Scientific Programming

Volume 2019, Article ID 7865197, 11 pages

https://doi.org/10.1155/2019/7865197

## Influences of Three-Way Concept Lattice Caused by Variations of Attribute Values

School of Computer Science and Technology, Henan Polytechnic University, Jiaozuo 454000, Henan, China

Correspondence should be addressed to Huilai Zhi; moc.621@ialiuhihz

Received 10 May 2019; Revised 3 August 2019; Accepted 19 August 2019; Published 30 September 2019

Academic Editor: Emiliano Tramontana

Copyright © 2019 Huilai Zhi and Shulin Hu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Three-way concept lattices have been widely used in various types of applications. As the construction of three-way concept lattices is rather time consuming, especially for large formal contexts, it is not applicable to construct the lattices from the beginning when changes are made to the contexts. Motivated by this problem, the influences of three-way concept lattices caused by variations of attribute values are explored in this study. Specifically, we discuss two types of changes. One is changing the value of a specific incidence relation from 0 to 1, and the other is from 1 to 0. Furthermore, two types of three-way concept lattices are investigated. One is the object-induced three-way concept lattice, and the other is the attribute-induced three-way concept lattice. Both the mathematical proofs and the examples show the effectiveness of our proposed methods.

#### 1. Introduction

Formal concept analysis (FCA) is a mathematical theory put forward by Wille in 1982 [1, 2]. Concept lattice is one of the main outcomes of FCA and describes a domain by using a set of concepts. At present, FCA has been used as an effective tool in data analysis and knowledge discovery. For instance, it has been applied in the fields of CT analysis [3], expert systems [4, 5], data mining [6, 7], data clustering [8, 9], software engineering [10], etc. However, it is worth noting that although both positive attributes and negative attributes play equal roles in many knowledge-based activities [11, 12, 13], classical concepts concern only positive attributes while left negative ones aside. As a result, we may lose many useful information.

By taking both positive and negative attributes into consideration, Qi et al. [14, 15] proposed the theory of three-way concept analysis (3WCA). In essence, 3WCA generalizes classical FCA by absorbing the theory of three decisions [16–21]. In 3WCA, there are two types of concept lattices. One is object-induced three-way concept lattice, and the other is attribute-induced three-way concept lattice. It has been proved that as a generalization of classical concepts, three-way concepts can provide more details than that of classical concepts [15]. On the basis of an object-induced three-way concept, we divide the attributes of the intent of this concept into three disjoint parts: positive parts, negative parts, and the rest, based on which we can make further decisions.

Nowadays, three-way concept analysis has attracted more and more attentions. Li and Wang [22] proposed three-way approximate concepts under the framework of three-way concept analysis. Shivhare and Cherukuri [23] applied the idea of three-way concept analysis to simulate a cognitive learning process. Mouliswaran et al. [24] designed a role-based access control scheme by using three-way formal concept analysis. Ren and Wei [25] studied the attribute reduction of three-way concept lattices. Li et al. [26] described a three-way cognitive concept learning process from a multigranularity perspective. Zhi and Li [27] studied granular description based on positive and negative attributes. Furthermore, in order to manage fuzziness, Singh [28, 29] proposed three-way fuzzy concepts. In addition, Singh also generalized three-way concepts to bipolar fuzzy concepts [30] and m-polar fuzzy concepts [31] to derive useful concepts from fuzzy contexts for decision making.

As three-way concept lattice is of vital importance in 3WCA-based applications, the construction of three-way concept lattices is an important topic and deserves our continuous efforts. For instance, Qi et al. [14, 15] elaborated on the relationship between three-way concept lattices and classical concept lattices and constructed three-way concept lattices by combining the sublattices of classical concept lattices.

Compared with the construction of three-way concept lattices, how to effectively update the three-way concept lattices to cater the continuous changing of formal context has not received enough attentions. However, this is very important both in theoretical research and applications. Let’s consider the following example.

*Example 1. *Consider five European students, each of which wants to learn some courses about Chinese history. At present, the available courses are about five dynasties of Chinese history, i.e., Tang Dynasty, Song Dynasty, Yuan Dynasty, Ming Dynasty, and Qing Dynasty. Before they come to China, each student has chosen their interested dynasties. In Table 1, their interests are listed and a “” indicates a student favors a specific dynasty. For instance, as the Yuan Dynasty had a great influence on European history, four students have chosen this dynasty. By using this table, we want to analysis the main interests of the students.

However, after they came to China, some of them changed their minds. For instance, the charming of Xi’an attracted another two students to choose Tang Dynasty. At the same time, three students removed Yuan Dynasty from their curriculums. Accordingly, Table 1 is updated to Table 2, and we have to update our analysis result. Furthermore, if the analysis is based on three-way concept lattices, we must initially update the three-way concept lattice and then update the analysis result. Then, incremental approaches are needed to solve this problem, and studies have shown their effectiveness to cater dynamic systems [32, 33, 34].

Motivated by the above problems, we will investigate the approach to update the three-way concept lattice caused by variations of attribute values. Specifically, this problem can be formulated as follows: given a formal context and its three-way concept lattice, if the attribute values are changed, how to update the corresponding three-way concept lattice. As in dynamic information systems, the values of attributes keep changing from time to time and the construction of a concept lattice is rather time consuming, the problem discussed in this paper is extremely important in both theoretical research and applications.

The rest of this paper is organized as follows. Section 2 briefly reviews some basic notions in 3WCA. In Section 3, the influences of changing attribute values on the object-induced three-way concept lattice are studied, and the algorithms for updating the object-induced three-way concept lattice are proposed. In Section 4, we investigate the effect of changing attribute values on the attribute-induced three-way concept lattice. Finally, conclusions are provided in Section 5.