Σ01 and Π01 equivalence structures

We study computability theoretic properties of 01 and 0 1 equivalence structures and how they di er from computable equivalence structures or equivalence structures that belong to the Ershov di erence hierarchy. This includes the complexity of isomorphisms between 01 equivalence structures and between 01 equivalence structures. Any 0 1 equivalence structure A with in nitely many in nite equivalence classes is isomorphic to a computable structure. On the other hand, there are 01 equivalence structures with nitely many in nite equivalence classes, which are not isomorphic to any computable structure. If 01 equivalence structures A1 and A2 are isomorphic to a computable structure A that is computably categorical or relatively 02 categorical, then A1 and A2 are 0 2 isomorphic. If B is a computably categorical computable equivalence structure and A is a 01 structure which is isomorphic to B, then A and B are 0 2 isomorphic. If B is a computable equivalence structure which is not computably categorical, then in several cases we construct a 01 structure A which is isomorphic to B but is not 02 isomorphic to B. The simplest case is when B consists of in nitely many classes of sizes 1 or 2, and no other classes; if B is 02 categorical, then we show that the 0 1 structure A is moreover not 02 isomorphic to any 0 1 structure. The spectrum question is to determine the possible sets (or degrees of sets) that can compose the elements in equivalence classes of size k, for some xed k, in a computable equivalence structure. For example, for any in nite c.e. set B, there is a computable equivalence structure with in nitely many equivalence classes of size 1, in nitely many classes of size 2, and no other equivalence classes, such that B = fx : card([x]) = 2g. We explore the connection between the complexity of the character (A) and the theory Th(A). If Th(A) is decidable, then the character (A) is computable. If an equivalence structure B has a computable character, then there is a decidable structure A isomorphic to B.