Pregroup Grammars and Context-free Grammars

Pregroup grammars were introduced by Lambek [20] as a new formalism of type-logical grammars. They are weakly equivalent to context-free grammars [8]. The proof in one direction uses the fact that context-free languages are closed under homomorphism and inverse homomorphism. Here we present a direct construction of a context-free grammar and a push-down automaton, equivalent to a given pregroup grammar. The size of the resulting contextfree grammar (push-down automaton) is polynomial in the size of the pregroup grammar, while the construction proposed in Béchet [4] is exponential. We also obtain similar results for pregroup grammars based on a full system of Compact Bilinear Logic in the form of [11]. First, we recall some basic notions related to type logics, pregroups, Compact Bilinear Logic and pregroup grammars. Context-free grammars and push-down automata will not be defined; the reader is referred to any standard textbook on mathematical linguistics.