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Decidable Fragments of Calculi Used in CatLog

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Natural Language Processing in Artificial Intelligence — NLPinAI 2021

Part of the book series: Studies in Computational Intelligence ((SCI,volume 999))

Abstract

CatLog is a categorial grammar parser/theorem-prover developed by Glyn Morrill and his co-authors. CatLog is based on an extension of Lambek calculus. A distinctive feature of this extension is the usage of brackets for controlled non-associativity and a subexponential modality whose contraction rule interacts with bracketing in a sophisticated way. We consider two variants of the calculus, appearing in different versions of CatLog. Both systems are, unfortunately, undecidable in general. We consider fragments where the usage of subexponential is restricted by so-called bracket non-negative/non-positive conditions, prove that these fragments are decidable, and pinpoint their place in the complexity hierarchy. We also consider a more complicated, but more practically interesting problem of inducing (guessing) brackets. For this problem, we prove one decidability and one undecidability result, and leave some open questions for further research.

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Acknowlegdement

We are grateful to Glyn Morrill for a number of very helpful interactions we benefited from at various stages of our work. We would also like to thank the reviewers for their efforts.

The work of Max Kanovich was partially supported by EPSRC Programme Grant EP/R006865/1: “Interface Reasoning for Interacting Systems (IRIS).” The part by Stepan G. Kuznetsov was prepared within the framework of the Academic Fund Program at HSE University in 2021–2022 (grant № 21-04-027). The work of Stepan L. Kuznetsov and the early part of the work of Andre Scedrov (until July 2020) was performed within the framework of the HSE University Basic Research Program. The work of Stepan L. Kuznetsov was also partially supported by the Council of the President of Russia for Support of Young Russian Researchers and Leading Research Schools of the Russian Federation (grant MK-1184.2021.1.1) and by the Russian Foundation for Basic Research (grant № 20-01-00435).

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Authors and Affiliations

  1. University College London, Gower Street, London, UK

    Max I. Kanovich

  2. Computer Science Department, HSE University, 11 Pokrovsky Blvd., Moscow, Russia

    Max I. Kanovich & Stepan L. Kuznetsov

  3. Mathematics Department, HSE University, 6 Usacheva Street, Moscow, Russia

    Stepan G. Kuznetsov

  4. Steklov Mathematical Institute of RAS, 8 Gubkina Street, Moscow, Russia

    Stepan L. Kuznetsov

  5. Department of Mathematics, University of Pennsylvania, 209 South 33rd Street, Philadelphia, PA, USA

    Andre Scedrov

Authors
  1. Max I. Kanovich
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  2. Stepan G. Kuznetsov
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  3. Stepan L. Kuznetsov
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  4. Andre Scedrov
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Correspondence to Stepan L. Kuznetsov .

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Editors and Affiliations

  1. Department of Algebra and Logic, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Roussanka Loukanova

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Kanovich, M.I., Kuznetsov, S.G., Kuznetsov, S.L., Scedrov, A. (2022). Decidable Fragments of Calculi Used in CatLog. In: Loukanova, R. (eds) Natural Language Processing in Artificial Intelligence — NLPinAI 2021. Studies in Computational Intelligence, vol 999. Springer, Cham. https://doi.org/10.1007/978-3-030-90138-7_1

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