What information really matters in supervisor reduction?

To make a supervisor comprehensible to a designer has been a long-standing goal in the supervisory control community. One strategy is to reduce the size of a supervisor to generate a control equivalent version, whose size is optimistically much smaller than the original one so that a user or control designer can easily check whether a designed controller fulfils its objectives and requirements. After the first journal paper on this topic appeared in 1986 by Vaz and Wonham, which relied on the concept of control covers, Su and Wonham proposed in 2004 to use control congruences to ensure computational viability. This work was later adopted in supervisor localization theory, which aims for a control equivalent distributed implementation of a given centralized supervisor. Despite these publications some fundamental questions, which might have been addressed in the first place, have not yet been answered, namely what information is critical to ensure control equivalence, what information is responsible for size reduction, and whether partial observation makes the problem essentially different. In this paper we address these questions by showing that there exists a unified supervisor reduction theory, which is applicable to all feasible supervisors regardless of whether they are under full observation or partial observation. Our theory proposes a preorder (called leanness) over all control equivalent feasible supervisors based on their enabling, disabling and marking information such that, if a supervisor S1 is leaner than another supervisor S2, then the size of the minimal control cover defined over the state set of S1 is no bigger than that of S2.