Degrees of categoricity and treeable degrees

TL;DR

This paper characterizes the strong degrees of categoricity of computable structures above b0'' as precisely the treeable degrees that compute b0'', providing new examples and answering open questions in computable structure theory.

Contribution

It introduces the concept of treeable degrees as the degrees of categoricity above b0'' and constructs various structures to demonstrate the range of degrees that can be degrees of categoricity.

Findings

Degrees b0^{(\u03b1)} b0 b0^{(\u03b1+1)} for b1 > 2 are degrees of categoricity of rigid structures.

Every degree b0' b0 b0'' is a degree of categoricity.

There exists a degree b0' < b0 < b0'' that is not a degree of categoricity of a rigid structure.

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.