Rough approximations based on bisimulations

In recent years, rough set theory initiated by Pawlak has been intensively investigated. When the classical rough sets based on equivalence relations have been extended to generalized rough sets based on binary relations, the lower and upper rough approximations, which are the core concepts of rough set theory, have been generalized in several different ways. A common feature of these generalized approximations is that they use only “one step” information of the underlying relation to discern objects. By “one step” in a binary relation we mean that the ordered pair of the starting and end points of the step belongs to the relation. Motivated by a rich notion, bisimulation, appearing in various areas of computer science, we introduce a kind of lower and upper rough approximations for generalized rough sets in this paper. Our lower and upper approximations are based on bisimulations, in particular, bisimilarity, which is the largest bisimulation. Roughly speaking, bisimilar objects are regarded as indiscernible. We present some basic properties of the new lower and upper rough approximations and illustrate our motivation and the applicability of our results by examples. Moreover, we make a detailed comparison between the rough approximations based on the underlying relation and the rough approximations based on bisimilarity. In particular, we provide a necessary and sufficient condition for the consistency of the two kinds of rough approximations.