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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
Citations:
Zbl 1184.03026
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