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Lambek Calculus with Optional Divisions

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Selected Reflections in Language, Logic, and Information (ESSLLI 2019)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14354))

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Abstract

In this paper, we introduce an extension of the Lambek calculus by optional divisions (\(\textrm{L}_{opt}\)). Namely, the right optional division \(A \angle B\) is defined as \(A \wedge (A/B)\), and the left one is defined as \(A \wedge (B \backslash A)\). A possible linguistic motivation to consider the new operations is describing verbs with optional arguments, e.g. reads in Tim reads the book. \(\textrm{L}_{opt}\) is a fragment of the multiplicative-additive Lambek calculus, so it would be interesting to compare the two calculi. The main part of the paper is devoted to the following grammar result: finite intersections of context-free languages can be generated by grammars over the calculus \(\textrm{L}_{opt}\). The proof involves introducing a useful normal form for Lambek grammars, namely, the notion of an interpretable grammar.

The study was supported by the Theoretical Physics and Mathematics Advancement Foundation “BASIS”; the Interdisciplinary Scientific and Educational School of Moscow University “Brain, Cognitive Systems, Artificial Intelligence”.

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Acknowledgments

Thanks to my scientific advisor Prof. Mati Pentus for helping me in many ways, thanks to reviewers for valuable advice, and thanks to Aleksey Starchenko for fruitful discussions.

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Authors and Affiliations

  1. Department of Mathematical Logic and Theory of Algorithms, Faculty of Mathematics and Mechanics, Lomonosov Moscow State University, GSP-1, Leninskie Gory, Moscow, 119991, Russian Federation

    Tikhon Pshenitsyn

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  1. Tikhon Pshenitsyn
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Correspondence to Tikhon Pshenitsyn .

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Editors and Affiliations

  1. Institute of Logic and Computation, TU Wien, Vienna, Austria

    Alexandra Pavlova

  2. Department of Information Science and Media Studies, University of Bergen, Bergen, Norway

    Mina Young Pedersen

  3. University of Trento, Rovereto, Italy

    Raffaella Bernardi

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Pshenitsyn, T. (2024). Lambek Calculus with Optional Divisions. In: Pavlova, A., Pedersen, M.Y., Bernardi, R. (eds) Selected Reflections in Language, Logic, and Information. ESSLLI 2019. Lecture Notes in Computer Science, vol 14354. Springer, Cham. https://doi.org/10.1007/978-3-031-50628-4_12

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  • DOI: https://doi.org/10.1007/978-3-031-50628-4_12

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-50627-7

  • Online ISBN: 978-3-031-50628-4

  • eBook Packages: Computer ScienceComputer Science (R0)

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