Contributions to the theory of rough sets

V. Wiktor Marek, Mirosław Truszczyński

Research output: Contribution to journalArticlepeer-review

37 Scopus citations


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 ≤kn). We prove that Pawlak's rough sets are characterized as ≤kn-greatest approximations. We show that for any consistent (that is, satisfiable) theory T in the language of inclusion-exclusion there exists a ≤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.

Original languageEnglish
Pages (from-to)389-409
Number of pages21
JournalFundamenta Informaticae
Issue number4
StatePublished - Sep 1999

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Algebra and Number Theory
  • Information Systems
  • Computational Theory and Mathematics


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