Optimal state-space partition for control of discrete-event systems with static specifications

The sensors distinguish among elements of the observation space, which is a partition of the state space, and the controller assigns a control pattern to each observation value. We study the coarsest partition to satisfy the static specification. If the sensors are reliable, then we can reduce the sensor cost by implementing the coarsest partition as the observation space. We show that, unfortunately, the coarsest partition does not uniquely exist in general. We then present a necessary and sufficient condition for the unique existence of the coarsest partition