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Nonassociative Lambek Calculus with Additives and Context-Free Languages

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Languages: From Formal to Natural

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 5533))

Abstract

We study Nonassociative Lambek Calculus with additives ∧ , ∨, satisfying the distributive law (Distributive Full Nonassociative Lambek Calculus DFNL). We prove that categorial grammars based on DFNL, also enriched with assumptions, generate context-free languages. The proof uses proof-theoretic tools (interpolation) and a construction of a finite model, earlier employed in [11] in the proof of Finite Embeddability Property (FEP) of DFNL; our paper is self-contained, since we provide a simplified version of the latter proof. We obtain analogous results for different variants of DFNL, e.g. BFNL, which admits negation ¬ such that ∧ , ∨ ,¬ satisfy the laws of boolean algebra, and HFNL, corresponding to Heyting algebras with an additional residuation structure. Our proof also yields Finite Embeddability Property of boolean-ordered and Heyting-ordered residuated groupoids. The paper joins proof-theoretic and model-theoretic techniques of modern logic with standard tools of mathematical linguistics.

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Author information

Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Adam Mickiewicz University in Poznań Chair of Logic and Computation, Poland

    Wojciech Buszkowski & Maciej Farulewski

  2. University of Warmia and Mazury in Olsztyn, Poland

    Wojciech Buszkowski

Authors
  1. Wojciech Buszkowski
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  2. Maciej Farulewski
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Editor information

Editors and Affiliations

  1. Computer Science Department, TECHNION - Israel Institute of Technology, 32000, Technion City, Haifa, Israel

    Orna Grumberg

  2. Department of Computer Science, Technion, Israel Institute of Technology, 32000, Haifa, Israel

    Michael Kaminski

  3. Department of Computer Science, The Technion, 32000, Haifa, Israel

    Shmuel Katz

  4. Department of Computer Science, University of Haifa, Israel

    Shuly Wintner

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Buszkowski, W., Farulewski, M. (2009). Nonassociative Lambek Calculus with Additives and Context-Free Languages. In: Grumberg, O., Kaminski, M., Katz, S., Wintner, S. (eds) Languages: From Formal to Natural. Lecture Notes in Computer Science, vol 5533. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01748-3_4

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  • DOI: https://doi.org/10.1007/978-3-642-01748-3_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-01747-6

  • Online ISBN: 978-3-642-01748-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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