Four-valued extension of rough sets

  • Authors:

  • Aida Vitória;Andrzej Szałas;Jan Małuszyński

  • Affiliations:

  • Department of Science and Technology, Linköping University, Norrköping, Sweden;Institute of Informatics, Warsaw University, Warsaw, Poland and Department of Computer and Information Science, Linköping University, Linköping, Sweden;Department of Computer and Information Science, Linköping University, Linköping, Sweden

  • Venue:
  • RSKT'08 Proceedings of the 3rd international conference on Rough sets and knowledge technology
  • Year:
  • 2008

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Abstract

Rough set approximations of Pawlak [15] are sometimes generalized by using similarities between objects rather than elementary sets. In practical applications, both knowledge about properties of objects and knowledge of similarity between objects can be incomplete and inconsistent. The aim of this paper is to define set approximations when all sets, and their approximations, as well as similarity relations are four-valued. A set is four-valued in the sense that its membership function can have one of the four logical values: unknown (u), false (f), inconsistent (i), or true (t). To this end, a new implication operator and set-theoretical operations on four-valued sets, such as set containment, are introduced. Several properties of lower and upper approximations of four-valued sets are also presented.