# 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.*

## Keywords

*index set*

*structure categorical relative to n-decidable presentations*

*n-decidable structure categorical relative to m-decidable presentations*

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