On embedding Lambek calculus into commutative categorial grammars

TL;DR

This paper explores embedding Lambek calculus into commutative tensor grammars derived from linear logic, enabling a unified, simple, and geometrically meaningful framework for different grammatical formalisms.

Contribution

It introduces a tensor grammar framework based on classical linear logic that can encode both ACG and Lambek grammars, including noncommutative operations, in a simple and intuitive way.

Findings

Tensor terms encode proof-nets and simplify syntax.

The system can represent ACG and Lambek grammars as fragments.

Enriching with unary operators allows noncommutative encoding.

Abstract

We consider tensor grammars, which are an example of \commutative" grammars, based on the classical (rather than intuitionistic) linear logic. They can be seen as a surface representation of abstract categorial grammars ACG in the sense that derivations of ACG translate to derivations of tensor grammars and this translation is isomorphic on the level of string languages. The basic ingredient are tensor terms, which can be seen as encoding and generalizing proof-nets. Using tensor terms makes the syntax extremely simple and a direct geometric meaning becomes transparent. Then we address the problem of encoding noncommutative operations in our setting. This turns out possible after enriching the system with new unary operators. The resulting system allows representing both ACG and Lambek grammars as conservative fragments, while the formalism remains, as it seems to us, rather simple and intuitive.