Covering-Based Generalized Rough Fuzzy Sets

Covering-Based Generalized Rough Fuzzy Sets Tao Feng1,2 , Jusheng Mi1, , and Weizhi Wu3 1 College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang, Hebei, 050016, P.R. China [email protected] (J.S. Mi) 2 College of Science, Hebei University of Science and Technology, Shijiazhuang, Hebei 050018 P.R. China Fengtao [email protected] 3 Information College, Zhejiang Ocean University, Zhoushan, Zhejiang 316004, P.R. China [email protected]

Abstract. This paper presents a general framework for the study of covering-based rough fuzzy sets in which a fuzzy set can be approximated by some elements in a covering of the universe of discourse. Some basic properties of the covering-based lower and upper approximation operators are examined. The concept of reduction of a covering is also introduced. By employing the discrimination matric of the covering, we provide an approach to find the reduct of a covering of the universe. It is proved that the reduct of a covering is the minimal covering that generates the same covering-based fuzzy lower (or upper) approximation operator, so this concept is also a technique to get rid of redundancy in data mining. Furthermore, it is shown that the covering-based fuzzy lower and upper approximations determine each other. Keywords: Rough fuzzy sets, reduction, covering, covering-based lower and upper approximations.

1

Introduction

Rough set theory [1,2], proposed by Pawlak in 1982, is an extension of set theory for the study of intelligent systems characterized by insufficient and incomplete information. Using the concepts of lower and upper approximations in rough set theory, the knowledge hidden in the system may be discovered and expressed in the form of decision rules. A partition or an equivalent relation plays an important role in Pawlak’s original rough set model. However, the requirement of an equivalence relation seems to be a very restrictive condition that may limit the applications of rough set theory. To address this issue, several interesting and meaningful extensions of equivalence relation have been proposed in the literature such as tolerance relations [3,4], similarity relations [4,5], neighborhood systems [6] and others [7]. Particularly, Zakowski [8] has used coverings of a universe for establishing

Corresponding author.

G. Wang et al. (Eds.): RSKT 2006, LNAI 4062, pp. 208–215, 2006. c Springer-Verlag Berlin Heidelberg 2006

Covering-Based Generalized Rough Fuzzy Sets

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the covering generalized rough set theory, and an extensive body of research works has been developed [9,10]. The covering generalized rough set theory is a model with promising potential for applications to data mining. Meanwhile, generalizations of rough set to fuzzy environment have also been discussed in a number of studies [11,12,13,14,15,16]. For example, by using an equivalence relation on the universe, Dubois and Prade introduced the lower and upper approximations of fuzzy sets in a Pawlak approximation space to obtain an extended notion called rough fuzzy set [11]. Alternatively, a fuzzy similarity relation can be used to replace an equivalence relation. The result is a deviation of rough set theory called fuzzy rough set [11,12]. Based on arbitrary fuzzy relations, fuzzy partitions on the universe, and Boolean subalgebras of the power set of the universe, extended notions called rough fuzzy sets and fuzzy rough sets have been obtained [13,14,15,16]. Alternatively, a rough fuzzy set is the approximation of a fuzzy set in a crisp approximation space. The rough fuzzy set model may be used to deal with knowledge acquisition in information systems with fuzzy decisions [17]. And a fuzzy rough set is the approximation of a crisp set or a fuzzy set in a fuzzy approximation space. The fuzzy rough set model may be used to unravel knowledge hidden in fuzzy decision systems. This paper extends Pawlak’s rough sets on the basis of a covering of the universe. In the next section, we review basic properties of rough approximation operators and give some basic notions of fuzzy sets. In Section 3, the model of covering-based generalized rough fuzzy sets is proposed. In the proposed model, fuzzy sets are approximated by some elements in a covering of the universe. The concepts of minimal descriptions and the covering boundary approximation set family are also introduced. Some basic properties of the covering-based fuzzy approximation operators are examined. In Section 4, we study the reduction of a covering of the universe. By employing the discrimination matric of a covering, we present an approach to find a reduct of the covering. This technique can be used to reduce the redundant information in data mining. It is proved that the reduct of a covering is the minimal covering that generates the same coveringbased fuzzy lower or upper approximation operator. We then conclude the paper with a summary in Section 5.

2

Preliminaries

Let U be a finite and nonempty set called the universe of discourse. The class of all subsets (fuzzy subsets, respectively) of U will be denoted by P(U ) (F (U ), respectively). For any A ∈ F(U ), the α−level and the strong α−level of A will be denoted by Aα and Aα+ respectively, that is, Aα = {x ∈ U : A(x) ≥ α} and Aα+ = {x ∈ U : A(x) > α}, where α ∈ I = [0, 1], the unit interval. Let R be an equivalence relation on U . Then R generates a partition U/R = {[x]R : x ∈ U } on U , where [x]R denotes the equivalence class determined by x with respect to (wrt.) R, i.e., [x]R = {y ∈ U : (x, y) ∈ R}. For any subset X ∈ P(U ), we can describe X in terms of the elements of U/R. In rough set theory, Pawlak introduced the following two sets:

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R(X) = {x ∈ U : [x]R ⊆ X}; R(X) = {x ∈ U : [x]R ∩ X = ∅}. R(X) and R(X) are called the lower and upper approximations of X respectively. The following Theorem [2,18] summarizes the basic properties of the lower and upper approximation operators R and R. Theorem 1. Let R be an equivalence relation on U , then the lower and upper approximation operators, R and R, satisfy the following properties: for any X, Y ∈ P(U ), (1) (2) (3) (4) (5) (6) (7) (8)

R(U ) = U = R(U ); R(∅) = ∅ = R(∅); R(X) ⊆ X ⊆ R(X); R(X ∩ Y ) = R(X) ∩ R(Y ), R(X ∪ Y ) = R(X) ∪ R(Y ); R(R(X)) = R(R(X)) = R(X), R(R(X)) = R(R(X)) = R(X); R(∼ X) =∼ R(X), R(∼ X) =∼ R(X); X ⊆ Y =⇒ R(X) ⊆ R(Y ), R(X) ⊆ R(Y ); ∀K ∈ U/R, R(K) = K, R(K) = K.

Where ∼ X is the complement of X in U . For the relationship between crisp sets and fuzzy sets, it is well-known that the representation theorem holds [16]. Definition 1. A set-valued mapping H : I → P(U ) is said to be nested if for all α, β ∈ I, α ≤ β =⇒ H(β) ⊆ H(α). The class of all P(U )-valued nested mapping on I will be denoted by N (U ). Theorem 2. Let H ∈ N (U ). Define a function f : N (U )→ F (U ) by: A(x) := f (H)(x) = ∨α∈I (α ∧ H(α)(x)), x ∈ U, where H(α)(x) is the characteristic function of H(α). Then f is a surjective homomorphism, and the following properties hold: (1) (2) (3) (4)

3

Aα+ ⊆ H(α) ⊆ Aα ; Aα = ∩λα H(λ); A = ∨α∈I (α ∧ Aα+ ) = ∨α∈I (α ∧ Aα ).

Concepts and Properties of Covering-Based Generalized Approximations

In [8,9,10], the authors introduced the concept of covering-based approximations. Any subset of a universal set U can be approximated by the elements of a covering of U . A covering C ⊆ P(U ) of U is a family of subsets of U , in which


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none of them is empty and ∪C = U . The ordered pair U, C is then called a covering-based approximation space. Let U, C be a covering-based approximation space, x ∈ U . The set family md(x) = { K ∈ C : x ∈ K ∧ (∀S ∈ C ∧ x ∈ S ∧ S ⊆ K =⇒ K = S)} is called the minimal description of x. In what follows, the universe of discourse U is considered to be finite. C ⊆ P(U ) is always a covering of U . We now study the approximations of a fuzzy set A ∈ F(U ) with respect to a covering C of U . Definition 2. For a fuzzy set A ∈ F(U ), the set family C∗ (A) = {αK : K ∈ C , K ⊆ A0+ , α = ∧{A(x) : x ∈ K}} is called the covering-based fuzzy lower approximation set family of A. Define A∗ (x) = ∨αK∈C∗ (A) αK(x), ∀x ∈ U , we call A∗ the covering-based fuzzy lower approximation of A. The set family Bn(A) = {αK : There exists x ∈ U, K ∈ md(x), A(x) − A∗ (x) > 0, α = A(x)} is called the covering-based boundary approximation set family of A. The set family C ∗ (A) = {αK : αK ∈ C∗ (A)} ∪ {αK : αK ∈ Bn(A)} is called the covering-based fuzzy upper approximation set family of A. Denote A∗ (x) = ∨αK∈C ∗ (A) αK(x), ∀x ∈ U , then A∗ is called the coveringbased upper approximation of A. If C ∗ (A) = C∗ (A), then A is said to be definable, otherwise it is rough. The following properties can be proved by the definitions: Proposition 1. The covering-based fuzzy approximation set family operators C∗ and C ∗ satisfy the following properties: ∀ X, Y ∈ F(U ), (1) (2) (3) (4)

C∗ (∅) = C ∗ (∅) = ∅; C∗ (U ) = C ∗ (U ) = C, ∗ C∗ (X) ⊆ C (X); C∗ (X∗ ) = C∗ (X) = C ∗ (X∗ ); X ⊆ Y ⇒ C∗ (X) ⊆ C∗ (Y ).

Proposition 2. If C is a partition of the universal set U , then for all X ∈ F(U ), X∗ is the lower approximation of X defined by Dubois and Prade in [11]. Proposition 3. For all X ∈ F(U ), C ∗ (X) = C∗ (X) if and only if there are some elements of C, say K1 , K2 , ..., Kn , such that X(x) = ∨ni=1 αi Ki (x), αi = ∧{X(x) : x ∈ Ki }. Proposition 4. ∀X ∈ F(U ), X∗ = X if and only if C ∗ (X) = C∗ (X).

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Proposition 5. ∀X ∈ F(U ), X∗ = X ∗ if and only if C ∗ (X) = C∗ (X). Corresponding to the properties of Pawlak’s approximation operators listed in Section 2, we have the following results. Proposition 6. For a covering C of U , the covering-based lower and upper approximation operators have the following properties: (1L) (2L) (3L) (4L) (5L) (6L)

4

U∗ = U ; (1H) (2H) ∅∗ = ∅; X∗ ⊆ X; (3H) (X∗ )∗ = X∗ ; (4H) X ⊆ Y =⇒ X∗ ⊆ Y∗ ; ∀K ∈ C, K∗ = K; (6H)

U∗ = U; ∅∗ = ∅; X ⊆ X ∗; (X ∗ )∗ = X ∗ ; ∀K ∈ C, K ∗ = K.

Reduction of Coverings

After dropping any of the members of a partition, the remainder is no longer a partition, thus, there is no redundancy problem for a partition. As for a covering, it could still be a covering by dropping some of its members. Furthermore, the resulting new covering might still produce the same covering-based lower and/or upper approximation. Hence, a covering may have redundant members and a procedure is needed to find its smallest covering that induces the same covering lower and upper approximations. Definition 3. Let C be a covering of a universe U and K ∈ C. If K is a union of some elements in C − {K}, we say that K is a reducible element of C, otherwise K is an irreducible element of C. If every element of C is irreducible, then C is called irreducible; otherwise C is reducible. Let C be a covering of a universe U . If K is a reducible element of C, then it is easy to see that C − {K} is still a covering of U . Proposition 7. Let C be a covering of U , K ∈ C be a reducible element of C, and K1 ∈ C − {K}. Then K1 is a reducible element of C if and only if it is a reducible element of C − {K}. Proposition 7 guarantees that, after deleting reducible elements in a covering, the remainder will not change the reducible property of the every element in C. Now we propose an approach to deleting reducible elements of a covering by employing its discrimination matric. Definition 4. (Discrimination function) Let C = {A1 , · · · , An } be a covering of U , Ai , Aj ∈ C, define 1, Aj ⊂ Ai f (Ai , Aj ) = 0, otherwise Then the binary function f (·, ·) is called discrimination function of C.


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Let |Ai | be the cardinality of Ai . Arranging the sequence A1 , A2 , . . . , An by the cardinality of Ai satisfying |A1 | ≤ |A2 | ≤ · · · ≤ |An |, then we have the following discrimination matric: A1 A1 0 A2 f (A2 , A1 ) .. .. . .

A2

· · · An

0 .. .

An f (An , A1 ) f (An , A2 ) · · · 0 For any i, if there exists j such that f (Ai , Aj ) = 0, and | {Aj : f (Ai , Aj ) = 0}| = |Ai |, then Ai is a reducible element, thus we can delete Ai . Example 1. Let U = {1, 2, 3, 4, 5}, C = {A1 , A2 , A3 , A4 , A5 }, where A1 = {1}, A2 = {3, 4, 5}, A3 = {2, 3, 4}, A4 = {1, 3, 4, 5}, A5 = {2, 3, 4, 5}, then we have ∪Ai = U , so C is a covering of U . Because the cardinalities of Ai s satisfy |A1 | ≤ |A2 | ≤ |A3 | ≤ |A4 | ≤ |A5 |. We have the following discrimination matric:

A1 A2 A3 A4 A5

A1 0 0 0 1 0

A2 A3 A4 A5 0 0 0 1 0 0 1 1 0 0

For A4 , f (A4 , A1 ) = 0, f (A4 , A2 ) = 0, and |A1 ∪ A2 | = 4 = |A4 |. For A5 , f (A5 , A2 ) = 0, f (A5 , A3 ) = 0, and |A2 ∪ A3 | = 4 = |A5 |. Therefore, A4 , A5 are reducible elements, we can delete them from the covering. Definition 5. For a covering C of a universe U , an irreducible covering is called the reduct of C, and denoted by REDU CT (C). Proposition 7 guarantees that a covering has only one reduct. We can obtain the reduct of a covering through the above discrimination matric method. Proposition 8. Let C be a covering of U , and K a reducible element of C, then C − {K} and C have the same md(x) for all x ∈ U . Particularly C and REDU CT (C) have the same md(x) for all x ∈ U . Proposition 9. Suppose C is a covering of U , K is a reducible element of C, X ∈ F(U ), then the covering-based fuzzy lower approximation of X generated by the covering C and the covering C − {K}, respectively, are same. Proof. Suppose the covering lower approximations of X generated by the covering C and the covering C − {K} are X1 , X2 respectively. From the definition of covering lower approximation, we have that X2 (x) ≤ X1 (x) ≤ X(x), for all

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x ∈ U . On the other hand, from Proposition 6 and Corollary 1, there exists K1 , K2 , . . . , Kn ∈ C, such that X1 (x) = ∨ni=1 αi Ki (x), αi = ∧{X(x) : x ∈ Ki }. If none of K1 , K2 , . . . , Kn ∈ C is equal to K, then they all belong to C − {K}, and the corresponding αi are the same. Thus, X2 (x) = ∨ni=1 αi Ki (x), αi = ∧{X(x) : x ∈ Ki }. If there is an element of {K1 , K2 , . . . Kn } that is equal to K, say K1 = K. Because K is a reducible element of C, K can be expressed as the union of some elements T1 , T2 , . . . , Tm ∈ C −{K}, that is, T1 ∪T2 ∪. . .∪Tm = K1 . Thus n m n X1 (x) = ∨m j=1 α1 Tj (x) ∨ ∨i=2 αi Ki (x) ≤ ∨j=1 βj Tj (x) ∨ ∨i=2 αi Ki (x)

where βj = ∧{X(x) : x ∈ Tj }, T1 , T2 , . . . , Tm , K2 , . . . , Kn ∈ C − {K}. So X1 (x) ≤ X2 (x), thus X1 = X2 . Proposition 10. Suppose C is a covering of U , K is a reducible element of C, and X ∈ F(U ), then the covering-based fuzzy upper approximations of X generated by the covering C and the covering C − {K}, respectively, are same. Proof. It follows from Definition 2 and Proposition 8. Combining Corollaries 5 and 6, we have the following conclusion. Theorem 3. Let C be a covering of U , then C and REDU CT (C) generate the same covering-based fuzzy lower and upper approximations. Proposition 11. If two irreducible coverings of U generate the same coveringbased fuzzy lower approximations for all X ∈ F(U ), then the two coverings are same. Proof. It can be induced directly from Proposition 12 in [10]. From Theorem 3 and Propositions 8 and 11, we have: Theorem 4. Let C1 , C2 be two coverings of U , C1 , C2 generate the same coveringbased fuzzy lower approximations if and only if they generate the same coveringbased fuzzy upper approximations. Theorem 4 shows that the covering lower approximation and the covering upper approximation determine each other.

5

Conclusion

We have developed in this paper a general framework for the study of the covering-based generalized rough fuzzy set model. In our proposed model, fuzzy sets can be approximated by a covering of the universe. The properties of the covering-based fuzzy approximation operators have been studied in detail. We have also presented an approach to obtaining the reduct of a covering by employing the discrimination matric. We have shown that the reduct of a covering is the minimal covering that generates the same covering-based fuzzy lower and upper approximations, and furthermore, the covering-based fuzzy lower and upper approximations determine each other. Another issue should be studied in the future is how to approximate a fuzzy set on the basis of a fuzzy covering of the universe of discourse.


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Acknowledgements This work was supported by Science Foundation of Hebei Normal University (L2005Z01) and Natural Science Foundation of Hebei Province (A2006000129).

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