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 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.