Contributions to the Theory of Rough Sets

We study properties of rough sets, that is, approximations to sets of records in a database or, more formally, to subsets of the universe of an information system. A rough set is a pair 〈L, U〉 such that L, U are definable in the information system and L ⊆ U. In the paper, we introduce a language, called the language of inclusion-exclusion, to describe incomplete specifications of (unknown) sets. We use rough sets in order to define a semantics for theories in the inclusion-exclusion language. We argue that our concept of a rough set is closely related to that introduced by Pawlak. We show that rough sets can be ordered by the knowledge ordering (denoted $$\preceq$$ kn)- We prove that Pawlak's rough sets are characterized as $$\preceq$$ kn-greatest approximations. We show that for any consistent (that is, satisfiable) theory T in the language of inclusion-exclusion there exists a $$\preceq$$ kn-greatest rough set approximating all sets X that satisfy T. For some classes of theories in the language of inclusion-exclusion, we provide algorithmic ways to find this best approximation. We also state a number of miscellaneous results and discuss some open problems. (This is an extended version of the first part of the presentation made by the authors at the RSCTC98, Rough Sets and Current Trends in Computing, an international meeting held in Warsaw, Poland, in June 1998. Address for correspondence: Department of Computer Science, University of Kentucky, Lexington, KY 40506-0046.)