Abstract
Rough sets theory and fuzzy sets theory are important mathematical tools to deal with uncertainties. Rough fuzzy sets and fuzzy rough sets as generalizations of rough sets have been introduced. Type-2 fuzzy set provides additional degree of freedom, which makes it possible to directly handle high uncertainties. In this paper, the rough type-2 fuzzy set model is proposed by combining the rough set theory with the type-2 fuzzy set theory. The rough type-2 fuzzy approximation operators induced from the Pawlak approximation space are defined. The rough approximations of a type-2 fuzzy set in the generalized Pawlak approximation space are also introduced. Some basic properties of the rough type-2 fuzzy approximation operators and the generalized rough type-2 fuzzy approximation operators are discussed. The connections between special crisp binary relations and generalized rough type-2 fuzzy approximation operators are further examined. The axiomatic characterization of generalized rough type-2 fuzzy approximation operators is also presented. Finally, the attribute reduction of type-2 fuzzy information systems is investigated.
1. Introduction
Rough sets theory [1, 2] proposed by Pawlak is a mathematical tool to deal with uncertainties and imprecision. Currently, rough set theory has been found to have very successful applications in many fields such as expert systems, machine learning, pattern recognition, decision analysis, and knowledge discovery. In Pawlak rough set model, the equivalence relation plays an important role and is a very restrictive requirement, which may limit its applications in real applications. Hence, the generalization ofrough set model is an important branch of rough set theory study. Afterward, several authors proposed generalized rough set model, variable precision rough set model, covering rough set model, rough fuzzy set model and fuzzy rough set model, and so forth.
Fuzzy set theory (type-1 fuzzy sets) was proposed by Zadeh in 1965 [3]. The type-1 fuzzy set theory has been widely applied in various fields [4–6]. Since rough set theory is highly complementary with fuzzy set theory, it is desirable to study the combination of fuzzy sets and rough sets. Dubois and Prade proposed rough fuzzy set model and fuzzy rough set model in [7]. Afterwards, many interesting achievements are obtained on the basis of [7]. Currently, the knowledge representation of fuzzy rough sets and the attribute reduction using fuzzy rough sets are two main branches on the researches of fuzzy rough sets. On the knowledge representation of fuzzy rough sets, researchers mainly focus on the construction of fuzzy rough approximation operators and the axiomatic characterization of fuzzy rough sets. At present, there exist many valuable results on the knowledge representation of fuzzy rough sets [8–12]. The definitions for generalized fuzzy lower and upper approximation operators determined by a residual implication were introduced in [8]. The lower and upper approximations of rough sets and fuzzy rough sets were, respectively, characterized by using outer and inner products in [9]. A general framework for the study of fuzzy rough sets was presented in [10]. The lattice and topological structures of fuzzy rough sets were proposed in [11]. The axiomatic approach of fuzzy rough sets was shown in [12]. On the attribute reduction using fuzzy rough sets, researchers mainly focus on the development of fast reduction algorithms. Jensen and Shen did a pioneering work on attribute reduction using fuzzy rough sets in [13]. A fuzzy rough QuickReduct algorithm was presented in [13], and this novel approach has been applied to aid classification of web content, with very promising results. However, the QuickReduct algorithm has been proved to have some shortcomings in [14]. Hence, some improved algorithms were proposed in [14, 15].
As a generalization of type-1 fuzzy sets, intuitionistic fuzzy sets were proposed by Atanassov [16, 17]. By setting the degree of membership, the degree of nonmembership, and the degree of hesitation, intuitionistic fuzzy sets depict the nature of ambiguity and solve the hesitant information in judging problems. Cornelis et al. [18] proposed intuitionistic fuzzy rough sets by combining intuitionistic fuzzy sets and rough sets. Afterwards, the theory and application studies of intuitionistic fuzzy rough sets have been expanded in [19–22]. The concept of interval-valued intuitionistic fuzzy sets was firstly proposed in [23]. Interval-valued intuitionistic fuzzy sets provide more flexible and effective way to deal with uncertainties because both membership degree and nonmembership degree of interval-valued intuitionistic fuzzy sets are denoted by interval numbers of . In [24], an interval-valued intuitionistic fuzzy rough model has been proposed by combining the classical Pawlak rough set theory with the interval-valued intuitionistic fuzzy set theory. Furthermore, in [25], the rough approximations of an interval-valued intuitionistic fuzzy set in the classical Pawlak approximation space and the generalized Pawlak approximation space have been presented, respectively.
Zadeh in 1975 proposed type-2 fuzzy sets which can enhance the system’s ability to deal with uncertainties [26]. The applications of type-2 fuzzy sets were limited due to the computational complexity. However, interval type-2 fuzzy sets, as a special case of general type-2 fuzzy sets, have been extensively studied in practical applications. There are some achievements about interval type-2 fuzzy rough sets by combining the characteristics of rough sets and interval type-2 fuzzy sets [27–29]. General type-2 fuzzy sets may be better than the interval type-2 fuzzy sets to deal with uncertainties because general type-2 fuzzy sets can obtain more degree of freedom [30]. In order to simplify the calculation for general type-2 fuzzy sets, the representation of general type-2 fuzzy sets is also an important research issue. Liu in [31] firstly proposed the -plane representation of the general type-2 fuzzy sets and claimed that the -plane representation can greatly reduce the computational workload. Currently, -plane method has been extensively studied [32–35] because it can take advantage of the interval type-2 fuzzy sets theory to study the general type-2 fuzzy sets.
Although general type-2 fuzzy sets and rough sets have important applications, a few results about the combination of general type-2 fuzzy set theory and rough set theory were presented [36–38]. In [36], we proposed general type-2 fuzzy rough sets based on general type-2 fuzzy relations. However, there are no contributions on the definitions of approximation operators of general type-2 fuzzy sets in the Pawlak approximation space and the generalized Pawlak approximation space. Moreover, it can be seen that the attribute reduction of information systems is an important application of rough set theory. The traditional rough set model can effectively handle these information systems in which both condition attributes and decision attributes are clear. In some cases, condition attributes may be clear and decision attributes may be interval type-2 fuzzy sets on objects set. That is to say, every decision attribute value is an interval on . At this time, the rough approximations of an interval type-2 fuzzy set in the generalized Pawlak approximation space were also introduced in [39]. Therefore, the interval type-2 rough fuzzy set model proposed in [39] may be suitable to deal with these information systems with interval values. However, in highly uncertain situations, we would encounter these information systems where all condition attributes are clear and all decision attributes are general type-2 fuzzy sets on objects set; that is, every decision attribute value is a type-1 fuzzy set on . To provide certain theoretical basis for dealing with such problem, we propose rough type-2 fuzzy set model by combining the classical Pawlak rough set theory with the general type-2 fuzzy set theory in this paper. The rest of our work is organized as follows. In Section 2, the basic definitions and terminologies on type-2 fuzzy sets are reviewed briefly. In Section 3, the rough type-2 fuzzy approximation operators based on the clear equivalence relation are firstly defined. The generalized rough type-2 fuzzy approximation operators based on the clear generalized binary relation are also derived in Section 3. In Section 4, the generalized rough type-2 fuzzy approximation operators are characterized by axioms. The attribute reduction method of type-2 fuzzy information systems is presented in Section 5. The last section concludes this paper.
2. Preliminaries
In this section, the basic definitions and terminologies on type-2 fuzzy sets with some modified notations are recalled.
Definition 1 (see [40]). A type-2 fuzzy set, denoted by , is expressed aswhere is the primary membership of ; is the secondary membership function; is a secondary membership grade.
In the following sections, the class of all type-2 fuzzy sets of the universe of discourse is denoted as , and the class of all crisp sets of the universe of discourse is denoted as . Currently, it is difficult to deal with the type-2 fuzzy sets whose secondary membership functions are not normal and convex. For simplicity, we only study these type-2 fuzzy sets whose secondary membership function is normal and convex in this paper.
Definition 2 (see [32]). An -plane for type-2 fuzzy set , which is denoted by , is defined as follows:where denote an -cut of the secondary membership function .
Definition 3 (see [32]). The -plane representation (theorem) for type-2 fuzzy set is
Theorem 4 (see [32]). Let and be -plane of and , respectively; one haswhereObviously, , , , and hold.
Theorem 5 (see [36]). Let be -plane of ; one haswhere .
Definition 6 (see [36]). Let , and define if and hold for any and . If and , then .
3. Rough Type-2 Fuzzy Sets
In this section, we introduce rough type-2 fuzzy approximation operators and generalized rough type-2 fuzzy approximation operators induced from the Pawlak approximation space and the generalized Pawlak approximation space, respectively, and discuss their properties.
3.1. Rough Type-2 Fuzzy Approximation Operators Based on the Equivalence Relation
Definition 7. Let be a nonempty universe of discourse and let be the clear equivalence relation on . denotes the equivalence class. Then is called the Pawlak approximation space. For any , define the upper and lower rough type-2 fuzzy approximation operators and about by where , , and Clearly, if is degraded to type-1 fuzzy set, then the rough type-2 fuzzy approximation operators defined in Definition 7 reduce to the rough type-1 fuzzy approximation operators. In the following, we define type-2 fuzzy universe set and type-2 fuzzy empty set . Obviously, and for any and .
Theorem 8. Let be a Pawlak approximation space and let and be the rough type-2 fuzzy lower and upper approximation operators about ; for any , the following properties hold:(1);(2), ;(3), ;(4), ;(5), ;(6), .
Proof. (1) For any and , we have Thus, . Similarly, . That is, .
(2) For any and ,Furthermore, Thus, . Similarly, we can prove .
(3) For any and , Similarly, we have . Thus, . Similarly, we can prove .
(4) For any and , since , we have and . Thus, . Furthermore, . Hence, . That is, . Similarly, we can prove .
(5) For any and ,Similarly, we have . Thus, . In additionSimilarly, we can obtain . Hence, . That is, . Similarly, we can prove .
(6) For any and , we haveThus, . Similarly, . That is to say, . The second equation can be proved in a similar way.
Definition 9. Let be a Pawlak approximation space and let and be the type-2 fuzzy sets on .(1) and are called lower rough equal denoted by if .(2) and are called upper rough equal denoted by if .(3) and are called rough equal denoted by if and are both lower rough equal and upper rough equal.
Theorem 10. Let be a Pawlak approximation space. Then the following properties hold for any :(1) and ;(2) and ;(3)if and , then ;(4)if and , then ;(5)if or , then ;(6)if or , then ;(7)if and , then ;(8)if and , then ;(9);(10).
Proof. By Definition 9, the proof procedure is trivial.
Theorem 11. Let be a Pawlak approximation space and let be type-2 fuzzy set on . Then, one has(1);(2).
Proof. By Theorem 8(5), this theorem can be easily proved.
3.2. Generalized Rough Type-2 Fuzzy Approximation Operators Based on the Generalized Binary Relation
Definition 12. Let be a nonempty finite universe of discourse and let be an arbitrary binary relation on ; ; then is called the generalized Pawlak approximation space. For any , the upper and lower generalized rough type-2 fuzzy approximation operators and about are, respectively, defined as follows:where , , and In particular, if is an equivalence relation of the universe , then degrades to the equivalence class . The lower approximation and the upper approximation reduce to the lower approximation and the upper approximation in the classical Pawlak approximation space, respectively.
In [36], a type-2 fuzzy singleton set and its complement are, respectively, defined as follows: Based on the above definition, we can obtain that and hold for any and .
Theorem 13. Let be a generalized Pawlak approximation space and let and be the generalized rough type-2 fuzzy lower and upper approximation operators about ; for any , the following properties hold:(1), ;(2), ;(3), ;(4), ;(5) for any and ;(6) for any and .
Proof. The proofs of (1)–(4) are analogous to Theorem 8, and therefore we omit them.
(5) For any and , we haveIf , then . Thus, . That is, . On the other hand, if , then . Thus, . Therefore, we have .
Similarly, we can obtain . That is to say, .
(6) The proof procedure is similar to (5).
Theorem 14. Let be a generalized Pawlak approximation space and let and be the generalized rough type-2 fuzzy lower and upper approximation operators about ; for any , the following statements are equivalent:(1) is reflexive;(2);(3).
Proof.
(1)(2). If is reflexive, then for any . For any ,Similarly, we can derive . Thus, .
(2)(3). We can obtain the conclusion according to Theorem 13(1).
(3)(1). For any , let ; thus .
If , then and hold for any and . Hence, and . We can obtain and . Therefore, . That is to say, there exists such that . Thus, .
We can conclude that is reflexive.
Theorem 15. Let be a generalized Pawlak approximation space and let and be the generalized rough type-2 fuzzy lower and upper approximation operators about ; the following statements are equivalent:(1) is symmetric.(2)For any and , and hold.(3)For any and , and hold.
Proof.
(1)(2). For any and If , then .
Since is symmetric, we have . Hence, . That is, .
Similarly, we can prove .
(2)(3). For any and Since , thus .
Similarly, we can prove .
(3)(1). For any and , if , in the following, we only should prove . Thus, .
We know that . Hence, . That is to say, there exists such that .
We can obtain . Therefore, is symmetric.
Theorem 16. Let be a generalized Pawlak approximation space and let and be the generalized rough type-2 fuzzy lower and upper approximation operators about ; for any , the following statements are equivalent:(1) is transitive;(2);(3).
Proof.
(1)(2). For any and Since is transitive, we have . Thus, . Similarly, . Hence, .
(2)(1). For any and , if and , then Furthermore, . Since , we have . That is, . Thus, there exists such that . That is to say, .
We can conclude that . Therefore, is transitive.
(2)(3). We can directly obtain the statement according to Theorem 13(1).
4. Axiomatic Characterization of Generalized Rough Type-2 Fuzzy Sets
In this section, the axiomatic characterization of generalized rough type-2 fuzzy approximation operators is presented. To characterize the generalized rough type-2 fuzzy approximation operators by axioms, we define a constant type-2 fuzzy set , where is secondary membership function and is not related to . Clearly, and hold for any , where and are functions related to .
Similar to [36], we define two special type-2 fuzzy sets denoted by and for any , , and , respectively. The two special type-2 fuzzy sets satisfy the following properties: , , , and for any and . Obviously, and are constant type-2 fuzzy sets.
Definition 17. Let be two operators. They are called dual operators if for all (L1);(H1).
Lemma 18. Let be a generalized Pawlak approximation space and let and be the generalized rough type-2 fuzzy lower and upper approximation operators about ; for any and constant type-2 fuzzy set , the following statements hold:(1);(2).
Proof. (1) For any and ,Similarly, we can obtain . Thus, .
(2) The proof procedure is similar to (1).
Lemma 19 (see [36]). For any , , and , the following statements hold:(1);(2);(3);(4).
Theorem 20. Let be two dual operators. Then there exists a crisp binary relation on such that, for all , and hold if and only if and satisfy the axioms: for all and any constant type-2 fuzzy set ,(L2);(L3);(L4) for any .
Proof. “” It follows immediately from Theorem 13 and Lemma 18.
“” By employing and axiom (L4), we can define a crisp relation on :Thus, . Then, we can obtainSimilarly, . Thus, .
We have according to Theorem 13(1).
Theorem 21. Let be two dual operators. Then there exists a crisp binary relation on such that, for all , and hold if and only if and satisfy the axioms: for all and any constant type-2 fuzzy set ,(H2);(H3);(H4) for any .
Proof. “” It follows immediately from Theorem 13 and Lemma 18.
“” By employing and axiom (H4), we can define a crisp relation on :Thus, . Then, we can obtainSimilarly, . Thus, .
We have according to Theorem 13(1).
Definition 22. Suppose that are two dual operators. If satisfies axioms (L2), (L3), and (L4) or equivalently satisfies axioms (H2), (H3), and (H4), then the system is called a generalized rough type-2 fuzzy set algebra, and and are called generalized type-2 fuzzy approximation operators.
Theorem 23. Let be a pair of dual generalized type-2 fuzzy approximation operators. Then there exists a reflexive relation on such that, for all , and hold if and only if and satisfy the following axioms: (L5);(H5).
Proof. “” It follows immediately from Theorem 14.
“” It follows immediately from Theorems 14, 20, and 21.
Theorem 24. Let be a pair of dual generalized type-2 fuzzy approximation operators. Then there exists a symmetric relation on such that, for all , and hold if and only if and satisfy the following axioms: (L6)For any and , and hold.(H6)For any and , and hold.
Proof. “” It follows immediately from Theorem 15.
“” It follows immediately from Theorems 15, 20, and 21.
Theorem 25. Let be a pair of dual generalized type-2 fuzzy approximation operators. Then there exists a transitive relation on such that, for all , and hold if and only if and satisfy the following axioms: (L7);(H7).
Proof. “” It follows immediately from Theorem 16.
“” It follows immediately from Theorems 16, 20, and 21.
Theorem 26. Let be a pair of dual generalized type-2 fuzzy approximation operators. Then there exists an equivalence relation on such that, for all , and hold if and only if satisfies the axioms (L5)–(L7) and satisfies the axioms (H5)–(H7).
Proof. “” It follows immediately from Theorems 14, 15, and 16.
“” It follows immediately from Theorems 14, 15, 16, 20, and 21.
5. The Attribute Reduction of Type-2 Fuzzy Information System
Let be an information system. Here, is the set of objects; that is, . is the attribute set; that is, . is the relation set of and ; that is, , , and is the domain of the attribute .
We call a decision information system, where is the classical information system, is the condition attribute set, and is the decision attribute set; that is, . is the relation set of and , , , and is the domain of the attribute .
is called a type-2 fuzzy information system, where is the classical information system, and is the type-2 fuzzy sets on .
Definition 27. Let be a type-2 fuzzy information system, andwhere is a crisp relation determined by attribute set . If for any and , then is called lower approximation consistent set of .
If is the lower approximation consistent set of and any subset of are not the lower approximation consistent sets of , then is called the lower approximation reduction of .
Definition 28. Let be a type-2 fuzzy information system; for any and ,is called discernibility matrix of .
Theorem 29. Let be the type-2 fuzzy information system. For any and , is the lower approximation consistent set of if and only if .
Proof. “” If , then .
In the following, we only should prove .
For any , . Similarly, we have .
Since is the lower approximation consistent set of , we can obtain Hence, and . That is, . That is to say, there must be which hold for .
“” Since forms a partition of , we haveIf , then . If , then .
Furthermore, we can obtain from .
Thus,Similarly, . That is, . Hence, is the lower approximation consistent set of .
Theorem 30. Let be the type-2 fuzzy information system. For any and , is the lower approximation consistent set of if and only if .
Proof.
“” If is the lower approximation consistent set of and , then we have . Thus, we can obtain from Theorem 29. That is, there exists such that . Hence, .
“” For any and , if holds, in the following, we only should prove .
If , then . That is, there exists such that . Thus, . That is to say, .
We have that is the lower approximation consistent set of from Theorem 29.
A discernibility function for is a Boolean function of Boolean variables corresponding to the attributes , respectively, and is defined as follows:where is the disjunction of all variables such that . In the sequel, we write instead of when no confusion can arise.
Theorem 31. Suppose ; then is a lower approximation reduction of if and only if is the minimal set satisfying .
Proof. The proof is straightforward by Theorem 30.
Let be the reduced disjunctive form of obtained from by applying the multiplication and absorption laws as many times as possible. Then, there exist and for such that , where every element in only appears one time. We have the following theorem.
Theorem 32. If is the collection of all lower approximation reductions of , then .
Proof. The proof is omitted since this theorem is similar to the one in [14].
The all lower approximation reductions of type-2 fuzzy information systems can be computed by discernibility matrix approach. In the following, we develop an algorithm to find all lower approximation reductions of type-2 fuzzy information system .
Step 1. Decide on how many -planes will be used. Call that number . Regardless of , and must always be used.
Step 2. For every , , and , compute and .
Step 3. For every , , compute .
Step 4. Compute using Definition 28.
Step 5. Define .
Step 6. Compute by .
Step 7. Output all lower approximation reductions .
Example 33. Table 1 gives a type-2 fuzzy information system, where the universe is , the condition attribute set is , and the decision attribute set is , where . denotes trapezoid function, the first parameter and fourth parameter of denote bottom left and right endpoint, respectively, and the second parameter and third parameter of denote top left and right endpoint, respectively. Furthermore, denotes triangular function, the first parameter and third parameter of denote bottom left and right endpoint, respectively, and the second parameter of denotes apex.
Obviously, the universe can be divided into three basic equivalence classes in accordance with the condition attribute set :In the following, we should decide on how many -planes will be used, where . Call that number . Regardless of , and must always be used. If , then . When , the approximation results of the type-2 fuzzy object are summarized in Table 2.
Similarly, we can compute the other approximation results for . The discernibility matrix of is presented in Table 3.
It is easy to compute and as the lower approximation reductions of .
6. Conclusions
The -plane representation method of type-2 fuzzy sets has been extensively studied and can greatly reduce the computational workload. Based on -plane theory, the rough type-2 fuzzy approximation operators and the generalized rough type-2 fuzzy approximation operators have been defined, and their properties have been derived. The connections between special binary relations and generalized rough type-2 fuzzy upper and lower approximation operators are examined. The axiomatics are provided to fully characterize the generalized rough type-2 fuzzy upper and lower approximation operators. The attribute reduction of the type-2 fuzzy information system has been investigated, and an attribute reduction algorithm to find all reductions of the type-2 fuzzy information system has also been developed. Further research will focus on the applications of the proposed rough type-2 fuzzy set model.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.