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Degrees of categoricity on a cone via \(\eta\)-systems. (English) Zbl 1386.03049

Let \(C_{\mathbf d}=\{\mathbf c:\mathbf c\geq\mathbf d\}\) be a cone of Turing degrees. A countable structure \(\mathcal A\) is \(\Delta^0_{\alpha}\) categorical on the cone \(C_{\mathbf d}\) if for all \(\mathbf c\geq\mathbf d\), whenever \(\mathcal B\) and \(\mathcal C\) are \(\mathbf c\)-computable copies of \(\mathcal A\), there exists a \(\Delta^0_{\alpha}(\mathbf c)\)-computable isomorphism between \(\mathcal B\) and \(\mathcal C\). Every countable structure is \(\Delta^0_{\alpha}\) categorical on a cone for some \(\alpha\).
The first result of the paper (Theorem 1.7) concerns the difficulty of computing isomorphism between two copies of a structure: if \(\alpha\) is such that a countable structure \(\mathcal A\) is \(\Delta^0_{\alpha}\) categorical on a cone, then for every copy \(\mathcal B\) of \(\mathcal A\), there is a degree \(\mathbf d\) that is \(\Sigma^0_{\alpha-1}\) if \(\alpha\) is a successor, or \(\Delta^0_{\alpha}(\mathcal B)\) if \(\alpha\) is a limit, such that \(\mathbf d\) is the least degree of an isomorphism between \(\mathcal A\) and \(\mathcal B\). A structure \(\mathcal A\) has degree of categoricity \(\mathbf d\) relative to \(\mathbf c\) if \(\mathbf d\geq\mathbf c\) is the least degree such that there is a \(\mathbf d\)-computable isomorphisms between any two \(\mathbf c\)-computable copies of \(\mathcal A\). If in addition there exist two \(\mathbf c\)-computable copies of \(\mathcal A\) such that for every isomorphism \(f\) between them \(f\oplus\mathbf c\geq_T\mathbf d\) then \(\mathcal A\) has strong degree of categoricity \(\mathbf d\) relative to \(\mathbf c\). A structure \(\mathcal A\) has a (strong) degree of categoricity on the cone \(C_{\mathbf d}\) if for every \(\mathbf c\geq\mathbf d\), \(\mathcal A\) has a (strong) degree of categoricity relative to \(\mathbf c\). It is proved that every \(\Delta^0_2\) categorical on a cone structure \(\mathcal A\) has \(\Delta^0_1\)-complete or \(\Delta^0_2\)-complete degree of categoricity on a cone.
A more general Theorem 1.5 states that every countable structure has a strong degree of categoricity on a cone, and this degree of categoricity is \(\Delta^0_{\alpha}\)-complete, where \(\alpha\) is the least ordinal such that \(\mathcal A\) is \(\Delta^0_{\alpha}\) categorical on a cone. The proof uses a modification of A. Montalbán’s \(\eta\)-systems [J. Symb. Log. 79, No. 4, 1315–1335 (2014; Zbl 1353.03050)].

MSC:

03D45 Theory of numerations, effectively presented structures
03C57 Computable structure theory, computable model theory
03D28 Other Turing degree structures

Citations:

Zbl 1353.03050
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References:

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