Abstract: Multi-Granulation Rough Set (MGRS) is a new and interesting topic in rough set theory. In this study, three kinds of Covering Multi-Granulation Fuzzy Rough Sets (CMGFRS) have been first proposed, which extend the covering rough sets from single-granulation to multi-granulation. Firstly, the basic properties of these covering multi-granulation fuzzy rough sets are discussed, it is shown that basic properties of MGRS are special cases of those of CMGFRS. Then, the conditions for two CMGFRS generate identical lower and upper approximations of a set in a covering approximation space are studied. Finally, this study explored the relationship between the three types of CMGFRS.
INTRODUCTION
Covering rough set is an interesting and meaningful extension direction of Pawlak’s rough sets. Since, Zakowski (1983) first proposed the concept of covering-based rough sets, extensive research works have been developed. Based on the mutual correspondence of the concepts of extension and intension, Bonikowski et al. (1998) have formulated the necessary and sufficient conditions for the existence of operations on rough sets, which are analogous to classical operations on sets. Pomykala (1987) studied the covering rough set model from the point view of topology. Mordeson (2001) studied algebraic structural properties of certain subsets for a type of covering-based rough sets. Tsang et al. (2004) put forward with a new type of upper approximations based on covering. Zhu and Wang (2003, 2007) and Zhu (2007a,b, 2009a, b) did systematic research on six types of covering-based rough sets. Yao and Yao (2012) proposed a framework for the study of covering based rough set approximations.
When a problem involves incomplete, uncertain, or vague information, it may be difficult to differentiate distinct elements and one is forced to consider granules. In the view of granular computing (Yao, 2005), an equivalence relation on a universe can be regarded as a granulation and a partition on the universe as a granulation space. Based on these studies, Qian and Liang (2006) and Qian et al. (2010) introduced the multi-granulation rough set, where the approximations are defined using multi-equivalence relations which should be used because of user requirements or problem specification. When two attribute sets in information systems possesses a contradiction or inconsistent relationship, MGRS will display its advantages for knowledge discovery. We find that the binary relations in the MGRS are definitely induced by partitions. By weakening the requirement of equivalence relations, we can have more general granulation and approximation structures based on coverings of the universe. To make the MGRS suitable for larger scopes, this paper proposed three kinds of Covering Multi-Granulation Fuzzy Rough Sets (CMGFRS), that is, the relations in there are induced by coverings.
PRELIMINARIES
In this section, we review some basic notions and notations related in the sequel of our work.
In the following, T stands for FC or SC or TC if there is no extra explanation.
Let U be a universe of discourse and C a family of nonempty subsets of U. If ∪C = U, C is called a covering of U. There is no doubt that a partition of U is certainly a covering of U.
Definition 1 Bonikowski et al. (1998): The pair (U, C) is called a covering approximation space, where U is a universe of discourse and C a covering of U. For any x∈U, the set md(x) is called the minimal description of x, where:
Definition 2 Zhu (2009a): Let (U, C) be a covering approximation space, for any X⊆U, the covering lower and upper approximations of X can be defined as following:
Definition 3: Let (U, C) be a covering approximation space, for any X⊆U, the covering lower and upper approximations of X can be defined as following:
Definition 4: Let (U, C) be a covering approximation space, for any X⊆U, the covering lower and upper approximations of X can also be defined as following:
Theorem 1 Zhu and Wang (2003): Let C be a covering of U and K∈C. If K is a union of some sets in C-{K}, we say K is a reducible element of C. If all the reducible elements have been discarded, then the remainder of C is called the reduction of C on U, denoted by reduct(C). For any x∈U, C and reduct(C) have the same md(x).
Definition 5 Qian and Liang (2006): Let K = (U, C) is a knowledge base, C a family of equivalence relations on U, for any X⊆U, P, Q∈C, the lower and upper approximations of X can be defined by the following:
where, C X is the complement of X in U.
COVERING MULTI-GRANULATION FUZZY ROUGH SETS AND THEIR PROPERTIES
Here, we study three types of covering multi-granulation fuzzy rough sets. Firstly, we shall discuss the basic properties of CMGFRS, and then we present the similarities and differences between CMGFRS and MGRS. Secondly, for the same type CMGFRS we explore the conditions under which two coverings generate identical CMGFRS lower and upper approximations.
Definition 6 Liu and Wang (2011): Let (U, C) be a covering approximation space, C a family of coverings on U and C1, C2∈C. For any X∈F(U), the lower and upper approximations of X in F(U) with respect to C1, C2 can be defined as follows: for any x∈U,
If
Definition 7: Let (U, C) be a covering approximation space, C a family of coverings on U and C1, C2∈C. For any X∈F(U), the lower and upper approximations of X in F(U) with respect to C1, C2 can be defined as follows: for any x∈U,
If
Definition 8: Let (U, C) be a covering approximation space, C a family of coverings on U and C1, C2∈C. For any X∈F(U), the lower and upper approximations of X in F(U) with respect to C1, C2 can be defined as follows: for any X∈U,
If
Proposition 1: Let (U, C) be a covering approximation space, C1, C2∈C. For any X∈F(U), the following properties hold:
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Proof: It is straightforward according the definitions, we will omit them here.
Remark 1: In general, the following two properties do not hold:
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Proposition 2: Let K = (U, C) be acovering approximation space and C1, C2∈C. Then, for and X, Y∈F(U), the following properties hold:
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| • | If X⊆Y, then |
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Definition 9: Let (U, C) be a covering approximation space, C1, C2 ∈ C. Reduct (C1) = {C11, C12, …, C1p}, reduct (C2) = {C21, C22, …, C2q}. For any x ∈ U, x ∈ C1i (1≤i≤p) and x ∈ C2j (1≤j≤q) there is C1i ⊆ C2j holds that is reduct (C1) ⊆ reduct (C2), then we say that C1 is finer than C2, denoted by C1 ◁ C2.
Theorem 2: Let (U, C) be a covering approximation space, C1, C2 ∈ C. Then, for any X ∈ F (U), the following properties hold.
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Theorem 3: Let (U, C) be a covering approximation space, C1, C2, ∈ C and reduct (C1), reduct (C2) are the reductions of C1, C2, respectively, then we say that T1+T2 and Tredcut (C1)+Tredcut (C2) generate identical CMGFRS lower and upper approximation operators.
Definition 10 (Zhu and Wang, 2003): Let C be a covering of U and K an element of C. If there exists another element K′ of C such that K⊂K, we say that K is an immured element of covering C.
Definition 11 (Zhu and Wang, 2003): Let C be a covering of U. When we remove all immured elements from C, the set of all remaining elements is still a covering of U and this new covering has no immured element. We call this new covering an exclusion of C, denoted by exclusion (C).
Theorem 4: Let (U, C) be a covering approximation space, C1, C2, C3, C4 ∈ C and reduct (C2), reduct (C1), reduct (C3), reduct (C4) are the reductions of C1, C2, C3, C4, respectively, if at least one of the following is satisfied:
| • | Reduct (C1) = reduct (C3) and exclusion (C1) = exclusion (C3) |
| • | Reduct (C1) = reduct (C4) and exclusion = (C1) = exclusion (C4) |
| • | Reduct (C2) = reduct (C3) and exclusion (C2) = exclusion (C3) |
| • | Reduct (C2) = reduct (C4) and exclusion (C2) = exclusion (C4) |
then T1+T2 and T3+T4 generate identical covering multi-granulation fuzzy rough lower approximation operator.
Remark 2: The reverse of theorem 4 does not hold, i.e., T1+T2 and T3+T4 may generate identical covering multi-granulation fuzzy rough lower approximation operator but their reductions and exclusions may not the same.
Theorem 5: Let (U, C) be a covering approximation space, C1, C2, C3, C4 and reduct (C2), reduct (C1), reduct (C3), reduct (C4), are the reductions of C1, C2, C3, C4, respectively if at least one of the following is satisfied:
| • | Reduct (C1) = reduct = (C3) and reduct (C2) = reduct (C4) |
| • | Reduct (C1) = reduct (C4) and reduct (C2) = reduct (C3) then T1+T2 and T3+T4 generate the same first type of covering multi-granulation rough lower and upper approximation operators |
Remark 3: The reverse of theorem 5 does not hold, i.e., T1+T2 and T3+T4 may generate identical covering multi-granulation fuzzy rough upper approximation operator, but their reductions are not the same.
RELATIONSHIP AMONG THREE TYPES OF CMGFRS
We have proposed three types of covering multi-granulation fuzzy rough sets in last section. In this section, we will study the interrelations among them.
According to Definition 6, 7 and 8, one can easily get the following theorem.
Theorem 6: Let (U, C) be a covering approximation space, C1, C2∈C. For any X∈F(U), the following formulas hold:
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Definition 12 (Zhu, 2007a): Let C be a covering of a set U. C is called unary if for any x∈U, |md(x)| = 1, where |.| is cardinality of a set.
Theorem 7: Let (U, C) be a covering approximation space, C1, C2∈C. For any X⊆U, if C1 and C2 are both unary, then three kinds of lower (upper) approximation of CMGFRS are identical.
Remark 4: The reverse of theorem 7 does not hold.
CONCLUSION
This study has proposed three types of covering multi-granulation fuzzy rough sets. To compare with MGRS, it is shown that the properties of MGRS are special cases of those of CMGFRS. Then, it studied the sufficient conditions for C1, C2 and C3, C4 to generate the identical covering multi-granulation lower or upper approximations. Finally, the relationships among the three types of covering multi-granulation rough sets have been explored. Of course, there are several issues are worthy of further investigation. For example, topological properties of CMGFRS may be a potential topic for future research.
ACKNOWLEDGMENTS
The research is supported by the National Natural Science Foundation of China under grant Nos: 60970061, 61075056, China National Natural Science Foundation of Youth Science Foundation under Grant No.: 61103067, China Postdoctoral Science Foundation under Grant Nos.: 2011M500626, 2011M500815.