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DEGREES OF CATEGORICITY AND SPECTRAL DIMENSION

  • NIKOLAY A. BAZHENOV (a1), ISKANDER SH. KALIMULLIN (a2) and MARS M. YAMALEEV (a3)
Abstract

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.

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  • ISSN: 0022-4812
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