Concept Formation: Rough Sets and Scott Systems

Abstract

The paper addresses the problem of concept formation knowledge granulation in the settings of rough set theory. The original version of rough set theory implicitly accommodates a lot of well-established philosophical assumptions about concept formation as presented by A. Rand. However, as suggested by S. Hawking and L. Mlodinow, one has also to consider the dynamics of the universe of objects and different scales at which concepts may be formed. These both aspects have already been discussed separately in rough set theory. Different forms of dynamics have been addressed explicitly --especially the case of extending the universe by new objects; in contrast, different scales of description have been addressed implicitly, mainly within the Granular Computing GrC paradigm. Following the example of Life, the famous game invented by J. Conway, we describe the corresponding dynamics in Pawlak information systems using a GrC driven methodology. Having dynamics discussed, we address the problem of concept formation at zoom-out scales of description. To this end, we build Scott systems as information systems describing the universe at a coarser scale than the original scale of Pawlak systems. We regard these systems as a special type of classifications, which have already been studied in the context of rough sets by A. Skowron et al.