  In this paper, we study two problems: determining action model equivalence
and minimizing the event space of an action model under certain structural
relationships. The Kripke model equivalence is perfectly caught by the
structural relationship called bisimulation. In this paper, we propose the
generalized action emulation perfectly catching the action model equivalence.
Previous structural relationships sufficient for the action model equivalence,
i.e. the bisimulation, the propositional action emulation, the action
emulation, and the action emulation of canonical action models, can be
described by various restricted versions of the generalized action emulation.
We summarize four critical properties of the atom set over preconditions, and
prove that any formula set satisfying these properties can be used to restrict
the generalized action emulation to determine the action model equivalence by
an iteration algorithm. We also construct a new formula set with these four
properties, which is generally more efficient than the atom set. The technique
of the partition refinement has been used to minimize the world space of a
Kripke model under the bisimulation. Applying the partition refinement to
action models allows one to minimize their event spaces under the bisimulation.
The propositional action emulation is weaker than bismulation but still
sufficient for the action model equivalence. We prove that it is
PSPACE-complete to minimize the event space of an action model under the
propositional action emulation, and provide a PSPACE algorithm for it. Finally,
we prove that minimize the event space under the action model equivalence is
PSPACE-hard, and propose a computable method based on the canonical formulas of
modal logics to solve this problem.
