Computational Coverage of TLG: Nonlinearity

13 pagesPublished: July 8, 2015

Abstract

We study nonlinear connectives (exponentials) in the context of Type Logical Grammar
(TLG). We devise four conservative extensions of the
Displacement calculus with brackets, \DbC, \DbCM, \DbCb and \DbCbMr which contain the universal and existential exponential modalities of linear logic (\LL). These modalities
do not exhibit the same structural properties as in \LL, which in TLG are especially adapted for linguistic purposes. The universal modality \univexp
for TLG allows only the commutative and contraction rules, but not weakening, whereas the existential modality \exstexp allows the so-called (intuitionistic) mingle rule, which
derives a restricted version of weakening called \emph{expansion}. We provide a Curry-Howard labelling for both exponential connectives. As it turns out,
controlled contraction by \univexp gives a way to account for the so-called parasitic gaps, and controlled Mingle \exstexp iterability, in particular iterated
coordination. Finally, the four calculi are proved to be Cut-Free but decidability is only proved for $\DbCb$, whereas
for the rest the question of decidability remains open.

Keyphrases: computational linguistics, cut elimination, exponentials, linear logic, sublinear logic

In: Makoto Kanazawa, Larry Moss and Valeria de Paiva (editors). NLCS'15. Third Workshop on Natural Language and Computer Science, vol 32, pages 51--63