  In our previous work, we proposed the logic obtained from full
non-associative Lambek calculus by adding a sort of linear-logical modality. We
call this logic non-associative non-commutative intuitionistic linear logic
($\mathbf{NACILL}$, for short). In this paper, we establish the decidability
and undecidability results for various extensions of $\mathbf{NACILL}$.
Regarding the decidability results, we show that the deducibility problems for
several extensions of $\mathbf{NACILL}$ with the rule of left-weakening are
decidable. Regarding the undecidability results, we show that the provability
problems for all the extensions of non-associative non-commutative classical
linear logic by the rules of contraction and exchange are undecidable.
