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Three-way dual concept analysis
International Journal of Approximate Reasoning 114 (2019) 151–165
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International Journal of Approximate Reasoning www.elsevier.com/locate/ijar
Three-way dual concept analysis Huilai Zhi a , Jianjun Qi b,∗ , Ting Qian c , Ling Wei d a
School of Computer Science and Technology, Henan Polytechnic University, Jiaozuo 454000, Henan, PR China School of Computer Science and Technology, Xidian University, Xi’an 710071, PR China College of Science, Xi’an Shiyou University, Xi’an, 710065, PR China d School of Mathematics, Northwest University, Xi’an 710069, PR China b c
a r t i c l e
i n f o
Article history: Received 11 March 2019 Received in revised form 6 August 2019 Accepted 26 August 2019 Available online 2 September 2019 Keywords: Formal concept analysis Three-way concept analysis Three-way dual concept analysis
a b s t r a c t Three-way concept analysis is a mathematical theory, which combines formal concept analysis and three-way decision. The existing models, i.e., three-way concept lattices and three-way object oriented concept lattices, have been used successfully in many fields. However, these two models are established to cater some specific applications and cannot fulfill some special requirements such as the needs to seek potential collaborators in international import and export transactions. In this paper, a novel type of three-way concept lattices is presented based on dual concept analysis, which enables ones to characterize specific set by pointing out the attributes that are not possessed by at least one object in the complement of this specific set. And then, the connections between three-way dual concept lattices and classical dual concept lattices are explored. Besides, the relationships among four types of three-way concept analysis models, i.e., three-way concept lattices, three-way dual concept lattices, three-way object oriented concept lattices and three-way property oriented concept lattices, are investigated. © 2019 Elsevier Inc. All rights reserved.
1. Introduction Formal concept analysis (FCA), a mathematical framework founded on lattice theory [43], is used effectively in various types of data analysis activities [19,24,61]. Formal context and formal concept are two cornerstones in FCA. A formal context describes a domain in an ideal case. For an object of a given context, one can precisely points out which attributes this object possesses and which attributes this object does not possess. Formally, a concept is composed of two parts, i.e., the extent and the intent. Moreover, there is an affirmative relationship between them. That is, each object of the extent possesses all the attributes in the intent and each attribute of the intent is shared by all the objects in the extent. In classical FCA, in order to describe a specific target set, only positive attributes are concerned, while the negative ones are ignored. However, many researchers have pointed out that both positive attributes and negative attributes play equally important roles in many fields, such as knowledge creation and communication, concept representation and rule acquisition [13,37,39,44]. Concretely, give a specific target set, we describe it by pointing out what features it has and what features it does not have. In the settings of data mining and FCA, we call them positive attributes and negative attributes respectively. Three-way decision (3WD) [47,51,54] is a theory which promotes thinking, problem solving, and information processing in threes. Initially, This theory was introduced for interpreting the three types of classification rules in rough set theory. Since
*
Corresponding author. E-mail addresses: [email protected] (H. Zhi), [email protected] (J. Qi), [email protected] (T. Qian), [email protected] (L. Wei).
https://doi.org/10.1016/j.ijar.2019.08.010 0888-613X/© 2019 Elsevier Inc. All rights reserved.
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Table 1 Companies and their specific traducements.
1 2 3 4 5
a
b
c
×
×
× × ×
×
×
d
e
× × ×
× ×
its proposal, three-way decision has gained increasing attentions from the communities of rough set theory [3,5,12,15,23, 25,26,34]. Besides, three-way decision was frequently used in feature selection [35,56] and data clustering [1,57]. What is more, plenty of studies have shown that three-way decision has played an active and important role in decision makings [11,27,31,41,45,46,55]. Inspired by the theory of three-way decision, Qi et al. [29] proposed three-way concept analysis (3WCA). The significant improvement is that one can characterize any specific set by both positive and negative attributes, and as a result three-way concepts can provide more information than that of classical concepts [30]. In 3WCA, there are two kinds of three-way concept lattices, one is the object-induced three-way concept lattice, and the other is attribute-induced three-way concept lattice. Formally, the intent of an object-induced three-way concept has a positive part and a negative part. The positive part is composed of the attributes which are possessed by all the objects in the extent, while the negative part is composed of the attributes which are not possessed by all the objects in the extent. Dually, attribute-induced three-way concepts can be defined as well [30]. At present, 3WCA has attracted more and more attentions. For instance, Qian et al. [32] adopted dividing and conquering strategy to improve the efficiency of constructing three-way concept lattices. Recently, Qian et al. [33] further carried out a theoretical study on the object (property) oriented concept lattices based on three-way decision. Singh [40] constructed three-way fuzzy concept lattices and extracted useful patterns to support medical diagnoses. Shivhare and Cherukuri [38] exploited the spirit of three-way concept analysis to simulate a cognitive memory process. Li et al. [19,20] explored the three-way cognitive learning process via multi-granularity, which can be viewed as a generalization of the classical concept learning scheme. Zhi and Li [60] proposed an approach to granule description based on positive and negative attributes. Besides, as two important examples of partially-known formal concepts [36,52], ill-known formal concepts [6] and approximate concepts [18,22] can be viewed as a sort of three-way concepts to some extent. The existing 3WCA models, i.e., three-way concept lattices and three-way object oriented concept lattices, are established in specific perspectives. Concretely, three-way concept lattices are based on common attribute analysis and characterize any specific set by pointing out both the positive and negative common attributes. Three-way object oriented concept analysis, proposed by Wei and Qian [42], is based on necessary attribute analysis and characterizes any specific set by pointing out both the positive and negative necessary attributes. As mentioned above, these two models have been used successfully in many fields. However, these two models cannot completely fulfill some specific requirements. Let’s consider the following Example 1. Example 1. Consider five different companies from five different countries, each of which deals in some specific commodity traducements. The details about these companies are shown in Table 1, where {1, 2, 3, 4, 5} is the set of companies, {a, b, c , d, e } is the set of commodities and a cross denotes a company is engaged in a specific kind of commodity transaction. More concretely, companies 1, 2, 3, 4 and 5 are from China, Australia, Thailand, Philippines and New Zealand, respectively; a, b, c, d and e represent tea, rice, food, spice and wool, respectively. In international import and export transactions, businessmen must have a good knowledge about what commodities their potential customers cannot produce. In other words, if all their potential customers can produce a certain type of commodities, then there is no chance to export this type of commodities. Otherwise, if there exists at least one customer who cannot produce this type of commodities, then the businessmen can consider producing this type of commodities. Usually, there are several middlemen between the manufactures and the ultimate users. For instance, there is a middleman who wants to make transactions between oceania companies and asian companies. Obviously, companies 1, 3 and 4 are from asian, and companies 2 and 5 are from oceania. On one hand, if the middleman wants to sell the commodities produced by asian companies to the oceania companies, he must know what commodities the oceania companies cannot produce. Based on Table 1, we can see that the middleman has a chance to sell tea, rice and spice to the oceania companies. It is worth noting that although company 2 can produce spice, the middleman still has an opportunity to sell spice to company 5. On the other hand, if the middleman wants to sell the commodities produced by oceania companies to the asian companies, he must have a knowledge about what commodities the oceania companies can produce. Based on Table 1, we can see that oceania companies can produce food, spice and wool. As a result, the middleman has a chance to sell these three types of commodities to the asian companies. In what follows, we formulate the above discussions with mathematical languages. Let X = {1, 3, 4}, Y = {2, 5}, A = {a, b, d} and B = {c , d, e }. Then, X denotes the set of asian companies, Y represents the set of oceania companies, and there is a useful pattern ( X , ( A , B )), which states that there exists at least one element of the complement of X , i.e., there exists
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y ∈ Y , who does not possess at least one attribute in A and possess at least one attribute in B. And meanwhile, X is composed of the elements who do not possess at least one attribute in A c or possesses at least one attribute in B c . Actually, there are still other similar patterns in Table 1, such as ({2, 5}, ({a, b, c , d, e }, {a, b, c , d})), ({1, 4}, ({a, b, d, e }, {b, c , d, e })), and ({1, 5}, ({a, b, c , e }, {b, c , d, e })) etc. In all, there are 19 such patterns embodied in Table 1. After a serious analysis, we can see that Example 1 shows us a useful and interesting pattern which totally differs from the existing ones, including three-way concepts [29,30] and three-way object oriented concepts [42]. Actually, similar requirements frequently arise in a variety of applications, such as international cooperation and international education. In international cooperation, one must have a good knowledge about their potential collaborators, including advantages and disadvantages. It is the advantages that make a will for us to cooperate with them. And it is the disadvantages that make a desire for them to cooperate with us. In the field of international education, we can only teach others what they do not know and meanwhile we can only learn something from the others what they already specialize in. Generally speaking, this problem can be formulated in the following way: given a formal context, how to effectively derive the patterns in the format of ( X , ( A , B )), where A is the set of attributes that are not possessed by at least one object in X c , B the set of attributes that are possessed by at least one object in X c , and X is the set of objects that who do not possess at least one attribute in A c or possesses at least one attribute in B c . In applications, the specific set X can represent a coalition of countries, companies and persons, et al. And the complement of the specific set X c can denote their potential collaborators. As the existing three-way concept lattices cannot effectively solve this problem, there is a vital need to seek a novel 3WCA model. Motivated by the above stated problem, we propose a new type of three-way concept analysis model based on dual concept analysis [10,49,50]. The rest of this paper is organized as follows. Section 2 briefly reviews some basic notions in FCA. Section 3 introduces a novel type of three-way concept analysis model, i.e., object-induced three-way dual concept lattices. Besides, the relationships between object-induced three-way dual concept lattices and dual concept lattices are also investigated. Section 4 explores the relationships among four types of object-induced three-way concept analysis models, i.e., object-induced three-way concept lattices, object-induced three-way dual concept lattices, object-induced three-way object oriented concept lattices and object-induced three-way property oriented concept lattices. Finally, conclusions are provided in Section 5. 2. Related theoretical foundations In this section, we briefly review some basic notions in FCA to make the paper self-contained. 2.1. Basic notions of FCA Let U and V be two finite and nonempty sets. In the settings of FCA, the elements of U are called objects, and the elements of V are called attributes. The relationships between the objects and the attributes are described by a binary relation R. For a pair of elements x ∈ U and a ∈ V , if (x, a) ∈ R, also written as xRa, we say that x has the attribute a, or a is possessed by the object x. Let R (x) = {a ∈ V | xRa} be the set of attributes possessed by the object x. Dually, let R (a) = {x ∈ U | xRa} be the set of objects that possess the attribute a. The complement of a binary relation R is defined by:
R c = U × V − R = {(x, a) | ¬(xRa)}. Then, we call the triplet K = (U , V , R ) a formal context and K c = (U , V , R c ) the complement of K . If there exists x ∈ U such that x has all the attributes in V or x doesn’t have any attribute in V , we think the object x is meaningless. Similarly, we are not interested in such attribute that is in a relation with all the objects or not in a relation with any object. We call this formal context canonical, and the formal context we discuss in this paper are all canonical. Let K = (U , V , R ) be a formal context. A set-theoretic operator ∗ : 2U → 2 V is defined as:
X ∗ = {a ∈ V | x ∈ X ⇒ xRa} a ∈ V | X ⊆ R (a)} = { R (x), = x∈ X
which associates a set of attributes X ∗ to X . Dually, ∗ : 2 V → 2U is defined as:
A ∗ = {x ∈ U | a ∈ A ⇒ xRa)} x ∈ U | A ⊆ R (x)} = { R (a), = a∈ A
which describes a mapping from A ∗ to A.
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A formal concept is a pair ( X , A ) such that X ∗ = A and A ∗ = X . The sets X and A are called the extent and the intent of ( X , A ), respectively. Given a formal context K , the set of all formal concepts is naturally equipped with a partial order ≤ defined as: ( X 1 , A 1 ) ≤ ( X 2 , A 2 ) ⇔ X 1 ⊆ X 2 ⇔ A 2 ⊆ A 1 . Ganter and Wille [10] have proved that all the formal concepts ordered with ≤ form a complete lattice, which is the so-called the concept lattice and is denoted by L ( K ). The infimum and supremum in the lattice are respectively given by:
( X 1 , A 1 ) ∧ ( X 2 , A 2 ) = ( X 1 ∩ X 2 , ( A 1 ∪ A 2 )∗∗ ), ( X 1 , A 1 ) ∨ ( X 2 , A 2 ) = (( X 1 ∪ X 2 )∗∗ , A 1 ∩ A 2 ). 2.2. Formal concept analysis with a possibility theoretic view In FCA, besides the sufficiency operator (·)∗ (which corresponds to strong possibility), we can define the other three operators, i.e., (·)3 , (·)2 and (·)# , which are called possibility operator, necessity operator and dual sufficiency operator [4,7–9,49,50], respectively. Let K = (U , V , R ) be a formal context and X ∈ 2U . X 3 is the set of attributes that are satisfied by at least one object in X:
X 3 = {a ∈ V | X ∩ R (a) = ∅} = {a ∈ V | ∃x ∈ X , xRa}. X 2 is the set of attributes such that any object that satisfies one of them is necessarily in X :
X 2 = {a ∈ V | R (a) ⊆ X } = {a ∈ V | ∀x ∈ U (xRa ⇒ x ∈ X )}. X # is the set of attributes that are not possessed by at least one object in X c :
X # = {a ∈ V | X ∪ R (a) = U } = {a ∈ V | ∃x ∈ X c , xR c a}. Dually, for A ∈ 2 V , we can define A 3 , A 2 and A # respectively. A 3 is the set of objects that possess at least one attribute in A:
A 3 = {x ∈ U | A ∩ R (x) = ∅} = {x ∈ U | ∃a ∈ A , xRa}. A 2 is the set of objects such that any attribute that is satisfied by one of them is necessarily in A:
A 2 = {x ∈ U | R (x) ⊆ A } = {x ∈ U | ∀a ∈ V (xRa ⇒ a ∈ A )}. A # is the set of objects that do not possess at least one attribute in A c :
A # = {x ∈ U | A ∪ R (x) = V } = {x ∈ U | ∃a ∈ A c , xR c a}. Proposition 1. [49,50] Let (U , V , R ) be a formal context and X ⊆ U . Then the following properties hold. (i) (ii) (iii) (iv)
∗c
X ∗ = X c#c , X # = X c . 3c 2c X2 = Xc , X3 = Xc . ## ∗∗ X ⊆X⊆X . X 23 ⊆ X ⊆ X 32 . 2
c Proposition 2. Let (U , V , R ) be a formal context and X ⊆ U . Then X ∗R = X 3 Rc = X Rc . c
Proof.
X ∗R = {a ∈ V | X ⊆ R (a)} = {a ∈ V | X ∩ R c (a) = ∅} = V − {a ∈ V | X ∩ R c (a) = ∅} = V − X Rc 3 c = X3 Rc X ∗R = {a ∈ V | X ⊆ R (a)} = {a ∈ V | R c (a) ⊆ X c } = X cR c 2 The proof is completed.
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2.3. Three-way concept analysis In this subsection, we introduce two types of three-way concepts as well as their corresponding lattices. Definition 1. ([29,30]) Let K = (U , V , R ) be a formal context. For X ∈ 2U and ( A , B ) ∈ 2 V × 2 V , two associated three-way operators : 2U → 2 V × 2 V and : 2 V × 2 V → 2U are respectively defined as:
X = ( X ∗ , X ∗ ) and ( A , B ) = A ∗ ∩ B ∗ , where X ∗ = {a ∈ V | ∀x ∈ X , xR c a} and A ∗ = {x ∈ U | ∀a ∈ A , xR c a}. Besides, if X = ( A , B ) and ( A , B ) = X , then ( X , ( A , B )) is called an object-induced three-way concept. Furthermore, all the object-induced three-way concepts form a complete lattice, which is the so-called object-induced three-way concept lattice, and is denoted by O E L ( K ). What is more, by using duality principle, we can further define attribute-induced three-way concept lattice. Interested reader can refer to [30]. Three-way concepts and 3WD are in full compliance in terms of the semantics of tripartition. In short, applying the idea of 3WD to FCA, we can formulate three-way concepts and three-way concept lattices. Actually, three-way concepts and formal concepts in FCA [10] share a common framework and both of them are equipped with extensions and intensions. Nevertheless, as a generalization of a formal concept, the extension (or the intension) of a three-way concept is constituted with two parts: positive one and negative one. By using these two parts, we can respectively describe the semantics “jointly possessed” and “jointly not possessed” in a formal context. In addition, on the basis of a three-way concept, we can further divide the object (or attribute) universe into three regions, i.e., positive region, negative region and boundary region. And then, according to these three regions we make three-way decisions to solve our specific problems. Definition 2. ([42]) Let K = (U , V , R ) be a formal context. For X ∈ 2U and ( A , B ) ∈ 2 V × 2 V , two associated three-way operators : 2U → 2 V × 2 V and : 2 V × 2 V → 2U are respectively defined as:
X = ( X 2 , X 2 ) and ( A , B ) = A 3 ∪ B 3 , where X 2 = {a ∈ V | R c (a) ⊆ X } and B 3 = {x ∈ U | B ∩ R c (x) = ∅}. The pair ( X , ( A , B )) is called an object-induced three-way object oriented concept if X = ( A , B ) and ( A , B ) = X . Furthermore, all the object-induced three-way object oriented concepts form a complete lattice, which is the so-called object-induced three-way object oriented concept lattice, and is denoted by O E O L ( K ). Similar to three-way concepts, the semantics of three-way object oriented concepts can be described as well and the details are omitted here. 3. Object-induced three-way dual concept lattice In this section, we propose object-induced three-way dual concept lattices and study the relationships between objectinduced three-way dual concept lattices and classical dual concept lattices. 3.1. The constructions of object-induced three-way dual concept lattices Let M be a nonempty and finite set and 2 M × 2 M be the Cartesian product of 2 M and 2 M . For ( A 1 , A 2 ), ( B 1 , B 2 ) ∈ 2 M × 2 M , ≤ is defined as:
( A 1 , A 2 ) ≤ ( B 1 , B 2 ) ⇔ A 1 ⊆ B 1 and A 2 ⊆ B 2 . Then, it follows that
( A 1 , A 2 ) = ( B 1 , B 2 ) ⇔ ( A 1 , A 2 ) ≤ ( B 1 , B 2 ) and ( B 1 , B 2 ) ≤ ( A 1 , A 2 ). Besides, ∩ and ∪ are defined respectively as:
( A 1 , A 2 ) ∩ ( B 1 , B 2 ) = ( A 1 ∩ B 1 , A 2 ∩ B 2 ) and ( A 1 , A 2 ) ∪ ( B 1 , B 2 ) = ( A 1 ∪ B 1 , A 2 ∪ B 2 ). In what follows, we define two associated negative dual sufficiency operators in preparing the constructions of objectinduced three-way dual concept lattices.
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Definition 3. Let K = (U , V , R ) be a formal context. For X ∈ 2U and A ∈ 2 V , two negative derivation operators # : 2U → 2 V and # : 2 V → 2U are respectively defined as:
X # = {a ∈ V | ∃x ∈ X c , xRa} and A # = {x ∈ U | ∃a ∈ A c , xRa}. X # is the set of attributes that are possessed by at least one object in X c and A # is the set of objects that possess at least one attribute in A c . From the computational perspective, we still have the following results:
X # = {a ∈ V | X ∪ R c (a) = U } and A # = {x ∈ U | A ∪ R c (x) = V }. It is obvious that the operators # : 2U → 2 V and # : 2 V → 2U are exactly the operators # : 2U → 2 V and # : 2 V → 2U defined on K c = (U , V , R c ), and form a Galois connection between (2U , ⊆) and (2 V , ⊆). Definition 4. Let K = (U , V , R ) be a formal context. For X ∈ 2U and ( A , B ) ∈ 2 V × 2 V , two associated three-way operators : 2U → 2 V × 2 V and : 2 V × 2 V → 2U are respectively defined as:
X = ( X # , X # ) and ( A , B ) = A # ∪ B # . X is composed of two related parts, i.e., X # and X # . ( A , B ) is the maximal set containing objects that do not possess at least one attribute in A c or possess at least one attribute in B c . Definition 5. Let K = (U , V , R ) be a formal context. For X ∈ 2U and ( A , B ) ∈ 2 V × 2 V , if X = ( A , B ) and ( A , B ) = X , then ( X , ( A , B )) is called an object-induced three-way dual concept. Proposition 3. Let K = (U , V , R ) be a formal context. For X , X 1 , X 2 ∈ 2U and ( A , B ), ( A 1 , B 1 ), ( A 2 , B 2 ) ∈ 2 V × 2 V , the following properties hold. (i) (ii) (iii) (iv) (v)
X 1 ⊆ X 2 ⇒ X 1 ≥ X 2 ; ( A 1 , B 1 ) ≤ ( A 2 , B 2 ) ⇒ ( A 1 , B 1 ) ⊇ ( A 2 , B 2 ) . X ⊇ X ; ( A , B ) ≥ ( A , B ) . ( X 1 ∩ X 2 ) = X 1 ∪ X 2 and (( A 1 , B 1 ) ∩ ( A 2 , B 2 )) = ( A 1 , B 1 ) ∪ ( A 2 , B 2 ) . and form a Galois connection between (2U , ⊆) and (2 V × 2 V , ≤). ( X , X ) and (( A , B ) , ( A , B ) ) are two object-induced three-way dual concepts of K .
Proof. (i) If X 1 ⊆ X 2 , then it follows that X 1# ⊇ X 2# and X 1# ⊇ X 2# . That is, ( X 1# , X 1# ) ≥ ( X 2# , X 2# ), which implies that X 1 ≥ X 2 .
# # # # # # # If ( A 1 , B 1 ) ≤ ( A 2 , B 2 ), it implies that A # 1 ⊇ A 2 and B 1 ⊇ B 2 . Then, we have A 1 ∪ B 1 ⊇ A 2 ∪ B 2 , i.e., ( A 1 , B 1 ) ⊇ ( A 2 , B 2 ) . (ii) As X ⊇ X ## and X ⊇ X ## , it follows that X ⊇ X ## ∪ X ## , i.e., X ⊇ X . As A # ⊆ A # ∪ B # , it follows that A ## ⊇ ( A # ∪ B # )# , which leads to A ⊇ A ## ⊇ ( A # ∪ B # )# . Besides, as B # ⊆ A # ∪ B # , it follows that B ## ⊇ ( A # ∪ B # )# , which implies that B ⊇ B ## ⊇ ( A # ∪ B # )# . To sum up, we have ( A , B ) ≥ (( A # ∪ B # )# , ( A # ∪ B # )# ), i.e., ( A , B ) ≥ ( A , B ) .
(iii) Firstly, we have
( X 1 ∩ X 2 )# = {a ∈ V | ∃x ∈ ( X 1 ∩ X 2 )c , xR c a} = {a ∈ V | ∃x ∈ X 1c ∪ X 2c , xR c a} = {a ∈ V | ∃x ∈ X 1c , xR c a} ∪ {a ∈ V | ∃x ∈ X 2c , xR c a} = X 1# ∪ X 2# . Likewise, we can prove that ( X 1 ∩ X 2 )# = X 1# ∪ X 2# . To sum up, ( X 1 ∩ X 2 ) = (( X 1 ∩ X 2 )# , ( X 1 ∩ X 2 )# ) = ( X 1# ∪ X 2# , X 1# ∪ X 2# ) =
( X 1# , X 1# ) ∪ ( X 2# , X 2# ) = X 1 ∪ X 2 .
Analogously, we can prove that (( A 1 , B 1 ) ∩ ( A 2 , B 2 )) = ( A 1 , B 1 ) ∪ ( A 2 , B 2 ) . (iv) This proposition will be proved if we can prove X ⊇ ( A , B ) ⇔ ( A , B ) ≥ X . Sufficiency: X ⊇ ( A , B ) ⇒ X ≤ ( A , B ) ≤ ( A , B ). Necessity: ( A , B ) ≥ X ⇒ ( A , B ) ⊆ X ⊆ X . Then, this proposition is proved. (v) By the properties of a Galois connection, it follows immediately. 2
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Given a formal context K , the set of all object-induced three-way dual concepts is naturally equipped with a partial order ≤ defined as:
( X 1 , ( A 1 , B 1 )) ≤ ( X 2 , ( A 2 , B 2 )) ⇔ X 1 ⊆ X 2 ⇔ ( A 1 , B 1 ) ≥ ( A 2 , B 2 ). It is straightforward that all the object-induced three-way dual concepts contained in K with ≤ form a poset, which is denoted henceforth by O E D L ( K ). Theorem 1. Let K = (U , V , R ) be a formal context. The poset O E D L ( K ) is a complete lattice, in which the infimum and supremum are respectively given by:
( X 1 , ( A 1 , B 1 )) ∧ ( X 2 , ( A 2 , B 2 )) = (( X 1 ∩ X 2 ) , ( A 1 , B 1 ) ∪ ( A 2 , B 2 )) ( X 1 , ( A 1 , B 1 )) ∨ ( X 2 , ( A 2 , B 2 )) = ( X 1 ∪ X 2 , (( A 1 , B 1 ) ∩ ( A 2 , B 2 )) ) Proof. By Proposition 3 (iii), it follows that ( X 1 ∩ X 2 ) = X 1 ∪ X 2 = ( A 1 , B 1 ) ∪ ( A 2 , B 2 ) and (( A 1 , B 1 ) ∩ ( A 2 , B 2 )) = ( A 1 , B 1 ) ∪ ( A 2 , B 2 ) = X 1 ∪ X 2 . Moreover, by Proposition 3 (v), we can conclude that (( X 1 ∩ X 2 ) , ( A 1 , B 1 ) ∪ ( A 2 , B 2 )) and ( X 1 ∪ X 2 , (( A 1 , B 1 ) ∩ ( A 2 , B 2 )) ) are two object-induced three-way dual concepts of K . Then, this theorem is proved. 2 Henceforth, the complete lattice O E D L ( K ) is called the object-induced three-way dual concept lattice of K . Let ( X , ( A , B )) be an object-induced three-way concept. Then, A and B characterize X c in two complemented angles, i.e., A points out the attributes that are not possessed by at least one object in X c and B shows the attributes that are possessed by at least one object in X c . Besides, we can further show that A c contains the common attributes possessed by X c , while B c contains the common attributes which are not possessed by X c . What is more, it is easy to show that A c ∩ B c = ∅. It is worth noting that although three-way is used as a prefix to name ( X , ( A , B )) ∈ O E D L ( K ), the two parts A and B may overlap with each other, which means the pair cannot trisect V in a similar way as that of three-way concepts [29,30] and three-way object oriented concepts [42]. However, 3WD is a fast developing theory and has developed into a new stage with a wide sense. Yao has pointed out that 3WD promotes thinking, problem solving, and information processing in threes, that is, using three parts, three elements, three components, three perspectives, three views, three levels, three generations, three periods, three stages, three steps, triangles, triads, triplets, and many others [53]. Yao also stressed that it is the wide sense that increases the power of three-way decision [53]. Apparently, three-way dual concepts, which have the form ( X , ( A , B )), are built by using three perspectives or views. That are, a positive view A, a negative view B and the rest (the ones which are neither A nor B). What is more, although A and B may overlap with each other in the set-theoretic view, A and B are disjoint with each other in the semantic view. That is, A points out the attributes that are not possessed by at least one object in X c and B shows the attributes that are possessed by at least one object in X c . As a result, to a certain extent, A can be viewed as a set of positive attributes and B can be view as set of negative attributes in building a three-way dual concept. As object-induced three-way dual concept lattice and classical concept lattice are actually in full compliance, we put forward an incremental algorithm (i.e., Algorithm 1) with a similar idea originating from the existing algorithms for building classical concept lattices [14,17,28,59]. Algorithm 1 Building an object-induced three-way dual concept lattice. Require: A canonical formal context K = (U , V , R ). Ensure: Object-induced three-way dual concept lattice O E D L ( K ). 1: Initialize O E D L ( K ) = {(∅, ( V , V ))}. 2: Initialize i = 1. 3: While i ≤ |U | 4: Randomly select an object x from U . 5: U ← U − {x}. 6: Let n be the number of object-induced three-way dual concepts of O E D L ( K ). 7: Sort all the concepts of O E D L ( K ) into a sequence ( X 1 , ( A 1 , B 1 )), · · · , ( X n , ( A n , B n )) such that | X 1 | ≤ · · · ≤ | X n |. 8: Initialize j = 1. 9: While j ≤ n do 10: If x∗ c ∩ A cj = ∅ or x∗ ∩ B cj = ∅ 11: Update ( X j , ( A j , B j )) to ( X j ∪ {x}, ( A j , B j )). 12: If there dose not exist an object-induced three-way dual concept with the intent ( A j ∪ x∗ c , B j ∪ x∗ ) 13: Then add ( X j , ( A j ∪ x∗ c , B j ∪ x∗ )) to O E D L ( K ) 14: End If 15: End If 16: j++. 17: End While 18: i++. 19: End While 20: Return O E D L ( K ).
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Table 2 A formal context K of Example 2. a 1 2 3 4 5
× × ×
b
× × × ×
c
d
×
×
× ×
× ×
e
× ×
Fig. 1. O E D L ( K ) of Example 2.
Let N be the number of the object-induced three-way dual concepts of K . Then the time complexity of Algorithm 1 is O (|U |2 | V | N ). Besides, the lines from 6 to 17 actually present a strategy to update three-way dual concept lattices when adding a new object. Example 2. Table 2 shows a formal context K = (U , V , R ) which describes a crowd of software development engineers, where U = {1, 2, 3, 4, 5} denotes five engineers and V = {a, b, c , d, e } represents five professional skills, namely, Java programming, C programming, computer graphics, algorithm analysis and mathematical modeling, respectively. In the subsequent discussions, in the representation of a concept, we omit the braces and comma for convenience. For example, we use (25, bcde , abcde ) instead of its standard format ({2, 5}, ({b, c , d, e }, {a, b, c , d, e })). Fig. 1 shows the object-induced three-way dual concept lattice O E D L ( K ). As already stated in this section, each object-induced three-way dual concept denotes a meaningful pattern. For instance, from the object-induced three-way concept (245, bcd, abce ), we can get the following information. (i) There is someone of engineers 1 and 3 who does not master C programming or computer graphics or algorithm analysis, which implies that the engineers 2, 4 and 5 have a chance to teach someone of engineers 1 and 3 C programming or computer graphics or algorithm analysis. (ii) There is someone of engineers 1 and 3 who masters Java programming or C programming or computer graphics or mathematical modeling, which implies that the engineers 2, 4 and 5 have a chance to learn Java programming or C programming or computer graphics or mathematical modeling from someone of engineers 1 and 3. However, by using three-way concept analysis [29,30], we get different statements from a totally different perspective. (iii) Both engineers 1 and 3 have the skills of Java programming and mathematical modeling. Comparing with item (ii), we are unable to be aware of that there is someone of engineers 1 and 3 who is capable of C programming or computer graphics. (iv) Neither engineer 1 nor engineer 3 masters algorithm analysis. Comparing with item (i), we cannot realize that there is someone of engineers 1 and 3 who does not master C programming or computer graphics.
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From the above discussions, we can conclude that three-way dual concept analysis, as a generalization of classical concept analysis by absorbing the spirit of 3WD, provides us useful patterns in data analysis. In essence, formal concept analysis and its generalized ones, including three-way dual concept analysis and three-way concept analysis, aim to describe granules from a certain perspective. No matter what we want to do with the granules, the first thing is to describe them in a suitable way. At most times, given a specific granule, we cannot point out why it comes into a granule. In other words, we can not point out the intrinsic features of the discussed granules. This case dues to our limited cognitive abilities to a large extent other than the specific granule’s beyond description. If we consider this problem in multiple perspectives, we may find the answers. 3.2. The connections between object-induced three-way dual concept lattices and classical dual concept lattices In this subsection, we investigate the relationships between object-induced three-way dual concept lattices and classical dual concept lattices. Proposition 4. Let K = (U , V , R ) be a formal context, D L ( K ) be the dual concept lattice of K and D L ( K c ) be the dual concept lattice of K c . Then the following properties hold. (i) If ( X , A ) ∈ D L ( K ) and ( X , B ) ∈ D L ( K c ), then ( X , ( A , B )) ∈ O E D L ( K ). (ii) If ( X , A ) ∈ D L ( K ) and there does not exist a dual concept ( X , B ) ∈ D L ( K c ), then ( X , ( A , C )) ∈ O E D L ( K ), where ( Z , C ) is the greatest dual concept in D L ( K c ) such that Z ⊂ X . (iii) If ( X , B ) ∈ D L ( K c ) and there does not exist a dual concept ( X , A ) ∈ D L ( K ), then ( X , (C , B )) ∈ O E D L ( K ), where ( Z , C ) is the greatest dual concept in D L ( K ) such that Z ⊂ X . Proof. (i) If ( X , A ) ∈ D L ( K ) and ( X , B ) ∈ D L ( K c ), it follows that X # = A, A # = X , X # = B and B # = X . Then, X = ( X # , X # ) = ( A , B ) and ( A , B ) = A # ∪ B # = X , i.e., ( X , ( A , B )) ∈ O E D L ( K ). (ii) If ( X , A ) ∈ D L ( K ) and ( Z , C ) ∈ D L ( K c ), it follows that X # = A, A # = X , Z # = C and C # = Z . As Z ⊂ X , it follows that # A ∪ C # = X ∪ Z = X , i.e., ( A , C ) = X . Furthermore, as ( Z , C ) is the greatest dual concept in D L ( K c ) with C ⊂ X , we can conclude that X # = C . Then, it follows that X = ( X # , X # ) = ( A , C ). Therefore, the proof is completed. (iii) If ( X , B ) ∈ D L ( K c ) and ( Z , C ) ∈ D L ( K ), it follows that X # = B, B # = X , Z # = C and C # = Z . As Z ⊂ X , it follows that C # ∪ B # = Z ∪ X = X , i.e., (C , B ) = X . Furthermore, as ( Z , C ) is the greatest dual concept in D L ( K ) with Z ⊂ X , we can conclude that X # = C . Then, it follows that X = ( X # , X # ) = (C , B ). Hence, the proposition is proved. 2 Proposition 5. Let K = (U , V , R ) be a formal context. For ( X , ( A , B )) ∈ O E D L, we have ( A # , A ) ∈ D L ( K ) and ( B # , B ) ∈ D L ( K c ) such that A # ⊆ X and B # ⊆ X . Proof. As ( X , ( A , B )) ∈ O E D L ( K ), it follows A = X # and B = X # . Thus, ( A # , A ) = ( X ## , X # ) is a concept of D L ( K ) and ( B # , B ) = ( X ## , X # ) is a concept of D L ( K c ). Moreover, as A # ∪ B # = X , we have A # ⊆ X and B # ⊆ X . Then, the proof is completed. 2 Propositions 4 and 5 exactly manifest that object-induced three-way dual concept lattices embody dual concept lattices and thus provide more useful information than that of dual concept lattices. Furthermore, by light of the above two propositions, we propose Algorithm 2 to build object-induced three-way dual concept lattice of a formal context. Before proceeding, we define a binary relation to facilitate our discussions. For (( X 1 , A 1 ), (Y 1 , B 1 )), (( X 2 , A 2 ), (Y 2 , B 2 )) ∈ D L ( K ) × D L ( K c ), we define (( X 1 , A 1 ), (Y 1 , B 1 )) I (( X 2 , A 2 ), (Y 2 , B 2 )) ⇔ X 1 ∩ Y 1 = X2 ∩ Y 2 . Then, it is easy to show that I is an equivalent relation on D L ( K ) × D L ( K c ). In addition, for (( X , A ), (Y , B )) ∈ D L ( K ) × D L ( K c ), the equivalent class induced by I is denoted as [(( X , A ), (Y , B ))] I . In Algorithm 2, we use two dual lattices D L ( K ) and D L ( K c ) to build an object-induced three-way dual concept lattice. So, the complexity is determined by the constructions of dual concept lattices. It is not difficult to show that dual concept lattices follow the theoretical framework of formal concept lattices. Then, we can get an algorithm for dual concept lattices similarly as that of building formal concept lattices. To the best of our knowledge, the best time complexity of constructing a formal concept lattice is O (|U |2 | V | N ) (N is the number of concepts of the constructed concept lattice) [17]. Example 3. Table 3 shows a formal context K = (U , V , R ) with U = {1, 2, 3, 4, 5} and V = {a, b, c , d, e }. Fig. 2 shows dual concept lattice D L ( K ) and Fig. 3 shows its dual concept lattice D L ( K c ). By using Algorithm 2, we derive the object-induced three-way dual concept lattice O E D L ( K ), which is shown in Fig. 4. In what follows, we take several examples to illustrate Propositions 4 and 5.
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Algorithm 2 Building an object-induced three-way dual concept lattice by using two classical dual concept lattices. Require: A formal context K = (U , V , R ). Ensure: Object-induced three-way dual concept lattice O E D L ( K ). 1: Initialize O E D L ( K ) = ∅. 2: Let be a set of object sets and initialize = ∅. 3: Build dual concept lattices D L ( K ) and D L ( K c ). 4: For each ( X , A ) ∈ D L ( K ) 5: If there is a concept ( X , B ) ∈ D L ( K c ) 6: Then add ( X , ( A , B )) to O E D L ( K ) . 7: D L ( K c ) ← D L ( K c ) − {( X , B )}. 8: Else find the greatest dual concept ( Z , C ) in D L ( K c ) with Z ⊂ X . 9: Add ( X , ( A , C )) to O E D L ( K ). 10: End If 11: Add X to . 12: End For 13: For each ( X , B ) ∈ D L ( K c ) 14: Find the greatest dual concept ( Z , C ) in D L ( K ) with Z ⊂ X . 15: Add ( X , (C , B )) to O E D L ( K ) and add X to . 16: End For 17: For each ( X , A ) ∈ D L ( K ) and (Y , B ) ∈ D L ( K c ) 18: If Z = X ∩ Y ∈ / 19: Establish equivalent class [(( X , A ), (Y , B ))] I . 20: End For 21: For each established equivalent class [(( X , A ), (Y , B ))] I 22: Find the greatest element (( X g , A g ), (Y g , B g )) in [(( X , A ), (Y , B ))] I . 23: Add ( X ∩ Y , ( A g , B g )) to O E D L ( K ). 24: End For 25: Return O E D L ( K ).
Table 3 A formal context K of Example 3. a 1 2 3 4 5
× × ×
b
× × ×
c
× × ×
Fig. 2. D L ( K ) of Example 3.
d
×
×
e
× × ×
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Fig. 3. D L ( K c ) of Example 3.
Fig. 4. O E D L ( K ) of Example 3.
(i) As (1245, bd) ∈ D L ( K ), (1245, ace ) ∈ D L ( K c ) and (1245, bd, ace ) ∈ O E D L ( K ), it follows that (i) of Proposition 4 is verified. (ii) As (145, abd) ∈ D L ( K ), (15, abce ) ∈ D L ( K c ) and there does not exist another dual concept ( X , A ) ∈ D L ( K c ) such that (15, abce ) < ( X , A ) and X is a subset of the extent of (145, abd), and (145, abd, abce ) ∈ O E D L ( K ), it follows that (ii) of Proposition 4 is verified. (iii) As (1235, be ) ∈ D L ( K c ), (135, acd) ∈ D L ( K ) and there does not exist another dual concept ( X , A ) ∈ D L ( K ) such that (135, acd) < ( X , A ) and X is a subset of the extent of (1235, be ), and (1235, acd, be ) ∈ O E D L ( K ), it follows that (iii) of Proposition 4 is verified. (iv) For any object-induced three-way dual concept ( X , ( A , B )) ∈ O E D L ( K ), we can show that ( A # , A ) ∈ D L ( K ) or # ( B , B ) ∈ D L ( K c ). Therefore, Proposition 5 is verified.
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4. The relationships among four types of object-induced three-way concept lattices In the previous sections, we have introduced object-induced three-way concept lattices, object-induced three-way dual concept lattices and object-induced three-way object oriented concept lattices. Analogously, we can further define objectinduced three-way property oriented concept lattices. And then, we discuss the relationships among these four types of lattices. Definition 6. Let K = (U , V , R ) be a formal context. For X ∈ 2U and ( A , B ) ∈ 2 V × 2 V , two associated three-way operators : 2U → 2 V × 2 V and : 2 V × 2 V → 2U are respectively defined as:
X = ( X 3 , X 3 ) and ( A , B ) = A 2 ∩ B 2 , where X 3 = {a ∈ V | X ∩ R c (a) = ∅} and B 2 = {x ∈ U | R c (x) ⊆ B }. The pair ( X , ( A , B )) is called an object-induced three-way property oriented concept if X = ( A , B ) and ( A , B ) = X . Given a formal context K , the set of all object-induced three-way property oriented concepts is naturally equipped with a partial order ≤ defined as:
( X 1 , ( A 1 , B 1 )) ≤ ( X 2 , ( A 2 , B 2 )) ⇔ X 1 ⊆ X 2 ⇔ ( A 1 , B 1 ) ≤ ( A 2 , B 2 ). It is straightforward that all the object-induced three-way property oriented concepts contained in K with ≤ form a poset, which is denoted henceforth by O E P L ( K ). Further more, the infimum and supremum in the lattice are respectively given by:
( X 1 , ( A 1 , B 1 )) ∧ ( X 2 , ( A 2 , B 2 )) = ( X 1 ∩ X 2 , (( A 1 , B 1 ) ∩ ( A 2 , B 2 )) ) ( X 1 , ( A 1 , B 1 )) ∨ ( X 2 , ( A 2 , B 2 )) = (( X 1 ∪ X 2 ) , ( A 1 , B 1 ) ∪ ( A 2 , B 2 )). It is worth noting that for a three-way property oriented concept ( X , ( A , B )), A and B may overlap with each other. And this is the case similar to that of three-way dual concepts. Also, this can be explained in a wide sense of 3WD as that of three-way dual concepts. As this paper mainly focuses on three-way dual concepts, we omit the details here. Proposition 6. Let (U , V , R ) be a formal context. For X ⊆ U , the following properties hold. (i) X ∗ = X c#c , X # = X c ∗c .
(ii) X 2 = X c 3c , X 3 = X c 2c . c c2 ∗ (iii) X ∗R = X 3 Rc , X R = X Rc .
Proof. It can be proved analogously as that of Proposition 1.
2
Although Proposition 1 and Proposition 6 hold a similar meaning, they are both useful in our discussions and cannot replace each other. Concretely, Proposition 1 is about positive operators and Proposition 6 concerns negative operators. Theorem 2. Let K = (U , V , R ) be a formal context. For X ∈ 2U and ( A , B ) ∈ 2 V × 2 V , the following propositions are equivalent. (i) (ii) (iii) (iv)
( X , ( A , B )) is an object-induced three-way concept of K . ( X c , ( A c , B c )) is an object-induced three-way dual concept of K . ( X c , ( B , A )) is an object-induced three-way object oriented concept of K . ( X , ( B c , A c )) is an object-induced three-way property oriented concept of K .
Proof. (i) ⇔ (ii): applying Proposition 1 (i), Proposition 6 (i) and duality principle, we get:
( X , ( A , B )) is an object-induced three-way concept of K ⇔ X ∗ = A , X ∗ = B and A ∗ ∩ B ∗ = X ⇔ X c#c = A , X c#c = B and ( A ∗ ∩ B ∗ )c = A c ⇔ X c# = A c , X c# = B c and A ∗c ∪ B ∗c = A c ⇔ X c# = A c , X c# = B c and A c # ∪ B c# = A c ⇔ ( X c , ( A c , B c )) is an object-induced three-way dual concept of K
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(i) ⇔ (iii): applying Proposition 2, Proposition 6 (iii) and duality principle, we get:
( X , ( A , B )) is an object-induced three-way concept of K ⇔ X ∗R = B , X ∗R = A and A ∗R ∩ B ∗R = X ⇔ X cR2c = B , X cR2c = A and ( A ∗R ∩ B ∗R )c = X c
c
⇔ {a ∈ V | R (a) ⊆ X c } = B , {a ∈ V | R c (a) ⊆ X c } = A and B ∗R ∪ A ∗R c = X c
⇔ X cR 2 = B , X cR 2 = A and {x ∈ U | ∀b ∈ B , xR c b}c ∪ {x ∈ U | ∀a ∈ A , xRa}c = X c ⇔ X cR 2 = B , X cR 2 = A and {x ∈ U | ∃b ∈ B , xRb} ∪ {x ∈ U | ∃a ∈ A , xR c a} = X c ⇔ X cR 2 = B , X cR 2 = A and B 3 ∪ A 3 = X c ⇔ ( X c , ( B , A )) is an object-induced three-way object oriented concept of K
(iii) ⇔ (iv): applying Proposition 1 (ii), Proposition 6 (ii) and duality principle, we get:
( X c , ( B , A )) is an object-induced three-way object oriented concept of K ⇔ X c2 = B , X c2 = A and B 3 ∪ A 3 = X c ⇔ X c2c = B c , X c2c = A c and ( B 3 ∪ A 3 )c = X ⇔ X 3 = B c , X 3 = A c and B 3c ∩ A 3c = X ⇔ X 3 = B c , X 3 = A c and B c2 ∩ A c2 = X ⇔ ( X , ( B c , A c )) is an object-induced three-way property oriented concept of K The proof is completed.
2
Corollary 1. Let K = (U , V , R ) be a formal context. Then | O E L ( K )| = | O E D L ( K )| = | O E O L ( K )| = | O E P L ( K )|. 5. Conclusion In many applications, such as international cooperation, if we want to do something with our potential collaborators, we must have a good knowledge about both of their strength and weakness. In applications, the strength may refer to the cases such as one has some kinds of natural resources, professional skills, domain knowledge and the like, while the weakness means the opposite. However, the existing formal concept analysis theory does not work for such cases. Generally speaking, this is a task to find a meaningful pattern ( X , ( A , B )), where A is the set of attributes that are not possessed by at least one object in X c , B is the set of attributes that are possessed by at least one object in X c , and at the same time, X is composed of the elements who do not possess at least one attribute in A c or possesses at least one attribute in B c . In this study, we call the pattern ( X , ( A , B )) a three-way dual concept and call the task of finding such patterns three-way dual concept analysis. In order to show the significance of three-way dual concept analysis, we use two illustrative examples. One is about international import and export transactions, by which we stress the necessity to propose and define three-way dual concepts. The other is about knowledge sharing, by which we emphasize the strength of three-way dual concepts. Briefly speaking, the proposal of three-way dual concept and its semantic explanation are the main contributions of this paper. Besides, an incremental algorithm for building a three-way dual concept lattice is presented. And then, a batch algorithm is also described by exploring the connections between object-induced three-way dual concept lattices and classical dual concept lattices. What is more, the relationships among four types of three-way concept analysis models, i.e., three-way concept lattices, three-way dual concept lattices, three-way object oriented concept lattices and three-way property oriented concept lattices, are investigated. It is worth noting that in this study only object-induced three-way concepts are formally presented. By using duality principle, interested readers can further explore the attribute-induced three-way dual concepts analogously. Building three-way concept lattices are time consuming and sometimes brings insurmountable difficulties in real applications, especially in this big data era. Then, how to use the spirit of three-way concept analysis, rather than directly resort to three-way concept lattices, is an important and interesting topic. Besides, how to combine two or more concept analysis methods to improve granule description accuracy is another promising research direction. For instance, we may combine common attribute analysis and necessary attribute analysis to characterize a domain. What is more, the studies in this paper is about complete contexts. However, in some circumstances, the relationships between objects and attributes are partially known and incomplete contexts are needed to manage such cases [2,16,21,58, 61]. Although interval-set algebra has been proposed to solve this problem [48,52], the representation of three-way concepts in incomplete formal contexts is still an open problem and deserves our further investigations. Declaration of competing interest The authors declare that they have no conflicts of interest.
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Acknowledgements This work was supported by the National Natural Science Foundation of China (Nos. 61502150, 11801440 and 61772021) and the Innovation Talent Promotion Plan of Shaanxi Province for Young Sci-Tech New Star (Program No. 2017KJXX-60). References [1] M.K. Afridi, N. Azam, J.T. Yao, E. Alanazi, A three-way clustering approach for handling missing data using GTRS, Int. J. Approx. Reason. 98 (2018) 11–24. [2] P. Burmeister, R. Holzer, On the treatment of incomplete knowledge in formal concept analysis, in: Proceedings of ICCS 2000, in: LNCS (LNAI), vol. 1867, 2000, pp. 385–398. [3] J. Chen, J. Mi, B. Xie, Y. Lin, A fast attribute reduction method for large formal decision contexts, Int. J. Approx. Reason. 106 (2019) 1–17. [4] Y.H. Chen, Y.Y. Yao, A multiview approach for intelligent data analysis based on data operators, Inf. Sci. 178 (2008) 1–20. [5] G. Chiaselotti, F. Infusino, Notions from rough set theory in a generalized dependency relation context, Int. J. Approx. Reason. 98 (2018) 25–61. [6] Y. Djouadi, D. Dubois, P. Prade, Différentes extensions floues de lanalyse formelle de concepts, in: Actes Renc. Franc. sur la Logique Floue et ses Applications Cépadues edn., 2009, pp. 141–148. [7] Y. Djouadi, H. Prade, Possibility-theoretic extension of derivation operators in formal concept analysis over fuzzy lattices, Fuzzy Optim. Decis. Mak. 10 (2011) 287–309. [8] D. Dubois, F. Dupin de Saint Cyr, H. Prade, A possibility-theoretic view of formal concept analysis, Fundam. Inform. 75 (1–4) (2007) 195–213. [9] D. Dubois, H. Prade, Possibility theory and formal concept analysis in information systems, in: Proceedings of IFSA/EUSFLAT’09, Lisbon, Portugal, 2009, pp. 1021–1026. [10] B. Ganter, R. Wille, Formal Concept Analysis: Mathematical Foundations, Springer, Berlin, 1999. [11] Y. Fang, F. Min, Cost-sensitive approximate attribute reduction with three-way decisions, Int. J. Approx. Reason. 104 (2019) 148–165. [12] A. Ferone, Feature selection based on composition of rough sets induced by feature granulation, Int. J. Approx. Reason. 101 (2018) 276–292. [13] B. Ganter, S.O. Kuznetsov, Hypotheses and version spaces, conceptual structures for knowledge creation and communication, in: Formal Concept Analysis: Mathematical Foundations, Springer, Berlin, 2003, pp. 83–95. [14] R. Godin, R. Missaoui, H. Alaoui, Incremental concept formation algorithms based on Galois (concept) lattices, Comput. Intell. 11 (2) (1995) 246–267. [15] S.E. Han, Marcus-Wyse topological rough sets and their applications, Int. J. Approx. Reason. 106 (2019) 214–227. [16] M. Krupka, J. Lastovicka, Concept lattices of incomplete data, in: Proceedings of ICFCA 2012, in: LNCS (LNAI), vol. 7278, 2012, pp. 180–194. [17] S.O. Kuznetsov, S.A. Obiedkov, Comparing performance of algorithms for generating concept lattices, J. Exp. Theor. Artif. Intell. 14 (2–3) (2002) 189–216. [18] J. Li, C. Mei, Y. Lv, Incomplete decision contexts: approximate concept construction, rule acquisition and knowledge reduction, Int. J. Approx. Reason. 54 (2013) 149–165. [19] J. Li, C.L. Mei, W.H. Xu, Y.H. Qian, Concept learning via granular computing: a cognitive viewpoint, Inf. Sci. 298 (2015) 447–467. [20] J. Li, C. Huang, J. Qi, Y. Qian, W. Liu, Three-way cognitive concept learning via multi-granularity, Inf. Sci. 378 (2017) 244–263. [21] W. Lipski Jr., On semantic issues connected with incomplete information databases, ACM Trans. Database Syst. 4 (1979) 269–296. [22] M.Z. Li, G.Y. Wang, Approximate concept construction with three-way decisions and attribute reduction in incomplete contexts, Knowl.-Based Syst. 91 (2016) 165–178. [23] W. Li, X. Jia, L. Wang, B. Zhou, Multi-objective attribute reduction in three-way decision-theoretic rough set model, Int. J. Approx. Reason. 105 (2019) 327–341. [24] J.M. Ma, M.J. Cai, C.J. Zou, Concept acquisition approach of object-oriented concept lattices, Int. J. Mach. Learn. Cybern. 8 (1) (2017) 123–134. [25] L. Ma, The investigation of covering rough sets by Boolean matrices, Int. J. Approx. Reason. 100 (2018) 69–84. [26] M. Ma, M.K. Chakraborty, Covering-based rough sets and modal logics. Part II, Int. J. Approx. Reason. 95 (2018) 113–123. [27] X.A. Ma, X.R. Zhao, Cost-sensitive three-way class-specific attribute reduction, Int. J. Approx. Reason. 105 (2019) 153–174. [28] L. Nourine, O. Raynaud, A fast algorithm for building lattices, Inf. Process. Lett. 71 (1999) 199–204. [29] J.J. Qi, L. Wei, Y.Y. Yao, Three-way formal concept analysis, in: Rough Sets and Knowledge Technology, in: Lecture Notes in Computer Science, vol. 8818, 2014, pp. 732–741. [30] J.J. Qi, T. Qian, L. Wei, The connections between three-way and classical concept lattices, Knowl.-Based Syst. 91 (2016) 143–151. [31] J. Qian, C. Liu, X. Yue, Multigranulation sequential three-way decisions based on multiple thresholds, Int. J. Approx. Reason. 105 (2019) 396–416. [32] T. Qian, L. Wei, J. Qi, Constructing three-way concept lattices based on apposition and subposition of formal contexts, Knowl.-Based Syst. 166 (2017) 39–48. [33] T. Qian, L. Wei, J. Qi, A theoretical study on the object (property) oriented concept lattices based on three-way decisions, Soft Comput. (2019), https:// doi.org/10.1007/s00500-019-03799-6. [34] Y. Qian, X. Liang, Q. Wang, J. Liang, B. Liu, A. Skowron, Y. Yao, J. Ma, C. Dang, Local rough set: a solution to rough data analysis in big data, Int. J. Approx. Reason. 97 (2018) 38–63. [35] M.S. Raza, U. Qamar, Feature selection using rough set-based direct dependency calculation by avoiding the positive region, Int. J. Approx. Reason. 92 (2018) 175–197. [36] R.S. Ren, L. Wei, Y.Y. Yao, An analysis of three types of partially-known formal concepts, Int. J. Mach. Learn. Cybern. 9 (11) (2018) 1767–1783. [37] J.M. Rodriguez-Jimenez, P. Cordero, M. Enciso, A. Mora, Negative attributes and implications in formal concept analysis, Proc. Comput. Sci. 31 (2014) 758–765. [38] R. Shivhare, A.K. Cherukuri, Three-way conceptual approach for cognitive memory functionalities, Int. J. Mach. Learn. Cybern. 8 (1) (2017) 21–34. [39] P.K. Singh, Ch. Aswani Kumar, Bipolar fuzzy graph representation of concept lattice, Inf. Sci. 288 (2014) 437–448. [40] P.K. Singh, Medical diagnoses using three-way fuzzy concept lattice and their Euclidean distance, Comput. Appl. Math. 37 (3) (2018) 3283–3306. [41] B. Sun, W. Ma, B. Li, X. Li, Three-way decisions approach to multiple attribute group decision making with linguistic information-based decisiontheoretic rough fuzzy set, Int. J. Approx. Reason. 93 (2018) 424–442. [42] L. Wei, T. Qian, The three-way object oriented concept lattice and the three-way property oriented concept lattice, in: Proceedings of the 2015 International Conference on Machine Learning and Cybernetics, 2015, pp. 854–859. [43] R. Wille, Restructuring lattice theory: an approach based on hierarchies of concepts, in: I. Rival (Ed.), Ordered Sets, Reidel Publishing Company, Dordrecht–Boston, 1982, pp. 445–470. [44] X.D. Wu, C.Q. Zhang, S.C. Zhang, Efficient mining of both positive and negative association rules, ACM Trans. Inf. Syst. 22 (3) (2004) 381–405. [45] Y.T. Yan, Z.B. Wu, X.Q. Du, J. Chen, S. Zhao, Y.P. Zhang, A three-way decision ensemble method for imbalanced data oversampling, Int. J. Approx. Reason. 107 (2019) 1–16. [46] X. Yang, T. Li, H. Fujita, D. Liu, A sequential three-way approach to multi-class decision, Int. J. Approx. Reason. 104 (2019) 108–125.
H. Zhi et al. / International Journal of Approximate Reasoning 114 (2019) 151–165
165
[47] J.T. Yao, N. Azam, Web-based medical decision support systems for three-way medical decision making with game-theoretic rough sets, IEEE Trans. Fuzzy Syst. 23 (1) (2015) 3–15. [48] Y.Y. Yao, Interval-set algebra for qualitative knowledge representation, in: Proceedings of the 5th International Conference on Computing and Information, Sudbury, 1993, pp. 370–374. [49] Y.Y. Yao, A comparative study of formal concept analysis and rough set theory in data analysis, in: Proceedings of 4th International Conference on Rough Sets and Current Trends in Computing, RSCTC 2004, Uppsala, Sweden, 2004, pp. 59–68. [50] Y.Y. Yao, Concept lattices in rough set theory, in: Proceedings of 23rd International Meeting of the North American Fuzzy Information Processing Society, 2004, pp. 796–801. [51] Y.Y. Yao, An outline of a theory of three-way decisions, in: Proceedings of RSCTC 2012, in: LNCS (LNAI), vol. 7413, Springer, Heidelberg, 2012, pp. 1–17. [52] Y.Y. Yao, Interval sets and three-way concept analysis in incomplete contexts, Int. J. Mach. Learn. Cybern. 8 (1) (2017) 3–20. [53] Y.Y. Yao, Three-way decision and granular computing, Int. J. Approx. Reason. 103 (2018) 107–123. [54] Y. Zhang, J.T. Yao, Gini objective functions for three-way classifications, Int. J. Approx. Reason. 81 (2017) 103–114. [55] H.Y. Zhang, S.Y. Yang, Three-way group decisions with interval-valued decision-theoretic rough sets based on aggregating inclusion measures, Int. J. Approx. Reason. 110 (2019) 31–45. [56] Y. Zhang, D. Miao, Z. Zhang, J. Xu, S. Luo, A three-way selective ensemble model for multi-label classification, Int. J. Approx. Reason. 103 (2018) 394–413. [57] Y. Zhang, D. Miao, J. Wang, Z. Zhang, A cost-sensitive three-way combination technique for ensemble learning in sentiment classification, Int. J. Approx. Reason. 105 (2019) 85–97. [58] Y.X. Zhao, J.H. Li, W.Q. Liu, W.H. Xu, Cognitive concept learning from incomplete information, Int. J. Mach. Learn. Cybern. 8 (1) (2017) 159–170. [59] H. Zhi, J. Li, Influence of dynamical changes on concept lattice and implication rules, Int. J. Mach. Learn. Cybern. 9 (2018) 705–805. [60] H. Zhi, J. Li, Granule description based on positive and negative attributes, Granul. Comput. 4 (2019) 337–350. [61] H. Zhi, J. Li, Granule description based knowledge discovery from incomplete formal contexts via necessary attribute analysis, Inf. Sci. 485 (2019) 347–361.
Contents lists available at ScienceDirect
International Journal of Approximate Reasoning www.elsevier.com/locate/ijar
Three-way dual concept analysis Huilai Zhi a , Jianjun Qi b,∗ , Ting Qian c , Ling Wei d a
School of Computer Science and Technology, Henan Polytechnic University, Jiaozuo 454000, Henan, PR China School of Computer Science and Technology, Xidian University, Xi’an 710071, PR China College of Science, Xi’an Shiyou University, Xi’an, 710065, PR China d School of Mathematics, Northwest University, Xi’an 710069, PR China b c
a r t i c l e
i n f o
Article history: Received 11 March 2019 Received in revised form 6 August 2019 Accepted 26 August 2019 Available online 2 September 2019 Keywords: Formal concept analysis Three-way concept analysis Three-way dual concept analysis
a b s t r a c t Three-way concept analysis is a mathematical theory, which combines formal concept analysis and three-way decision. The existing models, i.e., three-way concept lattices and three-way object oriented concept lattices, have been used successfully in many fields. However, these two models are established to cater some specific applications and cannot fulfill some special requirements such as the needs to seek potential collaborators in international import and export transactions. In this paper, a novel type of three-way concept lattices is presented based on dual concept analysis, which enables ones to characterize specific set by pointing out the attributes that are not possessed by at least one object in the complement of this specific set. And then, the connections between three-way dual concept lattices and classical dual concept lattices are explored. Besides, the relationships among four types of three-way concept analysis models, i.e., three-way concept lattices, three-way dual concept lattices, three-way object oriented concept lattices and three-way property oriented concept lattices, are investigated. © 2019 Elsevier Inc. All rights reserved.
1. Introduction Formal concept analysis (FCA), a mathematical framework founded on lattice theory [43], is used effectively in various types of data analysis activities [19,24,61]. Formal context and formal concept are two cornerstones in FCA. A formal context describes a domain in an ideal case. For an object of a given context, one can precisely points out which attributes this object possesses and which attributes this object does not possess. Formally, a concept is composed of two parts, i.e., the extent and the intent. Moreover, there is an affirmative relationship between them. That is, each object of the extent possesses all the attributes in the intent and each attribute of the intent is shared by all the objects in the extent. In classical FCA, in order to describe a specific target set, only positive attributes are concerned, while the negative ones are ignored. However, many researchers have pointed out that both positive attributes and negative attributes play equally important roles in many fields, such as knowledge creation and communication, concept representation and rule acquisition [13,37,39,44]. Concretely, give a specific target set, we describe it by pointing out what features it has and what features it does not have. In the settings of data mining and FCA, we call them positive attributes and negative attributes respectively. Three-way decision (3WD) [47,51,54] is a theory which promotes thinking, problem solving, and information processing in threes. Initially, This theory was introduced for interpreting the three types of classification rules in rough set theory. Since
*
Corresponding author. E-mail addresses: [email protected] (H. Zhi), [email protected] (J. Qi), [email protected] (T. Qian), [email protected] (L. Wei).
https://doi.org/10.1016/j.ijar.2019.08.010 0888-613X/© 2019 Elsevier Inc. All rights reserved.
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Table 1 Companies and their specific traducements.
1 2 3 4 5
a
b
c
×
×
× × ×
×
×
d
e
× × ×
× ×
its proposal, three-way decision has gained increasing attentions from the communities of rough set theory [3,5,12,15,23, 25,26,34]. Besides, three-way decision was frequently used in feature selection [35,56] and data clustering [1,57]. What is more, plenty of studies have shown that three-way decision has played an active and important role in decision makings [11,27,31,41,45,46,55]. Inspired by the theory of three-way decision, Qi et al. [29] proposed three-way concept analysis (3WCA). The significant improvement is that one can characterize any specific set by both positive and negative attributes, and as a result three-way concepts can provide more information than that of classical concepts [30]. In 3WCA, there are two kinds of three-way concept lattices, one is the object-induced three-way concept lattice, and the other is attribute-induced three-way concept lattice. Formally, the intent of an object-induced three-way concept has a positive part and a negative part. The positive part is composed of the attributes which are possessed by all the objects in the extent, while the negative part is composed of the attributes which are not possessed by all the objects in the extent. Dually, attribute-induced three-way concepts can be defined as well [30]. At present, 3WCA has attracted more and more attentions. For instance, Qian et al. [32] adopted dividing and conquering strategy to improve the efficiency of constructing three-way concept lattices. Recently, Qian et al. [33] further carried out a theoretical study on the object (property) oriented concept lattices based on three-way decision. Singh [40] constructed three-way fuzzy concept lattices and extracted useful patterns to support medical diagnoses. Shivhare and Cherukuri [38] exploited the spirit of three-way concept analysis to simulate a cognitive memory process. Li et al. [19,20] explored the three-way cognitive learning process via multi-granularity, which can be viewed as a generalization of the classical concept learning scheme. Zhi and Li [60] proposed an approach to granule description based on positive and negative attributes. Besides, as two important examples of partially-known formal concepts [36,52], ill-known formal concepts [6] and approximate concepts [18,22] can be viewed as a sort of three-way concepts to some extent. The existing 3WCA models, i.e., three-way concept lattices and three-way object oriented concept lattices, are established in specific perspectives. Concretely, three-way concept lattices are based on common attribute analysis and characterize any specific set by pointing out both the positive and negative common attributes. Three-way object oriented concept analysis, proposed by Wei and Qian [42], is based on necessary attribute analysis and characterizes any specific set by pointing out both the positive and negative necessary attributes. As mentioned above, these two models have been used successfully in many fields. However, these two models cannot completely fulfill some specific requirements. Let’s consider the following Example 1. Example 1. Consider five different companies from five different countries, each of which deals in some specific commodity traducements. The details about these companies are shown in Table 1, where {1, 2, 3, 4, 5} is the set of companies, {a, b, c , d, e } is the set of commodities and a cross denotes a company is engaged in a specific kind of commodity transaction. More concretely, companies 1, 2, 3, 4 and 5 are from China, Australia, Thailand, Philippines and New Zealand, respectively; a, b, c, d and e represent tea, rice, food, spice and wool, respectively. In international import and export transactions, businessmen must have a good knowledge about what commodities their potential customers cannot produce. In other words, if all their potential customers can produce a certain type of commodities, then there is no chance to export this type of commodities. Otherwise, if there exists at least one customer who cannot produce this type of commodities, then the businessmen can consider producing this type of commodities. Usually, there are several middlemen between the manufactures and the ultimate users. For instance, there is a middleman who wants to make transactions between oceania companies and asian companies. Obviously, companies 1, 3 and 4 are from asian, and companies 2 and 5 are from oceania. On one hand, if the middleman wants to sell the commodities produced by asian companies to the oceania companies, he must know what commodities the oceania companies cannot produce. Based on Table 1, we can see that the middleman has a chance to sell tea, rice and spice to the oceania companies. It is worth noting that although company 2 can produce spice, the middleman still has an opportunity to sell spice to company 5. On the other hand, if the middleman wants to sell the commodities produced by oceania companies to the asian companies, he must have a knowledge about what commodities the oceania companies can produce. Based on Table 1, we can see that oceania companies can produce food, spice and wool. As a result, the middleman has a chance to sell these three types of commodities to the asian companies. In what follows, we formulate the above discussions with mathematical languages. Let X = {1, 3, 4}, Y = {2, 5}, A = {a, b, d} and B = {c , d, e }. Then, X denotes the set of asian companies, Y represents the set of oceania companies, and there is a useful pattern ( X , ( A , B )), which states that there exists at least one element of the complement of X , i.e., there exists
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y ∈ Y , who does not possess at least one attribute in A and possess at least one attribute in B. And meanwhile, X is composed of the elements who do not possess at least one attribute in A c or possesses at least one attribute in B c . Actually, there are still other similar patterns in Table 1, such as ({2, 5}, ({a, b, c , d, e }, {a, b, c , d})), ({1, 4}, ({a, b, d, e }, {b, c , d, e })), and ({1, 5}, ({a, b, c , e }, {b, c , d, e })) etc. In all, there are 19 such patterns embodied in Table 1. After a serious analysis, we can see that Example 1 shows us a useful and interesting pattern which totally differs from the existing ones, including three-way concepts [29,30] and three-way object oriented concepts [42]. Actually, similar requirements frequently arise in a variety of applications, such as international cooperation and international education. In international cooperation, one must have a good knowledge about their potential collaborators, including advantages and disadvantages. It is the advantages that make a will for us to cooperate with them. And it is the disadvantages that make a desire for them to cooperate with us. In the field of international education, we can only teach others what they do not know and meanwhile we can only learn something from the others what they already specialize in. Generally speaking, this problem can be formulated in the following way: given a formal context, how to effectively derive the patterns in the format of ( X , ( A , B )), where A is the set of attributes that are not possessed by at least one object in X c , B the set of attributes that are possessed by at least one object in X c , and X is the set of objects that who do not possess at least one attribute in A c or possesses at least one attribute in B c . In applications, the specific set X can represent a coalition of countries, companies and persons, et al. And the complement of the specific set X c can denote their potential collaborators. As the existing three-way concept lattices cannot effectively solve this problem, there is a vital need to seek a novel 3WCA model. Motivated by the above stated problem, we propose a new type of three-way concept analysis model based on dual concept analysis [10,49,50]. The rest of this paper is organized as follows. Section 2 briefly reviews some basic notions in FCA. Section 3 introduces a novel type of three-way concept analysis model, i.e., object-induced three-way dual concept lattices. Besides, the relationships between object-induced three-way dual concept lattices and dual concept lattices are also investigated. Section 4 explores the relationships among four types of object-induced three-way concept analysis models, i.e., object-induced three-way concept lattices, object-induced three-way dual concept lattices, object-induced three-way object oriented concept lattices and object-induced three-way property oriented concept lattices. Finally, conclusions are provided in Section 5. 2. Related theoretical foundations In this section, we briefly review some basic notions in FCA to make the paper self-contained. 2.1. Basic notions of FCA Let U and V be two finite and nonempty sets. In the settings of FCA, the elements of U are called objects, and the elements of V are called attributes. The relationships between the objects and the attributes are described by a binary relation R. For a pair of elements x ∈ U and a ∈ V , if (x, a) ∈ R, also written as xRa, we say that x has the attribute a, or a is possessed by the object x. Let R (x) = {a ∈ V | xRa} be the set of attributes possessed by the object x. Dually, let R (a) = {x ∈ U | xRa} be the set of objects that possess the attribute a. The complement of a binary relation R is defined by:
R c = U × V − R = {(x, a) | ¬(xRa)}. Then, we call the triplet K = (U , V , R ) a formal context and K c = (U , V , R c ) the complement of K . If there exists x ∈ U such that x has all the attributes in V or x doesn’t have any attribute in V , we think the object x is meaningless. Similarly, we are not interested in such attribute that is in a relation with all the objects or not in a relation with any object. We call this formal context canonical, and the formal context we discuss in this paper are all canonical. Let K = (U , V , R ) be a formal context. A set-theoretic operator ∗ : 2U → 2 V is defined as:
X ∗ = {a ∈ V | x ∈ X ⇒ xRa} a ∈ V | X ⊆ R (a)} = { R (x), = x∈ X
which associates a set of attributes X ∗ to X . Dually, ∗ : 2 V → 2U is defined as:
A ∗ = {x ∈ U | a ∈ A ⇒ xRa)} x ∈ U | A ⊆ R (x)} = { R (a), = a∈ A
which describes a mapping from A ∗ to A.
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A formal concept is a pair ( X , A ) such that X ∗ = A and A ∗ = X . The sets X and A are called the extent and the intent of ( X , A ), respectively. Given a formal context K , the set of all formal concepts is naturally equipped with a partial order ≤ defined as: ( X 1 , A 1 ) ≤ ( X 2 , A 2 ) ⇔ X 1 ⊆ X 2 ⇔ A 2 ⊆ A 1 . Ganter and Wille [10] have proved that all the formal concepts ordered with ≤ form a complete lattice, which is the so-called the concept lattice and is denoted by L ( K ). The infimum and supremum in the lattice are respectively given by:
( X 1 , A 1 ) ∧ ( X 2 , A 2 ) = ( X 1 ∩ X 2 , ( A 1 ∪ A 2 )∗∗ ), ( X 1 , A 1 ) ∨ ( X 2 , A 2 ) = (( X 1 ∪ X 2 )∗∗ , A 1 ∩ A 2 ). 2.2. Formal concept analysis with a possibility theoretic view In FCA, besides the sufficiency operator (·)∗ (which corresponds to strong possibility), we can define the other three operators, i.e., (·)3 , (·)2 and (·)# , which are called possibility operator, necessity operator and dual sufficiency operator [4,7–9,49,50], respectively. Let K = (U , V , R ) be a formal context and X ∈ 2U . X 3 is the set of attributes that are satisfied by at least one object in X:
X 3 = {a ∈ V | X ∩ R (a) = ∅} = {a ∈ V | ∃x ∈ X , xRa}. X 2 is the set of attributes such that any object that satisfies one of them is necessarily in X :
X 2 = {a ∈ V | R (a) ⊆ X } = {a ∈ V | ∀x ∈ U (xRa ⇒ x ∈ X )}. X # is the set of attributes that are not possessed by at least one object in X c :
X # = {a ∈ V | X ∪ R (a) = U } = {a ∈ V | ∃x ∈ X c , xR c a}. Dually, for A ∈ 2 V , we can define A 3 , A 2 and A # respectively. A 3 is the set of objects that possess at least one attribute in A:
A 3 = {x ∈ U | A ∩ R (x) = ∅} = {x ∈ U | ∃a ∈ A , xRa}. A 2 is the set of objects such that any attribute that is satisfied by one of them is necessarily in A:
A 2 = {x ∈ U | R (x) ⊆ A } = {x ∈ U | ∀a ∈ V (xRa ⇒ a ∈ A )}. A # is the set of objects that do not possess at least one attribute in A c :
A # = {x ∈ U | A ∪ R (x) = V } = {x ∈ U | ∃a ∈ A c , xR c a}. Proposition 1. [49,50] Let (U , V , R ) be a formal context and X ⊆ U . Then the following properties hold. (i) (ii) (iii) (iv)
∗c
X ∗ = X c#c , X # = X c . 3c 2c X2 = Xc , X3 = Xc . ## ∗∗ X ⊆X⊆X . X 23 ⊆ X ⊆ X 32 . 2
c Proposition 2. Let (U , V , R ) be a formal context and X ⊆ U . Then X ∗R = X 3 Rc = X Rc . c
Proof.
X ∗R = {a ∈ V | X ⊆ R (a)} = {a ∈ V | X ∩ R c (a) = ∅} = V − {a ∈ V | X ∩ R c (a) = ∅} = V − X Rc 3 c = X3 Rc X ∗R = {a ∈ V | X ⊆ R (a)} = {a ∈ V | R c (a) ⊆ X c } = X cR c 2 The proof is completed.
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2.3. Three-way concept analysis In this subsection, we introduce two types of three-way concepts as well as their corresponding lattices. Definition 1. ([29,30]) Let K = (U , V , R ) be a formal context. For X ∈ 2U and ( A , B ) ∈ 2 V × 2 V , two associated three-way operators : 2U → 2 V × 2 V and : 2 V × 2 V → 2U are respectively defined as:
X = ( X ∗ , X ∗ ) and ( A , B ) = A ∗ ∩ B ∗ , where X ∗ = {a ∈ V | ∀x ∈ X , xR c a} and A ∗ = {x ∈ U | ∀a ∈ A , xR c a}. Besides, if X = ( A , B ) and ( A , B ) = X , then ( X , ( A , B )) is called an object-induced three-way concept. Furthermore, all the object-induced three-way concepts form a complete lattice, which is the so-called object-induced three-way concept lattice, and is denoted by O E L ( K ). What is more, by using duality principle, we can further define attribute-induced three-way concept lattice. Interested reader can refer to [30]. Three-way concepts and 3WD are in full compliance in terms of the semantics of tripartition. In short, applying the idea of 3WD to FCA, we can formulate three-way concepts and three-way concept lattices. Actually, three-way concepts and formal concepts in FCA [10] share a common framework and both of them are equipped with extensions and intensions. Nevertheless, as a generalization of a formal concept, the extension (or the intension) of a three-way concept is constituted with two parts: positive one and negative one. By using these two parts, we can respectively describe the semantics “jointly possessed” and “jointly not possessed” in a formal context. In addition, on the basis of a three-way concept, we can further divide the object (or attribute) universe into three regions, i.e., positive region, negative region and boundary region. And then, according to these three regions we make three-way decisions to solve our specific problems. Definition 2. ([42]) Let K = (U , V , R ) be a formal context. For X ∈ 2U and ( A , B ) ∈ 2 V × 2 V , two associated three-way operators : 2U → 2 V × 2 V and : 2 V × 2 V → 2U are respectively defined as:
X = ( X 2 , X 2 ) and ( A , B ) = A 3 ∪ B 3 , where X 2 = {a ∈ V | R c (a) ⊆ X } and B 3 = {x ∈ U | B ∩ R c (x) = ∅}. The pair ( X , ( A , B )) is called an object-induced three-way object oriented concept if X = ( A , B ) and ( A , B ) = X . Furthermore, all the object-induced three-way object oriented concepts form a complete lattice, which is the so-called object-induced three-way object oriented concept lattice, and is denoted by O E O L ( K ). Similar to three-way concepts, the semantics of three-way object oriented concepts can be described as well and the details are omitted here. 3. Object-induced three-way dual concept lattice In this section, we propose object-induced three-way dual concept lattices and study the relationships between objectinduced three-way dual concept lattices and classical dual concept lattices. 3.1. The constructions of object-induced three-way dual concept lattices Let M be a nonempty and finite set and 2 M × 2 M be the Cartesian product of 2 M and 2 M . For ( A 1 , A 2 ), ( B 1 , B 2 ) ∈ 2 M × 2 M , ≤ is defined as:
( A 1 , A 2 ) ≤ ( B 1 , B 2 ) ⇔ A 1 ⊆ B 1 and A 2 ⊆ B 2 . Then, it follows that
( A 1 , A 2 ) = ( B 1 , B 2 ) ⇔ ( A 1 , A 2 ) ≤ ( B 1 , B 2 ) and ( B 1 , B 2 ) ≤ ( A 1 , A 2 ). Besides, ∩ and ∪ are defined respectively as:
( A 1 , A 2 ) ∩ ( B 1 , B 2 ) = ( A 1 ∩ B 1 , A 2 ∩ B 2 ) and ( A 1 , A 2 ) ∪ ( B 1 , B 2 ) = ( A 1 ∪ B 1 , A 2 ∪ B 2 ). In what follows, we define two associated negative dual sufficiency operators in preparing the constructions of objectinduced three-way dual concept lattices.
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Definition 3. Let K = (U , V , R ) be a formal context. For X ∈ 2U and A ∈ 2 V , two negative derivation operators # : 2U → 2 V and # : 2 V → 2U are respectively defined as:
X # = {a ∈ V | ∃x ∈ X c , xRa} and A # = {x ∈ U | ∃a ∈ A c , xRa}. X # is the set of attributes that are possessed by at least one object in X c and A # is the set of objects that possess at least one attribute in A c . From the computational perspective, we still have the following results:
X # = {a ∈ V | X ∪ R c (a) = U } and A # = {x ∈ U | A ∪ R c (x) = V }. It is obvious that the operators # : 2U → 2 V and # : 2 V → 2U are exactly the operators # : 2U → 2 V and # : 2 V → 2U defined on K c = (U , V , R c ), and form a Galois connection between (2U , ⊆) and (2 V , ⊆). Definition 4. Let K = (U , V , R ) be a formal context. For X ∈ 2U and ( A , B ) ∈ 2 V × 2 V , two associated three-way operators : 2U → 2 V × 2 V and : 2 V × 2 V → 2U are respectively defined as:
X = ( X # , X # ) and ( A , B ) = A # ∪ B # . X is composed of two related parts, i.e., X # and X # . ( A , B ) is the maximal set containing objects that do not possess at least one attribute in A c or possess at least one attribute in B c . Definition 5. Let K = (U , V , R ) be a formal context. For X ∈ 2U and ( A , B ) ∈ 2 V × 2 V , if X = ( A , B ) and ( A , B ) = X , then ( X , ( A , B )) is called an object-induced three-way dual concept. Proposition 3. Let K = (U , V , R ) be a formal context. For X , X 1 , X 2 ∈ 2U and ( A , B ), ( A 1 , B 1 ), ( A 2 , B 2 ) ∈ 2 V × 2 V , the following properties hold. (i) (ii) (iii) (iv) (v)
X 1 ⊆ X 2 ⇒ X 1 ≥ X 2 ; ( A 1 , B 1 ) ≤ ( A 2 , B 2 ) ⇒ ( A 1 , B 1 ) ⊇ ( A 2 , B 2 ) . X ⊇ X ; ( A , B ) ≥ ( A , B ) . ( X 1 ∩ X 2 ) = X 1 ∪ X 2 and (( A 1 , B 1 ) ∩ ( A 2 , B 2 )) = ( A 1 , B 1 ) ∪ ( A 2 , B 2 ) . and form a Galois connection between (2U , ⊆) and (2 V × 2 V , ≤). ( X , X ) and (( A , B ) , ( A , B ) ) are two object-induced three-way dual concepts of K .
Proof. (i) If X 1 ⊆ X 2 , then it follows that X 1# ⊇ X 2# and X 1# ⊇ X 2# . That is, ( X 1# , X 1# ) ≥ ( X 2# , X 2# ), which implies that X 1 ≥ X 2 .
# # # # # # # If ( A 1 , B 1 ) ≤ ( A 2 , B 2 ), it implies that A # 1 ⊇ A 2 and B 1 ⊇ B 2 . Then, we have A 1 ∪ B 1 ⊇ A 2 ∪ B 2 , i.e., ( A 1 , B 1 ) ⊇ ( A 2 , B 2 ) . (ii) As X ⊇ X ## and X ⊇ X ## , it follows that X ⊇ X ## ∪ X ## , i.e., X ⊇ X . As A # ⊆ A # ∪ B # , it follows that A ## ⊇ ( A # ∪ B # )# , which leads to A ⊇ A ## ⊇ ( A # ∪ B # )# . Besides, as B # ⊆ A # ∪ B # , it follows that B ## ⊇ ( A # ∪ B # )# , which implies that B ⊇ B ## ⊇ ( A # ∪ B # )# . To sum up, we have ( A , B ) ≥ (( A # ∪ B # )# , ( A # ∪ B # )# ), i.e., ( A , B ) ≥ ( A , B ) .
(iii) Firstly, we have
( X 1 ∩ X 2 )# = {a ∈ V | ∃x ∈ ( X 1 ∩ X 2 )c , xR c a} = {a ∈ V | ∃x ∈ X 1c ∪ X 2c , xR c a} = {a ∈ V | ∃x ∈ X 1c , xR c a} ∪ {a ∈ V | ∃x ∈ X 2c , xR c a} = X 1# ∪ X 2# . Likewise, we can prove that ( X 1 ∩ X 2 )# = X 1# ∪ X 2# . To sum up, ( X 1 ∩ X 2 ) = (( X 1 ∩ X 2 )# , ( X 1 ∩ X 2 )# ) = ( X 1# ∪ X 2# , X 1# ∪ X 2# ) =
( X 1# , X 1# ) ∪ ( X 2# , X 2# ) = X 1 ∪ X 2 .
Analogously, we can prove that (( A 1 , B 1 ) ∩ ( A 2 , B 2 )) = ( A 1 , B 1 ) ∪ ( A 2 , B 2 ) . (iv) This proposition will be proved if we can prove X ⊇ ( A , B ) ⇔ ( A , B ) ≥ X . Sufficiency: X ⊇ ( A , B ) ⇒ X ≤ ( A , B ) ≤ ( A , B ). Necessity: ( A , B ) ≥ X ⇒ ( A , B ) ⊆ X ⊆ X . Then, this proposition is proved. (v) By the properties of a Galois connection, it follows immediately. 2
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Given a formal context K , the set of all object-induced three-way dual concepts is naturally equipped with a partial order ≤ defined as:
( X 1 , ( A 1 , B 1 )) ≤ ( X 2 , ( A 2 , B 2 )) ⇔ X 1 ⊆ X 2 ⇔ ( A 1 , B 1 ) ≥ ( A 2 , B 2 ). It is straightforward that all the object-induced three-way dual concepts contained in K with ≤ form a poset, which is denoted henceforth by O E D L ( K ). Theorem 1. Let K = (U , V , R ) be a formal context. The poset O E D L ( K ) is a complete lattice, in which the infimum and supremum are respectively given by:
( X 1 , ( A 1 , B 1 )) ∧ ( X 2 , ( A 2 , B 2 )) = (( X 1 ∩ X 2 ) , ( A 1 , B 1 ) ∪ ( A 2 , B 2 )) ( X 1 , ( A 1 , B 1 )) ∨ ( X 2 , ( A 2 , B 2 )) = ( X 1 ∪ X 2 , (( A 1 , B 1 ) ∩ ( A 2 , B 2 )) ) Proof. By Proposition 3 (iii), it follows that ( X 1 ∩ X 2 ) = X 1 ∪ X 2 = ( A 1 , B 1 ) ∪ ( A 2 , B 2 ) and (( A 1 , B 1 ) ∩ ( A 2 , B 2 )) = ( A 1 , B 1 ) ∪ ( A 2 , B 2 ) = X 1 ∪ X 2 . Moreover, by Proposition 3 (v), we can conclude that (( X 1 ∩ X 2 ) , ( A 1 , B 1 ) ∪ ( A 2 , B 2 )) and ( X 1 ∪ X 2 , (( A 1 , B 1 ) ∩ ( A 2 , B 2 )) ) are two object-induced three-way dual concepts of K . Then, this theorem is proved. 2 Henceforth, the complete lattice O E D L ( K ) is called the object-induced three-way dual concept lattice of K . Let ( X , ( A , B )) be an object-induced three-way concept. Then, A and B characterize X c in two complemented angles, i.e., A points out the attributes that are not possessed by at least one object in X c and B shows the attributes that are possessed by at least one object in X c . Besides, we can further show that A c contains the common attributes possessed by X c , while B c contains the common attributes which are not possessed by X c . What is more, it is easy to show that A c ∩ B c = ∅. It is worth noting that although three-way is used as a prefix to name ( X , ( A , B )) ∈ O E D L ( K ), the two parts A and B may overlap with each other, which means the pair cannot trisect V in a similar way as that of three-way concepts [29,30] and three-way object oriented concepts [42]. However, 3WD is a fast developing theory and has developed into a new stage with a wide sense. Yao has pointed out that 3WD promotes thinking, problem solving, and information processing in threes, that is, using three parts, three elements, three components, three perspectives, three views, three levels, three generations, three periods, three stages, three steps, triangles, triads, triplets, and many others [53]. Yao also stressed that it is the wide sense that increases the power of three-way decision [53]. Apparently, three-way dual concepts, which have the form ( X , ( A , B )), are built by using three perspectives or views. That are, a positive view A, a negative view B and the rest (the ones which are neither A nor B). What is more, although A and B may overlap with each other in the set-theoretic view, A and B are disjoint with each other in the semantic view. That is, A points out the attributes that are not possessed by at least one object in X c and B shows the attributes that are possessed by at least one object in X c . As a result, to a certain extent, A can be viewed as a set of positive attributes and B can be view as set of negative attributes in building a three-way dual concept. As object-induced three-way dual concept lattice and classical concept lattice are actually in full compliance, we put forward an incremental algorithm (i.e., Algorithm 1) with a similar idea originating from the existing algorithms for building classical concept lattices [14,17,28,59]. Algorithm 1 Building an object-induced three-way dual concept lattice. Require: A canonical formal context K = (U , V , R ). Ensure: Object-induced three-way dual concept lattice O E D L ( K ). 1: Initialize O E D L ( K ) = {(∅, ( V , V ))}. 2: Initialize i = 1. 3: While i ≤ |U | 4: Randomly select an object x from U . 5: U ← U − {x}. 6: Let n be the number of object-induced three-way dual concepts of O E D L ( K ). 7: Sort all the concepts of O E D L ( K ) into a sequence ( X 1 , ( A 1 , B 1 )), · · · , ( X n , ( A n , B n )) such that | X 1 | ≤ · · · ≤ | X n |. 8: Initialize j = 1. 9: While j ≤ n do 10: If x∗ c ∩ A cj = ∅ or x∗ ∩ B cj = ∅ 11: Update ( X j , ( A j , B j )) to ( X j ∪ {x}, ( A j , B j )). 12: If there dose not exist an object-induced three-way dual concept with the intent ( A j ∪ x∗ c , B j ∪ x∗ ) 13: Then add ( X j , ( A j ∪ x∗ c , B j ∪ x∗ )) to O E D L ( K ) 14: End If 15: End If 16: j++. 17: End While 18: i++. 19: End While 20: Return O E D L ( K ).
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Table 2 A formal context K of Example 2. a 1 2 3 4 5
× × ×
b
× × × ×
c
d
×
×
× ×
× ×
e
× ×
Fig. 1. O E D L ( K ) of Example 2.
Let N be the number of the object-induced three-way dual concepts of K . Then the time complexity of Algorithm 1 is O (|U |2 | V | N ). Besides, the lines from 6 to 17 actually present a strategy to update three-way dual concept lattices when adding a new object. Example 2. Table 2 shows a formal context K = (U , V , R ) which describes a crowd of software development engineers, where U = {1, 2, 3, 4, 5} denotes five engineers and V = {a, b, c , d, e } represents five professional skills, namely, Java programming, C programming, computer graphics, algorithm analysis and mathematical modeling, respectively. In the subsequent discussions, in the representation of a concept, we omit the braces and comma for convenience. For example, we use (25, bcde , abcde ) instead of its standard format ({2, 5}, ({b, c , d, e }, {a, b, c , d, e })). Fig. 1 shows the object-induced three-way dual concept lattice O E D L ( K ). As already stated in this section, each object-induced three-way dual concept denotes a meaningful pattern. For instance, from the object-induced three-way concept (245, bcd, abce ), we can get the following information. (i) There is someone of engineers 1 and 3 who does not master C programming or computer graphics or algorithm analysis, which implies that the engineers 2, 4 and 5 have a chance to teach someone of engineers 1 and 3 C programming or computer graphics or algorithm analysis. (ii) There is someone of engineers 1 and 3 who masters Java programming or C programming or computer graphics or mathematical modeling, which implies that the engineers 2, 4 and 5 have a chance to learn Java programming or C programming or computer graphics or mathematical modeling from someone of engineers 1 and 3. However, by using three-way concept analysis [29,30], we get different statements from a totally different perspective. (iii) Both engineers 1 and 3 have the skills of Java programming and mathematical modeling. Comparing with item (ii), we are unable to be aware of that there is someone of engineers 1 and 3 who is capable of C programming or computer graphics. (iv) Neither engineer 1 nor engineer 3 masters algorithm analysis. Comparing with item (i), we cannot realize that there is someone of engineers 1 and 3 who does not master C programming or computer graphics.
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From the above discussions, we can conclude that three-way dual concept analysis, as a generalization of classical concept analysis by absorbing the spirit of 3WD, provides us useful patterns in data analysis. In essence, formal concept analysis and its generalized ones, including three-way dual concept analysis and three-way concept analysis, aim to describe granules from a certain perspective. No matter what we want to do with the granules, the first thing is to describe them in a suitable way. At most times, given a specific granule, we cannot point out why it comes into a granule. In other words, we can not point out the intrinsic features of the discussed granules. This case dues to our limited cognitive abilities to a large extent other than the specific granule’s beyond description. If we consider this problem in multiple perspectives, we may find the answers. 3.2. The connections between object-induced three-way dual concept lattices and classical dual concept lattices In this subsection, we investigate the relationships between object-induced three-way dual concept lattices and classical dual concept lattices. Proposition 4. Let K = (U , V , R ) be a formal context, D L ( K ) be the dual concept lattice of K and D L ( K c ) be the dual concept lattice of K c . Then the following properties hold. (i) If ( X , A ) ∈ D L ( K ) and ( X , B ) ∈ D L ( K c ), then ( X , ( A , B )) ∈ O E D L ( K ). (ii) If ( X , A ) ∈ D L ( K ) and there does not exist a dual concept ( X , B ) ∈ D L ( K c ), then ( X , ( A , C )) ∈ O E D L ( K ), where ( Z , C ) is the greatest dual concept in D L ( K c ) such that Z ⊂ X . (iii) If ( X , B ) ∈ D L ( K c ) and there does not exist a dual concept ( X , A ) ∈ D L ( K ), then ( X , (C , B )) ∈ O E D L ( K ), where ( Z , C ) is the greatest dual concept in D L ( K ) such that Z ⊂ X . Proof. (i) If ( X , A ) ∈ D L ( K ) and ( X , B ) ∈ D L ( K c ), it follows that X # = A, A # = X , X # = B and B # = X . Then, X = ( X # , X # ) = ( A , B ) and ( A , B ) = A # ∪ B # = X , i.e., ( X , ( A , B )) ∈ O E D L ( K ). (ii) If ( X , A ) ∈ D L ( K ) and ( Z , C ) ∈ D L ( K c ), it follows that X # = A, A # = X , Z # = C and C # = Z . As Z ⊂ X , it follows that # A ∪ C # = X ∪ Z = X , i.e., ( A , C ) = X . Furthermore, as ( Z , C ) is the greatest dual concept in D L ( K c ) with C ⊂ X , we can conclude that X # = C . Then, it follows that X = ( X # , X # ) = ( A , C ). Therefore, the proof is completed. (iii) If ( X , B ) ∈ D L ( K c ) and ( Z , C ) ∈ D L ( K ), it follows that X # = B, B # = X , Z # = C and C # = Z . As Z ⊂ X , it follows that C # ∪ B # = Z ∪ X = X , i.e., (C , B ) = X . Furthermore, as ( Z , C ) is the greatest dual concept in D L ( K ) with Z ⊂ X , we can conclude that X # = C . Then, it follows that X = ( X # , X # ) = (C , B ). Hence, the proposition is proved. 2 Proposition 5. Let K = (U , V , R ) be a formal context. For ( X , ( A , B )) ∈ O E D L, we have ( A # , A ) ∈ D L ( K ) and ( B # , B ) ∈ D L ( K c ) such that A # ⊆ X and B # ⊆ X . Proof. As ( X , ( A , B )) ∈ O E D L ( K ), it follows A = X # and B = X # . Thus, ( A # , A ) = ( X ## , X # ) is a concept of D L ( K ) and ( B # , B ) = ( X ## , X # ) is a concept of D L ( K c ). Moreover, as A # ∪ B # = X , we have A # ⊆ X and B # ⊆ X . Then, the proof is completed. 2 Propositions 4 and 5 exactly manifest that object-induced three-way dual concept lattices embody dual concept lattices and thus provide more useful information than that of dual concept lattices. Furthermore, by light of the above two propositions, we propose Algorithm 2 to build object-induced three-way dual concept lattice of a formal context. Before proceeding, we define a binary relation to facilitate our discussions. For (( X 1 , A 1 ), (Y 1 , B 1 )), (( X 2 , A 2 ), (Y 2 , B 2 )) ∈ D L ( K ) × D L ( K c ), we define (( X 1 , A 1 ), (Y 1 , B 1 )) I (( X 2 , A 2 ), (Y 2 , B 2 )) ⇔ X 1 ∩ Y 1 = X2 ∩ Y 2 . Then, it is easy to show that I is an equivalent relation on D L ( K ) × D L ( K c ). In addition, for (( X , A ), (Y , B )) ∈ D L ( K ) × D L ( K c ), the equivalent class induced by I is denoted as [(( X , A ), (Y , B ))] I . In Algorithm 2, we use two dual lattices D L ( K ) and D L ( K c ) to build an object-induced three-way dual concept lattice. So, the complexity is determined by the constructions of dual concept lattices. It is not difficult to show that dual concept lattices follow the theoretical framework of formal concept lattices. Then, we can get an algorithm for dual concept lattices similarly as that of building formal concept lattices. To the best of our knowledge, the best time complexity of constructing a formal concept lattice is O (|U |2 | V | N ) (N is the number of concepts of the constructed concept lattice) [17]. Example 3. Table 3 shows a formal context K = (U , V , R ) with U = {1, 2, 3, 4, 5} and V = {a, b, c , d, e }. Fig. 2 shows dual concept lattice D L ( K ) and Fig. 3 shows its dual concept lattice D L ( K c ). By using Algorithm 2, we derive the object-induced three-way dual concept lattice O E D L ( K ), which is shown in Fig. 4. In what follows, we take several examples to illustrate Propositions 4 and 5.
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Algorithm 2 Building an object-induced three-way dual concept lattice by using two classical dual concept lattices. Require: A formal context K = (U , V , R ). Ensure: Object-induced three-way dual concept lattice O E D L ( K ). 1: Initialize O E D L ( K ) = ∅. 2: Let be a set of object sets and initialize = ∅. 3: Build dual concept lattices D L ( K ) and D L ( K c ). 4: For each ( X , A ) ∈ D L ( K ) 5: If there is a concept ( X , B ) ∈ D L ( K c ) 6: Then add ( X , ( A , B )) to O E D L ( K ) . 7: D L ( K c ) ← D L ( K c ) − {( X , B )}. 8: Else find the greatest dual concept ( Z , C ) in D L ( K c ) with Z ⊂ X . 9: Add ( X , ( A , C )) to O E D L ( K ). 10: End If 11: Add X to . 12: End For 13: For each ( X , B ) ∈ D L ( K c ) 14: Find the greatest dual concept ( Z , C ) in D L ( K ) with Z ⊂ X . 15: Add ( X , (C , B )) to O E D L ( K ) and add X to . 16: End For 17: For each ( X , A ) ∈ D L ( K ) and (Y , B ) ∈ D L ( K c ) 18: If Z = X ∩ Y ∈ / 19: Establish equivalent class [(( X , A ), (Y , B ))] I . 20: End For 21: For each established equivalent class [(( X , A ), (Y , B ))] I 22: Find the greatest element (( X g , A g ), (Y g , B g )) in [(( X , A ), (Y , B ))] I . 23: Add ( X ∩ Y , ( A g , B g )) to O E D L ( K ). 24: End For 25: Return O E D L ( K ).
Table 3 A formal context K of Example 3. a 1 2 3 4 5
× × ×
b
× × ×
c
× × ×
Fig. 2. D L ( K ) of Example 3.
d
×
×
e
× × ×
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Fig. 3. D L ( K c ) of Example 3.
Fig. 4. O E D L ( K ) of Example 3.
(i) As (1245, bd) ∈ D L ( K ), (1245, ace ) ∈ D L ( K c ) and (1245, bd, ace ) ∈ O E D L ( K ), it follows that (i) of Proposition 4 is verified. (ii) As (145, abd) ∈ D L ( K ), (15, abce ) ∈ D L ( K c ) and there does not exist another dual concept ( X , A ) ∈ D L ( K c ) such that (15, abce ) < ( X , A ) and X is a subset of the extent of (145, abd), and (145, abd, abce ) ∈ O E D L ( K ), it follows that (ii) of Proposition 4 is verified. (iii) As (1235, be ) ∈ D L ( K c ), (135, acd) ∈ D L ( K ) and there does not exist another dual concept ( X , A ) ∈ D L ( K ) such that (135, acd) < ( X , A ) and X is a subset of the extent of (1235, be ), and (1235, acd, be ) ∈ O E D L ( K ), it follows that (iii) of Proposition 4 is verified. (iv) For any object-induced three-way dual concept ( X , ( A , B )) ∈ O E D L ( K ), we can show that ( A # , A ) ∈ D L ( K ) or # ( B , B ) ∈ D L ( K c ). Therefore, Proposition 5 is verified.
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4. The relationships among four types of object-induced three-way concept lattices In the previous sections, we have introduced object-induced three-way concept lattices, object-induced three-way dual concept lattices and object-induced three-way object oriented concept lattices. Analogously, we can further define objectinduced three-way property oriented concept lattices. And then, we discuss the relationships among these four types of lattices. Definition 6. Let K = (U , V , R ) be a formal context. For X ∈ 2U and ( A , B ) ∈ 2 V × 2 V , two associated three-way operators : 2U → 2 V × 2 V and : 2 V × 2 V → 2U are respectively defined as:
X = ( X 3 , X 3 ) and ( A , B ) = A 2 ∩ B 2 , where X 3 = {a ∈ V | X ∩ R c (a) = ∅} and B 2 = {x ∈ U | R c (x) ⊆ B }. The pair ( X , ( A , B )) is called an object-induced three-way property oriented concept if X = ( A , B ) and ( A , B ) = X . Given a formal context K , the set of all object-induced three-way property oriented concepts is naturally equipped with a partial order ≤ defined as:
( X 1 , ( A 1 , B 1 )) ≤ ( X 2 , ( A 2 , B 2 )) ⇔ X 1 ⊆ X 2 ⇔ ( A 1 , B 1 ) ≤ ( A 2 , B 2 ). It is straightforward that all the object-induced three-way property oriented concepts contained in K with ≤ form a poset, which is denoted henceforth by O E P L ( K ). Further more, the infimum and supremum in the lattice are respectively given by:
( X 1 , ( A 1 , B 1 )) ∧ ( X 2 , ( A 2 , B 2 )) = ( X 1 ∩ X 2 , (( A 1 , B 1 ) ∩ ( A 2 , B 2 )) ) ( X 1 , ( A 1 , B 1 )) ∨ ( X 2 , ( A 2 , B 2 )) = (( X 1 ∪ X 2 ) , ( A 1 , B 1 ) ∪ ( A 2 , B 2 )). It is worth noting that for a three-way property oriented concept ( X , ( A , B )), A and B may overlap with each other. And this is the case similar to that of three-way dual concepts. Also, this can be explained in a wide sense of 3WD as that of three-way dual concepts. As this paper mainly focuses on three-way dual concepts, we omit the details here. Proposition 6. Let (U , V , R ) be a formal context. For X ⊆ U , the following properties hold. (i) X ∗ = X c#c , X # = X c ∗c .
(ii) X 2 = X c 3c , X 3 = X c 2c . c c2 ∗ (iii) X ∗R = X 3 Rc , X R = X Rc .
Proof. It can be proved analogously as that of Proposition 1.
2
Although Proposition 1 and Proposition 6 hold a similar meaning, they are both useful in our discussions and cannot replace each other. Concretely, Proposition 1 is about positive operators and Proposition 6 concerns negative operators. Theorem 2. Let K = (U , V , R ) be a formal context. For X ∈ 2U and ( A , B ) ∈ 2 V × 2 V , the following propositions are equivalent. (i) (ii) (iii) (iv)
( X , ( A , B )) is an object-induced three-way concept of K . ( X c , ( A c , B c )) is an object-induced three-way dual concept of K . ( X c , ( B , A )) is an object-induced three-way object oriented concept of K . ( X , ( B c , A c )) is an object-induced three-way property oriented concept of K .
Proof. (i) ⇔ (ii): applying Proposition 1 (i), Proposition 6 (i) and duality principle, we get:
( X , ( A , B )) is an object-induced three-way concept of K ⇔ X ∗ = A , X ∗ = B and A ∗ ∩ B ∗ = X ⇔ X c#c = A , X c#c = B and ( A ∗ ∩ B ∗ )c = A c ⇔ X c# = A c , X c# = B c and A ∗c ∪ B ∗c = A c ⇔ X c# = A c , X c# = B c and A c # ∪ B c# = A c ⇔ ( X c , ( A c , B c )) is an object-induced three-way dual concept of K
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(i) ⇔ (iii): applying Proposition 2, Proposition 6 (iii) and duality principle, we get:
( X , ( A , B )) is an object-induced three-way concept of K ⇔ X ∗R = B , X ∗R = A and A ∗R ∩ B ∗R = X ⇔ X cR2c = B , X cR2c = A and ( A ∗R ∩ B ∗R )c = X c
c
⇔ {a ∈ V | R (a) ⊆ X c } = B , {a ∈ V | R c (a) ⊆ X c } = A and B ∗R ∪ A ∗R c = X c
⇔ X cR 2 = B , X cR 2 = A and {x ∈ U | ∀b ∈ B , xR c b}c ∪ {x ∈ U | ∀a ∈ A , xRa}c = X c ⇔ X cR 2 = B , X cR 2 = A and {x ∈ U | ∃b ∈ B , xRb} ∪ {x ∈ U | ∃a ∈ A , xR c a} = X c ⇔ X cR 2 = B , X cR 2 = A and B 3 ∪ A 3 = X c ⇔ ( X c , ( B , A )) is an object-induced three-way object oriented concept of K
(iii) ⇔ (iv): applying Proposition 1 (ii), Proposition 6 (ii) and duality principle, we get:
( X c , ( B , A )) is an object-induced three-way object oriented concept of K ⇔ X c2 = B , X c2 = A and B 3 ∪ A 3 = X c ⇔ X c2c = B c , X c2c = A c and ( B 3 ∪ A 3 )c = X ⇔ X 3 = B c , X 3 = A c and B 3c ∩ A 3c = X ⇔ X 3 = B c , X 3 = A c and B c2 ∩ A c2 = X ⇔ ( X , ( B c , A c )) is an object-induced three-way property oriented concept of K The proof is completed.
2
Corollary 1. Let K = (U , V , R ) be a formal context. Then | O E L ( K )| = | O E D L ( K )| = | O E O L ( K )| = | O E P L ( K )|. 5. Conclusion In many applications, such as international cooperation, if we want to do something with our potential collaborators, we must have a good knowledge about both of their strength and weakness. In applications, the strength may refer to the cases such as one has some kinds of natural resources, professional skills, domain knowledge and the like, while the weakness means the opposite. However, the existing formal concept analysis theory does not work for such cases. Generally speaking, this is a task to find a meaningful pattern ( X , ( A , B )), where A is the set of attributes that are not possessed by at least one object in X c , B is the set of attributes that are possessed by at least one object in X c , and at the same time, X is composed of the elements who do not possess at least one attribute in A c or possesses at least one attribute in B c . In this study, we call the pattern ( X , ( A , B )) a three-way dual concept and call the task of finding such patterns three-way dual concept analysis. In order to show the significance of three-way dual concept analysis, we use two illustrative examples. One is about international import and export transactions, by which we stress the necessity to propose and define three-way dual concepts. The other is about knowledge sharing, by which we emphasize the strength of three-way dual concepts. Briefly speaking, the proposal of three-way dual concept and its semantic explanation are the main contributions of this paper. Besides, an incremental algorithm for building a three-way dual concept lattice is presented. And then, a batch algorithm is also described by exploring the connections between object-induced three-way dual concept lattices and classical dual concept lattices. What is more, the relationships among four types of three-way concept analysis models, i.e., three-way concept lattices, three-way dual concept lattices, three-way object oriented concept lattices and three-way property oriented concept lattices, are investigated. It is worth noting that in this study only object-induced three-way concepts are formally presented. By using duality principle, interested readers can further explore the attribute-induced three-way dual concepts analogously. Building three-way concept lattices are time consuming and sometimes brings insurmountable difficulties in real applications, especially in this big data era. Then, how to use the spirit of three-way concept analysis, rather than directly resort to three-way concept lattices, is an important and interesting topic. Besides, how to combine two or more concept analysis methods to improve granule description accuracy is another promising research direction. For instance, we may combine common attribute analysis and necessary attribute analysis to characterize a domain. What is more, the studies in this paper is about complete contexts. However, in some circumstances, the relationships between objects and attributes are partially known and incomplete contexts are needed to manage such cases [2,16,21,58, 61]. Although interval-set algebra has been proposed to solve this problem [48,52], the representation of three-way concepts in incomplete formal contexts is still an open problem and deserves our further investigations. Declaration of competing interest The authors declare that they have no conflicts of interest.
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Acknowledgements This work was supported by the National Natural Science Foundation of China (Nos. 61502150, 11801440 and 61772021) and the Innovation Talent Promotion Plan of Shaanxi Province for Young Sci-Tech New Star (Program No. 2017KJXX-60). References [1] M.K. Afridi, N. Azam, J.T. Yao, E. Alanazi, A three-way clustering approach for handling missing data using GTRS, Int. J. Approx. Reason. 98 (2018) 11–24. [2] P. Burmeister, R. Holzer, On the treatment of incomplete knowledge in formal concept analysis, in: Proceedings of ICCS 2000, in: LNCS (LNAI), vol. 1867, 2000, pp. 385–398. [3] J. Chen, J. Mi, B. Xie, Y. Lin, A fast attribute reduction method for large formal decision contexts, Int. J. Approx. Reason. 106 (2019) 1–17. [4] Y.H. Chen, Y.Y. Yao, A multiview approach for intelligent data analysis based on data operators, Inf. Sci. 178 (2008) 1–20. [5] G. Chiaselotti, F. Infusino, Notions from rough set theory in a generalized dependency relation context, Int. J. Approx. Reason. 98 (2018) 25–61. [6] Y. Djouadi, D. Dubois, P. Prade, Différentes extensions floues de lanalyse formelle de concepts, in: Actes Renc. Franc. sur la Logique Floue et ses Applications Cépadues edn., 2009, pp. 141–148. [7] Y. Djouadi, H. Prade, Possibility-theoretic extension of derivation operators in formal concept analysis over fuzzy lattices, Fuzzy Optim. Decis. Mak. 10 (2011) 287–309. [8] D. Dubois, F. Dupin de Saint Cyr, H. Prade, A possibility-theoretic view of formal concept analysis, Fundam. Inform. 75 (1–4) (2007) 195–213. [9] D. Dubois, H. Prade, Possibility theory and formal concept analysis in information systems, in: Proceedings of IFSA/EUSFLAT’09, Lisbon, Portugal, 2009, pp. 1021–1026. [10] B. Ganter, R. Wille, Formal Concept Analysis: Mathematical Foundations, Springer, Berlin, 1999. [11] Y. Fang, F. Min, Cost-sensitive approximate attribute reduction with three-way decisions, Int. J. Approx. Reason. 104 (2019) 148–165. [12] A. Ferone, Feature selection based on composition of rough sets induced by feature granulation, Int. J. Approx. Reason. 101 (2018) 276–292. [13] B. Ganter, S.O. Kuznetsov, Hypotheses and version spaces, conceptual structures for knowledge creation and communication, in: Formal Concept Analysis: Mathematical Foundations, Springer, Berlin, 2003, pp. 83–95. [14] R. Godin, R. Missaoui, H. Alaoui, Incremental concept formation algorithms based on Galois (concept) lattices, Comput. Intell. 11 (2) (1995) 246–267. [15] S.E. Han, Marcus-Wyse topological rough sets and their applications, Int. J. Approx. Reason. 106 (2019) 214–227. [16] M. Krupka, J. Lastovicka, Concept lattices of incomplete data, in: Proceedings of ICFCA 2012, in: LNCS (LNAI), vol. 7278, 2012, pp. 180–194. [17] S.O. Kuznetsov, S.A. Obiedkov, Comparing performance of algorithms for generating concept lattices, J. Exp. Theor. Artif. Intell. 14 (2–3) (2002) 189–216. [18] J. Li, C. Mei, Y. Lv, Incomplete decision contexts: approximate concept construction, rule acquisition and knowledge reduction, Int. J. Approx. Reason. 54 (2013) 149–165. [19] J. Li, C.L. Mei, W.H. Xu, Y.H. Qian, Concept learning via granular computing: a cognitive viewpoint, Inf. Sci. 298 (2015) 447–467. [20] J. Li, C. Huang, J. Qi, Y. Qian, W. Liu, Three-way cognitive concept learning via multi-granularity, Inf. Sci. 378 (2017) 244–263. [21] W. Lipski Jr., On semantic issues connected with incomplete information databases, ACM Trans. Database Syst. 4 (1979) 269–296. [22] M.Z. Li, G.Y. Wang, Approximate concept construction with three-way decisions and attribute reduction in incomplete contexts, Knowl.-Based Syst. 91 (2016) 165–178. [23] W. Li, X. Jia, L. Wang, B. Zhou, Multi-objective attribute reduction in three-way decision-theoretic rough set model, Int. J. Approx. Reason. 105 (2019) 327–341. [24] J.M. Ma, M.J. Cai, C.J. Zou, Concept acquisition approach of object-oriented concept lattices, Int. J. Mach. Learn. Cybern. 8 (1) (2017) 123–134. [25] L. Ma, The investigation of covering rough sets by Boolean matrices, Int. J. Approx. Reason. 100 (2018) 69–84. [26] M. Ma, M.K. Chakraborty, Covering-based rough sets and modal logics. Part II, Int. J. Approx. Reason. 95 (2018) 113–123. [27] X.A. Ma, X.R. Zhao, Cost-sensitive three-way class-specific attribute reduction, Int. J. Approx. Reason. 105 (2019) 153–174. [28] L. Nourine, O. Raynaud, A fast algorithm for building lattices, Inf. Process. Lett. 71 (1999) 199–204. [29] J.J. Qi, L. Wei, Y.Y. Yao, Three-way formal concept analysis, in: Rough Sets and Knowledge Technology, in: Lecture Notes in Computer Science, vol. 8818, 2014, pp. 732–741. [30] J.J. Qi, T. Qian, L. Wei, The connections between three-way and classical concept lattices, Knowl.-Based Syst. 91 (2016) 143–151. [31] J. Qian, C. Liu, X. Yue, Multigranulation sequential three-way decisions based on multiple thresholds, Int. J. Approx. Reason. 105 (2019) 396–416. [32] T. Qian, L. Wei, J. Qi, Constructing three-way concept lattices based on apposition and subposition of formal contexts, Knowl.-Based Syst. 166 (2017) 39–48. [33] T. Qian, L. Wei, J. Qi, A theoretical study on the object (property) oriented concept lattices based on three-way decisions, Soft Comput. (2019), https:// doi.org/10.1007/s00500-019-03799-6. [34] Y. Qian, X. Liang, Q. Wang, J. Liang, B. Liu, A. Skowron, Y. Yao, J. Ma, C. Dang, Local rough set: a solution to rough data analysis in big data, Int. J. Approx. Reason. 97 (2018) 38–63. [35] M.S. Raza, U. Qamar, Feature selection using rough set-based direct dependency calculation by avoiding the positive region, Int. J. Approx. Reason. 92 (2018) 175–197. [36] R.S. Ren, L. Wei, Y.Y. Yao, An analysis of three types of partially-known formal concepts, Int. J. Mach. Learn. Cybern. 9 (11) (2018) 1767–1783. [37] J.M. Rodriguez-Jimenez, P. Cordero, M. Enciso, A. Mora, Negative attributes and implications in formal concept analysis, Proc. Comput. Sci. 31 (2014) 758–765. [38] R. Shivhare, A.K. Cherukuri, Three-way conceptual approach for cognitive memory functionalities, Int. J. Mach. Learn. Cybern. 8 (1) (2017) 21–34. [39] P.K. Singh, Ch. Aswani Kumar, Bipolar fuzzy graph representation of concept lattice, Inf. Sci. 288 (2014) 437–448. [40] P.K. Singh, Medical diagnoses using three-way fuzzy concept lattice and their Euclidean distance, Comput. Appl. Math. 37 (3) (2018) 3283–3306. [41] B. Sun, W. Ma, B. Li, X. Li, Three-way decisions approach to multiple attribute group decision making with linguistic information-based decisiontheoretic rough fuzzy set, Int. J. Approx. Reason. 93 (2018) 424–442. [42] L. Wei, T. Qian, The three-way object oriented concept lattice and the three-way property oriented concept lattice, in: Proceedings of the 2015 International Conference on Machine Learning and Cybernetics, 2015, pp. 854–859. [43] R. Wille, Restructuring lattice theory: an approach based on hierarchies of concepts, in: I. Rival (Ed.), Ordered Sets, Reidel Publishing Company, Dordrecht–Boston, 1982, pp. 445–470. [44] X.D. Wu, C.Q. Zhang, S.C. Zhang, Efficient mining of both positive and negative association rules, ACM Trans. Inf. Syst. 22 (3) (2004) 381–405. [45] Y.T. Yan, Z.B. Wu, X.Q. Du, J. Chen, S. Zhao, Y.P. Zhang, A three-way decision ensemble method for imbalanced data oversampling, Int. J. Approx. Reason. 107 (2019) 1–16. [46] X. Yang, T. Li, H. Fujita, D. Liu, A sequential three-way approach to multi-class decision, Int. J. Approx. Reason. 104 (2019) 108–125.
H. Zhi et al. / International Journal of Approximate Reasoning 114 (2019) 151–165
165
[47] J.T. Yao, N. Azam, Web-based medical decision support systems for three-way medical decision making with game-theoretic rough sets, IEEE Trans. Fuzzy Syst. 23 (1) (2015) 3–15. [48] Y.Y. Yao, Interval-set algebra for qualitative knowledge representation, in: Proceedings of the 5th International Conference on Computing and Information, Sudbury, 1993, pp. 370–374. [49] Y.Y. Yao, A comparative study of formal concept analysis and rough set theory in data analysis, in: Proceedings of 4th International Conference on Rough Sets and Current Trends in Computing, RSCTC 2004, Uppsala, Sweden, 2004, pp. 59–68. [50] Y.Y. Yao, Concept lattices in rough set theory, in: Proceedings of 23rd International Meeting of the North American Fuzzy Information Processing Society, 2004, pp. 796–801. [51] Y.Y. Yao, An outline of a theory of three-way decisions, in: Proceedings of RSCTC 2012, in: LNCS (LNAI), vol. 7413, Springer, Heidelberg, 2012, pp. 1–17. [52] Y.Y. Yao, Interval sets and three-way concept analysis in incomplete contexts, Int. J. Mach. Learn. Cybern. 8 (1) (2017) 3–20. [53] Y.Y. Yao, Three-way decision and granular computing, Int. J. Approx. Reason. 103 (2018) 107–123. [54] Y. Zhang, J.T. Yao, Gini objective functions for three-way classifications, Int. J. Approx. Reason. 81 (2017) 103–114. [55] H.Y. Zhang, S.Y. Yang, Three-way group decisions with interval-valued decision-theoretic rough sets based on aggregating inclusion measures, Int. J. Approx. Reason. 110 (2019) 31–45. [56] Y. Zhang, D. Miao, Z. Zhang, J. Xu, S. Luo, A three-way selective ensemble model for multi-label classification, Int. J. Approx. Reason. 103 (2018) 394–413. [57] Y. Zhang, D. Miao, J. Wang, Z. Zhang, A cost-sensitive three-way combination technique for ensemble learning in sentiment classification, Int. J. Approx. Reason. 105 (2019) 85–97. [58] Y.X. Zhao, J.H. Li, W.Q. Liu, W.H. Xu, Cognitive concept learning from incomplete information, Int. J. Mach. Learn. Cybern. 8 (1) (2017) 159–170. [59] H. Zhi, J. Li, Influence of dynamical changes on concept lattice and implication rules, Int. J. Mach. Learn. Cybern. 9 (2018) 705–805. [60] H. Zhi, J. Li, Granule description based on positive and negative attributes, Granul. Comput. 4 (2019) 337–350. [61] H. Zhi, J. Li, Granule description based knowledge discovery from incomplete formal contexts via necessary attribute analysis, Inf. Sci. 485 (2019) 347–361.