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

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.

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