Algebra and Logic

, Volume 54, Issue 4, pp 336–341 | Cite as

Index Sets for n-Decidable Structures Categorical Relative to m-Decidable Presentations

Article

We say that a structure is categorical relative to n-decidable presentations (or autostable relative to n-constructivizations) if any two n-decidable copies of the structure are computably isomorphic. For n = 0, we have the classical definition of a computably categorical (autostable) structure. Downey, Kach, Lempp, Lewis, Montalb´an, and Turetsky proved that there is no simple syntactic characterization of computable categoricity. More formally, they showed that the index set of computably categorical structures is Π11-complete. Here we study index sets of n-decidable structures that are categorical relative to m-decidable presentations, for various m, n ∈ ω. If m ≥ n ≥ 0, then the index set is again Π11-complete, i.e., there is no nice description of the class of n-decidable structures that are categorical relative to m-decidable presentations. In the case m = n−1 0, the index set is Π40-complete, while if 0 ≤ m ≤ n−2, the index set is Π30-complete.

Keywords

index set structure categorical relative to n-decidable presentations n-decidable structure categorical relative to m-decidable presentations 

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© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Vienna University of Technology, Institute of Discrete Mathematics and GeometryViennaAustria
  2. 2.Sobolev Institute of MathematicsNovosibirskRussia
  3. 3.Novosibirsk State UniversityNovosibirskRussia
  4. 4.George Washington UnivWashingtonUSA
  5. 5.Kurt Gödel Research Center for Mathematical LogicUniversity of ViennaViennaAustria

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