Approximation Theories: Granular Computing vs Rough Sets

The goal of approximation in granular computing (GrC), in this paper, is to learn/approximate/express an unknown concept (a subset of the universe) in terms of a collection of available knowledge granules. So the natural operations are "and" and "or". Approximation theory for five GrC models is introduced. Note that GrC approximation theory is different from that of rough set theory (RST), since RST uses "or'" only. The notion of universal approximation theory (UAT) is introduced in GrC. This is important since the learning capability of fuzzy control and neural networks is based on UAT. Z. Pawlak had introduced point based and set based approximations. We use an example to illustrate the weakness of set based approximations in CrC.