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Lambek calculus is a logical foundation of categorial grammar, a linguisticparadigm of grammar as logic and parsing as deduction. Pentus (2010) gave apolynomial-time algorithm for determ- ining provability of bounded depthformulas in the Lambek calculus with empty antecedents allowed. Pentus'algorithm is based on tabularisation of proof nets. Lambek calculus withbrackets is a conservative extension of Lambek calculus with bracketmodalities, suitable for the modeling of syntactical domains. In this paper wegive an algorithm for provability the Lambek calculus with brackets allowingempty antecedents. Our algorithm runs in polynomial time when both the formuladepth and the bracket nesting depth are bounded. It combines a Pentus-styletabularisation of proof nets with an automata-theoretic treatment ofbracketing.

A polynomial time algorithm for the Lambek calculus with brackets of bounded order