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

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\u27 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

Spurious ambiguity and focalization

Spurious ambiguity is the phenomenon whereby distinct derivations in grammar may assign the same structural reading, resulting in redundancy in the parse search space and inefficiency in parsing. Understanding the problem depends on identifying the essential mathematical structure of derivations. This is trivial in the case of context free grammar, where the parse structures are ordered trees; in the case of type logical categorial grammar, the parse structures are proof nets. However, with respect to multiplicatives, intrinsic proof nets have not yet been given for displacement calculus, and proof nets for additives, which have applications to polymorphism, are not easy to characterize. In this context we approach here multiplicative-additive spurious ambiguity by means of the proof-theoretic technique of focalization.Peer ReviewedPostprint (published version

The Logic of Categorial Grammars: Lecture Notes

These lecture notes present categorial grammars as deductive systems, and include detailed proofs of their main properties. The first chapter deals with Ajdukiewicz and Bar-Hillel categorial grammars (AB grammars), their relation to context-free grammars and their learning algorithms. The second chapter is devoted to the Lambek calculus as a deductive system; the weak equivalence with context free grammars is proved; we also define the mapping from a syntactic analysis to a higher-order logical formula, which describes the semantics of the parsed sentence. The third and last chapter is about proof-nets as parse structures for Lambek grammars; we show the linguistic relevance of these graphs in particular through the study of a performance question. Although definitions, theorems and proofs have been reformulated for pedagogical reasons, these notes contain no personnal result but in the proofnet chapter

A dynamic programming approach to categorial deduction

Colloque avec actes et comité de lecture.We reduce the provability problem of any formula of the Lambek calculus to some context-free parsing problem. This reduction, which is based on non-commutative proof-net theory, allows us to derive an automatic categorial deduction algorithm akin to the well-known Cocke-Kasami-Younger parsing algorithm