Equivalence Relations on Classes of Computable Structures

Abstract

If $\mathcal{L}$ is a finite relational language then all computable $\mathcal{L}$-structures can be effectively enumerated in a sequence in such a way that for every computable $\mathcal{L}$-structure $\mathcal{B}$ an index n of its isomorphic copy can be found effectively and uniformly. Having such a universal computable numbering, we can identify computable structures with their indices in this numbering. If K is a class of $\mathcal{L}$-structures closed under isomorphism we denote by K c the set of all computable members of K . We measure the complexity of a description of K c or of an equivalence relation on K c via the complexity of the corresponding sets of indices. If the index set of K c is hyperarithmetical then (the index sets of) such natural equivalence relations as the isomorphism or bi-embeddability relation are $\Sigma^1_1$. In the present paper we study the status of these $\Sigma^1_1$ equivalence relations (on classes of computable structures with hyperarithmetical index set) within the class of $\Sigma^1_1$ equivalence relations as a whole, using a natural notion of hyperarithmetic reducibility.