Degrees of Categoricity and spectral Dimension.
JOURNAL OF SYMBOLIC LOGIC(2018)
摘要
A Turing degree d is the degree of categoricity of a computable structure S if d is the least degree capable of computing isomorphisms among arbitrary computable copies of S. A degree d is the strong degree of categoricity of S if d is the degree of categoricity of S, and there are computable copies A and B of S such that every isomorphism from A onto B computes d. In this paper, we build a c.e. degree d and a computable rigid structure M such that d is the degree of categoricity of M, but d is not the strong degree of categoricity of M. This solves the open problem of Fokina, Kalimullin, and Miller [13]. For a computable structure S, we introduce the notion of the spectral dimension of S, which gives a quantitative characteristic of the degree of categoricity of S. We prove that for a nonzero natural number N, there is a computable rigid structure M such that 0' is the degree of categoricity of M, and the spectral dimension of M is equal to N.
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关键词
categoricity spectrum,degree of categoricity,rigid structure,computable categoricity
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