On a Notion of Approximate Opacity for Discrete-Time Stochastic Control Systems

This paper investigates a confidentiality property called opacity for discrete-time stochastic control systems. In order to quantitatively evaluate the security guarantee, a notion of approximate initial-state opacity for stochastic control systems is proposed. Then, we introduce a new notion of so-called opacity-preserving stochastic simulation functions to quantify the distance between two systems in a probabilistic setting, while preserving approximate initial-state opacity across them. In addition, we show that for a class of stochastic control systems satisfying incremental input-to-state stability property, one can construct their finite abstractions (a.k.a finite Markov decision processes) together with a corresponding opacity-preserving stochastic simulation function between them.