  We study the $(1,q=-1)$ model coupled to topological gravity as a candidate
to describing $2d$ string theory at the self-dual radius. We define the model
by analytical continuation of $q>1$ topological recursion relations to $q=-1$.
We show that at genus zero the $q=-1$ recursion relations yield the
$W_{1+\infty}$ Ward identities for tachyon correlators on the sphere. A scheme
for computing correlation functions of $q=-1$ gravitational descendants is
proposed and applied for the computation of several correlators. It is
suggested that the latter correspond to correlators of discrete states of the
$c=1$ string. In a similar manner to the $q>1$ models, we show that there exist
topological recursion relations for the correlators in the $q=-1$ theory that
consist of only one and two splittings of the Riemann surface. Using a
postulated regularized contact, we prove that the genus one $q=-1$ recursion
relations for tachyon correlators coincide with the $W_{1+\infty}$ Ward
identities on the torus. We argue that the structure of these recursion
relations coincides with that of the $W_{1+\infty}$ Ward identities for any
genus.
