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DEGREES OF CATEGORICITY AND SPECTRAL DIMENSION

Published online by Cambridge University Press:  01 May 2018

NIKOLAY A. BAZHENOV ,
ISKANDER SH. KALIMULLIN  and
MARS M. YAMALEEV
NIKOLAY A. BAZHENOV
Affiliation:
LABORATORY OF COMPUTABILITY THEORY AND APPLIED LOGIC SOBOLEV INSTITUTE OF MATHEMATICS PR. AKAD. KOPTYUGA 4 NOVOSIBIRSK 630090, RUSSIA and DEPARTMENT OF MATHEMATICS AND MECHANICS NOVOSIBIRSK STATE UNIVERSITY UL. PIROGOVA 2 NOVOSIBIRSK 630090, RUSSIA E-mail: bazhenov@math.nsc.ru
ISKANDER SH. KALIMULLIN
Affiliation:
N.I. LOBACHEVSKY INSTITUTE OF MATHEMATICS AND MECHANICS KAZAN FEDERAL UNIVERSITY UL. KREMLEVSKAYA 18 KAZAN 420008, RUSSIA E-mail: iskander.kalimullin@kpfu.ru
MARS M. YAMALEEV
Affiliation:
N.I. LOBACHEVSKY INSTITUTE OF MATHEMATICS AND MECHANICS KAZAN FEDERAL UNIVERSITY UL. KREMLEVSKAYA 18 KAZAN 420008, RUSSIA E-mail: mars.yamaleev@kpfu.ru
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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.

Keywords

03D4503C57categoricity spectrumdegree of categoricityrigid structurecomputable categoricity
Type
Articles
Information
The Journal of Symbolic Logic , Volume 83 , Issue 1 , March 2018 , pp. 103 - 116
Copyright
Copyright © The Association for Symbolic Logic 2018 

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