We propose a categorial grammar based on classical multiplicative linear logic. This can be seen as an extension of abstract categorial grammars (ACG) and is at least as expressive. However, constituents of { \it linear logic grammars (LLG)} are not abstract ${ \lambda}$-terms, but simply tuples of words with labeled endpoints and supplied with specific { \it plugging instructions}: the sets of endpoints are subdivided into the { \it incoming} and the { \it outgoing} parts. We call such objects { \it word cobordisms}. A key observation is that word cobordisms can be organized in a category, very similar to the familiar category of topological cobordisms. This category is symmetric monoidal closed and compact closed and thus is a model of linear $\lambda$-calculus and classical, as well as intuitionistic linear logic. This allows us using linear logic as a typing system for word cobordisms. At least, this gives a concrete and intuitive representation of ACG. We think, however, that the category of word cobordisms, which has a rich structure and is independent of any grammar, might be interesting on its own right.
Categories: math.LO cs.CL cs.LO