Local attribute reductions of formal contexts
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ORIGINAL ARTICLE
Local attribute reductions of formal contexts Keyun Qin1 · Hong Lin1 · Yuting Jiang1 Received: 3 August 2018 / Accepted: 23 April 2019 / Published online: 30 April 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract Attribute reductions is an important topic in formal concept analysis. The existing attribute reduction approaches are dominated by global reductions and there is very limited investigation on local reductions. This paper is devoted to the study of decision rule specific reduction for formal decision context and concept specific reduction for formal context. The notion of decision rule specific reduction is proposed and the related reduction methods are presented. The relationships between existing reduction approaches and decision rule specific reduction approaches are analyzed. Accordingly, we make an analysis of attributes based on three-way classification by using reductions. Furthermore, the notion of concept specific reduction for formal context is proposed and the concept specific reduction methods are examined. The relationship between concept specific reduction and decision rule specific reduction is surveyed. Keywords Formal concept analysis · Concept lattice · Decision rule · Attribute reduction
1 Introduction Formal concept analysis (FCA) was introduced by Wille in 1982 [36]. It is formulated based on the notion of a formal context specifying which objects posses what properties or attributes. The basic notions of FCA are formal context, formal concept and the corresponding concept lattice. A formal concept is defined as a pair of a set of objects and a set of attributes connected by two set-theoretic operators. The set of all formal concepts forms a complete lattice, called concept lattice, which reflects the relationship of generalization and specialization among formal concepts. As a useful tool for knowledge description and summarization, FCA has been applied to many fields, such as information retrieval, data mining and knowledge discovery [1–3, 8, 9, 11, 13, 27, 29]. It has become increasingly popular among various methods of conceptual data analysis and knowledge representation. * Hong Lin [email protected] Keyun Qin [email protected] Yuting Jiang [email protected] 1
School of Mathematics, Southwest Jiaotong University, Chengdu 610031, Sichuan, China
The notions of attribute reduction play a fundamental role in FCA. By removing redundant attributes and objects, the efficiency of constructing concept lattice is improved and more compact knowledge is discovered. Ganter and Wille [10] proposed an efficient method, namely, clarification and reduction (CR method for short), by removing the reducible objects and attributes of a formal context. In such a reduced context, the corresponding concept lattice is isomorphic to the one induced from the initial context. Konecny [12] shows that CR method is also effective for attribute reduction of some extended concept lattice models, such as three-way concept lattice [28], generalized one-sided conce
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