Effective Categoricity of Automatic Equivalence and Nested Equivalence Structures

Abstract

We study automatic equivalence and nested equivalence structures. The goal is to compare and contrast these automatic structures with computable equivalence and nested equivalence structures. Equivalence structures $\mathcal {A}$ may be characterized by their characters $\chi ({\mathcal {A}})$ which encodes the number of equivalence classes of any given size. The characters of computably categorical, ${{\Delta }^{0}_{2}}$ categorical but not computably categorical, or ${{\Delta }^{0}_{3}}$ categorical but not ${{\Delta }^{0}_{2}}$ categorical have been determined. We show that every computably categorical equivalence structure has an automatic copy, but not every ${{\Delta }^{0}_{2}}$ categorical structure has an automatic copy. We construct an automatic equivalence structure which is ${{\Delta }^{0}_{2}}$ categorical but not computably categorical and another automatic equivalence structure which is not ${{\Delta }^{0}_{2}}$ categorical. We observe that the theory of an automatic equivalence structure is decidable and hence the character of any automatic equivalence structure is computable. On the other hand, there is a computable character which is not the character of any automatic equivalence structure. We show that any two automatic equivalence structures which are isomorphic are in fact computably isomorphic. Moreover, we show that for certain characters, there is always a exponential time isomorphism between two automatic equivalence structures with that character. Finally, we briefly consider nested equivalence structures and construct an automatic nested equivalence structure that is not ${{\Delta }^{0}_{3}}$ categorical but ${{\Delta }^{0}_{4}}$ categorical and an automatic nested equivalence structure that is not ${{\Delta }^{0}_{4}}$ categorical.