This paper investigates an important class of information-flow security property called opacity for stochastic control systems. Opacity captures whether a system's secret behavior (a subset of the system's behavior that is considered to be critical) can be kept from outside observers. Existing works on opacity for control systems only provide a binary characterization of the system's security level by determining whether the system is opaque or not. In this work, we introduce a quantifiable measure of opacity that considers the likelihood of satisfying opacity for stochastic control systems modeled as general Markov decision processes (gMDPs). We also propose verification methods tailored to the new notions of opacity for finite gMDPs by using value iteration techniques. Then, a new notion called approximate opacity-preserving stochastic simulation relation is proposed, which captures the distance between two systems' behaviors in terms of preserving opacity. Based on this new system relation, we show that one can verify opacity for stochastic control systems using their abstractions (modeled as finite gMDPs). We also discuss how to construct such abstractions for a class of gMDPs under certain stability conditions.
