A discernibility matrix for the topological reduction
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ORIGINAL ARTICLE
A discernibility matrix for the topological reduction Peirong Lin
Received: 14 September 2011 / Accepted: 14 December 2011 / Published online: 28 December 2011 Ó Springer-Verlag 2011
Abstract Discernibility matrices play an important role in the attribute reduction of information systems. The reduction of a family of general relations, which preserves the topological base, is an extension of the attribute reduction of information systems. In this paper, we construct a new discernibility matrix for the topological reduction of a family of general relations. Keywords Discernibility matrix General relation Reduction Topological base
1 Introduction Rough set theory was originally proposed by Pawlak [11, 12] as a mathematical tool to deal with imprecision and vagueness in information systems. So far, rough set theory has been successfully applied in many fields such as process control [2], knowledge acquisition [7, 13, 20], machine learning [3, 8, 14] and decision analysis [1, 21, 22]. The classical rough set theory is based on equivalence relations. However, equivalence relation is still restrictive for many applications. To make rough sets more suitable for application, equivalence relations are extended to more general relations [24], or partitions are extended to coverings [25, 28]. Topology is an efficient mathematical tool to study rough sets [4–6, 23, 28].
P. Lin (&) Department of Computer Sciences and Engineering, Zhangzhou Normal University, Fujian 363000, People’s Republic of China e-mail: [email protected]
The concept of attributes reduction is one of most important topics in the research on rough set theory [17–19]. The discernibility matrix [15] is an important instrument to study the attribute reduction in information systems. Many types of attribute reduction in information systems have been proposed. For example, b-reduct was proposed by Ziarko [26] in the variable precision rough set model. Nguyen et al. introduced the notions of a-reduct and a-relative reduct for decision tables [10]. In [9], the concepts of b lower distribution reduct and b upper distribution reduct based on variable precision rough sets were defined. Slezak [16] presented a new concept of attribute reduction that keeps the class membership distribution unchanging in the information system. In recent years, more attention has been paid to attribute reduction in a family of general relations. The reduction for a family of general relations, which preserves a topological base, was first defined by Lashin [6]. However, the reduction of a family of general relations, which is obtained by the discernibility matrix, may not coincide with the definitions of the minimal reducts and core in [6]. In this paper, we will fill this gap by constructing a new discernibility matrix. The remainder of this paper is structured as follows. In Sect. 2, we briefly introduce some basic definitions for the topological reduction of a family of general relations. In Sect. 3, we study the relation reduction for a family of general relati
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