Abstract Working toward showing the decidability of the $\forall \exists $ -theory of the ${\Sigma ^0_2}$ -enumeration degrees, we prove that no so-called Ahmad pair of ${\Sigma ^0_2}$ -enumeration degrees can join to ${\mathbf 0}_e'$ .

Abstract A Turing degreedis the degree of categoricity of a computable structure ${\cal S}$ ifdis the least degree capable of computing isomorphisms among arbitrary computable copies of ${\cal S}$ . A degreedis the strong degree of categoricity of ${\cal S}$ ifdis 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 ${\cal B}$ computesd. In this paper, we build a c.e. degreedand a computable rigid structure ${\cal M}$ such thatdis the degree of categoricity of ${\cal M}$ , butdis 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 numberN, 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 toN.