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.

[1]  J. Lambek.  The Mathematics of Sentence Structure , 1958 .

[2]  Wojciech Buszkowski.  The Equivalence of Unidirectional Lambek Categorial Grammars and Context-Free Grammars , 1985, Math. Log. Q..

[3]  W. Buszkowski.  Generative Power of Categorial Grammars , 1988 .

[4]  Emmon W. Bach,et al.  Categorial Grammars and Natural Language Structures , 1988 .

[5]  Dirk Roorda,et al.  Resource Logics : Proof-Theoretical Investigations , 1991 .

[6]  Mati Pentus,et al.  Lambek grammars are context free , 1993, [1993] Proceedings Eighth Annual IEEE Symposium on Logic in Computer Science.

[7]  H. Wansing.  Substructural logics , 1996 .

[8]  Joachim Lambek,et al.  An Algebraic Analysis of Clitic Pronouns in Italian , 2001, LACL.

[9]  Joachim Lambek,et al.  Type Grammars as Pregroups , 2001, Grammars.

[10]  Wojciech Buszkowski,et al.  Lambek Grammars Based on Pregroups , 2001, LACL.

[11]  Wojciech Buszkowski.  Sequent systems for compact bilinear logic , 2003, Math. Log. Q..

[12]  Maria Bulinska.  The Pentus Theorem for Lambek Calculus with Simple Nonlogical Axioms , 2005, Stud Logica.

[13]  Mati Pentus,et al.  Lambek calculus is NP-complete , 2006, Theor. Comput. Sci..

[14]  Annie Foret,et al.  Fully Lexicalized Pregroup Grammars , 2007, WoLLIC.

[15]  Peter Jipsen,et al.  Residuated lattices: An algebraic glimpse at sub-structural logics , 2007 .

[16]  Nissim Francez,et al.  Pushdown automata with cancellation and commutation-augmented pregroups grammars , 2007, LATA.

[17]  Anne Preller,et al.  Free compact 2-categories , 2007, Mathematical Structures in Computer Science.

[18]  Anne Preller.  Linear Processing with Pregroups , 2007, Stud Logica.

[19]  Denis Béchet,et al.  Parsing Pregroup Grammars and Lambek Calculus Using Partial Composition , 2007, Stud Logica.

[20]  Wojciech Buszkowski,et al.  Type Logics and Pregroups , 2007, Stud Logica.