Rough Sets: Probabilistic versus Deterministic Approach

The issue of knowledge representation and the method of inferring decision rules are of fundamental nature in the design of intelligent systems. When knowledge of the system is sufficient and precise (without uncertainty), many problems in artificial intelligence can be successfully modelled by techniques such as first order logic (Kowalski, 1979; Barr & Feigenbanm, 1981). For instance, the logical structure of a computer program can be precisely described by a set of decision rules (Hurley, 1981). On the other hand, it is rather difficult, if possible at all, to describe unambiguously the knowledge and the decision-making process of a human expert such as a physician or a business manager. The knowledge acquired under these circumstances is often imprecise and incomplete. Many methods have been proposed to deal with the uncertain aspects inherent in a knowledge representation system. They vary from approaches based on subjective assignment to decision rules of some "certainty factors" (Shortliffe, 1976) to those based on fuzzy logic (Zadeh, 1981). Recently, the notion of rough sets (Pawlak, 1982) was introduced, which provides a systematic framework for the study of the problems arising from imprecise and insufficient knowledge. Some of the advantages in using the rough-set concepts to expert systems design have been demonstrated by Pawlak, Slowinski & Slowinski (1986). However, in the existing rough-set model the probabilistic information crucial to non-deterministic classification (recognition) problems is not taken into consideration. For this reason, a probabilistic model has been proposed (Wong & Ziarko, 1986a, b) which is a natural extension of the rough-set method. The main advantage of the probabilistic model is that it provides a unified approach for both deterministic and non-deterministic knowledge representation systems. Furthermore, an effective inductive algorithm can be developed for a variety of applications (Wong & Ziarko, 1986a). The main objective of this paper is to review and compare the fundamental results in the probabilistic and deterministic models of rough sets. In section 2, the basic ideas of rough sets are reviewed. We also present a method for simplifying a given knowledge representation system, which has been the subject of research for many years (Orlowska & Pawlak, 1984; Pawlak, 1984). In section 3, the probabilistic model is introduced and various concepts are explained. We conclude this section by