Computational Coverage of TLG: Nonlinearity

We study nonlinear connectives (exponentials) in the context of Type Logical Grammar(TLG). We devise four conservative extensions of theDisplacement calculus with brackets, \DbC, \DbCM, \DbCb and \DbCbMr which contain the universal and existential exponential modalities of linear logic (\LL). These modalitiesdo not exhibit the same structural properties as in \LL, which in TLG are especially adapted for linguistic purposes. The universal modality \univexpfor TLG allows only the commutative and contraction rules, but not weakening, whereas the existential modality \exstexp allows the so-called (intuitionistic) mingle rule, whichderives 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 iteratedcoordination. Finally, the four calculi are proved to be Cut-Free but decidability is only proved for $\DbCb$, whereasfor the rest the question of decidability remains open.