Rough Approximation Spaces and Belief Structures in Infinite Universes of Discourse

In rough set theory there exists a pair of approximation operators, the lower and upper approximations; whereas in Dempster-Shafer theory of evidence there exists a dual pair of uncertainty measures, the belief and plausibility functions. To represent uncertainty knowledge in various information systems in crisp and fuzzy environments, general types of belief structures and their inducing dual pairs of belief and plausibility functions in infinite universes of discourse are first introduced. Relationships between the belief and plausibility functions in the Dempser-Shafer theory of evidence and the lower and upper approximations in the rough set theory are then established. It is shown that the probabilities of lower and upper approximations induced from an approximation space yield a dual pair of belief and plausibility functions. And for any belief structure there must exist a probability approximation space such that the belief and plausibility functions defined by the given belief structure are, respectively, the lower and upper probabilities induced by the approximation space. The lower and upper approximations of a set characterize the non-numeric aspect of uncertainty of the available information and can be interpreted as the qualitative representation of the set, whereas the belief and plausibility measures of the set capture the numeric aspect of uncertainty of the available information and can be treated as the quantitative characterization of the set. Finally, the potential applications of the main results to knowledge discovery in intelligent information systems in various situations are explored.