Probabilistic opacity for Markov decision processes

Opacity is a security property that guarantees that no information leaks in a system.We study opacity in systems modeled by Markov decision processes.For classical MDP, the value of the probabilistic disclosure is computable.In POMDP, most of the problems are undecidable.However, with natural restrictions, one can recover decidability of almost sure disclosure. Opacity is a generic security property, that has been defined on (non-probabilistic) transition systems and later on Markov chains with labels. For a secret predicate, given as a subset of runs, and a function describing the view of an external observer, the value of interest for opacity is a measure of the set of runs disclosing the secret. We extend this definition to the richer framework of Markov decision processes, where non-deterministic choice is combined with probabilistic transitions, and we study related decidability problems with partial or complete observation hypotheses for the schedulers. We prove that all questions are decidable with complete observation and ω-regular secrets. With partial observation, we prove that all quantitative questions are undecidable but the question whether a system is almost surely non-opaque becomes decidable for a restricted class of ω-regular secrets, as well as for all ω-regular secrets under finite-memory schedulers.