Efficiency of Pregroups and the French Noun Phrase

We study mathematical and algorithmic properties of Lambek’s pregroups and illustrate them by the French noun phrase. An algorithm of complexity 3 n to solve the reduction problem in an arbitrary free pregroup as well as recognition by a pregroup grammar is presented. This algorithm is then specified to run in linear time. A sufficient condition for a language fragment that makes the linear algorithm complete is given. Introduction Pregroups are a recent mathematical tool introduced in [Lambek 99] for natural language processing. They arose as a simplification of the Lambek calculus, first published under the name of Syntactic Calculus in [Lambek 58]. A pregroup grammar consists of a free pregroup and a pregroup dictionary, i.e. a finite relation which associates finitely many elements of the pregroup, the so-called types, to each word. [Buszkowski] has shown that pregroup grammars are weakly equivalent to context-free grammars. Well known algorithms like [CYK] or [Earley] for context-free grammars solve the problem of the grammaticality of strings of words in time proportional to the cube of the length of the string. However, it would not only be clumsy and inefficient to translate a pregroup grammar to a context-free grammar (this grammar would include the whole dictionary in its set of rules), but it would also go against the pregroups spirit. Indeed, pregroups were conceived as a tool which “processes the words as you hear them”. For each word you hear, you choose a type from the (mental) dictionary. The grammaticality of a string of words from the dictionary is then decided by performing computations on the corresponding string of types in the pregroup. So far, much of the work with pregroups has been aimed at establishing the universality of the rules, which are the same whatever the language only the dictionary changes. Fragments of Arabic, Japanese, Latin [MathLing], Polish [Kislak], English [Lambek 99, 03], French [Barg Lamb], German [Lamb-Prel NP, SE], Italian [Cas-Lamb] among others have been analysed with the help of pregroups.