  Privacy is an increasing concern in cyber-physical systems that operates over
a shared network. In this paper, we propose a method for privacy verification
of cyber- physical systems modeled by Markov decision processes (MDPs) and
partially-observable Markov decision processes (POMDPs) based on barrier
certificates. To this end, we consider an opacity-based notion of privacy,
which is characterized by the beliefs in system states. We show that the belief
update equations can be represented as discrete-time switched systems, for
which we propose a set of conditions for privacy verification in terms of
barrier certificates. We further demonstrate that, for MDPs and for POMDPs,
privacy verification can be computationally implemented by solving a set of
semi-definite programs and sum-of-squares programs, respectively. The method is
illustrated by an application to privacy verification of an inventory
management system.
