Eliminating the unit constant in the Lambek calculus with brackets

We present a translation of the Lambek calculus with brackets and the unit constant, ${\mathbf{L}\mathbf{b}}_{\mathbf{1}}^{\mathit{\ast }}$, into the Lambek calculus with brackets allowing empty antecedents, but without the unit constant, ${\mathbf{L}\mathbf{b}}^{\mathit{\ast }}$. Using this translation, we extend previously known results for ${\mathbf{L}\mathbf{b}}^{\mathit{\ast }}$ to ${\mathbf{L}\mathbf{b}}_{\mathbf{1}}^{\mathit{\ast }}$: (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).