Mathematics : Rough Set Theory and Modal Logics

Rough Set Theory (RST) addresses imprecision that arises from a difficulty in describing reality. In everyday discourse, we place a grid over reality, the grid being typically induced by attributes. Then pieces of data having the same values for a set of attributes, cannot be distinguished. As a result, our concepts, generally, are not definable in terms of the grid. RST prescribes approximations to describe such concepts, and there may be several concepts with the same approximations describing them. RST thus serves as a means for reasoning with objects and concepts that are rendered indiscernible, due to incomplete information about the domain of discourse.

A major concern here is to look for appropriate formal logical frameworks to represent reasoning in RST. Inherent modalities point to the domain of modal logics. It is thus that we have some modal systems capturing 'rough truth', and different versions of 'rough modus ponens'. Modal logics also come in while studying dynamic aspects of RST. Data is presented in RST with the help of an information system, which may be complete, incomplete or non-deterministic. One then investigates sequences of information systems that evolve with time, or which arise from multiple sources (agents), and notions of information updates in the context. New temporal and quantified modal logics, and logics for information systems along with their 'dynamic' versions, have surfaced during this study.

Algebraic studies of structures that have arisen in the course of RST investigations constitute an important part of the research. Of special interest is a category-theoretic study of rough sets, and in fact, of concepts in a general framework of 'granulations'. Other applications of RST are also being studied, e.g. in dialogues between participants of a discourse, in communicative approximations, or in representing 'open universes'. Techniques for computation of 'minimal' sets of attributes required for classification (reducts) of objects also hold interest.

On another side, there is interest in the use of modal systems for reasoning with beliefs revealed by agents.