  To make a supervisor comprehensible to a layman has been a long-lasting 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 hopefully
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, in 2004 Su and
Wonham proposed to use control congruences to ensure computational viability.
This work is later adopted in the supervisor localization theory, which aims
for a control equivalent distributed implementation of a given centralized
supervisor. But after so many publications, some fundamental questions, which
should have been addressed in the first place, have not been answered yet,
namely what information is critical to ensure control equivalence, what
information is responsible for size reduction, and whether the partial
observation really makes things different. In this paper we will address these
fundamental questions by showing that there does exist 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
provides a partial order over all control equivalent feasible supervisors based
on their enabling, disabling and marking information, which can be used to
categorize the corresponding reduction rates. Based on this result we can see
that, given two control equivalent feasible supervisors, the one under full
observation can always result in a reduced supervisor no bigger than that
induced by a supervisor under partial observation.
