Modal Reasoning and Rough Set Theory
In this paper, we would like to present some modal logics with semantics based on rough set theory and related notions. In addition to surveying some well-known results about the links between modal logics and rough set theory, we also develop some new applied logics inspired I)y generalized rough set theory. Kcywor(ls: Rough set, modal logic, cpistcmic logic. 1 I n t r o d u c t i o n Tile rough set theory is invented by Pawlak[15, 16] to account for the definability of a concept in terms of some elementary ones in an approximation space. Pawlak claims that knowledge is deep-seated in the classificatory abilities of human beings and other species, so rough set theory is a framework for discussions about knowledge, ill particular when imprecise knowledge is of primary concern([16], p.2). Thus the theory is in particular effective in extracting knowledge from data tables and it has been successfully and widely applied to domains such as intelligent data analysis (data mining and knowledge discovery in database), decision making, machine learning, pattern recognition and conflict analysis[13]. An important common feature of these applications is the classification of objects and the representation of the classificatory knowledge in rough set notions. The most well-known relationship between rough set theory and logics is the connection of approximation space with possible world semantics for the modal epistemic logic $5. Recently, the relationship between more general rough set model and modal logics have been examined in [28, 27, 29]. The common semantic intuition behind them is to view the set of possible worlds as an approximation space. These results all show the cross fertilization between rough set theory and logics, so it is worthwhile to investigate the further relationship between different generalizations of rough set models and logic systems. Based on this background, the purpose of this paper is twofold. The first is to survey and present the current results in a uniform way. The second is to fill some missing links between the existing general rough set models and logical formalisms and develop further some applied logics inspired by rough set theory. In what follows, we will first review the rough set theory and some of its main generalizatious. Then different modal logics and their relationship with rough set notions are discussed. Finally, some concluding remarks are given.