Degrees of categoricity and treeable degrees

Abstract

We give a characterization of the strong degrees of categoricity of computable structures greater or equal to $\mathbf 0''$. They are precisely the \emph{treeable} degrees -- the least degrees of paths through computable trees -- that compute $\mathbf 0''$. As a corollary, we obtain several new examples of degrees of categoricity. Among them we show that every degree $\mathbf d$ with $\mathbf 0^{(\alpha)}\leq \mathbf d\leq \mathbf 0^{(\alpha+1)}$ for $\alpha$ a computable ordinal greater than $2$ is the strong degree of categoricity of a rigid structure. Using quite different techniques we show that every degree $\mathbf d$ with $\mathbf 0'\leq \mathbf d\leq \mathbf 0''$ is the strong degree of categoricity of a structure. Together with the above example this answers a question of Csima and Ng. To complete the picture we show that there is a degree $\mathbf d$ with $\mathbf 0'<\mathbf d<\mathbf 0''$ that is not the degree of categoricity of a rigid structure.