We present a translation of the Lambek calculus with brackets and the unit constant, $\mathbf{ Lb}^{ \boldsymbol{ *}}_{ \mathbf{ 1}}$, into the Lambek calculus with brackets allowing empty antecedents, but without the unit constant, $\mathbf{ Lb}^{ \boldsymbol{ *}}$. Using this translation, we extend previously known results for $\mathbf{ Lb}^{ \boldsymbol{ *}}$ to $\mathbf{ Lb}^{ \boldsymbol{ *}}_{ \mathbf{ 1}}$: (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).
Categories: cs.LO math.LO