Some Complexity and Expressiveness Results on Multimodal and Stratified Proof Nets

We introduce a multimodal stratified framework MS that generalizes an idea hidden in the definitions of Light Linear/Affine logical systems: “More modalities means more expressiveness”. MS is a set of building-rule schemes that depend on parameters.We interpret the values of the parameters as modalities. Fixing the parameters yields deductive systems as instances of MS, that we call subsystems. Every subsystem generates stratified proof nets whose normalization preserves stratification, a structural property of nodes and edges, like in Light Linear/Affine logical systems. A first result is a sufficient condition for determining when a subsystem is strongly polynomial time sound. A second one shows that the ability to choose which modalities are used and how can be rewarding. We give a family of subsystems as complex as Multiplicative Linear Logic— they are linear time and space sound— that can representChurch numerals and some common combinators on them.