  We consider the problem of computing the set of initial states of a dynamical
system such that there exists a control strategy to ensure that the
trajectories satisfy a temporal logic specification with probability 1
(almost-surely). We focus on discrete-time, stochastic linear dynamics and
specifications given as formulas of the Generalized Reactivity(1) fragment of
Linear Temporal Logic over linear predicates in the states of the system. We
propose a solution based on iterative abstraction-refinement, and turn-based
2-player probabilistic games. While the theoretical guarantee of our algorithm
after any finite number of iterations is only a partial solution, we show that
if our algorithm terminates, then the result is the set of satisfying initial
states. Moreover, for any (partial) solution our algorithm synthesizes witness
control strategies to ensure almost-sure satisfaction of the temporal logic
specification. We demonstrate our approach on an illustrative case study.
