The attribute reductions of three-way concept lattices.

Three-way concept analysis is a newly proposed area of formal concept analysis from which one can obtain both the inclusion decision and the exclusion decision. In general, given a context, some attributes may not be essential in three-way concept analysis, such as forming three-way concept lattice. So in this paper, we study the attribute reductions of three-way concept lattices in order to make the data easily be understood. Firstly, based on different criteria generated from object-induced three-way concept (OE-concept), four kinds of attribute reductions are proposed. The four reductions together embody different characteristics of a formal context and can be used in different occasions. Secondly, we discuss their relationships, including their advantages and disadvantages and the relationships among consistent sets and among the cores. Thirdly, based on attribute-induced three-way concept (AE-concept), we also give four attribute-induced three-way attribute reductions and discuss their relationships. Finally, the approaches to computing these attribute reductions are presented and the obtained results are demonstrated and verified by an empirical case. In this paper, we systematically investigate the attribute reductions of three-way concept lattices which enriches the study of formal concept analysis.

Introduction

The three-way decision (3WD), an extension of the two-way decision model with an added third option, is a common human practice in problem solving and is widely used in our daily life’s decision making process [1]. Because of its extensive usages, Yao [2] outlined the theory of three-way decision. The basic idea of 3WD is to divide the universe of discourse into three disjoint subsets by given criteria. These three disjoint parts are called positive, negative and boundary regions from which one can construct different rules to make three-way decision. One can construct rules of acceptance from positive region, rules of rejection from negative region and rules of non-commitment from boundary region.

In the past few years, more and more attentions have been paid to this newly proposed theory and the 3WD has been studied from the different views. Actually, 3WD plays a key role in knowledge-based systems and has been widely used in many fields and disciplines, including computer science, information science, management science, engineering, social science, medical decision-making, etc.[1], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13].

Formal concept analysis (FCA), as an efficient tool for decision making and knowledge discovery, was proposed by Wille in 1982 [14], and has subsequently been extended by Wille and Ganter [15] and other scholars. As an effective method for data analysis and knowledge processing, FCA has been used in various areas, such as data mining, information retrieval, and software engineering [16], [17], [18], [19], [20], [21], [22].

However, in FCA we can only get the two-way decision rather than the three-way decision. That is, based on a formal concept, one can only determine whether an object (attribute) certainly possesses (is shared by) all elements in the intent (extent) and such decision is called inclusion method. Actually, the formal context also offers us the information about whether an object (attribute) does not possesses (is not shared by) any attribute (object), which is called exclusion method, but this cannot be reflected in the formal concept. Since this kind of exclusion method is commonly used in our daily life, Qi et al. [23] applied 3WD to FCA and proposed three-way concept analysis (3WCA). Two key components in 3WCA are three-way concepts and three-way concept lattices. Similar with the formal concept in FCA, a three-way concept is also determined by two parts named extent and intent. However, the extent or intent of a three-way concept should be an orthopair studied by Ciucci [24]. That is to say, in 3WCA, “the extent (or the intent) of a three way concept is equipped with two parts: positive one and negative one. These two parts are used to express the semantics ‘jointly possessed’ and ‘jointly not possessed’ in a formal context.” [25] This newly proposed three-way concept combines inclusion method (positive region) with exclusion method (negative region). Thus based on a three-way concept, one can divide the object (attribute) universe into three regions to make three-way decision.

Being an important issue in knowledge discovery, attribute reduction has been extensively studied in different fields of soft computing since it can decrease the dimension and make the data easily be understood [26], [27], [28], [29], [30], [31], [32], [33], [34], [35]. It is also an interesting topic in FCA. Many significant results in attribute reduction have been obtained based on a formal context. For example, Zhang et al. [36] discussed the lattice reduction theory where the notion of “reducing the context” in [15] was developed. Within such framework, Zhang et al. [36] constructed judgment theorems of consistent sets, and developed approaches to attribute reduction based on discernibility matrix, which was given a further simplification by Qi [37] from the viewpoint of parent-child concepts. Based on the lattice reduction theory of formal contexts, Wei et al. [38] also studied the reduction theory of formal decision contexts under two different meanings of consistency. And Li et al. [39] have proposed the attribute reduction in order to preserve the rules in the formal decision context. On the other hand, Wang and Ma [40] proposed another attribute reduction in order to preserve the extents of meet-irreducible elements in the original formal concept lattice. Based on the irreducible elements, Li et al. [41] defined the join-irreducible attribute reduction in order to preserve the extents of join-irreducible elements in the original formal concept lattice. From the viewpoint of granular computing, granular reduction was proposed by Wu et al. [42] Additionally, Liu et al. [43] have studied the attribute reduction of object oriented formal concept lattices and property oriented formal concept lattices, and furthermore Wang et al. [44] discussed relations of attribute reductions between object and property oriented formal concept lattices.

Because of the extensive usages of 3WD and FCA, 3WCA will become a very useful tool for knowledge discovery and data analysis. In order to make the discovery and representation of implicit knowledge in three-way concept lattices easier and simpler, based on the different criteria generated from three-way concepts, we firstly propose different attribute reductions. Since the generalization and the specialization of the knowledge contained in the context are all reflected in the three-way concept lattices, if our reduction can preserve the lattice structure, then the basic knowledge reflected in the three-way concept lattices is preserved. Also, because every three-way concept is the join (meet) of join (meet)-irreducible elements, if our reduction can preserve the irreducible elements, we actually preserve the basic elements in lattice construction. Thus this kind of reduction is very important in lattice construction. From the perspective of granular computation, a three-way object or attribute concept can be regarded as a kind of granule, so a kind of attribute reduction which can preserve the extents of the three-way object (attribute) concepts is proposed. All these attribute reductions together embody different characteristics of a context. The basic ideas of these attribute reductions are similar with the classical attribute reductions mentioned in the previous paragraph. However, different from the attribute reductions in FCA, all the reductions in this paper depend on the three-way concept and the three-way concept can reflect more information than the classical formal concept, so the results and meaning of these attribute reductions in 3WCA are different with the classical ones in FCA. Then, we analyze the relationships among these three-way reductions. Finally, we give some approaches to these attribute reductions. Since the study we make is based on the three-way concept analysis, our work can be regarded as a three-way extension of the attribute reductions mentioned in previous paragraph.

This paper mainly discusses the different attribute reductions of three-way concept lattices and the relationships among them. The content is organized as follows. In Section 2, the basic knowledge of classical and three-way concept analysis is reviewed. Section 3 gives the definitions of four different OE-attribute reductions and discusses their relationships, including the relationships among the corresponding consistent sets and cores. Analogously, in Section 4, we define four different AE-attribute reductions and discuss the relationships among them. Then, the approaches to computing these attribute reductions are presented in Section 5. Finally, an empirical case is shown in Section 6 and this paper is concluded with a summary in Section 7.

Preliminaries

In order to make this paper self-contained, we firstly review some basic notions in formal concept analysis and three-way concept analysis.

Formal concept analysis

[15]

A formal context (G, M, I) consists of two sets G and M, and a relation I between G and M. The elements of G are called the objects and the elements of M are called the attributes of the context. In order to express that an object x is in a relation I with an attribute m, we write xIm or (x, m) ∈ I and read it as “the object x has the attribute m”.

A pair of operators are defined on X ⊆ G and A ⊆ M respectively in (G, M, I) [15]:

The operator * is same with the modal-style operator R in [45], which expresses the meaning of jointly possessing.

The properties of this pair of operators are shown in [15], so we do not describe here.

A formal concept of the context (G, M, I) is a pair (X, A) with and (X ⊆ G, A ⊆ M). We call X the extent and A the intent of the formal concept (X, A).

The formal concepts of a formal context (G, M, I) are ordered by

All formal concepts of (G, M, I) can form a complete lattice called the formal concept lattice of (G, M, I) and denoted by L(G, M, I). The infimum and supremum are given by

In order to distinguish from the following three-way concept analysis, the formal concept mentioned above will be called classical concept and other notions in the FCA also be named in the same way in the following parts of this paper.

Now, we give the definition of reduct as follows.

[36]

Let (G, M, I) be a formal context. If there exists an attribute set D ⊆ M such that L(G, D, I)≅L(G, M, I), then D is called a consistent set of (G, M, I). And further, if the set D is called a reduct of (G, M, I). The intersection of all the reducts is called the core of (G, M, I). Here .

[46]

Let L be a lattice. An element x ∈ L is join-irreducible if

1. x ≠ 0 (in case L has a zero)

implies or for all a, b ∈ L.

A meet-irreducible element is dually defined.

Three-way concept analysis

Applying the idea of three-way decision into formal concept analysis, Qi et al. [23] recently proposed three-way concept analysis. Firstly, based on the formal concept in Definition 2 and the operators defined on it, called positive operators here, another pair of two-way operators are given, named negative operators, as follows:

for X ⊆ G and A ⊆ M, we have

Combining the operators * and a pair of three-way operators are defined on X ⊆ G and A ⊆ M respectively: In relation to the above three-way operators, their inverse can be defined as follows [23]: for X, Y ⊆ G and A, B ⊆ M, Then the three-way concept is given.

[23]

Let (G, M, I) be a formal context. A pair (X, (A, B)) of an object subset X ⊆ G and two attribute subsets A, B ⊆ M is called an object-induced three-way concept, for short, an OE-concept, of (G, M, I), if and only if and . X is called the extent and (A, B) is called the intent of the OE-concept (X, (A, B)).

In particular, (x ⋖⋗, x ⋖) is an OE-concept for all x ∈ G, which is called OE-object concept. Here, for convenience, we write x ⋖ instead of {x}⋖ for any x ∈ G.

If (X, (A, B)) and (Y, (C, D)) are OE-concepts, then they can be ordered by

(X, (A, B)) ≤ (Y, (C, D))⇔X ⊆ Y⇔(C, D) ⊆ (A, B).

Here (C, D) ⊆ (A, B)⇔C ⊆ A and D ⊆ B. (X, (A, B)) is called a sub-concept of (Y, (C, D)), and (Y, (C, D)) is called a super-concept of (X, (A, B)). If there is no OE-concept (Z, (E, F)) such that (X, (A, B)) < (Z, (E, F)) < (Y, (C, D)), then (X, (A, B)) is called a child-concept of (Y, (C, D)), and (Y, (C, D)) is called a parent-concept of (X, (A, B)), which is denoted by (X, (A, B))≺(Y, (C, D)).

All the OE-concepts form a complete lattice, which is called the object-induced three-way concept lattice of (G, M, I) and written as OEL(G, M, I). The infimum and supremum are given by

Similarly, the dual attribute-induced three-way concept is given.

Let (G, M, I) be a formal context. A pair ((X, Y), A) of two object subsets X, Y ⊆ G and an attribute subset A ⊆ M is called an attribute-induced three-way concept, for short, an AE-concept, of (G, M, I), if and only if and . (X, Y) is called the extent and A is called the intent of the AE-concept ((X, Y), A).

In particular, (m ⋖, m ⋖⋗) is an AE-concept for all m ∈ M which is called AE-attribute concept.

If ((X, Y), A) and ((Z, W), B) are AE-concepts, then they can be ordered by

((X, Y), A) ≤ ((Z, W), B)⇔(X, Y) ⊆ (Z, W)⇔B ⊆ A.

((X, Y), A) is called a sub-concept of ((Z, W), B), and ((Z, W), B) is called a super-concept of ((X, Y), A). If there is no AE-concept ((U, V), C) such that ((X, Y), A) < ((U, V), C) < ((Z, W), B), then((X, Y), A) is called a child-concept of ((Z, W), B), and ((Z, W), B) is called a parent-concept of ((X, Y), A), which is denoted by ((X, Y), A)≺((Z, W), B).

All AE-concepts also form a complete lattice, which is called the attribute-induced three-way concept lattice of (G, M, I) and written as AEL(G, M, I). The infimum and supremum are given by

Four types of attribute reductions of an OE-concept lattice

In the classical formal concept analysis, the reduction is defined related to the extents of the formal concepts. In order to give and analyze the attribute reductions of an OE-concept lattice, we firstly give the necessary and sufficient condition for an object set to be an extent of an OE-concept.

Let (G, M, I) be a formal context. Here is the set of all extents of the OE-concepts on (G, M, I).

Suppose (X, (A, B)) is an OE-concept of (G, M, I). From Definition 5, we know that which means . Also from the Definition 5, we can get that is, . Thus, . On the contrary, if it’s obvious that is an OE-concept. □

The three-way concept lattices are the core structures in 3WCA, because the hierarchical knowledge of three-way concepts is all reflected in these three-way concept lattices. Thus based on the OE-concept lattice, we propose the OE-lattice reduction of (G, M, I).

Let (G, M, I) be a formal context. An attribute set D ⊆ M is called an object-induced three-way concept lattice (OE-lattice) consistent set of (G, M, I) if . Here, . Furthermore, if for any d ∈ D, then set D is called an OE-lattice reduct of (G, M, I).

From [15], we know that in a finite lattice, the set of meet-irreducible elements is infimum-dense and the set of join-irreducible elements is supremum-dense. So the meet-irreducible elements and join-irreducible elements are important in the structure of the lattice. Thus from this point of view, we define the object-induced three-way irreducible elements preserving reductions as follows.

Let (G, M, I) be a formal context. An attribute set D ⊆ M is called an object-induced three-way meet-irreducible elements preserving (OE-MIE-preserving) consistent set of (G, M, I) if . If set D is an OE-MIE-preserving consistent set of (G, M, I), and there is no proper subset E ⊂ D such that E is an OE-MIE-preserving consistent set of (G, M, I), then set D is called an OE-MIE-preserving reduct of (G, M, I).

Here, Ext(G, M, I) is the extent set of all the meet-irreducible elements in the OE-concept lattice OEL(G, M, I).

Let (G, M, I) be a formal context. An attribute set D ⊆ M is called an object-induced three-way join-irreducible elements preserving (OE-JIE-preserving) consistent set of (G, M, I) if . If set D is an OE-JIE-preserving consistent set of (G, M, I), and there is no proper subset E ⊂ D such that E is an OE-JIE-preserving consistent set of (G, M, I), then set D is called an OE-JIE-preserving reduct of (G, M, I).

Here, Ext(G, M, I) is the extent set of all the join-irreducible elements in the OE-concept lattice OEL(G, M, I).

Analogous to the classical formal concept, every OE-concept can be represented by the join of OE-object concepts. Hence, every OE-object concept can be regarded as an information granule. Thus from the viewpoint of granular computing, object-induced three-way granular (OE-granular) reduction is proposed.

Let (G, M, I) be a formal context. An attribute set D ⊆ M is called an object-induced three-way granular (OE-granular) consistent set of (G, M, I) if for all x ∈ G. If set D is an OE-granular consistent set of (G, M, I), and there is no proper subset E ⊂ D such that E is an OE-granular consistent set of (G, M, I), then set D is called an OE-granular reduct of (G, M, I).

Here, ⋖D and ⋗D are the three-way operators in (G, D, I).

The sets containing the above four kinds of consistent sets are denoted according to the above order by CS(OEL), CS(OEM), CS(OEJ) and CS(OEG), respectively; the sets containing the associated reducts are denoted by Red(OEL), Red(OEM), Red(OEJ) and Red(OEG); and the cores of reductions are denoted by Core(OEL), Core(OEM), Core(OEJ) and Core(OEG).

The four different object-induced three-way attribute reductions embody different characteristics of a formal context. The OE-lattice reduction preserves the OE-lattice of the context. Since the OE-lattice can reflect the hierarchical knowledge of the context, the OE-lattice reduction can preserve this kind of hierarchical knowledge of the context. But it needs to consider all the OE-concepts, which let OE-lattice reduction more difficult to be computed. The OE-MIE (JIE)-preserving reduction is to preserve the extents of the meet (join)-irreducible elements of the OE-concept lattice. Because every OE-concept is meet (join) of meet (join)-irreducible elements of the OE-lattice, the meet (join)-irreducible elements are the basic elements of lattice construction. Thus, although the OE-MIE (JIE)-preserving reduction cannot preserve all the hierarchical knowledge of the context, it still plays an important role in lattice construction. Also since OE-MIE (JIE)-preserving reduction only considers the irreducible elements, the results are easy to be computed. The OE-granular reduction is to preserve the extent of the OE-object concept which can be regarded as a kind of information granule from the perspective of granular computing. Thus OE-granular reduction is meaningful in granular computing. Since it only considers OE-object concepts, compared with other OE-reductions, the computation is the simplest. From Definition 7 to 10, we find that these different attribute reductions consider different perspectives or different information of the context and they can be used in different occasions. If we want to preserve all the knowledge of the context, we need to use OE-lattice reduction; if we want to preserve the basic elements of lattice construction, we need to use OE-MIE (JIE)-preserving reduction; if we want to preserve the information granules, we need to use OE-granular reduction. The above discussions are about the differences of these four OE-reductions which are summarized in Table 1. Now, let us discuss the relationships among these four OE-attribute reductions.

Table 1

The comparison of four OE-attribute reductions.

	Completeness of knowledge	Computation	Applicable occasion
OE-lattice reduction	No knowledge loss	Complicate	No knowledge loss simplification
OE-MIE-preserving reduction	No knowledge loss	Simple	No knowledge loss simplification
			/Lattice construction
OE-JIE-preserving reduction	Has knowledge loss	Simple	Lattice construction
OE-granular reduction	Has knowledge loss	Simple	Granular computing

Let L be a finite lattice. Every element is join (meet) of join-irreducible (meet-irreducible) elements.

Let (G, M, I) be a formal context, then and .

Firstly, we give the proof of .

If D ∈ CS(OEL), then holds by Definition 7. For any X ∈ Ext(G, M, I), from the definition of meet-irreducible element, we can get X ≠ Y ∩ Z ( ∀ Y, Z ∈ Ext(G, M, I) and Y ≠ X, Z ≠ X). And because of X ∈ Ext(G, D, I) and X ≠ Y ∩ Z ( ∀ Y, Z ∈ Ext(G, D, I) and Y ≠ X, Z ≠ X). So, according to the definition of meet-irreducible elements, X ∈ Ext(G, D, I). Thus Ext(G, M, I) ⊆ Ext(G, D, I), and vice versa. Therefore that is D ∈ CS(OEM). Thus CS(OEL) ⊆ CS(OEM).

Suppose D ∈ CS(OEM), we need to show . Since Ext(G, D, I) ⊆ Ext(G, M, I) holds for any D ⊆ M, we only need to prove Ext(G, M, I) ⊆ Ext(G, D, I). For any X ∈ Ext(G, M, I), there exist X ∈ Ext(G, M, I) (i ∈ τ) such that due to Lemma 1. Since and Ext(G, D, I) ⊆ Ext(G, D, I), we obtain . Thus, Ext(G, M, I) ⊆ Ext(G, D, I). Therefore, which means D ∈ CS(OEL). Hence, CS(OEM) ⊆ CS(OEL). So, we obtain .

The first equation of this theorem shows that D is an OE-lattice consistent set if and only if D is an OE-MIE-preserving consistent set. Therefore, the following parts are naturally obtained. □

Let (G, M, I) be a formal context, D ⊆ M, and X ⊆ G. We have X ⋖⊇X ⋖ ; consequently, ∀x ∈ G, x ⋖⊇x ⋖ . If D is an OE-lattice consistent set, then the two equations hold.

∀X ⊆ G, since D ⊆ M, we have X ⋖ ⊆ X ⋖. Then, X ⋖⊇X ⋖. Let x ⋖⊇x ⋖ holds naturally. Suppose set D is an OE-lattice consistent set. ∀X ⊆ G, since X ⋖ ∈ Ext(G, D, I), X ⋖ ∈ Ext(G, M, I), and there must be . Similarly, let also holds. □

Lemma 2 says that if D is an OE-lattice consistent set, then, ∀x ∈ G, holds. Then, D is also an OE-granular consistent set according to Definition 10. Therefore, we have the following results.

Let (G, M, I) be a formal context, if D ⊆ M is an OE-lattice consistent set, then D is an OE-granular consistent set. That is, CS(OEL) ⊆ CS(OEG). Also, we have Red(OEL) ⊆ CS(OEG) and Core(OEG) ⊆ Core(OEL).

Based on Lemma 2, CS(OEL) ⊆ CS(OEG) is easy to be obtained. Because Red(OEL) ⊆ CS(OEL), we have Red(OEL) ⊆ CS(OEG). That is, an OE-lattice reduct must be an OE-granular consistent set. Now we give the proof of Core(OEG) ⊆ Core(OEL).

From the definition of core, we know that and . By the first part of Theorem 3, we have CS(OEL) ⊆ CS(OEG). Thus, . That is, Core(OEG) ⊆ Core(OEL). □

In order to analyze the relationship between OE-granular reduction and OE-JIE-preserving reduction, we firstly discuss the relationship between OE-object concept and the join-irreducible element of OEL(G, M, I) in the following proposition.

Let (G, M, I) be a formal context, X ⊆ G, A ⊆ M and B ⊆ M. Then (X, (A, B)) is a join-irreducible element of OEL(G, M, I), if and only if (X, (A, B)) is an OE-object concept.

Necessity. Let (X, (A, B)) ∈ OEL(G, M, I). Then By the properties of * and we can obtain Thus, From Lemma 1, we know that in a finite lattice, every element is join of join-irreducible elements, so the join-irreducible element of OEL(G, M, I) must be an OE-object concept.

Sufficiency. Assume is an OE-object concept induced by x. From Lemma 1 we know that is a join of join-irreducible elements, and the necessity of Lemma 3 shows that the join-irreducible element of OEL(G, M, I) must be an OE-object concept, thus we have Here are join-irreducible elements of OEL(G, M, I). So, and which imply and . Also, it’s easy to get that and so . Thus and . That is . Therefore, according to the definition of join-irreducible element, we obtain that is a join-irreducible element of OEL(G, M, I). □

Lemma 3 shows that in an OE-lattice, the join-irreducible element is same with the object concept; therefore, if D ⊆ M can preserve the extent set of OE-object concepts, then it must preserve the extent set of join-irreducible elements. Thus, we obtain the following theorem.

Let (G, M, I) be a formal context, then and .

Based on the above analysis, we can sum up the relationships among these four kinds of OE-reductions in Fig. 1 , and its explanation is shown in Fig. 2 .

Fig. 1

The relationships among four kinds of OE-reductions.

Fig. 2

The explanation of Fig. 1.

Four types of attribute reductions of an AE-concept lattice

Similarly, we can give four attribute reductions of an AE-concept lattice. Analogously to the proofs of theorems in Section 3, the proofs in this section are omitted.

Let (G, M, I) be a formal context. An attribute set D ⊆ M is called an attribute-induced three-way concept lattice (attribute-induced three-way meet-irreducible elements preserving; attribute-induced three-way join-irreducible elements preserving) consistent set of (G, M, I) if (; ). Furthermore, if there is no proper subset E ⊂ D such that set E is an AE-lattice (AE-MIE-preserving; AE-JIE-preserving) consistent set of (G, M, I), then set D is called an AE-lattice (AE-MIE-preserving; AE-JIE-preserving) reduct of (G, M, I).

Here, Ext(G, M, I) is the extent set of all the AE-concepts, Ext(G, M, I) is the extent set of all the meet-irreducible elements and Ext(G, M, I) is the extent set of all the join-irreducible elements of the AE-concept lattice AEL(G, M, I).

In AE-concept lattice, the granule is different from the one in OE-concept lattice. Here, the AE-attribute concept can be regarded as granule, since every AE-concept can be represented by the meet of AE-attribute concepts. Then the AE-granular reduction is given as follows.

Let (G, M, I) be a formal context. An attribute set D ⊆ M is called an attribute-induced three-way granular (AE-granular) consistent set of (G, M, I) if for all m ∈ M. If set D is an AE-granular consistent set of (G, M, I), and there is no proper subset E ⊂ D such that E is an AE-granular consistent set of (G, M, I), then set D is called an AE-granular reduct of (G, M, I).

The sets containing the above four kinds of consistent sets are denoted according to the above order by CS(AEL), CS(AEM), CS(AEJ) and CS(AEG), respectively; the sets containing the associated reducts are denoted by Red(AEL), Red(AEM), Red(AEJ) and Red(AEG); and the cores of reductions are denoted by Core(AEL), Core(AEM), Core(AEJ) and Core(AEG).

Then, we give the relationships among four AE-attribute reductions, which are similar to the last section.

Let (G, M, I) be a formal context and set D ⊆ M. Then and .

Let (G, M, I) be a formal context, X ⊆ G, Y ⊆ G and A ⊆ M. Then ((X, Y), A)) is a meet-irreducible element of AEL(G, M, I), if and only if ((X, Y), A)) is an AE-attribute concept.

Lemma 4 shows that in AE-lattice, the meet-irreducible element is same with the attribute concept; therefore, if D ⊆ M can preserve the extent set of AE-attribute concepts, then it must preserve the extent set of meet-irreducible elements. Thus, we obtain the following theorem.

Let (G, M, I) be a formal context and set D ⊆ M. Then and .

Moreover, from Theorem 5 , we know that the AE-lattice consistent set and AE-MIE-preserving consistent set are same, so the following theorem is easy to get:

Let (G, M, I) be a formal context and set D ⊆ M. Then, and .

Finally, we present the relationship between AE-lattice reduction and AE-JIE-preserving reduction. Because Ext(G, M, I) ⊆ Ext(G, M, I), it is easy to see that if set D ⊆ M can preserve the extent set of all AE-concept, then it must preserve the extent set of join-irreducible elements of AEL(G, M, I). Thus the following proposition is obtained.

Let (G, M, I) be a formal context and set D ⊆ M. Then, CS(AEL) ⊆ CS(AEJ) and Core(AEJ) ⊆ Core(AEL).

Based on above analysis, we can sum up the relationships among the four kinds of AE-reductions in Fig. 3 , and its explanation is shown in Fig. 4 .

Fig. 3

The relationships among four kinds of AE-reductions.

Fig. 4

The explanation of Fig. 3.

Approaches to attribute reductions

Approaches to attribute reductions of an OE-concept lattice

Based on the similar idea of discernibility matrix proposed by Zhang et al. [36], we introduce the discernibility matrix and discernibility functions which can be used to find the four different OE-reductions.

Let (G, M, I) be a formal context, and (X, (A, B)), (Y, (C, D)) ∈ OEL(G, M, I). Then is called the OE-discernibility attributes set between (X, (A, B)) and (Y, (C, D)).

is called the OEL-discernibility matrix of (G, M, I).

where (X, (A, B)) is the join-irreducible element of OEL(G, M, I)).

Because we only need to use the non-empty sets in the attribute reduction, we also let Λ denote the set of the non-empty OE-discernibility attributes sets.

Let (G, M, I) be a formal context, B ⊆ M and X ∈ Ext(G, M, I). Then if and only if for any (Y, (C, D)) ∈ PC(X, X ⋖). Here PC(X, X ⋖) is a set of all the parent-OE-concepts of (X, X ⋖).

For simplicity, we firstly define (A, B) ⊂ (C, D) ≔ A ⊂ C and B ⊆ D , or A ⊆ C and B ⊂ D. And (A, B, C, D ⊆ M).

Necessity. Suppose and (Y, (C, D)) ∈ PC(X, X ⋖).

(i) If (Y, (C ∩ B, D ∩ B)) ∈ OEL(G, M, I), then from X ⊂ Y, we can get equivalently, obviously, .

(ii) If (Y, (C ∩ B, D ∩ B)) ∉ OEL(G, M, I), then X ⊂ Y ⊂ (C ∩ B, D ∩ B)⋗. Since((C ∩ B, D ∩ B)⋗, (C ∩ B, D ∩ B)⋗) ∈ OEL(G, M, I), we can get (C ∩ B, D ∩ B) ⊆ (C ∩ B, D ∩ B)⋗ ⊂ X ⋖. Moreover, it’s sure that (C ∩ B, D ∩ B) ⊂ X ⋖. Because if then which is a contradiction. Thus that is .

Sufficiency. Assume for any (Y, (C, D)) ∈ PC(X, X ⋖) and X ⋖ ≠ X. Then we have . Since (X ⋖, X ⋖) ∈ OEL(G, M, I), we obtain (X, X ⋖) < (X ⋖, X ⋖). Moreover, it’s sure that (X, X ⋖)≺(X ⋖, X ⋖). If there is (Z, (E, F)) ∈ OEL(G, M, I) such that (X, X ⋖)≺(Z, (E, F)) < (X ⋖, X ⋖), then X ⋖ ⊂ (E, F) ⊂ X ⋖, which implies . From the assumption, and because X ⋖ ⊆ X ⋖, we get which is impossible. Thus (X, X ⋖)≺(X ⋖, X ⋖). From assumption, we have that is, . So, which is impossible. Hence, . □

Theorem 9 gives us a necessary and sufficient condition of an OE-concept to hold its extent unchanged. Considering the definitions of the four different OE-attribute reductions and their relationships, we give the following methods to computing these reductions.

In order to give a more concise description of the following methods, we also use DIS((X, (A, B)), (Y, (C, D))) to denote the union of its components in the following definition. For instance, if then E ∪ F is also denoted as DIS((X, (A, B)), (Y, (C, D))). And we also use Λ to represent the OE-discernibility matrix of (G, M, I).

Let (G, M, I) be a formal context. The OEL-discernibility function, OEM-discernibility function, OEJ-discernibility function and OEG-discernibility function are defined as follows:

By absorption law and distributive law, the OE-discernibility function f can be transformed to a minimal disjunctive normal form, whose items can form all the OE-reducts of (G, M, I).

Table 2 shows a formal context (G 1, M 1, I 1), in which, the object set is and the attribute set is . And the OE-concept lattice of (G 1, M 1, I 1) is shown in Fig. 5 .

Table 2

A formal context

G1	a	b	c	d	e
1	×	×		×	×
2	×	×	×
3				×
4	×	×	×		×

Fig. 5

The OE-concept lattice of (G1, M1, I1).

There are 11 OE-concepts: (13, (d, c)), (24, (abc, d)), (1, (abde, c)), (3, (d, abce)), (2, (abc, de)), (4, (abce, d)), which are labeled as . According to Definition 13, the Λ of OEL(G 1, M 1, I 1) is shown in Table 3 .

Table 3

The OEL-discernibility matrix of the OE-concept lattice in Fig. 5.

	TC1	TC2	TC3	TC4	TC5	TC6	TC7	TC8	TC9	TC10	TC11
TC1
TC2	(d, c)
TC3	(⌀,e)
TC4	(ab,⌀)
TC5				(e,⌀)
TC6				(c, d)
TC7		(abe,⌀)			(d, c)
TC8		(⌀,abe)	(d, abc)
TC9			(abc, d)			(⌀,e)
TC10					(c, d)	(e,⌀)
TC11							(c, abde)	(abce, d)	(de, abc)	(d, abce)

Fig. 8

The AE-concept lattice of (G1, M1, I1).

More specifically, if TC is a child-concept of TC, then the OE-discernibility attributes set between TC and TC (DIS(TC)) is at the ith row and jth column of Table 3. Now we calculate the discernibility functions.

So the OE-lattice (OE-MIE-preserving) reducts are {a, c, e}, {b, c, e}, {a, d, e} and {b, d, e}. And the OE-JIE-preserving (OE-granular) reducts are {c, e} and {d, e}.

Considering the space limitation, we only present the OE-concept lattices of and in Figs. 6 and 7 , respectively. Here M 2 is OE-lattice (OE-MIE-preserving) reduct {a, c, e} and M 3 is OE-JIE-preserving (OE-granular) reduct {c, e}.

Fig. 6

The OE-concept lattice of .

Fig. 7

The OE-concept lattice of .

From Figs. 5–7, we can see that OE-lattice (OE-MIE-preserving) reduct must be OE-JIE-preserving (OE-granular) reduct, but OE-JIE-preserving (OE-granular) reduct may not be OE-lattice (OE-MIE-preserving) reduct.

Approaches to attribute reductions of an AE-concept lattice

In order to get the AE-attribute reductions, the AE-discernibility attributes set is defined as follows:

Let (G, M, I) be a formal context, and ((X, Y), A), ((Z, W), B) ∈ AEL(G, M, I). Then is called the AE-discernibility attributes set between ((X, Y), A) and ((Z, W), B).

is called the AE-discernibility matrix of (G, M, I).

where ((X, Y), A) is the AE-attribute concept of (G, M, I). Here, we also let Λ denote the set of the non-empty AE-discernibility attributes sets.

Any AE-concept ((X, Y), A) can be written as the following form ((X, Y)⋗⋖, (X, Y)⋗). Since operators ⋖ and ⋗ are dual, we can get the following theorem from Theorem 9 dually.

Let (G, M, I) be a formal context, B ⊆ M and (X, Y) ∈ Ext(G, M, I). Then if and only if for any ((Z, W), B)) ∈ PC((X, Y), A).

Now, we can give the method of computing the different AE-attribute reductions of (G, M, I)

Let (G, M, I) be a formal context. The AEL-discernibility function, AEM-discernibility function and AEG-discernibility function are defined as follows: By absorption law and distributive law, the AE-discernibility function f can be transformed to a minimal disjunctive normal form, whose items can form all the AE-reducts of (G, M, I).

Continued with Example 1

The AE-concept lattice of (G 1, M 1, I 1) is shown in Fig. 8 .

There are 11 AE-concepts: ((24, 13), c), ((124, 3), ab), ((14, 23), e), ((13, 24), d), ((24, 3), abc)), ((14, 3), abe), ((1, 2), de), ((4, 3), abce), which are labeled as . According to Definition 15, the Λ of AEL(G 1, M 1, I 1) is shown in Table 4.

Table 4

The AEL-discernibility matrix of the AE-concept lattice in Fig. 8.

	TC1	TC2	TC3	TC4	TC5	TC6	TC7	TC8	TC9	TC10	TC11
TC1
TC2	c
TC3	ab
TC4	e
TC5	d
TC6		ab	c
TC7			e	ab
TC8				d	e
TC9						e	c
TC10							d	ab
TC11									d	c

Then we calculate the discernibility functions. So the AE-lattice (AE-MIE-preserving and AE-granular) reducts are {a, c, e, d} and {b, c, e, d}.

Considering the space limitation, we only present the AE-concept lattice of in Fig. 9 . Here M 4 is AE-lattice (AE-MIE-preserving and AE-granular) reduct {a, c, e, d}.

Fig. 9

The AE-concept lattice of .

An empirical case

In this section, a real-life database is analyzed to demonstrate the application of the proposed attribute reduction methods. The chosen database is about the patients who suffer from severe acute respiratory syndrome (SARS) and their symptoms. Here we only use OE-reductions as examples and the AE-reductions can be studied similarly. The details are as follows:

Table 5 depicts a dataset of four patients who suffer from severe acute respiratory syndrome (SARS) [47]. Let G be the set of four patients and M be the set of five symptoms. For convenience, we denote the four patients by 1, 2, 3, 4, respectively, and the five symptoms (Fever, Cough, Headache, Difficulty Breathing and Diarrhea) by a, b, c, d, e, respectively. That is and .

Table 5

A SARS dataset.

Patient	Fever	Cough	Headache	Difficulty breathing	Diarrhea
1	Yes	No	Yes	No	Yes
2	Yes	No	No	Yes	No
3	No	Yes	No	No	No
4	No	No	Yes	No	Yes

Firstly, we can get all the formal concepts of Table 5 and the corresponding formal concept lattice is shown in Fig. 10 . There are 7 formal concepts: (14, ce), (12, a), (1, ace), (2, ad), (3, b), . Then, we can get 12 OE-concepts: (14, (ce, bd)), (12, (a, b)), (4, (ce, abd)), (1, (ace, bd)), (2, (ad, bce)), (3, (b, acde)), which are labeled as . The OE-lattice is shown in Fig. 11 .

Fig. 10

The formal concept lattice of (G, M, I).

Fig. 11

The OE-concept lattice of (G, M, I).

From classical formal concept, one can only get the information that the objects in the extent have the attributes in the intent of this formal concept (inclusion method). But from the OE-concept, one can obtain not only that the objects in the extent have the attributes in the first component of intent (inclusion method), but also that the objects in the extent don’t have the attributes in the second component of intent (exclusion method). For example, the formal concept (14, ce) only tells us that patients 1 and 4 belong to one class because they have same symptoms Headache and Diarrhea, but the OE-concept (14, (ce, bd)) tells us that patients 1 and 4 belong to one class not only because they have the symptoms Headache and Diarrhea but also because they don’t have the symptoms Cough and Difficulty Breathing. Thus three-way concepts can reflect more knowledge than the classical formal concepts.

However, in order to obtain all OE-concepts and the corresponding OE-concept lattice, one needs to consider all attributes. In this case, there are only 5 attributes, but from Fig. 11 we can see that the representations of all OE-concepts are a little bit complicated. Actually, in real life, we even meet more complicated data. Thus the attribute reductions are necessary to be considered.

Since the ordered hierarchical structure of all the OE-concepts is reflected by the OE-concept lattice, OE-concept lattice is a core structure in 3WCA. Hence, we show the OE-lattice attribute reduction firstly. According to Definition 13, the Λ of OEL(G, M, I) is shown in Table 6 .

Table 6

The OEL-discernibility matrix of the OE-concept lattice in Fig. 11.

	TC1	TC2	TC3	TC4	TC5	TC6	TC7	TC8	TC9	TC10	TC11	TC12
TC1
TC2	(⌀,d)
TC3	(⌀,b)
TC4		(⌀,a)
TC5		(ce, b)	(ce, d)
TC6			(a,⌀)
TC7	(⌀,ce)
TC8				(ce, b)	(⌀,a)
TC9					(a,⌀)	(ce, d)
TC10						(d, ce)	(ad, b)
TC11				(b, ce)			(b, ad)
TC12								(abd, ce)	(db, ace)	(bce, ad)	(acde, b)

Now we calculate the discernibility functions.

So the OE-lattice (OE-MIE-preserving) reducts are {a, b, d, c} and {a, b, d, e}. We only present the OE-concept lattice of in Fig. 12 . Here M 5 is the OE-lattice reduct {a, b, d, c}.

Fig. 12

The OE-concept lattice of .

From Fig. 12, we can see that the OE-lattice of is isomorphic to the OE-lattice of (G, M, I). Thus the classification of patients and the hierarchy of the three-way concepts don’t change, but we only need to consider smaller amount of attributes. For instance, only based on 4 symptoms Fever, Cough, Headacheand Difficulty Breathing, patients 1 and 4 can still be classified into one class. So after OE-lattice reduction, the knowledge is represented in an easier way.

Then, we calculate the OE-JIE-preserving (OE-granular) reducts.

The OE-JIE-preserving (OE-granular) reducts are {a, c}, {a, e} and {a, b, d}. The OE-concept lattice of is presented in Fig. 13 . Here M 6 is the OE-JIE-preserving (OE-granular) reduct {a, c}.

Fig. 13

The OE-concept lattice of .

From Fig. 13, we can see that the OE-granular reduct cannot preserve the structure of the OE-lattice. That means some knowledge may be lost after reduction. For example, now in Fig. 13, the patients 1, 2 and 4 can not belong to one class because the attribute Difficulty Breathing has been removed after reduction. But the extents of the OE-object concepts and the join-irreducible elements of OE-lattice in Fig. 11 don’t change. Since the join-irreducible elements are important in the lattice construction and the OE-object concepts are important in granular computing, these two reductions are important in lattice construction and granular computing.

Conclusion

In this paper, the attribute reductions of three-way concept lattices have been systematically studied. We have defined eight kinds of attribute reductions of three-way concept lattices, four OE-attribute reductions and four AE-attribute reductions, and discussed their advantages and disadvantages. Then we investigated the relationships among these newly proposed attribute reductions, the relationships among their consistent sets, as well as the relationships among their cores. Also, we have presented the approaches to computing most of these reductions. Finally, an empirical case is shown to illustrate our analysis. Since three-way concept analysis can be regarded as an extension of formal concept analysis and the three-way concept lattice can reflect more knowledge than the classical one, the study of the three-way concept lattice is a newly proposed but meaningful part of formal concept analysis. Also because the attribute reductions of three-way concept lattices are necessary in three-way concept analysis, the attribute reductions we discussed in this paper enrich the study of formal concept analysis.

The current approach to AE-JIE-preserving reduction is complicated and cannot be unified under the frame of the discernibility matrices and discernibility functions, so, in this paper, we omitted the discussion of it and focused on the relationships among different attribute reductions. Also, since the three-way concept analysis can be regarded as a generalization of the classical formal concept analysis, there must exist some relationships between these two theories. Hence, the relationship among the attribute reductions on these two theories should be studied in the future. Actually, for any kind of attribute reduction mentioned in this paper, the attributes can be classified into three types according to their roles in the reduction [36]. Let (G, M, I) be a formal context. For a certain kind of reduction, suppose the set {D|D is a reduct, i ∈ τ}(τ is an index set) includes all associated reducts of (G, M, I). Then attributes in M can be classified into three types: absolutely necessary attribute, also called core attribute (attribute in  ∩  D ), relatively necessary attribute (attribute in ), and absolutely unnecessary attribute (attribute in ). Of course, different attributes play different roles in different reductions. For further research, it is of significance to investigate their relationships based on different reductions. Also, in order to apply the theories we discussed in this paper in the real world, in the future, we will propose the corresponding algorithms of these eight different attribute reductions, which may help us to deal with the big data problems in our daily lives more conveniently. And the utility value will also be introduced and added to every concept and the new attribute reductions based on the utility value will be studied, which will make our study more suitable in practices.