Strong Degrees of Categoricity and Weak Density

Bazhenov, Nikolay; Kalimullin, I. Sh.; Yamaleev, M. M.

doi:10.1134/s1995080220090048

Cited by 2 publications

(1 citation statement)

References 27 publications

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“…Our results and questions are analogous to the study of degrees of computable categoricity initiated in [18] and the further developed in, e.g., [19][20][21][22][23]. Theorem 3 is similar to the following result on degrees of categoricity for structures with a finite automorphism group [24,25]: the degree of categoricity of a structure A with |Aut(A)| < ∞ can be directly decomposed into the degrees of isomorphisms between finitely many computable copies of the structure.…”

Section: Theoremmentioning

confidence: 55%

Punctual Categoricity Relative to a Computable Oracle

Kalimullin

Melnikov

2021

Lobachevskii J Math

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We are studying the punctual structures, i.e., the primitive recursive structures on the whole set of integers. The punctual categoricity relative to a computable oracle $$f$$ means that between any two punctual copies of a structure there is an isomorphism which togeteher with its inverse can be derived via primitive recursive schemes augmented with $$f$$. We will prove that the punctual categoricity relative to a computable oracle can hold only for finitely generated or locally finite structures. We will show that the punctual categoricity of finitely generated structures is exhaused by the computable oracles with primitive recursive graph. We also present an example of locally finite structure where the punctual categoricity is provided by a primitive recursively bounded computable oracle.

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Section: Theoremmentioning

confidence: 55%

Punctual Categoricity Relative to a Computable Oracle

Kalimullin

Melnikov

2021

Lobachevskii J Math

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Degrees of categoricity and treeable degrees

Csima

Rossegger

2023

J. Math. Log.

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In this paper, we give a characterization of the strong degrees of categoricity of computable structures greater or equal to [Formula: see text]. They are precisely the treeable degrees — the least degrees of paths through computable trees — that compute [Formula: see text]. As a corollary, we obtain several new examples of degrees of categoricity. Among them we show that every degree [Formula: see text] with [Formula: see text] for [Formula: see text] a computable ordinal greater than [Formula: see text] is the strong degree of categoricity of a rigid structure. Using quite different techniques we show that every degree [Formula: see text] with [Formula: see text] is the strong degree of categoricity of a structure. Together with the above example this answers a question of Csima and Ng. To complete the picture we show that there is a degree [Formula: see text] with [Formula: see text] that is not the degree of categoricity of a rigid structure.

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Strong Degrees of Categoricity and Weak Density

Cited by 2 publications

References 27 publications

Punctual Categoricity Relative to a Computable Oracle

Punctual Categoricity Relative to a Computable Oracle

Degrees of categoricity and treeable degrees

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