Opacity enforcing control synthesis

Given a finite transition system and a regular predicate, we address the problem of computing a controller enforcing the opacity of the predicate against an attacker (that partially observes the system), supposedly trying to push the system to reveal the predicate. Assuming that the controller can only control a subset of the events it observes (possibly different from the ones of the attacker), we show that an optimal control always exists and provide sufficient conditions under which it is regular and effectively computable. These conditions rely on the inclusion relationships between the observable alphabets of the attacker and the controller and the controllable alphabet.