Rough Set Theory: Ontological Systems, Entailment Relations and Approximation Operators

Abstract

The paper exhibits some new connections between ontological information systems equipped with approximation operators and entailment relations. The study is based on the conceptual framework of information quanta [4,5], where each system is defined as a relational structure and all approximation operators are defined in terms of Galois connections. We start our investigation with Scott systems, that is sets equipped with Scott entailment relations. Following the research by Vakarelov [2,11,12,13], we shall consider Scott systems induced by property systems and provide their characterisation in terms of Galois connections. We also recall how such connections allow one to define approximation operators from rough set theory (RST) [6,7] and derivation operators from formal concept analysis (FCA) [14,15]. Since we would like to have a uniform representation for both complete and incomplete Pawlak information systems, our attention is drawn by topological property systems, which additionally allows for a natural Galois-based generalisation of approximation operators. While considering more specific Scott systems, such like Tarski systems and standard Tarski systems, we obtain stronger connections to approximation operators from RST and FCA. Eventually, on the basis of these operators one can define Scott information systems with the trivial consistency predicate.