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The Multiplicative-Additive Lambek Calculus with Subexponential and Bracket Modalities

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Abstract

We give a proof-theoretic and algorithmic complexity analysis for systems introduced by Morrill to serve as the core of the CatLog categorial grammar parser. We consider two recent versions of Morrill’s calculi, and focus on their fragments including multiplicative (Lambek) connectives, additive conjunction and disjunction, brackets and bracket modalities, and the ! subexponential modality. For both systems, we resolve issues connected with the cut rule and provide necessary modifications, after which we prove admissibility of cut (cut elimination theorem). We also prove algorithmic undecidability for both calculi, and show that categorial grammars based on them can generate arbitrary recursively enumerable languages.

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Notes

  1. Methods of Gaifman and Buszkowski work only for context-free languages without the empty word. The empty word case was handled by Kuznetsov (2012a).

  2. Here \(E \mathop {/}F_1 \ldots F_n\) is used as a shortcut for \((E \mathop {/}F_n) \mathop {/}\ldots \mathop {/}F_1\), thus, B is a one-division formula.

  3. This part can be simplified a bit by modifying \({\mathcal {G}}\). Namely, we could introduce a new starting symbol \(s'\) with a rule \(s' \Rightarrow s\). The language generated by \({\mathcal {G}}\) will not change. After this transformation, the starting symbol \(s'\) will never appear in the derivation, except for its start, and therefore there would be always \(y_i \ne s'\), and \(\varDelta _i = y_i\).

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Acknowledgements

We are grateful to Glyn Morrill for a number of very helpful interactions we benefited from at various stages of this work. The work of Max Kanovich was partially supported by EPSRC Programme Grant EP/R006865/1: “Interface Reasoning for Interacting Systems (IRIS).” The work of Andre Scedrov and Stepan Kuznetsov was prepared within the framework of the HSE University Basic Research Program and partially funded by the Russian Academic Excellence Project ‘5–100.’ The work of Stepan 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, by the Young Russian Mathematics Award, and by the Russian Foundation for Basic Research Grant 20-01-00435.

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Authors and Affiliations

  1. University College London, Gowers St., London, UK

    Max Kanovich

  2. Steklov Mathematical Institute of the Russian Academy of Sciences, 8 Gubkina St., Moscow, Russia

    Stepan Kuznetsov

  3. National Research University Higher School of Economics, 11 Pokrovsky Blvd., Moscow, Russia

    Max Kanovich & Stepan Kuznetsov

  4. University of Pennsylvania, 209 South 33rd St., Philadelphia, PA, USA

    Andre Scedrov

  5. National Research University Higher School of Economics (until July 2020), 11 Pokrovsky Blvd., Moscow, Russia

    Andre Scedrov

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  1. Max Kanovich
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Correspondence to Stepan Kuznetsov.

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Kanovich, M., Kuznetsov, S. & Scedrov, A. The Multiplicative-Additive Lambek Calculus with Subexponential and Bracket Modalities. J of Log Lang and Inf 30, 31–88 (2021). https://doi.org/10.1007/s10849-020-09320-9

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