Complexity of \(\Sigma ^0_n\)-classifications for definable subsets

For a non-zero natural number n, we work with finitary \(\Sigma ^0_n\)-formulas \(\psi (x)\) without parameters. We consider computable structures \({\mathcal {S}}\) such that the domain of \({\mathcal {S}}\) has infinitely many \(\Sigma ^0_n\)-definable subsets. Following Goncharov and Kogabaev, we say that an infinite list of \(\Sigma ^0_n\)-formulas is a \(\Sigma ^0_n\)-classification for \({\mathcal {S}}\) if the list enumerates all \(\Sigma ^0_n\)-definable subsets of \({\mathcal {S}}\) without repetitions. We show that an arbitrary computable \({\mathcal {S}}\) always has a \({{\mathbf {0}}}^{(n)}\)-computable \(\Sigma ^0_n\)-classification. On the other hand, we prove that this bound is sharp: we build a computable structure with no \({{\mathbf {0}}}^{(n-1)}\)-computable \(\Sigma ^0_n\)-classifications.