Index Sets for n-Decidable Structures Categorical Relative to m-Decidable Presentations

Fokina, Ekaterina B.; Гончаров, С. С.; Harizanov, Valentina S.; Kudinov, Oleg V.; Turetsky, Daniel

doi:10.1007/s10469-015-9353-6

Cited by 9 publications

(3 citation statements)

References 23 publications

(24 reference statements)

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“…As a systematic framework, this strategy of looking at index sets of computable structures originated in [29] and has been used in many other applications, see [8,21,22,27,28,48].…”

Section: Decision Problemmentioning

confidence: 99%

Automatic and Polynomial-Time Algebraic Structures

Bazhenov¹,

Harrison-Trainor²,

Kalimullin³

et al. 2019

J. symb. log.

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A structure is automatic if its domain, functions, and relations are all regular languages. Using the fact that every automatic structure is decidable, in the literature many decision problems have been solved by giving an automatic presentation of a particular structure. Khoussainov and Nerode asked whether there is some way to tell whether a structure has, or does not have, an automatic presentation. We answer this question by showing that the set of Turing machines that represent automata-presentable structures is ${\rm{\Sigma }}_1^1 $-complete. We also use similar methods to show that there is no reasonable characterisation of the structures with a polynomial-time presentation in the sense of Nerode and Remmel.

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“…As a systematic framework, this strategy of looking at index sets of computable structures originated in [29] and has been used in many other applications, see [8,21,22,27,28,48].…”

Section: Decision Problemmentioning

confidence: 99%

Automatic and Polynomial-Time Algebraic Structures

Bazhenov¹,

Harrison-Trainor²,

Kalimullin³

et al. 2019

J. symb. log.

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“…have investigated index sets of n-decidable models; i.e. the complexity of classifying such models [7]. More results along these lines can be found in the survey by Fokina, Harizanov, and Melnikov [8].…”

Section: Introductionmentioning

confidence: 99%

On the complexity of the theory of a computably presented metric structure

Camrud¹,

Goldbring²,

McNicholl³

2021

Preprint

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We consider the complexity (in terms of the arithmetical hierarchy) of the various quantifier levels of the diagram of a computably presented metric structure. As the truth value of a sentence of continuous logic may be any real in [0, 1], we introduce two kinds of diagrams at each level: the closed diagram, which encapsulates weak inequalities of the form φ M ≤ r, and the open diagram, which encapsulates strict inequalities of the form φ M < r. We show that the closed and open Σ N diagrams are Π 0 N+1 and Σ N respectively, and that the closed and open Π N diagrams are Π 0 N and Σ 0 N+1 respectively. We then introduce effective infinitary formulas of continuous logic and extend our results to the hyperarithmetical hierarchy. Finally, we demonstrate that our results are optimal.

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“…They calculated, in a precise way, the complexity of the index sets for m-decidable structures autostable relative to n-constructivizations, where m, n ∈ ω. The following definition is motivated by the results ofFokina et al (2015).…”

mentioning

confidence: 99%

Autostability spectra for decidable structures

Bazhenov

2016

Math. Struct. Comp. Sci.

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We study autostability spectra relative to strong constructivizations (SC-autostability spectra). For a decidable structure $\mathcal{S}$, the SC-autostability spectrum of $\mathcal{S}$ is the set of all Turing degrees capable of computing isomorphisms among arbitrary decidable copies of $\mathcal{S}$. The degree of SC-autostability for $\mathcal{S}$ is the least degree in the spectrum (if such a degree exists).We prove that for a computable successor ordinal α, every Turing degree c.e. in and above 0(α) is the degree of SC-autostability for some decidable structure. We show that for an infinite computable ordinal β, every Turing degree c.e. in and above 0(2β+1) is the degree of SC-autostability for some discrete linear order. We prove that the set of all PA-degrees is an SC-autostability spectrum. We also obtain similar results for autostability spectra relative to n-constructivizations.

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Index Sets for n-Decidable Structures Categorical Relative to m-Decidable Presentations

Cited by 9 publications

References 23 publications

Automatic and Polynomial-Time Algebraic Structures

Automatic and Polynomial-Time Algebraic Structures

On the complexity of the theory of a computably presented metric structure

Autostability spectra for decidable structures

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