Index Sets for n-Decidable Structures Categorical Relative to m-Decidable Presentations
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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 1 1 -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 1 1 -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 4 0 -complete, while if 0 ≤ m ≤ n−2, the index set is Π 3 0 -complete.
Keywordsindex set structure categorical relative to n-decidable presentations n-decidable structure categorical relative to m-decidable presentations
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- 2.A. I. Mal’tsev, “Constructive algebras. 1,” Usp. Mat. Nauk, 16, No. 3, 3–60 (1961).Google Scholar
- 4.S. S. Goncharov, “Autostable models and algorithmic dimensions,” in Handbook of Recursive Mathematics, Vol. 1, Recursive Model Theory, Yu. L. Ershov et al. (eds.), Stud. Log. Found. Math., 138, Elsevier, Amsterdam (1998), pp. 261–287.Google Scholar
- 5.E. B. Fokina, V. Harizanov, and A. Melnikov, “Computable model theory,” in Turing’s Legacy: Developments from Turing’s Ideas in Logic, Lect. Notes Log., 42, R. Downey (ed.), Cambridge Univ. Press, Ass. Symb. Log., Cambridge (2014), pp. 124–194.Google Scholar
- 12.S. S. Goncharov and M. I. Marchuk, “Index sets of constructive models that are autostable under strong constructivizations,” Vestnik NGU, Mat., Mekh., Inf., 13, No. 4, 43–67 (2013).Google Scholar
- 15.S. S. Goncharov and M. I. Marchuk, “Index sets of constructive models of nontrivial signature autostable relative to strong constructivizations,” Dokl. AN, 461, No. 2, 140–142 (2015).Google Scholar
- 21.S. S. Goncharov and Yu. L. Ershov, Constructive Models, Sib. School Alg. Log. [in Russian], Nauch. Kniga, Novosibirsk (1999).Google Scholar