Classical linear logic, cobordisms and categorical semantics of categorial grammars
This work provides a novel categorical framework for formal linguistics, potentially aiding in the understanding of language semantics and offering representations for other formalisms, though it appears incremental in extending existing grammar models.
The authors tackled the problem of representing categorial grammars by proposing linear logic grammars (LLG) based on classical multiplicative linear logic, which extends abstract categorial grammars (ACG) and uses multiwords for a concrete representation. They showed that multiwords form a category similar to topological cobordisms, which is symmetric monoidal closed and compact closed, serving as a model for linear λ-calculus and classical linear logic.
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 $λ$-terms, but simply tuples of words with labeled endpoints, we call them {\it multiwords}. At least, this gives a concrete and intuitive representation of ACG. A key observation is that the class of multiwords has a fundamental algebraic structure. Namely, multiwords can be organized in a category, very similar to the category of topological cobordisms. This category is symmetric monoidal closed and compact closed and thus is a model of linear $λ$-calculus and classical linear logic. We think that this category is interesting on its own right. In particular, it might provide categorical representation for other formalisms. On the other hand, many models of language semantics are based on commutative logic or, more generally, on symmetric monoidal closed categories. But the category of {\it word cobordisms} is a category of language elements, which is itself symmetric monoidal closed and independent of any grammar. Thus, it might prove useful in understanding language semantics as well.