Attribute reduction in inconsistent formal decision contexts based on congruence relations
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ORIGINAL ARTICLE
Attribute reduction in inconsistent formal decision contexts based on congruence relations Jun-Yu Li1,2 • Xia Wang1,2 • Wei-Zhi Wu1,2 • You-Hong Xu1,2
Received: 1 July 2016 / Accepted: 16 August 2016 Ó Springer-Verlag Berlin Heidelberg 2016
Abstract In this paper, notions and methods of attribute reduction are investigated for an inconsistent formal decision context. Based on congruence relations defined on the object power set, we first introduce notions of distribution attribute reduct and maximum distribution attribute reduct for an inconsistent formal decision context, and discuss their relations in detail. We then define discernibility matrices and discernibility functions associated with the proposed attribute reducts, from which we can calculate all attribute reducts. Finally, we compare the proposed consistent sets with four types of consistent sets in previously published papers. The results show that a distribution consistent set belongs to any of those four types of consistent sets. Therefore, it has all the properties of those four types of consistent sets. Keywords Inconsistent formal decision context Attribute reduction Congruence relation Rough set Three-way concept analysis
& Xia Wang [email protected] Jun-Yu Li [email protected] Wei-Zhi Wu [email protected] You-Hong Xu [email protected] 1
School of Mathematics, Physics and Information Science, Zhejiang Ocean University, Zhoushan 316022, Zhejiang, People’s Republic of China
2
Key Laboratory of Oceanographic Big Data Mining and Application of Zhejiang Province, Zhoushan 316022, Zhejiang, People’s Republic of China
1 Introduction As two significant tools to extract information from databases, formal concept analysis [6, 54] and rough set theory [31] have closely connections though they deal with data from different perspectives. Many researches [5, 9, 10, 18, 19, 34, 42, 56–59] have been devoted to comparing and combining these two effective theories. In addition, a theory of three-way decisions was originally proposed by Yao [60, 61] to explain the three regions of probabilistic rough sets, and widely used to problemsolving and information-processing practice [62]. Recently it has been successfully introduced into formal concept analysis to construct three-way concept analysis [20, 34, 35, 61]. It is well known that one of the key problems of formal concept analysis and rough set theory is knowledge reduction, which reduces the computational complexity of extracting information from databases. Many types of approaches to knowledge reduction in rough set theory have been proposed [4, 8, 11, 26, 27, 32, 38–41, 43, 63, 64, 66] based on information systems, consistent decision information systems and inconsistent decision information systems respectively. Moreover, most of those approaches to knowledge reduction in rough set theory are based on binary relations such as equivalence relations, partial ordering relations and so on which define on the object (attribute) set or the object (attribute) power set. In recent years, mu
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