Classical linear logic, cobordisms and categorical semantics of categorial grammars

TL;DR

This paper introduces a categorial grammar based on classical multiplicative linear logic, representing language constituents as multiwords within a category akin to cobordisms, offering a new algebraic and categorical perspective on language structure and semantics.

Contribution

It extends abstract categorial grammars using linear logic, introduces multiwords as concrete representations, and models them within a cobordism-like category that is symmetric monoidal closed.

Findings

Multiwords form a category similar to topological cobordisms.

The category of multiwords is symmetric monoidal closed and compact closed.

This categorical framework models linear λ-calculus and classical linear logic.

Abstract

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, 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 $\lambda$-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.