Determination of action model equivalence and simplification of action model

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.