Attribute Reduction in Intuitionistic Fuzzy Concept Lattices
As an effective tool for knowledge discovery, concept lattice has been successfully applied to various fields. And one of the key problems of knowledge discovery is attribute reduction. In order to understand the problems better, the attribute reduction is necessary to perfect the theory as well as expand application of concept lattice. This paper introduces the intuitionistic fuzzy theory into the concept lattice theory and proposes a kind of intuitionistic fuzzy concept lattice. Then, an approach to attribute reduction based on the discernibility matrix is proposed and investigated, which makes the discovery of implicit knowledge easier and the representation simpler in data; furthermore, the theory of concept lattice is perfected. The theory of intuitionistic fuzzy concept lattice is useful and meaningful in view of the complexity and fuzziness of information in real world, and the potential value of dealing with information is expected in the future.
Concept lattice stems from the so-called formal concept analysis proposed by Wille in 1982 , which can be depicted by a Hasse diagram, where each node expresses a formal concept. A concept lattice is an ordered hierarchical structure of formal concepts that are defined by a binary relation between a set of objects and a set of attributes. Concretely, each formal concept is the pair (objects, attributes), which consists of two parts: the extension (objects covered by the concept) and intension (attributes describing the concept). As an effective tool for data analysis and knowledge processing, concept lattice has been applied to various fields, such as data mining, information retrieval, and software engineering [2–4].
The intuitionistic fuzzy (, for short) set theory initiated by Atanassov [5, 6] is also an important mathematical structure to cope with imprecise information. set, as an extension of Zadeh’s fuzzy set , considers both membership degree and nonmembership degree which are functions valued in , while a fuzzy set gives a membership degree only. The membership and nonmembership values induce an indeterminacy index, which models the hesitancy degree of how an object satisfies a particular property. So set theory can present vague information better. Recently, set theory has been successfully applied in decision analysis and pattern recognition [8–11].
In recent years, many new achievements on these topics have been achieved on theories such as construction of concept lattice [1, 2, 12–15] and acquisition of rules [16, 17]. For example, the paper  shows that an approach of creating fuzzy concept lattices proposed by Popescu was equivalent to the approach of Krajči called generalized concept lattices in some way by comparing this approach with several other approaches and give a straightforward generalization of Popescu’s approach to nonhomogeneous cases, and Li et al.  investigate the issue of rule acquisition in incomplete decision context. And although rough set theory and formal context analysis are different theories, they have much in common in terms of both goals and methodologies, and a formal context in formal concept analysis corresponds to an information system in rough set theory [18, 19]. Formal concept analysis abstracts the knowledge from a formal context through formal concepts, while rough set theory discovers the knowledge via lower and upper approximations, positive boundary, and negative regions from an information system [18, 20, 21]. In fact, there are strong connections between formal concept analysis and rough set theory, and some researchers have been devoted to comparing and combining these two useful theories [18, 22–24], based on which we can study the concept lattice in the similar way to the rough set. Similarly, the key to attribute reduction is to find the minimal subsets of attributes sets in concept lattice, which can determine a concept lattice isomorphic to the one determined by all attributes while the object set remains unchanged. It makes the discovery of implicit knowledge easier and the representation simpler in data and extends the theory of concept lattice; knowledge reduction in formal concept analysis has attracted much attention [16, 17, 25–40]. For instance, Ganter and Wille  develop a reduction method to remove the reducible attributes and objects of a formal context via some predefined arrow relations. The paper  proposed a method to reduce the size of the concept lattice of a formal context using -means clustering. Reduction approaches were presented to avoid the redundancy in the attributes from the perspectives of extension equivalence in the paper . The paper  investigated the issue of developing efficient knowledge reduction methods for real decision formal contexts and developed a corresponding heuristic algorithm to search for a minimal reduction.
Similarly, attribute reduction in concept lattice is to find the minimum attribute set which can assign the same concepts and hierarchy based on formal context keeping the same objects. Reduction approaches in both the papers [23, 36] are based on the equivalence relation between the objects and attributes. In most situations, however, it is fuzzy or intuitionistic fuzzy. At present, there are some achievements on knowledge reduction based on fuzzy formal context [15, 25, 28, 34], where Lifeng Li introduced and investigated the attribute reduction in fuzzy concept lattices based on the kind of transitive implication operator. However, the study on attribute reduction for intuitionistic fuzzy formal context has not been investigated perfectly, although the discernibility matrix was proposed by using of the cuts of sets and then established the method of the attribute reduction of concept lattice in paper . Because it leaves out some useful information about transforming the concept lattice based on the the formal context to a classical one only by using the cuts of the sets. Thus, for the requirement of knowledge-handling systems, combining set theory and formal concept analysis theory directly can result in a new hybrid mathematical structure by establishing two appropriate operators from other views.
Actually, the relation is an important type of data tables in formal concept analysis in real life. The paper combines the theory with the formal concept analysis, the main purpose of which is to study attribute reduction in concept lattices by introducing a pair of implication operators, and we establish approaches and theories of attribute reduction based on formal context, which is also suitable for classical formal context and fuzzy formal context.
The paper is organized as follows. Section 2 reviews basic definitions in formal concept analysis. We give the definitions and propositions in concept lattice with attributes in Section 3. In the next section, we discuss the corresponding definitions of attribute reduction in concept lattices, and then we divide the attributes into four types and investigate some related propositions and establish some propositions to determine the type of an attribute. In Section 5, the discernibility matrix and discernibility function in concept lattice are introduced, and then we discuss the approach to reduction as well as the corresponding characteristics. Finally, a simple conclusion is given in the paper.
To make this paper self-contained, the set theory and involved notions of formal concept analysis are introduced briefly. Detailed description of them can be found in corresponding references.
Definition 1 (Ganter and Wille ). A triple is called a formal context, if is an object set, where is called an object; is an attribute set, where is called an attribute; and is a binary relation between and .
In a formal context , if , that is, , we say that the object has the attribute , or that is fulfilled by . For convenience, we use “1” and “0” to represent and , respectively. Thus, a formal context can be represented by a table only with 0 and 1.
For a formal context , a pair of dual operators for and is defined as follows:
In fact, is the set of all the attributes shared by all the objects in , and is the set of all the objects that fulfill all the attributes in .
Meanwhile, the complement sets of and are denoted by and , where and .
Proposition 2 (Ganter and Wille ). Let be a formal context, and ; the following properties hold:(1).(2).(3).(4).(5).(6).
Definition 3 (Ganter and Wille ). Let be a formal context. A pair is called a formal concept (in brief a concept) if and for . Furthermore, and are called the extension and the intension of , respectively.
From the above discussions, it is clear that both and are concepts.
According to [23, 27], we have the corresponding account as follows.
For convenience, all concepts of a formal context are denoted by , and they are ordered by where and are concepts. Moreover, is called a subconcept of , and is called a superconcept of . And means that and hold at the same time. If and there does not exist a concept such that , then is called a child concept (immediate subconcept) of and is called a parent concept (immediate superconcept) of , and this is denoted by .
For any two concepts and of a formal context , it is easy to prove that both and are also concepts. Hence, if the meet and join are given by (Ganter and Wille ): then the concept lattice is complete lattice.
Definition 4 (Atanassov ). Let be a finite and non-empty set called universe. An set of has the following form: where and and and are, respectively, called the membership degree and nonmembership degree to of the object . Furthermore, they satisfy for any . In general, we use to denote all sets in the universe .
Definition 5 (Atanassov ). Let , , and for any . If both and , then we say is equal to , denoted by . The universe set and empty set are special set, where and .
Let us denote intersection and union of and by and , respectively. Moreover, we denote complement of by .
Definition 6 (Atanassov ). Let ; then
Many properties of these operators in set theory are similar to fuzzy set theory. Detailed description can be found easily in the corresponding references.
3. IF Concept Lattice
The definition of concept lattice with attributes is introduced, and some important properties are discussed in this section.
Definition 7. A triple is called an formal context, if is an object set, where is called an object; is an attribute set, where is called an attribute; and is an set of , where , , and .
The complement of is denoted by .
We denote ; then the set of is denoted by
Let , ; then
With respect to an formal context , for , and , , where .
A pair of operators is defined by where and denote . where and denote , if .
Similarly, , we use and instead of and , respectively, and for any denote
Example 8. An formal context is shown as in Table 1.
In this context, let and ,,,, where ; then from the definition we can obtain that
Proposition 9. Let be an formal context, , ; then the above operators have the following properties:(1).(2).(3).(4).(5).(6).
Proof. (1) Denote that and , so it can be known for any Since , it is true that . It follows that ; that is, . In addition, from the above definitions, we can obtain that if , , otherwise . So Since , we can obtain that . It follows that implies for any . Hence, if , then ; that is, .(2) On one hand, assume that ; then , where , according to Definition 7. If , then . Thus, ; that is, . On the other hand, assume that ; then we can denote , where , if . And we can obtain that , which follows that for any and , holds. So, we can obtain that . Hence, .(3) It is obvious from (1) and (2).(4) From (1) we can have , and from (2) we conclude .(5) It is obvious that . Furthermore, and , . Hence, . can be obtained similarly.(6) It can be easily proved from (1).
Definition 10. Let be an formal context. A pair is called an formal concept (in brief a concept) if and for , . and are called the extension and the intension of , respectively.
From the above, it is clear that both and are concepts.
IF concept lattice is referred to as all concepts of an IF formal context , and they are ordered by where and are concepts. is called a subconcept of , and is called a superconcept of .
And we denote the family of all IF concept lattices by .
Proposition 11. If and are two concepts of an IF formal context , then both and are also concepts.
Proof. It is straight from the definition and Proposition 9.
Hence, from the above, if the meet and join are given by then the IF concept lattice is complete lattice.
Example 12. In Example 8, we can find all concepts of the IF formal context by the definition, which are , , , , , , , , , , respectively, and we denote objects set by which is same to others, where
Furthermore, we can obtain the following IF concept lattice of the IF context (Figure 1).
4. Attribute Reduction in IF Concept Lattices
Definition 13. Let and be two IF concept lattices. If for any , there exists such that , then we say that is coarser than or is thinner than , denoted by
If and , we say that and are isomorphic with each other and denoted by .
Definition 14. Let be an IF formal context. The set of all extensions of is defined to be
Let be an IF formal context and . We denote , where is an IF set of ; that is, . Obviously, is also an IF formal context, we denote all concepts of by similarly. For any , it satisfies that if , , otherwise , and .
Proposition 15. Let be an IF formal context. If , and , then and hold.
Proposition 16. Let be an IF formal context. If and , then there must exist the following relation:
Proof. For any , and . From the above discussions, we know that is concept; thus we only need to verify .
According to Proposition 9 (2), we can obtain that . In addition to Proposition 15, . Therefore, .
Corollary 17. Let be an IF formal context. If , , then .
Definition 18. Let be an IF formal context, . We say that is a consistent set of , if . If is a consistent set, and for any , there exists , then is called an attribute reduction of . The intersection set of all reductions is called the core of .
Obviously, we can obtain the following propositions by the above definition.
Proposition 19. For any , there must exist a reduction of it.
Proof. This proposition is immediately obtained in a similar way to the .
Proposition 20. Let be an IF formal context. If and , then
Definition 21. Let be an IF formal context and suppose that is an index set and all the reductions are denoted by . Then, attributes can be divided into four types as follows:(1)absolutely necessary attribute (core attribute) ;(2)relatively necessary attribute ;(3)absolutely unnecessary attribute ;(4)unnecessary attribute .
In general, the reduction of is not unique. An example will be used to illustrate the above discussions as follows.
Example 22. For the IF formal context in Table 1, if we take out from the attributes set , then we can obtain a new IF formal context , where . And we can get all concepts of , they are , , , , , , , , , and marked by IFC1′, IFC2′, FC3′, IFC4′, IFC5′, IFC6′, IFC7′, IFC8′, IFC9′, and IFC10′, respectively, where
In addition, we can obtain concept lattice of context , as shown in Figure 2.
From Figures 1 and 2, we can find easily that and are isomorphic. So, is a consistent set of . In fact, we can find that , by calculation. Hence, is a reduction of .
If we take out from the attributes set , then we can obtain a new IF formal context , where . And we can get all concepts of , which are , , , , , , , , , and , respectively, where
Obviously, is isomorphic with .
Corollary 23. The core is the reduction The reduction is only one.
Assume that the core is the reduction, and the reduction is not unique; that is, there are two reductions: at least. Hence, the core of the reductions . For is the reduction, the proper subset of it (where it is the core of the reductions) must not be the reduction. This clearly contradicts the known conditions. So, if the core is the reduction, the reduction is only one.
Obviously, the following corollaries can be obtained by the above definitions and propositions.
Corollary 24. Let be an IF formal context; is a core attribute is not a consistent set.
Corollary 25. Let be an IF formal context, is an unnecessary attribute is a consistent set.
Since the reduction of an IF formal context satisfies the following conditions: (1) a consistent set; (2) is not a consistent set; it is helpful to give the necessary and sufficient conditions of consistent sets in order to get reductions more easily.
Proposition 26. Let be an IF formal context, .
Proof. Assume that is a consistent set; then we have according to Proposition 20. For any , it is easy to see that . Thus, by Definition 13, there exists such that . Hence, and , which concludes .
Conversely, suppose that for any . To prove that is a consistent set of , then it suffices to show that for any , there exists such that . Thus, suppose that ; then we can get that and . Taking , then we can obtain that , and so .
Corollary 27. Let be an IF formal context, , and . Then is a consistent set of , such that .
Proof. It can be certified easily by Proposition 20.
Proposition 28. Let be an IF formal context, , and . Then, is a consistent set of if and only if , such that .
Proof. Suppose that is a consistent set; then for any . For any , let , with , , for ; then . Thus, there exists such that then by Corollary 25.
Suppose that there exists such that for any . Let ; then and . Let with and , then . Hence, there exists such that , so . Now it follows from Proposition 9 (3) and Definition 10 that , and so , which means that is a consistent set.
The functions of attributes, which are closely related to consistent sets, vary from one to another. So, sufficient conditions to determine the type of attributes are useful in attribute reduction.
Proposition 29. Let be an IF formal context. Then, is an absolutely necessary attribute if there exists such that , and for any .
Proof. Suppose that is an unnecessary attribute, then is a consistent set; that is, . Let , then . Since , we know that and . Assume that and , then , . From , we know that . But since for any , . It follows that and so . That is to say, in IF formal context , every concept which contains also contains . But . Consistently, there is no concept in whose extent is equal to . Thus, is not a consistent set, which gets a contradiction. Therefore, is an absolutely necessary attribute.
Proposition 30. Let be an IF formal context. Then, is an unnecessary attribute if the following conditions hold: for any , if , then there exists such that . Moreover, if there exists such that , then .
Proof. Suppose that , . It suffices to prove that is consistent set. By Corollary 27, it remains to prove that for any , there exists such that . So suppose that , where , , .
If for any , , then let , , , , and so we can get that .
Otherwise, assume that there are , such that ; then there exist, according the condition, such that . Moreover, if there exists such that , then . Let , where then it follows that . So we know that if , that is, , then . Then, and . It follows that and so . If , that is, there exists such that . If , that is, , then ; that is, , and so . If , that is, then ; that is, , and so . Hence, we conclude that for any there exists such that .
Proposition 31. Let be an IF formal context. Then is an unnecessary attribute if there exists such that for any , implies that . Moreover, if is an absolutely necessary attribute, then is an absolutely unnecessary attribute.
Proof. We denote , , and . It suffices to prove that is consistent set. By Corollary 27, it remains to prove that for any , there exists , such that . So suppose that , where , , , .
If for any , , then let , , and it follows that .
Otherwise, there exists such that ; then . Denote to be the set whose elements satisfy the condition that . Then, and thus . Let , where then it follows that . So we know that if , that is, for all , then and . Then, and . It follows that and so . If , then or . If , then . If , then , and so, there exists such that . If , that is, , then ; that is, , and so that is, and so . Thus, we conclude that for any there exists such that . Therefore is an unnecessary attribute.
Moreover, suppose that is an absolutely necessary attribute and is a consistent set which contains . Since is an absolutely necessary attribute, we have ; thus is also a consistent set, that is, is not a reduction. Therefore, is an absolutely unnecessary attribute.
Corollary 32. Let be an IF formal context. Then, is an absolutely unnecessary attribute if for any , .
5. Approach to Reduction
In this section, discernibility matrix and discernibility function [18, 41] are introduced to compute all reductions for an IF formal context based on the conclusions discussed in Section 4, and we discuss the approach to reduction as well as the corresponding characteristics. Furthermore, we also show corresponding reduction algorithm.
Definition 33. Let be an IF formal context and , we define