Group Theory and Grammatical Description

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This paper presents a model for linguistic description based on group theory. A grammar in this model, or "G-grammar", is a collection of lexical expressions which are products of logical forms, phonological forms, and their inverses. Phrasal descriptions are obtained by forming products of lexical expressions and by cancelling contiguous elements which are inverses of each other. We show applications of this model to parsing and generation, long-distance movement, and quantifier scoping. We believe that by moving from the free monoid over a vocabulary V --- standard in formal language studies --- to the free group over V, deep affinities between linguistic phenomena and classical algebra come to the surface, and that the consequences of tapping the mathematical connections thus established could be considerable.