Concurrent secrets

Given a finite state system with partial observers and for each observer, a regular set of trajectories which we call a secret, we consider the question whether the observers can ever find out that a trajectory of the system belongs to some secret. We search for a regular control on the system, enforcing the specified secrets on the observers, even though they have full knowledge of this control. We show that an optimal control always exists although it is generally not regular. We state sufficient conditions for computing a finite and optimal control of the system enforcing the concurrent secret as desired.