Nonassociative Lambek Calculus with Additives and Context-Free Languages

Authors:
Wojciech Buszkowski;Maciej Farulewski
Affiliations:
Faculty of Mathematics and Computer Science, Adam Mickiewicz University in Poznań Chair of Logic and Computation, and University of Warmia and Mazury in Olsztyn,;Faculty of Mathematics and Computer Science, Adam Mickiewicz University in Poznań Chair of Logic and Computation,
Venue:
Languages: From Formal to Natural
Year:
2009

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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.