Making first order linear logic a generating grammar
This work addresses a foundational problem in computational linguistics and logic for researchers in language modeling and formal grammar, offering incremental improvements in representation and deductive systems.
The paper tackled the problem of representing categorial grammars in first-order multiplicative linear logic (MLL1) by showing equivalence to the extended tensor type calculus (ETTC), providing an alternative syntax, geometric representation, and an intrinsic deductive system for language modeling. They introduced a notationally enriched variation of ETTC, presenting cut-free sequent calculus and natural deduction formalisms for more concise computations.
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.