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 Π 11 -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 Π 11 -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 Π 04 -complete, while if 0 ≤ m ≤ n−2, the index set is Π 03 -complete.
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(E. B. Fokina and D. Turetsky) Supported by Austrian Science Fund FWF (projects V 206 and I 1238).
(S. S. Goncharov and O. V. Kudinov) Supported by RFBR (project No. 13-01-91001-ANF_a) and by the Grants Council (under RF President) for State Aid of Leading Scientific Schools (grant NSh-860.2014.1).
Translated from Algebra i Logika, Vol. 54, No. 4, pp. 520–528, July-August, 2015.
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Fokina, E.B., Goncharov, S.S., Harizanov, V. et al. Index Sets for n-Decidable Structures Categorical Relative to m-Decidable Presentations. Algebra Logic 54, 336–341 (2015). https://doi.org/10.1007/s10469-015-9353-6
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DOI: https://doi.org/10.1007/s10469-015-9353-6