Probabilistic Opacity for Markov Decision Processes
This work addresses security verification problems for systems modeled as MDPs, which is incremental as it extends existing opacity definitions from simpler models like Markov chains to more complex MDPs.
The authors extended the definition of opacity, a security property, to Markov decision processes (MDPs) by combining non-deterministic choice with probabilistic transitions, and they studied decidability problems under partial or complete observation hypotheses for schedulers. They proved that all questions are decidable with complete observation and ω-regular secrets, while with partial observation, quantitative questions are undecidable, but the question of whether a system is almost surely non-opaque becomes decidable for a restricted class of ω-regular secrets and under finite-memory schedulers.
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.