  The use of exponentials in linear logic greatly enhances its expressive
power. In this paper we focus on nonassociative noncommutative multiplicative
linear logic, and systematically explore modal axioms K, T, and 4 as well as
the structural rules of contraction and weakening. We give sequent systems for
each subset of these axioms; these enjoy cut elimination and have analogues in
more structural logics. We then appeal to work of Bulinska extending work of
Buszkowski to show that several of these logics are PTIME decidable and
generate context free languages as categorial grammars. This contrasts
associative systems where similar logics are known to generate all recursively
enumerable languages, and are thus in particular undecidable.
