On Decidable Categoricity and Almost Prime Models

Abstract

The complexity of isomorphisms for computable and decidable structures plays an important role in computable model theory. Goncharov [26] defined the degree of decidable categoricity of a decidable model $$\mathcal {M} $$ to be the least Turing degree, if it exists, which is capable of computing isomorphisms between arbitrary decidable copies of $$\mathcal {M} $$ . If this degree is $$\mathbf {0} $$ , we say that the structure $$\mathcal {M} $$ is decidably categorical. Goncharov established that every computably enumerable degree is the degree of categoricity of a prime model, and Bazhenov showed that there is a prime model with no degree of categoricity. Here we investigate the degrees of categoricity of various prime models with added constants, also called almost prime models. We relate the degree of decidable categoricity of an almost prime model $$\mathcal {M} $$ to the Turing degree of the set $$C(\mathcal {M}) $$ of complete formulas. We also investigate uniform decidable categoricity, characterizing it by primality of $$\mathcal {M} $$ and Turing reducibility of $$C(\mathcal {M}) $$ to the theory of $$\mathcal {M} $$ .