Covering rough sets based on neighborhoods: An approach without using neighborhoods

Rough set theory, a mathematical tool to deal with inexact or uncertain knowledge in information systems, has originally described the indiscernibility of elements by equivalence relations. Covering rough sets are a natural extension of classical rough sets by relaxing the partitions arising from equivalence relations to coverings. Recently, some topological concepts such as neighborhood have been applied to covering rough sets. In this paper, we further investigate the covering rough sets based on neighborhoods by approximation operations. We show that the upper approximation based on neighborhoods can be defined equivalently without using neighborhoods. To analyze the coverings themselves, we introduce unary and composition operations on coverings. A notion of homomorphismis provided to relate two covering approximation spaces. We also examine the properties of approximations preserved by the operations and homomorphisms, respectively.Comment: 13 pages; to appear in International Journal of Approximate Reasonin

Pawlak, Belnap and the magical number seven

We are considering the algebraic structure of the Pawlak-Brouwer-Zadeh lattice to distinguish vagueness due to imprecision from ambiguity due to coarseness. We show that a general class of many-valued logics useful for reasoning about data emerges from this context. All these logics can be obtained from a very general seven-valued logic which, interestingly enough, corresponds to a reasoning system developed by Jaina philosophers four centuries BC. In particular, we show how the celebrated Belnap four-valued logic can be obtained from the very general seven-valued logic based on the Pawlak-Brouwer-Zadeh lattice

Rough set theory and Galois connections in formal context

Se estudia la relación entre las Conexiones de Galois (CG) y la Rough Set Theory (RST) en Contextos Formales (CF). El objetivo de este trabajo es mostrar la fertilidad teórica que se manifiesta al expresar RST en base a CG en Contextos Formales, ya que las Conexiones de Galois participan en estructuras lógico matemáticas numerosas e interconectadas y forman parte de la base de la Informática teórica y aplicada. Se analizó la bibliografía específica y se destacó: a) la importancia del Teorema que conecta CG y RST, ya que asegura la ampliación de las aplicaciones de RST como se conjetura en el trabajo; b) El papel del Análisis de Conceptos Formales (Formal Concept Analysis- FCA) en lo conjeturado en a). Las CG, en tanto que relaciones binarias generalizadas, constituyen un puente que conecta tres niveles principales en los sistemas de información: A) computacional; B) algorítmico; C) Implementacional, que se encuentra en el nivel físico del sistema. Además, las Conexiones de Galois permiten expresar la Relación de Indiscernibilidad de RST empleando retículos distributivos y complementados, es decir por medio de Álgebras de Boole.Fil: Galardo, Osvaldo Jorge. Universidad Nacional de La Matanza; Argentina

Pseudo-Kleene algebras determined by rough sets

We study the pseudo-Kleene algebras of the Dedekind-MacNeille completion of the ordered set of rough set determined by a reflexive relation. We characterize the cases when PBZ and PBZ*-lattices can be defined on these pseudo-Kleene algebras.Comment: 24 pages, minor update to the initial versio

Three-valued logics, uncertainty management and rough sets

This paper is a survey of the connections between three-valued logics and rough sets from the point of view of incomplete information management. Based on the fact that many three-valued logics can be put under a unique algebraic umbrella, we show how to translate three-valued conjunctions and implications into operations on ill-known sets such as rough sets. We then show that while such translations may provide mathematically elegant algebraic settings for rough sets, the interpretability of these connectives in terms of an original set approximated via an equivalence relation is very limited, thus casting doubts on the practical relevance of truth-functional logical renderings of rough sets