Reconciling Lambek's restriction, cut-elimination, and substitution in the presence of exponential modalities

The Lambek calculus can be considered as a version of non-commutative intuitionistic linear logic. One of the interesting features of the Lambek calculus is the so-called "Lambek's restriction," that is, the antecedent of any provable sequent should be non-empty. In this paper we discuss ways of extending the Lambek calculus with the linear logic exponential modality while keeping Lambek's restriction. Interestingly enough, we show that for any system equipped with a reasonable exponential modality the following holds: if the system enjoys cut elimination and substitution to the full extent, then the system necessarily violates Lambek's restriction. Nevertheless, we show that two of the three conditions can be implemented. Namely, we design a system with Lambek's restriction and cut elimination and another system with Lambek's restriction and substitution. For both calculi we prove that they are undecidable, even if we take only one of the two divisions provided by the Lambek calculus. The system with cut elimination and substitution and without Lambek's restriction is folklore and known to be undecidable.