Constructing three-way concept lattices based on apposition and subposition of formal contexts

Knowledge-Based Systems 116 (2017) 39–48

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Constructing three-way concept lattices based on apposition and subposition of formal contexts Ting Qian a,b, Ling Wei a,∗, Jianjun Qi c a

School of Mathematics, Northwest University, Xi’an 710069, PR China College of Science, Xi’an Shiyou University, Xi’an 710065, PR China c School of Computer Science and Technology, Xidian University, Xi’an 710071, PR China b

a r t i c l e

i n f o

Article history: Received 16 February 2016 Revised 29 October 2016 Accepted 31 October 2016 Available online 2 November 2016 Keywords: Three-way concept lattice Three-way decision Apposition Subposition

a b s t r a c t Three-way concept analysis provides a new model to make three-way decisions. Its basic structure can be shown by the three-way concept lattices. Thus, how to construct three-way concept lattices is an important issue in the three-way concept analysis. This paper proposes approaches to create the threeway concept lattices of a given formal context. First, we can transform the given formal context and its complementary context into new formal contexts which are isomorphic to the given formal context and its complementary context respectively. And then, Type I-combinatorial context and Type IIcombinatorial context are defined, which are apposition and subposition of these new formal contexts, respectively. Second, we prove that the concept lattice of Type I-combinatorial context is isomorphic to object-induced three-way concept lattice and the concept lattice of Type II-combinatorial context is isomorphic to attribute-induced three-way concept lattice of the given formal context. And then, the approaches of creating the three-way concept lattices are proposed based on the concept lattices of Type I-combinatorial context and Type I-combinatorial context. Finally, we give the corresponding algorithms of constructing three-way concept lattices based on the above approaches and conduct several experiments to illustrate the efficient of proposed algorithms. © 2016 Elsevier B.V. All rights reserved.

1. Introduction Based on the mathematization of concept and conceptual hierarchy, formal concept analysis (FCA), proposed by Wille in 1982 [29], is a field of applied mathematics. Its foundation includes a formal context, formal concepts and corresponding concept lattice. Most of the researches on FCA are concentrating on the following topics: construction and pruning algorithms of concept lattices [1,6,9,20,23]; acquisition of rules [13,14,19]; reduction of concept lattices [25,30,36]. Recently, studying FCA by combining it with other theories together is a hot topic in FCA research [7,8,12,17,18,21,26–28,37]. For example, Qi et al. first proposed three-way concept analysis by combining FCA with threeway decisions [21,22]. Three-way decisions are widely used in real-world decisionmaking. They are used in different fields and disciplines by different names and notations. Observing this phenomenon, Yao [31] proposed an outline of the theory of three-way decisions ∗

Corresponding author. Fax: +86 29 88308433. E-mail addresses: qiant20 0 [email protected] (T. Qian), [email protected] (L. Wei), [email protected] (J. Qi). http://dx.doi.org/10.1016/j.knosys.2016.10.033 0950-7051/© 2016 Elsevier B.V. All rights reserved.

that provides an unified and discipline-independent framework for decision-making with three decisions, namely, acceptance, rejection and non-commitment. And after that many recent studies [4,10,11,15,16,21,22,24,31–35,38] investigate in various areas. For example, Yu [35] proposed a tree-based incremental overlapping clustering method by using three-way decisions. As a new model to make three-way decisions, three-way concept analysis can be taken as a generalization of formal concept analysis. Similar to a formal concept in FCA, a three-way concept is also constituted of an extent and an intent. The difference is that the extent (or the intent) in a three-way concept is equipped with two parts: the positive one and the negative one. These two parts are used to express the semantics “jointly possessed” and “jointly not possessed” in a formal context respectively. On the basis of three-way concept, one can divide the object (or attribute) universe into three regions to make three-way decisions [21]. More specifically, “jointly possessed” means acceptance and “jointly not possessed” means rejection in the three-way decision. The basic structure of three-way concept analysis can be shown by the three-way concept lattices. In [21] and [22], the theoretical foundations and frames of two kinds of three-way concept lattices were built and the ideology of the theory of three-way concept

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T. Qian et al. / Knowledge-Based Systems 116 (2017) 39–48

lattices was discussed clearly. However, the authors did not consider the problem of constructing three-way concept lattices at that time. This paper will discuss this problem and give a method to create the three-way concept lattices. As far as we know, there is one-to-one correspondence between a formal context and a complete lattice [5]. In addition, the three-way concept lattices are complete lattices. So we want to find some specific formal contexts whose concept lattices can correspond to the three-way concept lattices. If such formal contexts exist, then on one hand, the well-developed formal concept construction methods and techniques can be applied to construct three-way concept and threeway concept lattices. On the other hand, based on the relations between the new formal contexts and the original formal context, the deeper researches of the relations between formal concept lattices and three-way concept lattices can be studied. In fact, the information of “jointly possessed” and “jointly not possessed” in a formal concept is provided by a formal context and its complementary context, respectively. Therefore, we consider to construct the above specific formal contexts by using the related formal contexts of the formal context and its complementary context. Luckily, we constructed such above mentioned formal contexts called Type I-combinatorial context and Type II-combinatorial context, and this paper will show the idea and method of constructions. This paper is organized as follows. In Section 2, we briefly review some basic notions related to FCA and three-way concept lattices. In Section 3, some novel formal contexts and Type Icombinatorial context based on the object set are defined, and the related conclusions are given. In Section 4, the other novel formal contexts and Type II-combinatorial context based on the attribute set, are defined and some related conclusions are given, which are similar to results in Section 3. In Section 5, the corresponding algorithms of acquisition approaches to three-way concept lattices are presented. Furthermore, the related experiments are conducted. Finally, conclusions are drawn in Section 6. 2. Preliminaries This section gives some necessary operators, definitions and symbols in this paper. Some set-theoretic operators are introduced first. Let S be a non-empty finite set, P (S ) be its power set and DP (S ) = P (S ) × P (S ). Set-theoretic operators on DP (S ), such as intersection ∩ , union ∪ and complement c, are defined componentwise using standard set operators. For two pairs of subsets (A, B ), (C, D ) ∈ DP (S ), we define (A, B ) ∩ (C, D ) = (A ∩ C, B ∩ D ), (A, B ) ∪ (C, D ) = (A ∪ C, B ∪ D ), (A, B )c = (Ac , Bc ), and (A, B)⊆(C, D)⇔A⊆C and B⊆D. 2.1. Formal concept analysis The section reviews some basic notions and properties in FCA. Definition 2.1 [5]. 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 g is in a relation I with an attribute m, we write gIm or (g, m) ∈ I and read it as “the object g has the attribute m”. With respect to a formal context (G, M, I), Wille and Ganter [5] defined a pair of dual operators for any A ⊆ G and B ⊆ M by:

Table 1 A formal context (G, M, I). G

a

b

1 2 3 4

× ×

× ×

c ×

×

×

×

d

e

×

×

×

Fig. 1. B (G, M, I ) in Example 2.1.

Let (G, M, I) be a formal context. ∀A1 , A2 , A ⊆ G, ∀B1 , B2 , B ⊆ M, the following properties hold: 1. 2. 3. 4. 5. 6.

A1 ⊆ A2 ⇒ A∗2 ⊆ A∗1 , B1 ⊆ B2 ⇒ B∗2 ⊆ B∗1 . A ⊆ A∗∗ , B ⊆ B∗∗ . A∗ = A∗∗∗ , B∗ = B∗∗∗ . A ⊆ B∗ ⇔B ⊆ A∗ . (A1 ∪ A2 )∗ = A∗1 ∩ A∗2 , (B1 ∪ B2 )∗ = B∗1 ∩ B∗2 . (A1 ∩ A2 )∗ ⊇ A∗1 ∪ A∗2 , (B1 ∩ B2 )∗ ⊇ B∗1 ∪ B∗2 .

If A∗ = B and B∗ = A, then (A, B) is called a formal concept, where A is called the extent of the formal concept, B is called the intent of the formal concept. The family of all formal concepts of (G, M, I) form a complete lattice, which is called the concept lattice and is denoted by B (G, M, I ). For any (A1 , B1 ), (A2 , B2 ) ∈ B (G, M, I ), the partial order is defined by:

( A1 , B1 )  ( A2 , B2 ) ⇔ A1 ⊆ A2 ( ⇔ B1 ⊇ B2 ). The infimum and supremum of (A1 , B1 ) and (A2 , B2 ) are defined by:

(A1 , B1 ) ∧ (A2 , B2 ) = (A1 ∩ A2 , (B1 ∪ B2 )∗∗ ), (A1 , B1 ) ∨ (A2 , B2 ) = ((A1 ∪ A2 )∗∗ , B1 ∩ B2 ), respectively. Remark. Since we discuss different formal context with different I⊆(G × M), A∗ and B∗ are denoted in the (G, M, I) by AI and BI respectively in the sequel. Example 2.1. A formal context (G, M, I) is shown in Table 1. G = {1, 2, 3, 4} is an object set, M = {a, b, c, d, e} is an attribute set. The corresponding concept lattice B (G, M, I ) is shown in Fig. 1, in which, every set is denoted directly by listing its elements except G, M and ∅.

A∗ = {m ∈ M|gIm for all g ∈ A}, B∗ = {g ∈ G|gIm for all m ∈ B}.

Definition 2.2 [3]. Let L and K be lattices. A map f: L → K is said to be a homomorphism if f is join-preserving and meet-preserving, that is, for all a, b ∈ L, f (a ∨ b) = f (a ) ∨ f (b) and f (a ∧ b) = f (a ) ∧ f (b). A bijective homomorphism is a lattice isomorphism. Reference [3] also shows that f is lattice isomorphism if and only if f is order-isomorphism.

We say a formal context is canonical if ∀g ∈ G, g∗ = ∅, g∗ = M, and ∀m ∈ M, m∗ = ∅, m∗ = G. All the formal contexts we study in this paper are finite and canonical.

Definition 2.3 [5]. An isomorphism between contexts K1 = (G, M, I ) and K2 = (H, N, J ) is a pair (α , β ) of bijective maps α : G → H, β : M → N with gIm⇔α (g)Jβ (m).

T. Qian et al. / Knowledge-Based Systems 116 (2017) 39–48

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Reference [5] also shows that isomorphic contexts have isomorphic concept lattices. Definition 2.4 [5]. Let K = (G, M, I ), K1 = (G1 , M1 , I1 ) and K2 = (G2 , M2 , I2 ) be contexts. We will use the abbreviations G˙ j = { j} × G j , M˙ j = { j} × M j and I˙j = {(( j, g)( j, m ))|(g, m ) ∈ I j } for j ∈ {1, 2} in the following definitions. It is: Kc = (G, M, (G × M ) \ I ) is called the complementary context to K, if G=G1 =G2 , then K1 |K2 = (G, M˙ 1 ∪ M˙ 2 , I1˙ ∪ I2˙ ) is called the apposition of K1 and K2 . As well K as dually, if M = M1 = M2 , then K1 = (G˙1 ∪ G˙2 , M, I1˙ ∪ I2˙ ) is called 2 the subposition of K1 and K2 . Surely, if G = G1 = G2 and M1 ∩ M2 = ∅, we also call (G, M1 ∪ M2 , I1 ∪ I2 ) the apposition of K1 and K2 . Dually, if M = M1 = M2 , and G1 ∩ G2 = ∅, we also call (G1 ∪ G2 , M, I1 ∪ I2 ) the subposition of K1 and K2 . 2.2. Three-way concept analysis In real-life, exclusion method is also commonly used when making decisions. One may determine if an object (an attribute) does not possesses (is not shared by) any elements in the intent (the extent) of a concept. Combining inclusion method (acceptance) with exclusion method (rejection) induces a kind of threeway decision. However, this is not supported by FCA. Therefore, Qi et al. proposed three-way concept analysis in references [21] and [22]. For consistence, the two operators ∗ defined above are called positive operators in references [21] and [22]. Correspondingly, these two references also define a pair of negative operators as follows. Definition 2.5 [21,22]. Let (G, M, I) be a formal context, Ic = (G × M ) \ I. Define a pair of operators, ∗ : P (G ) → P (M ) and ∗ : P (M ) → P (G ), called negative operators, for X⊆G and A⊆M, we have

{v ∈ M|∀x ∈ X (¬(xIv ))} = {v ∈ M|∀x ∈ X (xIc v )} = {v ∈ M |X ⊆ I c v},

X∗ =

A∗ = {u ∈ G|∀a ∈ A(¬(uIa ))} = {u ∈ G|∀a ∈ A(uIc a )} = {u ∈ G|A ⊆ uIc }. Combining negative operators with positive operators, threeway operators are defined as follows. Definition 2.6 [21,22]. Let (G, M, I) be a formal context. Given X⊆G and A, B⊆M, a pair of object-induced three-way operators,  : P (G ) → DP (M ) and  : DP (M ) → P (G ), are defined by X  = (X ∗ , X ∗ ); (A, B ) = {x ∈ G|x ∈ A∗ and x ∈ B∗ } = A∗ ∩ B∗ . We abbreviated them as OE-operators. Similarly, references [21] and [22] also give a pair of attribute-induced three-way operators, for A⊆M and X, Y⊆G,  : P (M ) → DP (G ) and  : DP (G ) → P (M ), are defined by A = (A∗ , A∗ ); (X, Y ) = {v ∈ M|v ∈ X ∗ and v ∈ Y ∗ } = X ∗ ∩ Y ∗ , which are abbreviated as AE-operators. Based on the above three-way operators, three-way concepts and three-way concept lattices are shown in the following. Definition 2.7 [21,22]. 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 OEconcept, of (G, M, I), if and only if X  = (A, B ) and (A, B ) = X. X is called the extent and (A, B) is called the intent of the OE-concept (X, (A, B)). Given an OE-concept (X, (A, B)) and X = ∅, M can be divided into the following three regions.

Fig. 2. OEL(G, M, I) in Example 2.2.

P OSXM = A, NEGM X = B, BNDM X = M \ ( A ∪ B ). POSM is the positive region, in which every attribute is defiX nitely shared by all objects in X. NEGM is the negative region, in X which every attribute is not possessed definitely by any object in X. Those attributes possessed by some, but not all, objects in X belong to the boundary region BNDM [21,22]. X 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 ). 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

(X, (A, B )) ∧ (Y, (C, D )) = (X ∩ Y, ((A, B ) ∪ (C, D )) ), (X, (A, B )) ∨ (Y, (C, D )) = ((X ∪ Y ) , (A, B ) ∩ (C, D )). Definition 2.8 [21,22]. 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 AEconcept, of (G, M, I), if and only if (X, Y ) = A and A = (X, Y ). (X, Y) is called the extent and A is called the intent of the AE-concept ((X, Y), A). Given an AE-concept ((X, Y), A) and A = ∅, G can be divided into the following three regions.

P OSAG = X, NEGGA = Y,

BNDGA = G \ (X ∪ Y ). The positive region POSGA contains objects having all attributes in A. the negative region NEGGA is the set of objects not possessing any attribute in A. The boundary region BNDGA includes objects possessing some, but not all, attributes in A [21,22]. 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. All the 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

((X, Y ), A ) ∧ ((Z, W ), B ) = ((X, Y ) ∩ (Z, W ), (A ∪ B ) ), ((X, Y ), A ) ∨ ((Z, W ), B ) = (((X, Y ) ∪ (Z, W )) , A ∩ B ). Example 2.2. Consider the formal context in Example 2.1. The corresponding three-way concept lattices are shown in Figs. 2 and 3.

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T. Qian et al. / Knowledge-Based Systems 116 (2017) 39–48 Table 2 K2 = (G, M, Ic ). G

a

b

c

d

e

×

× × ×

×

1 2 3 4

×

×

× ×

Table 3 1 = (G, M × {0},  ). K Fig. 3. AEL(G, M, I) in Example 2.2.

3. Type I-combinatorial context based on the object set In this section, for using acquisition methods of concept lattices to construct object-induced three-way concept lattices, we need to find some formal context, whose concept lattice is isomorphic to the object-induced three-way concept lattice of original formal context. We will describe this process detailedly. Suppose K1 = (G, M, I ) is a formal context and K2 = (G, M, Ic ) is its complementary context with Ic = (G × M ) \ I. In order to distinguish the attribute sets in K1 and K2 , we add tags to them. And 1 and K 2 , respectively as folthen, K1 and K1 are translated into K lows.  1 = (G, M × {0}, ) with g(m, 0) ⇔gIm for any g ∈ G and (m, K 0) ∈ M × {0}.   2 = (G, {0} × M, ) with g (0, m) ⇔gIc m for any g ∈ G and (0, K m) ∈ {0} × M. In the following, we will show the relationship between Ki and i , i = 1, 2. K Theorem 3.1. Suppose K1 = (G, M, I ) is a formal context and K2 = (G, M, Ic ) is its complementary context. Then we have the following i are isomorphic contexts, B (Ki ) and results for i = 1, 2 : Ki and K i ) are isomorphic concept lattices. B (K Proof. We only prove the result for i = 1. Suppose α : G → G, for any g ∈ G, α (g) = g. Suppose β : M → M × {0}, for any m ∈ M, β (m ) = (m, 0 ). It is obvious that α and β are bijective 1 , we can get gIm⇔g(m, maps. According to the definition of K 0)⇔α (g)β (m). By the definition of an isomorphism between con1 are isomorphic contexts. Since texts, we can obtain K1 and K isomorphic contexts have isomorphic concept lattices, B (K1 ) and 1 ) are isomorphic concept lattices naturally. B (K  We easily obtain the relations among operators I, Ic ,  and  through Theorem 3.1 and the definitions of  and .  Corollary 3.1. Suppose K1 = (G, M, I ) is a formal context and K2 = (G, M, Ic ) is its complementary context, X⊆G and A⊆M. Then the following results hold: 



(1) X = X I × {0} and X = {0} ×X I ,  c (2) (A × {0} ) = AI and ({0} × A ) = AI . c

1 and Now, we can obtain a new formal context by combining K 2 . K Definition 3.1. Let K1 = (G, M, I ) be a formal context and K2 = (G, M, Ic ) is its complementary context. Type I-combinatorial con1 and text based on the object set is defined by the apposition of K 2 , which is denoted by KO , i.e., KO = K 1 |K 2 = (G, M × {0} ∪ {0} × K    M, ∪ ). For simplicity, we denote  ∪ by ∼ . From definition of B (KO ) and Corollary 3.1, we give the relations among operators I, Ic and ∼ .

G

(a, 0)

(b, 0)

(c, 0)

1 2 3 4

× ×

× ×

×

×

×

×

(d, 0)

(e, 0)

×

×

×

Table 4 2 = (G, {0} × M,  ). K G 1 2 3 4

(0, a)

(0, b)

(0, c)

(0, d)

(0, e)

×

× × ×

× ×

×

× ×

Corollary 3.2. Suppose K1 = (G, M, I ) is a formal context and K2 = (G, M, Ic ) is its complementary context, X⊆G and A, B⊆M. Then the following results hold: (1) X ∼ = (X I × {0} ) ∪ ({0} × X I ), c (2) {(A × {0} ) ∪ ({0} × B )}∼ = AI ∩ BI . c

We can easily obtain the property of formal concepts based on the above relations among operators. Theorem 3.2. Suppose (A, B ) ∈ B (KO ), B1 = {x|(x, y ) ∈ B} and B2 = {y|(x, y ) ∈ B}. Then we have B = (B1 × {0} ∪ {0} × B2 ) \ {(0, 0 )}. In the following, we use an example to demonstrate the new contexts defined above. Example 3.1. Consider Table 1 as the original formal context K1 = (G, M, I ). According to the definitions of new contexts above, K2 , 1 , K 2 and KO are shown in Tables 2–5, respectively. K The corresponding concept lattice of KO = (G, (M × {0} ) ∪ ({0} × M ), ∼ ) is shown in Fig. 4. We choose the concept F ∈ B (KO ) in Fig. 4 to explain Theorem 3.2. For this concept, we obtain A = {3}, B = {(d, 0 ), (0, a ), (0, b), (0, c ), (0, e )}, B1 = {d, 0} and B2 = {a, b, c, e, 0} according to the notions in Theorem 3.2. We can calculate ({d, 0} × {0} ∪ {0} × {a, b, c, e, 0} ) \ {(0, 0 )} = {(d, 0 ), (0, a ), (0, b), (0, c ), (0, e ), (0, 0 )} \ {(0, 0 )} = {(d, 0 ), (0, a ), (0, b), (0, c ), (0, e )}. Therefore, we easily obtain (B1 × {0} ∪ {0} × B2 ) \ {(0, 0 )} = B. In the following, we will prove KO is just a kind of formal context we need, i.e., its concept lattice corresponds to objectinduced three-way concept lattice. The conclusions are reflected by Theorem 3.3 and Theorem 3.4. Theorem 3.3. Suppose K1 = (G, M, I ) is a formal context and K2 = (G, M, Ic ) is its complementary context. Then B (KO ) and OEL(G, M, I) are isomorphic lattices. Proof. For any (A, B ) ∈ B (KO ), we define ϕ (A, B ) = (A, (B1 , B2 )), where B1 = B1 \ {0} and B2 = B2 \ {0}. Now, we prove ϕ is an isomorphism of B (KO ) and OEL(G, M, I). First, we have to demonstrate ϕ : B (KO ) → OEL(G, M, I ) is a map, i.e., for any (A, B ) ∈ B (KO ), ϕ (A, B ) = (A, (B1 , B2 )) ∈ OEL(G, M, I ).

T. Qian et al. / Knowledge-Based Systems 116 (2017) 39–48

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Table 5 2 . 1 |K KO = K G

(a, 0)

(b, 0)

1 2 3 4

× ×

× ×

(c, 0)

(d, 0)

(e, 0)

×

×

×

(0, b)

(0, c)

×

(0, d)

(0, e)

×

× × ×

×

× ×

×

(0, a)

×

×

× ×

Fig. 5. OEL(G, M, I) in Example 3.2.

Fig. 4. B (KO ) in Example 3.1.

c

On the one hand, we need to prove B1 = AI , B2 = AI . According to Theorem 3.2, we obtain (B1 × {0} ) ∪ ({0} × B2 ) = B = A∼ . Suppose m ∈ B1 , we have m ∈ gI for all g ∈ A, which yields m ∈ ∩gI = AI . Therefore, B1 ⊆AI . Conversely, suppose m ∈ AI , we have m ∈ gI for all g ∈ A. With the definition of ∼, this yields g ∼ (m, 0). That is (m, 0) ∈ g∼ . Hence, (m, 0 ) ∈ ∩g∼ = A∼ . Since (B1 × {0} ) ∪ ({0} × B2 ) = B = A∼ , we get m ∈ B1 . Therefore, B1 ⊇AI . c Then B1 = AI . Analogous to the above proof, we have B2 = AI . c I I On the other hand, we need to prove A = B1 ∩ B2 . Since B1 = c c c c c c AI , B2 = AI , we have BI1 = AII , BI2 = AI I . Since A ⊆ AII , A ⊆ AI I , we c Ic c c II I I I I I can get A ⊆ A ∩ A = B1 ∩ B2 . Suppose g ∈ B1 ∩ B2 , then, for any m ∈ B1 , we get gIm, i.e., m ∈ gI ; for any n ∈ B2 , we get gIc n, i.e., c n ∈ gI . By the definition of ∼, we can get g ∼ (m, 0) and g ∼ (0, n). With the definition of ∼, we have g ∼ (m, n) for any (m, n ) ∈ (B1 × c {0} ) ∪ ({0} × B2 ) = B. Hence, g ∈ B∼ = A. Therefore, A ⊇ BI1 ∩ BI2 . To c sum up, A = BI1 ∩ BI2 . Combining the above arguments, we obtain that ϕ : B (KO ) → OEL(G, M, I ) is a map. Second, we have to demonstrate ϕ is a bijective map. On the one hand, we need to prove ϕ is an injective map. Suppose (A, B), (C, D ) ∈ B (KO ) and (A, B) = (C, D). Since B (KO ) is a concept lattice, we have A = C, B = D. Thus, (A, (B1 , B2 )) = (C, (D1 , D2 )), i.e., ϕ (A, B) = ϕ (C, D). On the other hand, we need to prove ϕ is a subjective map. For any (A, (B1 , B2 )) ∈ OEL(G, M, I), we denote B = (B1 × {0} ) ∪ ({0} × B2 ), now we only need to demonstrate (A, B ) ∈ B (KO ) and ϕ (A, B ) = (A, (B1 , B2 )). For all (m, n) ∈ B, we consider the following cases. Case one: (m, n) ∈ B1 × {0}. Since (A, (B1 , B2 )) ∈ OEL(G, c c M, I), we get AI = B1 , AI = B2 and A = BI1 ∩ BI2 . Since m ∈ B1 , we I I get m ∈ A , i.e., for any g ∈ A, m ∈ g . Hence g ∼ (m, 0) by the definition of ∼ . Because of the arbitrariness of g, we get (m, n) ∈

A∼ . Case two: (m, n) ∈ {0} × B2 . For this case, following the above proof, we can prove (m, n) ∈ A∼ . Combining the above arguments, we have that B⊆A∼ . Similarly, for all (m, n) ∈ A∼ , i.e., for any g ∈ A, g ∼ (m, n), we also consider the following cases. Case one: (m, n ) = (m, 0 ). For any g ∈ A, g ∼ (m, n). Hence we have m ∈ gI by the definition of ∼ . According to arbitrariness of g, we obtain m ∈ AI . AI = B1 yields m ∈ B1 . Hence, (m, n ) = (m, 0 ) ∈ B1 × {0} ⊆ B. Case two: (m, n ) = (0, n ). We can also prove (m, n ) = (0, n ) ∈ B similarly. Combining the above arguments, we have that B⊇A∼ . To sum up, we get B = A∼ . Similarly, we can prove B∼ = A. Therefore, (A, B ) ∈ B (KO ). According to the definition of ϕ , it is obvious that ϕ (A, B ) = (A, (B1 , B2 )). Third, we have to demonstrate ϕ and ϕ −1 are order preserving maps. In fact, for any (A, B ), (C, D ) ∈ B (KO ), (A, B) ≤ (C, D)⇔A⊆C⇔(A, (B1 , B2 )) ≤ (C, (D1 , D2 ))⇔ϕ (A, B) ≤ ϕ (C, D). To sum up, we obtain that B (KO ) and OEL(G, M, I) are isomorphic lattices. The proof process of Theorem 3.3 induces the following theorem immediately. It presents a novel approach to constructing object-induced three-way concept lattices. In addition, it is the theoretical basis for the corresponding algorithm in Section 5.  Theorem 3.4. Suppose K1 = (G, M, I ) is a formal context and K2 = (G, M, Ic ) is its complementary context. Then OEL(G, M, I ) = {(A, (B1 , B2 ))|(A, B ) ∈ B (KO )}, where B1 = {x|(x, y ) ∈ B and x = 0}, B2 = {y|(x, y ) ∈ B and y = 0}. Example 3.2. We continue Example 3.1 and show the following results. We also take F ∈ B (KO ) in Fig. 4 as an example to find an object-induced three-way concept. Through Theorems 3.3 and 3.4, we get A = {3}, B1 = {d} and B2 = {a, b, c, e}. Thus, the corresponding object-induced three-way concept is (3, (d, abce)). The corresponding three-way concept lattice of (G, M, I) gotten from Theorem 3.4 is shown in Fig. 5 which is same with the object-induce three-way concept lattice shown in Fig. 2.

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T. Qian et al. / Knowledge-Based Systems 116 (2017) 39–48

From reference [5], we know the diagram of a concept lattice of an apposition can be adequately represented by a nested 1 ), B (K 2 ) and B (KO ) are line diagram. In addition, we know B (K isomorphic with B (K1 ), B (K2 ) and OEL(K1 ) by Theorem 3.1 and Theorem 3.3, respectively. Therefore, the relationships among B (K1 ), B (K2 ) and OEL(K1 ) we obtained from this viewpoint are consistent with the relationships among them in references [21] and [22]. 4. Type II-combinatorial context based on the attribute set

Table 6 1 = (G × {0}, M,  ). K G

a

b

(1, 0) (2, 0) (3, 0) (4, 0)

× ×

× ×

×

×

×

×

b

c

Theorem 4.1. Suppose K1 = (G, M, I ) is a formal context and K2 = (G, M, Ic ) is its complementary context. Then we have the following i are isomorphic contexts. B (Ki ) and results for i = 1, 2: Ki and K i ) are isomorphic concept lattices. B (K Similar to Section 3, we can easily obtain the relations among these operators who are I, Ic ,  and  through Theorem 4.1 and the definitions of  and . Corollary 4.1. Suppose K1 = (G, M, I ) is a formal context and K2 = (G, M, Ic ) is its complementary context, X⊆G and A⊆M. Then the following results hold: (1) A = AI × {0} and A = {0} × AI . c (2) (X × {0} ) = X I and ({0} × X ) = X I . c

1 and K 2 , we can obtain a new In the following, combining K formal context. Definition 4.1. Let K1 = (G, M, I ) be a formal context and K2 = (G, M, Ic ) is its complementary context. Type II-combinatorial con1 text based on the attribute set is defined by the subposition of K 2 , which is denoted by KA , i.e., KA = K1 = (G × {0} ∪ {0} × and K  K 2

G, M,  ∪  ). For simplicity, we denote  ∪  by . Through the definition of KA and Corollary 4.1, we will obtain the analogous conclusions for Type II-combinatorial context based on the attribute set. Corollary 4.2. Suppose K1 = (G, M, I ) is a formal context and K2 = (G, M, Ic ) is its complementary context, X, Y⊆G and A⊆M. Then the following results hold: (1) A = (AI × {0} ) ∪ ({0} × AI ), c (2) {(X × {0} ) ∪ ({0} × Y )} = X I ∩ Y I . c

Theorem 4.2. Suppose (A, B ) ∈ B (KA ), A1 = {x|(x, y ) ∈ A} and A2 = {y|(x, y ) ∈ A}. Then we have A = (A1 × {0} ∪ {0} × A2 ) \ {(0, 0 )}. In the following, we will show KA is exactly the other kind of formal context we need, i.e., its concept lattice corresponds to attribute-induced three-way concept lattice. The conclusion is reflected by Theorem 4.3.

d

e

×

×

×

Table 7 2 = ({0} × G, M,  ). K G

Similar to Section 3, to use acquisition methods of concept lattices to construct attribute-induced three-way concept lattices, we also need to find some formal context, whose concept lattice is isomorphic to the attribute-induced three-way concept lattice of original formal context. In this process, since the proof of each theorem is similar to that of the corresponding theorem in Section 3, we omit them here. In the following, we will describe this process in detail. Suppose K1 = (G, M, I ) is a formal context and K2 = (G, M, Ic ) is its complementary context. In order to distinguish the object sets in K1 and K2 , we also add tags to them. And then, K1 and K1 are 1 and K 2 , respectively as follows. translated into K 1 = (G × {0}, M,  ) with (g, 0)m ⇔gIm for any (g, 0) ∈ G × K {0} and m ∈ M. 2 = ({0} × G, M,  ) with (0, g)m ⇔gIc m for any (0, g) ∈ {0} K × G and m ∈ M.

c

(0, (0, (0, (0,

a 1) 2) 3) 4)

d

e

× ×

× × ×

d

e

×

×

× ×

×

×

Table 8 1 KA = K . K 2

G

a

b

c

(1, 0) (2, 0) (3, 0) (4, 0) (0, 1) (0, 2) (0, 3) (0, 4)

× ×

× ×

×

×

×

× ×

×

×

×

×

× ×

× × ×

Theorem 4.3. Suppose K1 = (G, M, I ) is a formal context and K2 = (G, M, Ic ) is its complementary context. Then we have the following results: (1) B (KA ) and AEL(G, M, I) are isomorphic lattices. (2) AEL(G, M, I ) = {((A1 , A2 ), B )|(A, B ) ∈ B (KA )}, where {x|(x, y ) ∈ A and x = 0}, A2 = {y|(x, y ) ∈ A and y = 0}.

A1 =

Remark. Theorem 4.3(2) presents a novel construction approach of the attribute-induced three-way concept lattice, and based on it, we will present corresponding algorithm in Section 5. From reference [5], we know the diagram of concept lattice of an subposition can be adequately represented by a nested line di1 ), B (K 2 ) and B (KA ) are isoagram. In addition, we know B (K morphic with B (K1 ), B (K2 ) and AEL(K1 ) by Theorem 4.1 and Theorem 4.3, respectively. Therefore, the relationships among B (K1 ), B (K2 ) and AEL(K1 ) we obtained from this viewpoint are consistent with the relationships among them in references [21] and [22]. Example 4.1. We also consider Table 1 as K1 = (G, M, I ). And by Example 3.1, we obtain K2 which is shown in Table 2. According 1 , K 2 and KA are shown to the definitions of new contexts above, K in Tables 6–8, respectively. The corresponding concept lattice of KA is shown in Fig. 6. We choose the concept E ∈ B (KA ) in Fig. 6 to explain Theorem 4.2. We obtain A = {(2, 0 ), (4, 0 ), (0, 1 ), (0, 3 )}, B = {c} by this concept, A1 = {2, 4, 0} and A2 = {1, 3, 0} according to the notions in Theorem 4.2. We can calculate ({2, 4, 0} × {0} ∪ {0} × {1, 3, 0} ) \ { ( 0, 0 )} = { ( 2, 0 ), ( 4, 0 ), ( 0, 1 ), ( 0, 3 ), ( 0, 0 )} \ { ( 0, 0 )} = {(2, 0 ), (4, 0 ), (0, 1 ), (0, 3 )}. Therefore, we easily obtain (A1 × {0} ∪ {0} × A2 ) \ {(0, 0 )} = A. We also take E ∈ B (KA ) in Fig. 6 as an example to find a attribute-induced three-way concept. Through Theorem 4.3, we obtain A1 = {2, 4}, A2 = {1, 3} and B = {c}. Thus, the corresponding attribute-induced three-way concept is ((24, 13), c). Similarly, we can obtain all the attributeinduced three-way concepts of (G, M, I) and the corresponding three-way concept lattice is shown in Fig. 7. It is easy to see that it is same with Fig. 3 in Section 2.

T. Qian et al. / Knowledge-Based Systems 116 (2017) 39–48

45

Algorithm 1 . 1. Input the formal context (G, M, I ). 2. Set OEL(G, M, I ) = ∅. 3. Set ∼= ∅, while (g ∈ G and (m, n ) ∈ (M × {0} ) ∪ ({0} × M ))

{

if (m, n ) ∈ M × {0} and (g, m ) ∈ I

{

∼=∼ ∪{(g, (m, n ))}

}

if (m, n ) ∈ {0} × M and (g, n ) ∈ /I

{

∼=∼ ∪{(g, (m, n ))}

}

}

kO = (G, (M × {0} ) ∪ ({0} × M ), ∼ ). 4. Use the algorithm Inclose 2 [1] to calculate the set of all concepts of kO = (G, (M × {0} ) ∪ ({0} × M ), ∼ ), denote it by B (kO ). 5. While ((A, B ) ∈ B (kO ))

{

Fig. 6. B (KA ) in Example 4.1.

}

B1 = {x|(x, y ) ∈ B and x = 0}, B2 = {y|(x, y ) ∈ B and y = 0}, OEL(G, M, I ) = OEL(G, M, I ) ∪ {(A, (B1 , B2 ))}

6. Output OEL(G, M, I ).

Table 9 K1 . G

a

b

1 2 3 4

× × ×

×

c

× ×

In this section, we will give the corresponding algorithms to construct the three-way concept lattices based on the methods of constructing the three-way concept lattices mentioned in Sections 3 and 4. And then we conduct experiments based on different data in order to illustrate our algorithms’ validity and practicality.

formal concepts into object-induced three-way concepts. The first part would take O(|G||M|) time at most. The complexity of invoking Inclose 2 is the same with the complexity of Inclose 2. The complexity of Inclose 2 is bounded by O(|B (KO )||G||M|2 ) (or O(|B (KO )||G|2 |M| ) if the main iteration of the algorithm is over objects instead of attributes, as in the Inclose 2 variants presented here). The complexity of the third part obviously has an upper bound of O(|B (KO )||M| ). Therefore, the total complexity of Algorithm 1 can be estimated by the complexity of Inclose 2. In addition, this algorithm is very flexible, since in step 4, we can change Inclose 2 which was proposed by Andrews in 2011 [1] to any other concept lattice construction algorithms. Correspondingly, time complexity of Algorithm 1 will change along with the substitutes of Inclose 2 in step 4. In the following, we give a simple example to describe the above algorithm detailedly.

5.1. Algorithms and illustrations

Example 5.1. K1 is shown in Table 9, where G = {1, 2, 3, 4} is an object set, M = {a, b, c} is an attribute set.

Fig. 7. AEL(G, M, I) in Example 4.1.

5. Algorithms and experiments

Algorithm 1 is proposed based on Theorem 3.4. We first proposed this algorithm to construct the object-induced three-way concept lattice. In Algorithm 1, step 3 constructs a new formal context and step 5 realizes the conversion between formal concepts and objectinduced three-way concepts. Time complexity of Algorithm 1 depends on step 4. The execution time of Algorithm 1 is composed of three parts: constructing a new formal context, marking all formal concepts of the formal context by invoking Inclose 2 and translating the

1. Input K1 . 2. Construct the new formal context KO , which is shown in Table 10. 3. Use the algorithm Inclose 2 [1] to calculate all concepts of KO , which are shown in Fig. 8. 4. Calculate all the three-way concepts of K1 which are shown in Fig. 9. The following Algorithm 2 is proposed based on Theorem 4.3. It provides a novel acquisition approach of the attribute-induced three-way concept lattice. The complexity issues of constructing

46

T. Qian et al. / Knowledge-Based Systems 116 (2017) 39–48 Table 10 KO .

Algorithm 2 .

G

(a, 0)

(b, 0)

1 2 3 4

× × ×

×

(c, 0)

(0, a)

× ×

× ×

(0, b)

×

(0, c) × × ×

1. Input the formal context (G, M, I ). 2. Set AEL(G, M, I ) = ∅. 3. Set = ∅. while ((g, h ) ∈ (G × {0} ) ∪ ({0} × G ) and m ∈ M

{

if (g, h ) ∈ G × {0} and (g, m ) ∈ I

{

= ∪{((g, h ), m )}

}

if (g, h ) ∈ {0} × G and (h, m ) ∈ /I

{

= ∪{((g, h ), m )}

}

}

kA = ((G × {0} ) ∪ ({0} × G ), M,  ). 4. Use the algorithm Inclose 2 [1] to calculate the set of all concepts of kA = ((G × {0} ) ∪ ({0} × G ), M,  ), and denote it by B (kA ). 5. While ((A, B ) ∈ B (kA ))

{

}

A1 = {x|(x, y ) ∈ A and x = 0}, A2 = {y|(x, y ) ∈ A and y = 0}, AEL(G, M, I ) = AEL(G, M, I ) ∪ {((A1 , A2 ), B )}

6. Output AEL(G, M, I ).

Fig. 8. B (KO ) in Example 5.1.

Table 11 KA . G

a

b

(1, 0) (2, 0) (3, 0) (4, 0) (0, 1) (0, 2) (0, 3) (0, 4)

× × ×

×

c

× × × ×

×

× × ×

tal complexity of Algorithm 2 changes along with the difference of algorithms constructing concept lattices in step 4. Similarly to Algorithm 1, Algorithm 2 is examined by Example 5.2. Example 5.2. K1 in Table 9 is a formal context.

Fig. 9. OEL(K1 ) in Example 5.1.

the attribute-induced three-way concept lattice of (G, M, I) can be analyzed similarly. Its execution time is also composed of three parts: constructing a new formal context, marking all formal concepts of the new formal context by invoking Inclose 2 and translating the formal concepts into attribute-induced three-way concepts. The first part would take O(|G||M|) time at most. The complexity of invoking Inclose 2 is the same with the complexity of Inclose 2 which is bounded by O(|B (KA )||G||M|2 ) (or O(|B (KA )||G|2 |M| ) if the main iteration of the algorithm is over objects instead of attributes, as in the Inclose 2 variants presented here). The complexity of the third part obviously has an upper bound of O(|B (KA )||G| ). Therefore, the total complexity of Algorithm 2 can be also estimated by the complexity of Inclose 2. Similarly, the to-

1. Input K1 . 2. Construct the new formal context KA shown as Table 11. 3. Use the algorithm Inclose 2 [1] to calculate all concepts of KA as follows in Fig. 10. 4. Calculate all the three-way concepts of K1 , which are shown in Fig. 11. To sum up, the examples shows that our algorithms are feasible and valid. 5.2. Experiments Algorithm 1 and Algorithm 2 are similar, so we only conduct some experiments to compare Algorithm 1 with the algorithm of the object-induced three-way concept lattice by its definition(denote it as D-Algorithm). The experiments were run on an idle 64-bit system (Intel Core i8-4790, 3.60 GHz, 16 GB RAM), and all tests are divided into two groups. In the first group of experiments, we have used the following formal contexts which are several real-world benchmark datasets

T. Qian et al. / Knowledge-Based Systems 116 (2017) 39–48

47

Table 12 Performance (in seconds). Data

|G|

|M |

FR

|B |

|OEL|

Running time T1

TD

1 2 3 4 5

4 4 8 130 101

5 3 8 6 28

0.550 0.500 0.406 0.496 0.300

6 6 19 23 354

8 11 44 122 29793

0.001 0.001 0.002 0.005 0.047

0.015 0.015 0.093 410.930 > 290

Table 13 Performance of random datasets.

A=({(1,0),(2,0),(3,0),(4,0),(0,1),(0,2),(0,3),(0,4)},Ø) B=({(1,0),(2,0),(3,0),(0,4)},a) C=({(1,0),(4,0),(0,2),(0,3)},b) D=({(3,0),(0,1),(0,2),(0,4)},c) E=({(1,0)},ab) F=({(3,0),(0,4)},ac) G=({(0,2)},bc) H=(Ø,abc) Fig. 10. B (KA ) in Example 5.1.

FR

|OEL|

Running time (s) T1

TD

0.300 0.500 0.700

31394 13338 34092

0.094 0.062 0.297

148.085 143.115 142.069

perimental results show that Algorithm 1 outperforms D-Algorithm significantly. In the second group of experiments, we have used contexts that are randomly generated datasets with different fill ratios (i.e. |I|/|G||M|). There are 20 objects and 50 attributes in each dataset. The experimental results have been shown in Table 13. It shows that Algorithm 1 outperforms D-Algorithm significantly no matter under the fill ratios. 6. Conclusion

Fig. 11. AEL(K1 ) in Example 5.1.

selected from the above sections, reference [5] and UCI Machine Learning Repository. The reason why we choose the datasets from the above sections is that we want to test the correctness of Algorithm 1. 1. 2. 3. 4.

The formal context in Example 2.1 in Section 2. The formal context in Example 5.1 in Section 5. Living Beings and Water [5]. Membership of developing countries in supranational group [5]. In this data, 130 developing countries are objects. And 6 properties (Group of 77, Non-aligned, Least developed countries, Most seriously affected countries, Organization of petrol exporting countries, African Caribbean and Pacific countries) are attributes. 5. Zoo database which comes from UCI Machine Learning Repository [2]. There are 101 objects and 17 attributes which consist of 15 Boolean attributes and 2 numerics attributes. We deal with 2 numerics attributes by nominal scales, thus we get other 13 attributes. That means, we have 28 attributes in total. The results are shown in the following Table 12, where FR is fill ratios, i.e. |I|/|G||M|, |B | presents the number of formal concepts, |OEL| presents the number of OE-concepts, T1 is the running time of Algorithm 1 and TD is the running time of D-Algorithm. The ex-

In this paper, we designed two kinds of new formal contexts whose concept lattices are isomorphic to three-way concept lattices of a given formal context. By means of existing methods of constructing concept lattices, we acquired three-way concept lattices. First, we proposed two new formal contexts (Type Icombinatorial context and Type II-combinatorial context) on the basis of an original formal context. By using these two contexts, we constructed three-way concepts by two-way operators and analyzed the relationships between these contexts and the original formal context. And then we achieved the relationships between three-way concept lattices and classical concept lattices of two new formal contexts. Moreover, we gave the corresponding algorithms and conducted some experiments. Based on the results of this paper, we can study the relationships between attribute reduction of three-way concept lattice and the attribute reductions of the above formal contexts in the future. Also, we can further study other construction methods of threeway concept lattices and find a better one. Acknowledgments This work is partially supported by the National Natural Science Foundation of China (grant nos. 11371014 and 11071281), the Natural Science Basic Research Plan in Shaanxi Province of China (program no. 2014JM8306) and the State Scholarship Fund of China (grant no. 201508610023). References [1] S. Andrews, In-CLOSE2, a High Performance Formal Concept Miner. Conceptual Structures for Discovering Knowledge, Proceedings of the 19th International Conference on Conceptual Structures (ICCS), Springer, 2011, pp. 50–62. [2] A. Asuncion, D.J. Newman, UCI Machine Learning Repository, University of California, Irvine, School of information and Computer Sciences, 2007. http: //www.ics.uci∼mlearn∼MLRepository.html [3] B.A. Davey, H. Priestley, Introduction to Lattices and Order, Cambridge University Press, 1990.

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