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Semilattices of Punctual Numberings

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Theory and Applications of Models of Computation (TAMC 2020)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12337))

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Abstract

The theory of numberings studies uniform computations for classes of mathematical objects. A large body of literature is devoted to investigations of computable numberings, i.e. uniform enumerations for families of computably enumerable sets, and the reducibility \(\le \) among these numberings. This reducibility, induced by Turing computable functions, aims to classify the algorithmic complexity of numberings.

The paper is inspired by the recent advances in the area of punctual algebraic structures. We recast the classical studies of numberings in the punctual setting—we study punctual numberings, i.e. uniform computations for families of primitive recursive functions. The reducibility \(\le _{pr}\) between punctual numberings is induced by primitive recursive functions. This approach gives rise to upper semilattices of degrees, which are called Rogers pr-semilattices. We prove that any infinite Rogers pr-semilattice is dense and does not have minimal elements. Furthermore, we give an example of infinite Rogers pr-semilattice, which is a lattice. These results exhibit interesting phenomena, which do not occur in the classical case of computable numberings and their semilattices.

The work was supported by Nazarbayev University Faculty Development Competitive Research Grants N090118FD5342. The first author was partially supported by the grant of the President of the Russian Federation (No. MK-1214.2019.1). The third author was partially supported by the program of fundamental scientific researches of the SB RAS No. I.1.1, project No. 0314-2019-0002.

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Acknowledgements

Part of the research contained in this paper was carried out while the first and the last authors were visiting the Department of Mathematics of Nazarbayev University, Nur-Sultan. The authors wish to thank Nazarbayev University for its hospitality.

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Authors and Affiliations

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

    Nikolay Bazhenov & Sergei Ospichev

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

    Nikolay Bazhenov & Sergei Ospichev

  3. Department of Mathematics, School of Sciences and Humanities, Nazarbayev University, 53 Qabanbaybatyr Avenue, Nur-Sultan, 010000, Kazakhstan

    Manat Mustafa

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  1. Nikolay Bazhenov
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Correspondence to Nikolay Bazhenov .

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Editors and Affiliations

  1. Department of Computer Science, Texas A&M University, College Station, TX, USA

    Jianer Chen

  2. School of Computer Science and Engineering, Central South University, Changsha, China

    Qilong Feng

  3. Department of Computer Science and Engineering, State University of New York at Buffalo, Buffalo, NY, USA

    Jinhui Xu

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Bazhenov, N., Mustafa, M., Ospichev, S. (2020). Semilattices of Punctual Numberings. In: Chen, J., Feng, Q., Xu, J. (eds) Theory and Applications of Models of Computation. TAMC 2020. Lecture Notes in Computer Science(), vol 12337. Springer, Cham. https://doi.org/10.1007/978-3-030-59267-7_1

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  • DOI: https://doi.org/10.1007/978-3-030-59267-7_1

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