Transversal and Function Matroidal Structures of Covering-Based Rough Sets

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1 Transversal and Function Matroidal Structures of Covering-Based Rough Sets Shiping Wang 1, William Zhu 2,,andFanMin 2 1 School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, China 2 Lab of Granular Computing, Zhangzhou Normal University, Zhangzhou, China williamfengzhu@gmail.com Abstract. In many real world applications, information blocks form a covering of a universe. Covering-based rough set theory has been proposed to deal with this type of information. It is more general and complex than classical rough set theory, hence there is much need to develop sophisticated structures to characterize covering-based rough sets. Matroids are important tools for describing graphs and linear independence of matrix theory. This paper establishes two matroidal structures of covering-based rough sets. Firstly, the transversal matroidal structure of a family of subsets of a universe is constructed. We also prove that the family of subsets of a universe is a covering if and only if the constructed transversal matroid is a normal one. Secondly, the function matroidal structure is constructed through the upper approximation number. Moreover, the relationships between the two matroidal structures are studied. Specifically, when a covering of a universe is a partition, these two matroidal structures coincide with each other. Keywords: Covering, Rough set, Matroid, Transversal matroid, Normal matroid, Upper approximation number. 1 Introduction In practical applications, many data are organized by a covering, instead of a partition of a universe. Covering-based rough sets [1 4] were proposed to deal with this type of data. They have not only enriched classical rough sets, but also broadened the application to real world problems. They have been applied to knowledge reduction [5 8] and decision rule synthesis [9 12]. Covering-based rough set theory is more general and complex than rough set theory, hence sophisticated structures and theories are much needed to describe it. Currently, the connections between covering-based rough sets and other theories are attracting increasing attention. For example, covering-based rough sets have been connected with matroid theory [13 16], with topology [17 20], and with fuzzy sets [21 24]. Many other important contributions are presented in [25 30]. Matroid theory [31] borrows extensively from matrix theory and graph theory, and has gained fruitful achievements on both theory and application. In theory, a number of Corresponding author. J.T. Yao et al. (Eds): RSKT 2011, LNCS 6954, pp , c Springer-Verlag Berlin Heidelberg 2011

2 Transversal and Function Matroidal Structures of Covering-Based Rough Sets 147 definitions are used to represent matroids equivalently, which lays sound foundations for matroid theory [32 35]. In application, matroids have been widely used in combinatorial optimization, network flows [36, 37], and so on. Therefore, matroids are more likely to enrich covering-based rough sets. In this paper, two matroidal structures of covering-based rough sets are constructed. One matroidal structure is induced by a family of subsets of a universe through transversal matroid. Specifically, when a family of subsets of the universe is a covering, a normal matroidal structure is induced. This lays a solid foundation for studying covering-based rough sets with normal matroids. Moreover, some special properties of transversal matroid induced by a covering are also studied. We connect the transversal with the upper approximation of covering-based rough sets. In fact, if a subset of a universe is a transversal of a covering, then the upper approximation of the subset is equal to the universe. We also define the spanning set to describe the transversal when a covering coincides with a partition. A subset of a universe is a transversal induced by a partition if and only if it is a minimal spanning set. The other matroidal structure, namely, function matroidal structure, is induced by a covering through the upper approximation number. That provides a quantitative tool to study covering-based rough sets. We also present the expression of the function matroid when a covering is a partition. Furthermore, the relationships between the two matroids are studied. Specially, these two matroids coincide with each other when the covering is a partition. The rest of this paper is arranged as follows. Section 2 reviews some fundamentals of covering-based rough sets and matroids. Section 3 establishes two matroidal structures of covering-based rough sets, and studies the relationship between them. Finally, Section 4 concludes this paper. 2 Basic Definitions This section recalls some basic concepts of covering-based rough sets and matroids. In real world application, data are organized by coverings, instead of partitions. For example, some overlapping subsets of a universe are often used to describe an attribute in an information system. And these overlapping subsets form a covering of the universe. When these subsets describing the attribute are disjoint, then the covering is degenerated to a partition. Definition 1. (Covering [2]) Let U be a finite universe of discourse and C a family of subsets of U. If none of subsets in C is empty and C = U,thenC is called a covering of U. The covering is a fundamental concept in covering-based rough sets, and it can characterize the practical problems with extensive coverage. In covering-based rough sets, a pair of approximation operators are used to approximate an object. We introduce a pair of widely used approximation operators in the following definition.

3 148 S. Wang and W. Zhu Definition 2. (Approximation [2]) Let C beacoveringofu. For all X U, X = {K C K X}, X = {K C K X }, are called the lower, upper approximations of X, respectively. Matroids are important structures that cope with discrete data. In application, they have been widely used in combinatorial optimization, network flows, greedy algorithm design, and so on. In theory, there are a number of equivalent ways to define a matroid, and there is no preferred or customary definition; in this respect, the matroid differs from many other mathematical structures, such as group and topology. Definition 3. (Matroid [31]) A matroid is a pair M =(E,I) where E is a finite set and I (independent sets) a collection of subsets of E with the following properties: (1) I; (2) If I I, and I I, theni I; (3) If I 1,I 2 I, and I 1 < I 2, then there exists e I 2 I 1 such that I 1 {e} I, where I denotes the cardinality of I. The concept of the independent set borrows from the concept of the linear independence set of vectors in linear algebra. In Definition 3, (1) ensures that the family of independent sets contains at least one element. (2) ensures that any subset of an independent set is also an independent set. (3) ensures that any independent set can be extended to a maximal independent set. In a vector set, any nonzero vector is linearly independent. Correspondingly, the normal matroid is proposed to characterize a matroid where any set having only one element is an independent set. Definition 4. (Normal matroid [31]) Let M = (E,I) be a matroid. M is called a normal matroid if I = E. 3 Matroidal Structures of Covering-Based Rough Sets Matroids largely borrow from matrix theory and graph theory. They represent the linear independence with high abstraction and serve as a useful tool for dealing with discrete data. In this section, two different approaches are used to establish matroidal structures of covering-based rough sets. One approach is the transversal theory, and the other one is the upper approximation number. 3.1 Matroidal Structure by Transversal Theory Transversal theory is a branch of matroid theory, and it reflects the relationships between collections of subsets of a nonempty set and their matroidal structures. It shows how to induce a matroid, namely, transversal matroid, by a family of subsets of a set. Therefore, the transversal matroid establishes a bridge between collections of subsets of a set and matroids. In this subsection, we study the special properties of the transversal matroid induced by a covering.

4 Transversal and Function Matroidal Structures of Covering-Based Rough Sets 149 Definition 5. (Transversal [31]) Let F = F(J) ={F j : j J} be a family of subsets of U. A transversal of F is a set T U for which there exists a bijection π : T J with t F π(t) for all t T. A partial transversal of F is a transversal of its subfamily. In fact, a transversal of a family of subsets of a universe is a subset, each element of which can come from different elements in the family. In order to illustrate the transversal, we give an example. Example 1. Let U = {a, b, c, d, e, f, g}, F = F J = {F 1,F 2,F 3 } and its index set J = {1, 2, 3},whereF 1 = {a, b, g}, F 2 = {c, d, g} and F 3 = {e, f, g}.thent = {a, c, e} is a transversal of F since there exists a bijection π : T J with t F π(t) for all t T,whereπ(a) =1, π(c) =2and π(e) =3. Similarly, suppose J = {2, 3} J = {1, 2, 3}.ThenT J = {d, e} is a transversal of F J, hence it is also a partial transversal of F. Through the partial transversal of a family of subsets of a universe, we establish the matroidal structure of the family. In fact, all partial transversals of a family of subsets of a universe construct the independent sets of a matroid. Proposition 1. [31] Let F = F(J) ={F j j J} be a family of subsets of U. Then M T (F) =(U, I T (F)) is a matroid where I T (F) is the family of all partial transversals of F. The above proposition shows that a collection of subsets of a universe generates a matroid. In our convenience, we say the matroid is the transversal matroid induced by the collection. Definition 6. Let F = F(J) = {F j j J} be a family of subsets of U. Wesay M T (F) =(U, I T (F)) is the transversal matroid induced by F. A covering of a universe is a special family of subsets of the universe. Therefore, Proposition 1 also shows that any covering induces a transversal matroid. Specifically, the special properties of the transversal matroid induced by a covering are studied in the following proposition. Firstly, a transversal of a covering is studied from the upper approximations of the covering. In fact, if a subset of a universe is a transversal, then its upper approximation is equal to the universe. Proposition 2. Let C beacoveringofu. For any T U, ift is a transversal of C, then T = U. Proof. According to Definition 5, it is straightforward. In the above proposition, we establish the relationship between the transversal of a covering and its upper approximation. The following proposition connects coverings with normal matroids. In fact, the transversal matroid induced by a family of subsets of a universe is normal if and only if the family is a covering. Proposition 3. Let F be a family of subsets of U, and M T (F) = (U, I T (F)) the transversal matroid induced by F. ThenF is a covering of U iff M T (F) is a normal matroid.

5 150 S. Wang and W. Zhu Proof. ( =): If M T is a normal matroid, then I T (F) =U. So{x} I T (F); this is, {x} is a partial transversal of F for all x U. Hence there exists F x F such that x F x. Therefore, U = x U {x} x U F x F U. This proves that U = F;thatis,F is a covering of U. (= ): If F is a covering of U, then there exists F x C such that x F x for all x U. Then {x} is a partial transversal of F; inotherwords,{x} I T (F) for all x U. Hence U = x U {x} I T (F) U. This proves that I T (F) =U; inother words, M T is a normal matroid. 3.2 Matroidal Structure by the Upper Approximation Number Many measurements [10, 38 40] have been proposed to enrich classical rough set theory and broaden its application. Hence it is much necessary to propose some measurements to quantify covering-based rough sets. In this subsection, the upper approximation number which serves as a quantitative tool is defined to induce a matroidal structure. Definition 7. [14, 16] Let C be a covering of U. For all X U, f C (X) = {K C K X }, is called the upper approximation number of X with respect to C. When there is no confusion, we omit the subscript C. The upper approximation number provides a tool to quantify covering-based rough sets. In order to illustrate it, an example and some properties are provided. Example 2. Let U = {a, b, c, d}, K 1 = {a, b}, K 2 = {a, c}, K 3 = {c, d}, andc = {K 1,K 2,K 3 }.LetX = {a}, Y = {d},andz = {a, c}.thenf(x) = {K 1,K 2 } = 2, f(y )= {K 3 } =1,andf(Z) = {K 1,K 2,K 3 } =3. Proposition 4. Let C beacoveringofu. The following properties hold: (1) f( ) =0; (2) For all X Y U, f(x) f(y ); (3) For all X, Y U, f(x Y )+f(x Y ) f(x)+f(y ). Proof. (1) and (2) are straightforward. (3): We need to prove f(x Y )+f(x Y ) f(x)+f(y ), i.e., {K C K (X Y ) } + {K C K (X Y ) } {K C K X } + {K C K Y }. In the following work, we use a counting method to prove it. For all S {K C K (X Y ) } = {K C (K X) (K Y ) }, then S X or S Y. SoS {K C K X } or S {K C K Y }. Similarly, for all S {K C K (X Y ) } = {K C (K X) (K Y ) }, S X and S Y, sos {K C K X } and S {K C K Y }.Thisprovesf(X Y )+f(x Y ) f(x)+f(y ). It is worth noting that (3) of Proposition 4 is called submodular property in matroid theory. And it plays an important role in the establishment of the matroidal structure of covering-based rough sets.

6 Transversal and Function Matroidal Structures of Covering-Based Rough Sets 151 Proposition 5. Let C be a covering of U. ThenM f (C) =(U, I f (C)) is a matroid where I f (C) ={I U for all I I,f(I ) I }. Proof. I f (C) since f( ) =0=. If A I f (C) and B A,thenf(A ) A for all A A. Obviously, f(b ) B for all B B A. SoB I f (C). For all A, B I f (C) and A < B, ifa {c} / I(C) for all c B A, then A f(a) f(a {c}) < A {c} A +1 f(a) +1. Hence f(a) = f(a {c}) = A. Forallc, d B A and c d, f(a {c, d}) + f(a) f(a {c}) +f(a {d}) =2f(A). Thisistosay,f(A {c, d}) f(a). Conversely, f(a) f(a {c, d}). Thus f(a {c, d}) = f(a). More generally, f(b) =f(a ( c B A {c})) = f(a). Because B I f (C), A = f(a) =f(b) B, which is contradictory with A < B. Thus there exists c B A such that A {c} I f (C). The above proposition shows that any covering induces a matroid through the upper approximation number. We say that the matroid is the function matroid induced the covering. Definition 8. Let C beacoveringofu. Then we say M f (C) = (U, I f (C)) is the function matroid induced by C. The function matroid is induced by a covering through the upper approximation number. In order to illustrate it, we provide an example. Example 3. Let U = {a, b, c, d}, andc = {{a, b, c}, {c, d}}. Then the function matroid induced by C is M f (C) =(U, I f (C)),whereI f (C) ={, {a}, {b}, {c}, {d}, {a, c}, {a, d}, {b, c}, {b, d}, {c, d}}. These results lay solid foundations for studying covering-based rough sets in the matroidal structure. 3.3 The Relationship between Proposed Two Matroidal Structures This subsection establishes the relationships between the two matroidal structures of covering-based rough sets. Specifically, the special properties of two matroidal structures are studied when the covering is a partition. Proposition 6. Let C = C J = {C j j J} be a covering of U. LetM T (C) = (U, I T (C)) be the transversal matroid, and M f (C) =(U, I f (C)) the function matroid induced by C.ThenI T (C) I f (C). Proof. For all T I T (C), according to Definition 5, there exist I J and a bijection π : T I such that t C π(t) for all t T.ForallT T, suppose I = {π(t) t T },andπ (t) =π(t) for all t T.Thenπ : T I is a bijection, and I I J. Hence f(t ) = {K C K T } I = T. This proves T I f (C).

7 152 S. Wang and W. Zhu The above proposition shows that the transversal matroid of a covering is contained in the function matroid. The following work is to explore the relationship between them when a covering coincides with a partition. For this purpose, two lemmas are presented. Lemma 1. Let P = {P j j J} be a partition of U, and M f (P) =(U, I T (P)) the transversal matroid induced by P. ThenI T (P) ={X U f P (X) = X }. Proof. For all X I T (P), according to Proposition 6, {x} I f (P), thenf P (X) X. Because P is a partition of U, f P (X) X. Hence f P (X) = X. Thisproves I T (P) {X U f P (X) = X }. Conversely, for all Y {X U f P (X) = X }, there exists P y P, such that y P y for all y Y, because f P (Y )= Y. SinceP is a partition of U,thenP y1 and P y2 are disjoint sets for all y 1,y 2 Y and y 1 y 2.Then there exists a bijection π : Y I, such that y P π(y) for all y Y,whereI J. Therefore, Y I T (P). This proves {X U f P (X) = X } I T (P). In fact, Lemma 1 presents the expression of the transversal matroid induced by a partition. This lays a sound foundation for studying classical rough sets by matroidal structures. From the viewpoint of the upper approximation, we can also present the expression of the transversal matroid induced by a partition. For this purpose, the minimal set of a family of subsets of a set is introduced. Definition 9. (Minimal set [31, 32]) Let F be a family of subsets of U.Then Min(F) ={F F : F F and F F imply F = F }, is called the minimal set of F. In the following definition, we define the spanning sets of a covering to characterize the transversal matroid from the standpoint of the upper approximation. Definition 10. (Spanning set) Let C be a covering of U. For any S U, ifs = U, then we call S a spanning set of C. And the family of all spanning sets of C is denoted by S(C), i.e.s(c) ={S U S = U}. Based on the spanning set, we establish a close connection between the transversal matroid induced by a partition and its upper approximations. In fact, the independent sets of the transversal matroid coincide with the minimal spanning sets. Proposition 7. Let P be a partition of U, and M f (P) =(U, I T (P)) the transversal matroid induced by P.ThenI T (P) =min(s(p)). Proof. For all T I T (P), according to Proposition 2, T = U,thenT S(P). Since P is a partition of U,thenT min(s(p)). Thus I T (P) min(s(p)). Conversely, for all S min(s(p)), thens S(P). Thus S = U = P P P.Since P is a partition of U, then for all P P, there exists s P S such that s P P.We denote T S = {s P P P}, thent S is a transversal of P and TS = U. Therefore, T S = S; inotherwords,s I T (P). This proves that min(s(p)) I T (P). Tosum up, this completes the proof.

8 Transversal and Function Matroidal Structures of Covering-Based Rough Sets 153 The above proposition shows that the transversal matroid induced by a partition can be represented by the upper approximations. That provides a new angle for us to study the relationships between transversal matroids and classical rough sets. Moreover, the expression of the function matroid induced by a partition is obtained in the following lemma. Lemma 2. Let P be a partition of U, and M f (C) =(U, I f (P)) the function matroid induced by P.ThenI f (P) ={X U f P (X) = X }. Proof. It is straightforward that I f (P) {X U f P (X) = X }. Conversely, for all Y {X U f P (X) = X }, f(y )= Y. Because P is a partition of U, forall y Y, there exists a unique P y P such that y P y. Therefore, for all Z Y, f(z) = {z Z z P z } = Z. This proves Y I f (P). Inotherwords,{X U f P (X) = X } I f (P). This completes the proof. Lemma 2 shows that the expression of the function matroid induced by a partition with weaker conditions is obtained. Based on Lemma 1, and Lemma 2, we establish the connection between the transversal matroid and the function matroid induced by a partition. Proposition 8. Let P be a partition of U, M T (P) =(U, I T (P)) the transversal matroid and M f (P) =(U, I f (P)) the function matroid induced by P. ThenI T (P) = I f (P); inotherwordsm T (P) =M f (P). Proof. According to Lemma 1 and Lemma 2, it is straightforward. Proposition 8 shows that the transversal matroid and the function matroid induced by a partition coincide with each other. 4 Conclusions Covering-based rough sets provide a systematic way to cope with overlapping data set. Matroids are important tools for dealing with the discrete data. In this paper, matroids are used to study covering-based rough sets. Specially, we establish two matroidal structures of covering-based rough sets. These results lay sound foundations for studying covering-based rough sets with matroidal approaches. Specifically, the relationships between these two matroidal structures are studied when a covering is a partition. In fact, these two matroidal structures induced by a partition coincide with each other. Moreover, the expressions of these two matroids induced by a partition are presented. That reflects the inherent relations between classical rough sets and matroids. Hopefully, this work can narrow the gap between covering-based rough set theory and its applications. Acknowledgments. This work is in part supported by National Nature Science Foundation of China under grant No /F

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