation  
     
    Bookmark   6 citations  
  • 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).
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   17 citations  
  • Degree Spectra of Relations on Structures of Finite Computable Dimension.Denis R. Hirschfeldt - 2002 - Annals of Pure and Applied Logic 115 (1-3):233-277.
    We show that for every computably enumerable degree a > 0 there is an intrinsically c.e. relation on the domain of a computable structure of computable dimension 2 whose degree spectrum is { 0 , a } , thus answering a question of Goncharov and Khoussainov 55–57). We also show that this theorem remains true with α -c.e. in place of c.e. for any α∈ω∪{ω} . A modification of the proof of this result similar to what was done in Hirschfeldt (...)
    Direct download (5 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  • Computably Categorical Structures and Expansions by Constants.Peter Cholak, Sergey Goncharov, Bakhadyr Khoussainov & Richard A. Shore - 1999 - Journal of Symbolic Logic 64 (1):13-37.
    Direct download (8 more)  
     
    Export citation  
     
    Bookmark   14 citations  
  • The Possible Turing Degree of the Nonzero Member in a Two Element Degree Spectrum.Valentina S. Harizanov - 1993 - 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.
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   9 citations