  In this paper, we are interested in path-dependent stochastic differential
equations (SDEs) which are controlled by Brownian motion and its delays. Within
this non-Markovian context, we give a H \"ormander-type criterion for the
regularity of solutions. Indeed, our criterion is expressed as a spanning
condition with brackets. A novelty in the case of delays is that noise can
"flow from the past" and give additional smoothness thanks to semi-brackets.
  The proof follows the general lines of Malliavin's probabilistic proof, in
the Markovian case. Nevertheless, in order to handle the non-Markovian aspects
of this problem and to treat anticipative integrals in a path-wise fashion, we
heavily invoke rough path integration.
