  This paper studies properties of the logic BV, which is an extension of
multiplicative linear logic (MLL) with a self-dual non-commutative operator. BV
is presented in the calculus of structures, a proof theoretic formalism that
supports deep inference, in which inference rules can be applied anywhere
inside logical expressions. The use of deep inference results in a simple
logical system for MLL extended with the self-dual non-commutative operator,
which has been to date not known to be expressible in sequent calculus. In this
paper, deep inference is shown to be crucial for the logic BV, that is, any
restriction on the ``depth'' of the inference rules of BV would result in a
strictly less expressive logical system.
