A Uniform Framework for Rough Approximations Based on Generalized Quantifiers

The rough set theory provides an effective tool for decision analysis in the way of extracting decision rules from information systems. The rule induction process is based on the definitions of lower and upper approximations of the decision class. The condition attributes of the information system constitute an indiscernibility relation on the universe of objects. An object is in the lower approximation of the decision class if all objects indiscernible with it are in the decision class and it is in the upper approximation of the decision class if some objects indiscernible with it are in the decision class. Various generalizations of rough set theory have been proposed to enhance the capability of the theory. For example, variable precision rough set theory is used to improve the robustness of rough set analysis and fuzzy rough set approach is proposed to deal with vague information. In this paper, we present a uniform framework for different variants of rough set theory by using generalized quantifiers. In the framework, the lower and upper approximations of classical rough set theory are defined with universal and existential quantifiers respectively, whereas variable precision rough approximations correspond to probability quantifiers. Moreover, fuzzy rough set approximations can be defined by using different fuzzy quantifiers. We show that the framework can enhance the expressive power of the decision rules induced by rough set-based decision analysis.