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Interval Extensions of Orders and Temporal Approximation Spaces

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Abstract

Studying the algorithmic properties of interval extensions of dense linear orders, in particular, the complexity degrees (namely, the \( s\Sigma \)-degree) of the extensions, we show that continuity is a necessary and sufficient condition for the equality between the complexity degrees of an order and its interval extension. We treat temporal approximation spaces over interval extensions as mathematical models of verb semantics in natural languages. We show that the continuity of order implies the effectiveness of checking the validity of \( \Delta_{0}^{DL} \)-formulas in spaces over \( sc \)-simple enrichments. As a corollary, we obtain an effective description of the intervals corresponding to various verb tenses in English.

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Acknowledgments

The author is grateful to the referees for the remarks and suggestions that helped to improve exposition substantially. The author is grateful to Professor Roussanka Loukanova for help and attention to this work.

Funding

The author was supported by the 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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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk, Russia

    A. I. Stukachev

  2. Novosibirsk State University, Novosibirsk, Russia

    A. I. Stukachev

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  1. A. I. Stukachev
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Correspondence to A. I. Stukachev.

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Translated from Sibirskii Matematicheskii Zhurnal, 2021, Vol. 62, No. 4, pp. 894–910. https://doi.org/10.33048/smzh.2021.62.415

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Stukachev, A.I. Interval Extensions of Orders and Temporal Approximation Spaces. Sib Math J 62, 730–741 (2021). https://doi.org/10.1134/S0037446621040157

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  • DOI: https://doi.org/10.1134/S0037446621040157

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