Mathematical Problems in Engineering

Volume 2018, Article ID 9546846, 11 pages

https://doi.org/10.1155/2018/9546846

## Three-Way Concept Analysis for Incomplete Formal Contexts

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

Correspondence should be addressed to Hao Chao; moc.361@1891oahoahc

Received 3 May 2018; Revised 20 July 2018; Accepted 9 August 2018; Published 12 September 2018

Academic Editor: Yuqiang Wu

Copyright © 2018 Huilai Zhi and Hao Chao. 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

Recently, incomplete formal contexts have received more and more attention from the communities of formal concept analysis. Different from a complete context where the binary relations between all the objects and attribute are known, an incomplete formal context has at least a pair of object and attribute with a completely unknown binary relation. Partially known formal concepts use interval sets to indicate the incompleteness. Three-way formal concept analysis is capable of characterizing a target set by combining positive and negative attributes. However, how to describe target set, by pointing out what attributes it has with certainty and what attributes it has with possibility and what attributes it does not has with certainty and what attributes it does not has with possibility, is still an open problem. This paper combines the ideas of three-way formal concept analysis and partially known formal concepts and presents a framework of approximate three-way concept analysis. At first, approximate object-induced and attribute-induced three-way concept lattices are introduced, respectively. And then, the relationship between approximate three-way concept lattice and classical three-way concept lattice are investigated. Finally, examples are presented to demonstrate and verify the obtained results.

#### 1. Introduction

*Formal concept analysis* (FCA), a mathematical framework founded on lattice theory [1], has become an effective data analysis tool in various applications [2–5]. Formal context and formal concept are two cornerstones in FCA. A formal context, which is composed of a nonempty set of objects and a nonempty set of attributes, describes a domain in an ideal case. Given any object of the context, one can precisely point out which attributes it possesses and which attributes it does not possess. Intuitively, a concept is composed of two parts, that is, a set of objects, which is known as the extent of the concept, and a set of attributes, which is known as the intent of the concept. The “jointly possessed” relationship between the extent and the intent of a concept is affirmative; i.e., each object of the extent has all the attributes in the intent with certainty and each attribute of the intent is enjoyed by all the objects in the extent with certainty.

In classical FCA, only positive attributes are concerned. As a result, given a set of objects, one cannot directly point out which attributes they do not possess. By incorporating the ideas of three-way decisions [6, 7], Qi et al. [8] proposed* three-way concept analysis* (3WCA), in which one can characterize target set by both positive and negative attributes. Studies have shown that three-way concepts can provide more details than that of classical concepts [9]. Intuitively, the intent of an object-induced three-way concept has positive part and negative part. Positive part points out which attributes are enjoyed by all the objects in the extent, while negative part points out which attributes are not enjoyed by all the objects in the extent. Dually, the extent of an attribute-induced three-way concept is also equipped with both positive part and negative part.

Since its the proposal, 3WCA has become a hot topic. For instance, Qian et al. [10] proposed a method of constructing three-way concept lattices by using divide and conquer strategy. Singh [11, 12] used three-way fuzzy concept lattice to facilitate medical diagnoses and also took uncertain attributes into consideration and put forward a method of decomposing a context based on positive, negative, and uncertain attributes. Li et al. [13] studied the cognitive learning process based on three-way concepts. Zhi and Li [14] studied granule description based on positive and negative attributes.

However, in many circumstances, we can not describe the relationships between objects and attributes precisely and completely. Incomplete formal contexts, introduced by Burmeister and Holzer [15], are a widely adopted approach to deal with the incompleteness of data. In an incomplete formal context, an object enjoys a set of attribute and does not enjoy another set of attributes; meanwhile, whether this object enjoys the rest of attributes is not known. Partially known formal concepts, which extend classical formal concepts and use interval set to manage the incompleteness [16–20], have been intensively studied. Ill-known formal concept [21] and approximate concepts [22, 23] are all examples of partially known formal concepts. However, in the aforementioned studies only positive attributes were considered while negative ones were ignored. In order to manage incompleteness and take both positive and negative attributes into consideration, it is appealing and interesting to combine 3WCA and partially known formal concepts together. And this is also the objective of this paper.

This paper is organized as follows. Section 2 briefly reviews some basic notions in FCA, 3WCA, and incomplete formal contexts. Section 3 introduces approximate object-induced three-way concept lattice. Besides, the relationships between approximate object-induced three-way concept lattice and classical object-induced three-way concept lattice [8, 9] is also investigated. By applying duality principle, Section 4 briefly presents approximate attribute-induced three-way concept lattice. Finally, conclusions are provided in Section 5.

#### 2. Related Theoretical Foundations

In this section, in order to make our paper self-contained, we briefly review some basic notions in FCA, 3WCA, and incomplete formal contexts.

##### 2.1. Formal Concept Analysis

The discussions of formal concept analysis always start with the notion of a formal context defined as follows.

*Definition 1 (see [24]). *A formal context is a triple , where and are two nonempty and finite sets, and is a binary relation on specifies the relationships between and . Moreover, and are called object set and attribute set, respectively, and, for any and , one uses to indicate that the object has the attribute and to indicate the opposite.

In order to define a formal concept, one has to define two derivation operators as follows.

*Definition 2 (see [24]). *Let be a formal context. For and ; the derivation operations and are defined by respectively.

By using the above two derivation operators, formal concepts can be defined as follows.

*Definition 3 (see [24]). *Let be a formal context, and . The pair is called a formal concept, if and .

Then, all the concepts contained in , ordered by form a complete lattice, which is known as the concept lattice of and denoted by .

##### 2.2. Three-Way Concept Analysis

Concept lattice uses positive attributes to characterize a target set. However, many studies have shown that positive attributes and negative attributes are of equal importance in describing a target [25–28]. By taking both positive and negative attributes into consideration, Qi et al. [8, 9] proposed* three-way concept analysis* (3WCA), which has become a hot topic in FCA [10, 20, 23].

*Definition 4 (see [24]). *Let be a formal context. For and , the negative derivation operations and are defined by respectively, where .

*Definition 5 (see [8, 9]). *Let be a formal context. For and , the three-way derivation operations and are defined by respectively.

*Definition 6 (see [8, 9]). *Let be a formal context, and . The pair is called an object-induced three-way concept, if and .

Then, all the object-induced three-way concepts contained in , ordered by form a complete lattice, which is called the object-induced three-way concept lattice ( for short) of and denoted by . The algorithm for building can be found in [9].

Dually, we can define attribute-induced three-way concept lattice ( for short) of and denote it by . Interested readers can refer to [8, 9] for more details.

Both object-induced three-way concepts and attribute-induced three-way concepts are collectively called three-way concepts. Likewise, both object-induced three-way concept lattices and attribute-induced three-way concept lattices are collectively called three-way concept lattices.

##### 2.3. Incomplete Formal Context

To establish a sound semantical basis, a possible world semantics of an incomplete formal context as a family of complete formal contexts was introduced by several researchers [15, 29, 30]. More appealingly, Djouadi [21] provided an elegant interpretation of an incomplete context in terms of two special completions.

*Definition 7 (see [15]). *An incomplete context is a quadruple consisting of a nonempty and finite set of objects, a nonempty and finite set of attributes, the set of values, and a mapping , where means has the attribute , means that the object does not have the attribute , and indicates that it is unknown whether or not the object has the attribute .

In the remainder of this paper, a formal context defined in Definition 1 is called a complete context in order to distinguish it from an incomplete context.

*Definition 8 (see [20]). *A complete formal context is called a completion of if satisfies the following two conditions:

Here, the binary is derived by changing each into either + or −.

*Definition 9 (see [20]). *In an incomplete context , the least completion and the greatest completion are defined by respectively.

*Example 10. *Table 1 depicts an incomplete context in which and .