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Definable Subsets of Polynomial-Time Algebraic Structures

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Computer Science – Theory and Applications (CSR 2020)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12159))

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Abstract

A structure S in a finite signature \(\sigma \) is polynomial-time if the domain of S, and the basic operations and relations of S are polynomial-time computable. Following the approach of semantic programming, for a given polynomial-time structure S, we consider the family B(S) containing all subsets of \(\mathrm {dom}(S)\), which are definable by a \(\varDelta _0\) formula with parameters. It is known that each of these sets is polynomial-time computable; hence, B(S), endowed with the standard set-theoretic operations, forms a natural Boolean algebra of polynomial-time languages, associated with S. We prove that up to isomorphism, the algebras B(S), where S is a polynomial-time structure, are precisely computable atomic Boolean algebras.

The work is supported by Mathematical Center in Akademgorodok under agreement No. 075-15-2019-1613 with the Ministry of Science and Higher Education of the Russian Federation.

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Acknowledgements

The author is grateful to the anonymous reviewers for their helpful suggestions.

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, 4 Acad. Koptyug Avenue, Novosibirsk, 630090, Russia

    Nikolay Bazhenov

  2. Novosibirsk State University, 2 Pirogova St., Novosibirsk, 630090, Russia

    Nikolay Bazhenov

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  1. Nikolay Bazhenov
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Correspondence to Nikolay Bazhenov .

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  1. University of Trier, Trier, Germany

    Henning Fernau

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Bazhenov, N. (2020). Definable Subsets of Polynomial-Time Algebraic Structures. In: Fernau, H. (eds) Computer Science – Theory and Applications. CSR 2020. Lecture Notes in Computer Science(), vol 12159. Springer, Cham. https://doi.org/10.1007/978-3-030-50026-9_10

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