  Lambek calculus is a logical foundation of categorial grammar, a linguistic
paradigm of grammar as logic and parsing as deduction. Pentus (2010) gave a
polynomial-time algorithm for determ- ining provability of bounded depth
formulas in the Lambek calculus with empty antecedents allowed. Pentus'
algorithm is based on tabularisation of proof nets. Lambek calculus with
brackets is a conservative extension of Lambek calculus with bracket
modalities, suitable for the modeling of syntactical domains. In this paper we
give an algorithm for provability the Lambek calculus with brackets allowing
empty antecedents. Our algorithm runs in polynomial time when both the formula
depth and the bracket nesting depth are bounded. It combines a Pentus-style
tabularisation of proof nets with an automata-theoretic treatment of
bracketing.
