  This paper addresses the problem of verifying discrete-time stochastic
systems against omega-regular specifications using finite-state abstractions.
Omega-regular properties allow specifying complex behavior and encompass, for
example, linear temporal logic. We focus on a class of systems with mixed
monotone dynamics. This class has recently been show to be amenable to
efficient reachable set computation and models a wide-range of physically
relevant systems. In general, finite-state abstractions of continuous state
stochastic systems give rise to augmented Markov Chains wherein the
probabilities of transition between states are restricted to an interval. We
present a procedure to compute a finite-state Interval-valued Markov Chain
abstraction of discrete-time, mixed monotone stochastic systems subject to
affine disturbances given a rectangular partition of the state-space. Then, we
suggest an algorithm for performing verification against omega-regular
properties in IMCs. Specifically, we aim to compute bounds on the probability
of satisfying the specification of interest from any initial state in the IMC.
This is achieved by solving a reachability problem on sets of so-called winning
and losing components in the Cartesian product between the IMC and a Rabin
automaton representing the specification. Next, the verification of IMCs may
yield a set of states whose acceptance status is undecided with respect to the
specification, requiring a refinement of the abstraction. We describe a
specification-guided approach that compares the best-case and worst-case
behaviors of accepting paths in the IMC and targets the appropriate states
accordingly. Finally, we show a case study.
