Eliminating the unit constant in the Lambek calculus with brackets

Abstract

We present a translation of the Lambek calculus with brackets and the unit constant, ${\mathbf{Lb}}_1^{\ast }$, into the Lambek calculus with brackets allowing empty antecedents, but without the unit constant, ${\mathbf{Lb}}^{\ast }$. Using this translation, we extend previously known results for ${\mathbf{Lb}}^{\ast }$ to ${\mathbf{Lb}}_1^{\ast }$: (1) languages generated by categorial grammars based on the Lambek calculus with brackets are context-free (Kanazawa 2017); (2) the polynomial-time algorithm for deciding derivability of bounded depth sequents (Kanovich et al. 2017).