Reasoning about relational granulation in modal logics

It is well known that the Kripke model for the modal logic system S5 can be interpreted as an approximation space in rough set theory. In this paper, we generalize the interpretation to relational granulation. We consider two multimodal logics for reasoning about relational granulation in open world and closed world environments respectively. In an open world environment, two objects are granulated into the same equivalence class only if they have the same relationship with other objects, while in a closed world environment; two objects are granulated into the same equivalence class if and only if they have the same relationship with other objects. Such equivalence relations are represented by derived modalities from modal operators representing the relationships between objects.