  By allowing torsion into the gravitational dynamics one can promote the
cosmological constant, $\Lambda$, to a dynamical variable in a class of
quasi-topological theories. In this paper we perform a mini-superspace
quantization of these theories in the connection representation. If $\Lambda$
is kept fixed, the solution is a delta-normalizable version of the Chern-Simons
(CS) state, which is the dual of the Hartle and Hawking and Vilenkin
wave-functions. We find that the CS state solves the Wheeler-DeWitt equation
also if $\Lambda$ is rendered dynamical by an Euler quasi-topological
invariant, {\it in the parity-even branch of the theory}. In the absence of an
infra-red (IR) cut-off, the CS state suggests the marginal probability
$P(\Lambda)=\delta(\Lambda)$. Should there be an IR cutoff (for whatever
reason) the probability is sharply peaked at the cut off. In the parity-odd
branch, however, we can still find the CS state as a particular (but not most
general) solution, but further work is needed to sharpen the predictions. For
the theory based on the Pontryagin invariant (which only has a parity-odd
branch) the CS wave function no longer is a solution to the constraints. We
find the most general solution in this case, which again leaves room for a
range of predictions for $\Lambda$.
