On embedding Lambek calculus into commutative categorial grammars
This work addresses a foundational challenge in formal linguistics for researchers in categorial grammar and linear logic, though it appears incremental as it builds on existing commutative frameworks.
The paper tackles the problem of embedding noncommutative Lambek calculus into commutative categorial grammars by enriching tensor grammars with new unary operators, resulting in a system that conservatively represents both ACG and Lambek grammars while maintaining simplicity.
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.