A Polynomial-Time Algorithm for the Lambek Calculus with Brackets of Bounded Order

Authors Max Kanovich, Stepan Kuznetsov, Glyn Morrill, Andre Scedrov

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Max Kanovich
Stepan Kuznetsov
Glyn Morrill
Andre Scedrov

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Max Kanovich, Stepan Kuznetsov, Glyn Morrill, and Andre Scedrov. A Polynomial-Time Algorithm for the Lambek Calculus with Brackets of Bounded Order. In 2nd International Conference on Formal Structures for Computation and Deduction (FSCD 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 84, pp. 22:1-22:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


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 determining provability of bounded depth formulas in L*, 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 in Lb*, 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.
  • Lambek calculus
  • proof nets
  • Lambek calculus with brackets
  • categorial grammar
  • polynomial algorithm


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