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Degrees of categoricity and spectral dimension. (English) Zbl 1447.03008
Summary: A Turing degree $$\mathbf d$$ is the degree of categoricity of a computable structure $$\mathcal S$$ if $$\mathbf d$$ is the least degree capable of computing isomorphisms among arbitrary computable copies of $$\mathcal S$$. A degree $$\mathbf d$$ is the strong degree of categoricity of $$\mathcal S$$ if $$\mathbf{d}$$ is the degree of categoricity of $$\mathcal S$$, and there are computable copies $$\mathcal A$$ and $$\mathcal B$$ of $$\mathcal S$$ such that every isomorphism from $$\mathcal A$$ onto $$\mathcal B$$ computes $$\mathbf d$$. In this paper, we build a c.e. degree $$\mathbf d$$ and a computable rigid structure $$\mathcal M$$ such that $$\mathbf d$$ is the degree of categoricity of $$\mathcal M$$, but $$\mathbf d$$ is not the strong degree of categoricity of $$\mathcal M$$. This solves the open problem of E. B. Fokina et al. [Arch. Math. Logic 49, No. 1, 51–67 (2010; Zbl 1184.03026)].
For a computable structure $$\mathcal S$$, we introduce the notion of the spectral dimension of $$\mathcal S$$, which gives a quantitative characteristic of the degree of categoricity of $$\mathcal S$$. We prove that for a nonzero natural number $$N$$, there is a computable rigid structure $$\mathcal M$$ such that $$0^\prime$$ is the degree of categoricity of $$\mathcal M$$, and the spectral dimension of $$\mathcal M$$ is equal to $$N$$.

##### MSC:
 03D45 Theory of numerations, effectively presented structures 03C57 Computable structure theory, computable model theory 03C35 Categoricity and completeness of theories 03D28 Other Turing degree structures
Zbl 1184.03026
Full Text:
##### References:
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