  We study black-box testing for stochastic systems and arbitrary
$\omega$-regular specifications, explicitly including liveness properties. We
are given a finite-state probabilistic system that we can only execute from the
initial state. We have no information on the number of reachable states, or on
the probabilities; further, we can only partially observe the states. The only
action we can take is to restart the system. We design restart strategies
guaranteeing that, if the specification is violated with non-zero probability,
then w.p.1 the number of restarts is finite, and the infinite run executed
after the last restart violates the specification. This improves on previous
work that required full observability. We obtain asymptotically optimal upper
bounds on the expected number of steps until the last restart. We conduct
experiments on a number of benchmarks, and show that our strategies allow one
to find violations in Markov chains much larger than the ones considered in
previous work.
