On the Recognizing Power of the Lambek Calculus with Brackets

TL;DR

This paper proves that all languages recognized by the Lambek calculus with brackets are context-free, using adaptations of existing methods and observations, including handling variants with empty antecedents.

Contribution

It demonstrates that the Lambek calculus with brackets recognizes only context-free languages, extending previous results and adapting proof techniques for variants with empty antecedents.

Findings

Lambek calculus with brackets recognizes only context-free languages

Adaptation of Pentus's method to the Lambek calculus with brackets

Handling of the variant with empty antecedents using restricted multiplicative unit

Abstract

Every language recognized by the Lambek calculus with brackets is context-free. This is shown by combining an observation by J\"ager with an entirely straightforward adaptation of the method Pentus used for the original Lambek calculus. The case of the variant of the calculus allowing sequents with empty antecedents is slightly more complicated, requiring a restricted use of the multiplicative unit.