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Proof-Theoretic Aspects of Hybrid Type-Logical Grammars

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Formal Grammar (FG 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11668))

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Abstract

This paper explores proof-theoretic aspects of hybrid type-logical grammars, a logic combining Lambek grammars with lambda grammars. We prove some basic properties of the calculus, such as normalisation and the subformula property and also present a proof net calculus for hybrid type-logical grammars. In addition to clarifying the logical foundations of hybrid type-logical grammars, the current study opens the way to variants and extensions of the original system, including but not limited to a non-associative version and a multimodal version incorporating structural rules and unary modes.

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Notes

  1. 1.

    The rule for the \(/\) directly parallels that for \(\backslash \), modulo directionality.

  2. 2.

    As is usual in the lambda calculus, we do not distinguish alpha-equivalent lambda terms.

  3. 3.

    For natural deduction, rule permutations are a problem only for the \(\bullet E\) and the \(\Diamond E\) rules.

  4. 4.

    To ensure confluence of ‘ / ’ and ‘\(\backslash \)’ in the presence of \(\epsilon \) we can add the side condition to the [ / I] and \([\backslash I]\) contractions that the component to which the par link is attached has at least one hypothesis other than the auxiliary conclusion of the par link. This forbids empty antecedent derivations and restores confluence.

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Author information

Authors and Affiliations

  1. LIRMM, Université de Montpellier, CNRS, Montpellier, France

    Richard Moot

  2. Ohio State University, Columbus, USA

    Symon Jory Stevens-Guille

Authors
  1. Richard Moot
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  2. Symon Jory Stevens-Guille
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Correspondence to Richard Moot .

Editor information

Editors and Affiliations

  1. University of Trento, Povo, Italy

    Raffaella Bernardi

  2. Universität Leipzig, Leipzig, Germany

    Greg Kobele

  3. LORIA, Inria Nancy, Villers-lès-Nancy, France

    Sylvain Pogodalla

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Moot, R., Stevens-Guille, S.J. (2019). Proof-Theoretic Aspects of Hybrid Type-Logical Grammars. In: Bernardi, R., Kobele, G., Pogodalla, S. (eds) Formal Grammar. FG 2019. Lecture Notes in Computer Science(), vol 11668. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-59648-7_6

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  • DOI: https://doi.org/10.1007/978-3-662-59648-7_6

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-59647-0

  • Online ISBN: 978-3-662-59648-7

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