Attribute reduction of SE-ISI concept lattices for incomplete contexts
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Attribute reduction of SE-ISI concept lattices for incomplete contexts Zhen Wang1,4 · Ling Wei1,4 · Jianjun Qi2,4 · Ting Qian3,4 Published online: 6 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Three-way concept analysis in incomplete contexts lays the theory dealing with the data in incomplete contexts, especially three kinds of partially known formal concepts including SE-ISI formal concept, ISE-SI formal concept and ISE-ISI formal concept. Generally speaking, not every attribute is essential in an incomplete context since the purpose of research is different. Thus, we propose four kinds of attribute reduction of SE-ISI concept lattices based on different criteria. Then, we discuss the relationships among the four kinds of attribute reduction, including the relationships among the consistent sets and relationships among the reducts. Finally, based on discernibility matrices and discernibility functions, the approaches to obtaining these attribute reduction are presented. Keywords Incomplete context · Concept lattice · SE-ISI formal concept · Attribute reduction · Discernibility matrix
1 Introduction Formal concept analysis (FCA), an efficient tool for decision making and knowledge discovery, was proposed by Wille (1982) and Ganter and Wille (1999). Formal context, formal concept and concept lattice are three basic notions of FCA. Formal context is the data foundation of FCA. Based on a formal context, formal concept, a pair of extent and intent, Communicated by A. Di Nola.
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Ling Wei [email protected] Zhen Wang [email protected] Jianjun Qi [email protected] Ting Qian [email protected]
1
School of Mathematics, Northwest University, Xi’an 710127, Shaanxi, People’s Republic of China
2
School of Computer Science and Technology, Xidian University, Xi’an 710071, Shaanxi, People’s Republic of China
3
College of Science, Xi’an Shiyou University, Xi’an 710065, Shaanxi, People’s Republic of China
4
Institute of Concepts, Cognition and Intelligence, Northwest University, Xi’an 710127, Shaanxi, People’s Republic of China
is obtained by a pair of derivation operators. All the formal concepts can form a complete lattice called a concept lattice, which is the basic structure of FCA. As an effective mathematical tool for conceptual data analysis and knowledge processing, both the theoretical researches and practical applications of FCA have been promoted. For the promotion of theoretical researches, many scholars have studied attribute reduction (Zhang et al. 2005; Wang and Ma 2006; Wei et al. 2008; Wang and Zhang 2008; Wu et al. 2009; Liu et al. 2009; Qi 2009; Li and Wu 2011; Medina 2012; Shao et al. 2013; Liang et al. 2013; Li et al. 2013a; Kumar et al. 2015; Ganter and Obiedkov 2016; Shao and Li 2016; Dias and Vieira 2017; Chen et al. 2018; Li and Zhang 2019), rules acquisition (Wille 1989; Missaoui et al. 1994; Li et al. 2013b; Shao et al. 2014), concept lattice construction (Nourine and Raynaud 1999; Djouadi and Prade 2010; Qian et al. 2017) and so
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