  The Lambek calculus is a well-known logical formalism for modelling natural
language syntax. The original calculus covered a substantial number of
intricate natural language phenomena, but only those restricted to the
context-free setting. In order to address more subtle linguistic issues, the
Lambek calculus has been extended in various ways. In particular, Morrill and
Valentin (2015) introduce an extension with so-called exponential and bracket
modalities. Their extension is based on a non-standard contraction rule for the
exponential that interacts with the bracket structure in an intricate way. The
standard contraction rule is not admissible in this calculus. In this paper we
prove undecidability of the derivability problem in their calculus. We also
investigate restricted decidable fragments considered by Morrill and Valentin
and we show that these fragments belong to the NP class.
