Nonassociative Lambek Calculus with Additives and Context-Free Languages

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 mathema...

• Type: Conference
• Year: 2009
• Where: BIRTHDAY
• Authors: Wojciech Buszkowski, Maciej Farulewski
• Keywords: Applied Computing, BIRTHDAY 2009, Finite Embeddability Property, Full Nonassociative Lambek, Nonassociative Lambek Calculus