Making first order linear logic a generating grammar

TL;DR

This paper demonstrates that a specific fragment of first order multiplicative linear logic, relevant for language modeling, is equivalent to the extended tensor type calculus (ETTC), providing a visual and deductive framework.

Contribution

It introduces a non-trivial enriched variation of ETTC with a formal sequent calculus and natural deduction, enhancing the expressiveness and transparency of the calculus.

Findings

Equivalence between MLL1 fragment and ETTC established

Introduction of a concise enriched ETTC variation

Development of a cut-free sequent calculus and natural deduction

Abstract

It is known that different categorial grammars have surface representation in a fragment of first order multiplicative linear logic (MLL1). We show that the fragment of interest is equivalent to the recently introduced extended tensor type calculus (ETTC). ETTC is a calculus of specific typed terms, which represent tuples of strings, more precisely bipartite graphs decorated with strings. Types are derived from linear logic formulas, and rules correspond to concrete operations on these string-labeled graphs, so that they can be conveniently visualized. This provides the above mentioned fragment of MLL1 that is relevant for language modeling not only with some alternative syntax and intuitive geometric representation, but also with an intrinsic deductive system, which has been absent. In this work we consider a non-trivial notationally enriched variation of the previously introduced ETTC, which allows more concise and transparent computations. We present both a cut-free sequent calculus and a natural deduction formalism.