Published online by Cambridge University Press: 01 May 2018
A Turing degree d is the degree of categoricity of a computable structure ${\cal S}$ if d is the least degree capable of computing isomorphisms among arbitrary computable copies of
${\cal S}$. A degree d is the strong degree of categoricity of
${\cal S}$ if d is the degree of categoricity of
${\cal S}$, and there are computable copies
${\cal A}$ and
${\cal B}$ of
${\cal S}$ such that every isomorphism from
${\cal A}$ onto
${\cal B}$ computes d. In this paper, we build a c.e. degree d and a computable rigid structure
${\cal M}$ such that d is the degree of categoricity of
${\cal M}$, but d is not the strong degree of categoricity of
${\cal M}$. This solves the open problem of Fokina, Kalimullin, and Miller [13].
For a computable structure ${\cal S}$, we introduce the notion of the spectral dimension of
${\cal S}$, which gives a quantitative characteristic of the degree of categoricity of
${\cal S}$. We prove that for a nonzero natural number N, there is a computable rigid structure
${\cal M}$ such that
$0\prime$ is the degree of categoricity of
${\cal M}$, and the spectral dimension of
${\cal M}$ is equal to N.