A Set Theory for Rough Sets. Toward A Formal Calculus of Vague Statements

Issue title: Theory and Applications of Soft Computing (TASC04)

Article type: Research Article

Authors: Polkowski, Lech

Affiliations: Polish-Japanese Institute of Information Technology, Koszykowa 86, 02008 Warsaw, Poland Department of Mathematics and Computer Science, University of Warmia and Mazury Żołnierska 14a, 10561 Olsztyn, Poland. E-mail: polkow@pjwstk.edu.pl

Abstract: Rough set theory is a paradigm for approximate reasoning based on a formal mathematical basis, viz., it assumes that concepts are divided into exact and non-exact (rough) ones by means of a topological structure induced by a representation of knowledge as a classification. A classification in its most simple form is an equivalence relation on a universe of objects; the classification induces a partition topology and concepts (subsets of the universe) that are clopen are exact whereas other concepts are rough. In consequence, rough sets are represented as pairs of exact sets of the form (interior, closure). In the paper, we propose a set theory RZF that represents formally exact and rough sets as satisfying or not a certain dichotomy based on a new form of membership in a set; this membership acquires a mereological character as it is based on containment. As a result, we propose a new form of set theory suitable as a set theory for rough sets. Logical models for reasoning in the framework of rough set theory were proposed and studied by many researchers, among them Orłowska, Orłowska and Pawlak, Rasiowa and Skowron, Vakarelov. We exploit here models of RZF as interpretation domains for rough mereological logics: intensional logics whose truth value assignment is based on rough inclusions – basic predicates of rough mereology, a paradigm for approximate reasoning introduced by Polkowski and Skowron. An application for those logics is proposed in semantic interpretation of vague statements forming the domain of Calculus of Perceptions proposed by Zadeh.

Journal: Fundamenta Informaticae, vol. 71, no. 1, pp. 49-61, 2006