Equivalence relations on classes of computable structures

Abstract

If ℒ is a finite relational language then all computable ℒ-structures can be effectively enumerated in a sequence {A n}nεω in such a way that for every computable ℒ-structure β 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 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 ∑11. In the present paper we study the status of these equivalence relations (on classes of computable structures with hyperarithmetical index set) within the class of ∑11 equivalence relations as a whole, using a natural notion of hyperarithmetic reducibility. © 2009 Springer Berlin Heidelberg.