A Granular Reduction Algorithm Based on Covering Rough Setsdownloads.hindawi.com/journals/jam/2012/970576.pdf · algorithm for covering granular reduction. The quintuple U, C,CL,CH

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2012, Article ID 970576, 13 pagesdoi:10.1155/2012/970576

Research ArticleA Granular Reduction Algorithm Based onCovering Rough Sets

Tian Yang,1 Zhaowen Li,2 and Xiaoqing Yang3

1 College of Science, Central South University of Forestry and Technology, Changsha 410004, China2 College of Mathematics and Computer Science, Guangxi University for Nationalities,Nanning 530006, China

3 School of Economics and Management, Changsha University of Science and Technology,Changsha 410004, China

Correspondence should be addressed to Zhaowen Li, math [email protected]

Received 31 March 2012; Revised 12 July 2012; Accepted 16 July 2012

Academic Editor: Chong Lin

Copyright q 2012 Tian Yang et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

The granular reduction is to delete dispensable elements from a covering. It is an efficient methodto reduce granular structures and get rid of the redundant information from information systems.In this paper, we develop an algorithm based on discernabilitymatrixes to compute all the granularreducts of covering rough sets. Moreover, a discernibility matrix is simplified to the minimalformat. In addition, a heuristic algorithm is proposed as well such that a granular reduct isgenerated rapidly.

1. Introduction

With the development of technology, the gross of information increases in a surprising way.It is a great challenge to extract valuable knowledge from the massive information. Rough settheory was raised by Pawlak [1, 2] to deal with uncertainty and vagueness, and it has beenapplied to the information processing in various areas [3–8].

One of the most important topics in rough set theory is to design reduction algorithms.The reduction of Pawlak’s rough sets is to reduce dispensable elements from a family ofequivalence relations which induce the equivalence classes, or a partition.

Covering generalized rough set [9–19] and binary relation generalized rough set [20–26] are two main extensions of Pawlak’s rough set. The reduction theory of covering roughsets [10, 11, 15, 23, 27, 28] plays an important role in practice. A partition is no longer apartition if any of its elements is deleted, while a covering may still be a covering withinvariant set approximations after dropping some elements. Therefore, there are two types

2 Journal of Applied Mathematics

of reduction on covering rough sets. One is to reduce redundant coverings from a family ofcoverings, referred to as the attribute reduction. The other is to reduce redundant elementsfrom a covering, noted as the granular reduction. It is to find the minimal subsets of a coveringwhich generate the same set approximations with the original covering. Employed to reducegranular structures and databases as well as interactive with the attribute reduction, we thinkthe granular reduction should be ignored by nomeans. In this paper, we devote to investigategranular reduction of covering rough sets.

In order to compute all attribute reducts for Pawlak’s rough sets, discernibilitymatrix is initially presented [29]. Tsang et al. [15] develop an algorithm of discerni-bility matrices to compute attribute reducts for one type of covering rough sets.Zhu and Wang [17] and Zhu [18] build one type of granular reduction for two coveringrough set models initially. In addition, Yang et al. systematically examine the granularreduction in [30] and the relationship between reducts and topology in [31]. Unfortunately,no effective algorithm for granular reduction has hitherto been proposed.

In this paper, we bridge the gap by constructing an algorithm based on discernibilitymatrixes which is applicable to all granular reducts of covering rough sets. This algorithmcan reduce granular structures and get rid of the redundant information from informationsystems. Then a discernibility matrix is simplified to the minimal format. Meanwhile, basedon a simplification of discernibility matrix, a heuristic algorithm is proposed as well.

The remainder of this paper proceeds as follows. Section 2 reviews the relevantbackground knowledge about the granular reduction. Section 3 constructs the algorithmbased on discernibility matrix. Section 4 simplifies the discernibility matrix and proposes aheuristic algorithm. Section 5 concludes the study.

2. Background

Our aim in this section is to give a glimpse of rough set theory.Let U be a finite and nonempty set, and let R be an equivalence relation on U. R

generates a partition U/R = {[x]R | x ∈ X} on U, where [x]R is an equivalence class of xgenerated by the equivalence relation R. We call it elementary sets of R in rough set theory.For any set X, we describe X by the elementary sets of R, and the two sets

R∗ = ∪{[x]R | [x]R ⊆ X}, R∗ = ∪{[x]R | [x]R ∩X /=∅} (2.1)

are called the lower and upper approximations of X, respectively. If R∗(X) = R∗(X), X is anR-exact set. Otherwise, it is an R-rough set.

Let R be a family of equivalence relations, and let A ∈ R, denoted as IND(R) =∩{R : R ∈ R}. A is dispensable in R if and only if IND(R) = IND(R − A). Otherwise, Ais indispensable in R. The family R is independent if every A ∈ R is indispensable in R.Otherwise, R is dependent. Q ∈ P is a reduct of P if Q is independent and IND(Q) = IND(P).The sets of all indispensable relations in P are called the core of P, denoted as CORE(P).Evidently, CORE(P) = ∩RED(P), where RED(P) is the family of all reducts of P. Thediscernibility matrix method is proposed to compute all reducts of information systems andrelative reducts of decision systems [29].

C is called a covering of U, where U is a nonempty domain of discourse, and C is afamily of nonempty subsets of U and ∪C = U.

Journal of Applied Mathematics 3

It is clear that a partition ofU is certainly a covering ofU, so the concept of a coveringis an extension of the concept of a partition.

Definition 2.1 (minimal description [9]). Let C be a covering of U,

MdC(x) = {K ∈ C | x ∈ K ∧ (∀S ∈ C ∧ x ∈ S ∧ S ⊆ K ⇒ K = S)} (2.2)

is called the minimal description of x. When there is no confusion, we omit the C from thesubscript.

Definition 2.2 (neighborhood [9, 19]). Let C be a covering of U, and NC(x) = ∩{C ∈ C |x ∈ C} is called the neighborhood of x. Generally, we omit the subscript C when there is noconfusion.

Minimal description and neighborhood are regarded as related information granulesto describe x, which are used as approximation elements in rough sets (as shown inDefinition 2.3). It shows that N(x) = ∩{C ∈ C | x ∈ C} = ∩Md(x). The neighborhood ofx can be seen as the minimum description of x, and it is the most precise description (moredetails are referred to [9]).

Definition 2.3 (covering lower and upper approximation operations [19]). Let C be a coveringof U. The operations CLC : P(U) → P(U) and CL′C : P(U) → P(U) are defined as follows:for all X ∈ P(U),

CLC(X) = ∪{K ∈ C | K ⊆ X} = ∪{K | ∃x, s.t. (K ∈ Md(x)) ∧ (K ⊆ X)},CL′C(X) = {x | N(x) ⊆ X} = ∪{N(x) | N(x) ⊆ X}.

(2.3)

We call CLC the first, the second, the third, or the fourth covering lower approximationoperations and CL′C the fifth, the sixth, or the seventh covering lower approximationoperations, with respect to the covering C.

The operations FH, SH, TH, RH, IH, XH, and VH : P(U) → P(U) are defined asfollows: for all X ∈ P(U),

FHC(X) = CL(X) ∪ (∪{Md(x) | x ∈ X − CL(X)}),SHC(X) = ∪{K | K ∈ C, K ∩X /=∅},

THC(X) = ∪{Md(x) | x ∈ X},RHC(X) = CL(X) ∪ (∪{K | K ∩ (X − CL(X))/=∅}),

IHC(X) = CL(X) ∪ (∪{N(x) | x ∈ X − CL(X)} = ∪{N(x) | x ∈ X}),XHC(X) = {x | N(x) ∩X /=∅},

VHC(X) = ∪{N(x) | N(x) ∩X /=∅}.

(2.4)


FHC, SHC, THC, RHC, IHC, XHC, and VHC are called the first, the second, the third, thefourth, the fifth, the sixth, and the seventh covering upper approximation operations withrespect to C, respectively. We leave out C at the subscript when there is no confusion.

As shown in [32], every approximation operation in Definition 2.3 may be appliedin certain circumstance. We choose the suitable approximation operation according to thespecific situation. So it is important to design the granular reduction algorithms for all ofthese models.

More precise approximation spaces are proposed in [30]. As a further result, areasonable granular reduction of coverings is also introduced. Let MC = ∪{Md(x) | x ∈ U},NC = {N(x) | x ∈ U}. 〈U,MC〉 is the approximation space of the first and the third types ofcovering rough sets, 〈U,C〉 is the approximation space of the second and the fourth types ofcovering rough sets, and 〈U,NC〉 is the approximation space of the fifth, the sixth, and theseventh types of covering rough sets (referred to [30] for the details). In this paper, we designthe algorithm of granular reduction for the fifth, the sixth, and the seventh type of coveringrough sets.

Let C be a covering ofU, denoting a covering approximation space.MC denotes anM-approximation space. NC represents an N-approximation space. We omit C at the subscriptwhen there is no confusion (referred to [30] for the details).

3. Discernibility Matrixes Based on Covering Granular Reduction

In the original Pawlak’s rough sets, a family of equivalence classes induced by equivalencerelations is a partition. Once any of its elements are deleted, a partition is no longer a partition.The granular reduction refers to the method of reducing granular structures and to get rid ofredundant information in databases. Therefore, granular reduction is not applicable to theoriginal Pawlak’s rough sets. However, as one of the most extensions of Pawlak’s rough sets,a covering is still working even subject to the omission of its elements, as long as the setapproximations are invariant. The purpose of covering granular reduction is to find minimalsubsets keeping the same set approximations. It is meaningful and necessary to develop thealgorithm for covering granular reduction.

The quintuple (U,C, CL,CH) is called a covering rough set system (CRSS), where C isa covering ofU, CL and CH are the lower and upper approximation operations with respectto the covering C, and 〈U,AC〉 is the approximation space. According to the categories ofcovering approximation operations in [30], there are two kinds of situations as follows.

(1) If AC = C or AC = MC, then AC ⊆ C: thus; AC is the unique granular reduct of C.There is no need to develop an algorithm to compute granular reducts for the first,the second, the third, and the fourth type of the covering rough sets.

(2) IfAC = NC, generally,AC is not a subset of C. Consequently, an algorithm is neededto compute all granular reducts of C for the fifth, the sixth, and the seventh type ofcovering rough set models.

Next we examine the algorithm of granular reduction for the fifth, the sixth, and theseventh type of covering rough sets. Let C be a covering of U, since NC = {N(x) | x ∈ U},and NC is the collection of all approximation elements of the fifth, the sixth, or the seventhtype of lower/upper approximation operations. NC is called the N-approximation spaceof C. Given a pair of approximation operations, the set approximations of any X ⊆ U are


determined by the N-approximation spaces. Thus, for the fifth, the sixth, and the seventhtype of covering rough set models, the purpose of granular reduction is to find the minimalsubsets C′ of C such that NC = NC′ . The granular reducts based on the N-approximationspaces are called theN-reducts.Nred(C) is the set of allN-reducts of C, andNI(C) is the setof all N-irreducible elements of C (referred to [30] for the details).

In Pawlak’s rough set theory, for every pair of x, y ∈ U, if y belongs to the equivalenceclass containing x, we say that x and y are indiscernible. Otherwise, they are discernible. LetR = {R1, R2, . . . , Rn} be a family of equivalence relation onU, Ri ∈ R. Ri is indispensable in R

if and only if there is a pair of x, y ∈ U such that the relation between x and y is altered afterdeleting Ri from R. The attribute reduction of Pawlak’s rough sets is to find minimal subsetsof R which keep the relations invariant for any x, y ∈ U. Based on this statement, the methodof discernibility matrix to compute all reducts of Pawlak’s rough sets was proposed in [29]. Incovering rough sets, however, the discernibility relation between x, y ∈ U is different fromthat in Pawlak’s rough sets.

Let C be a covering on U, (x, y) ∈ U × U. Then we call (x, y) indiscernible ify ∈ N(x), that is, N(y) ⊆ N(x). Otherwise, (x, y) is discernible. When C is a partition,the new discernibility relation coincides with that in Pawlak’s. It is an extension of Pawlak’sdiscernibility relation. In Pawlak’s rough sets, (x, y) is indiscernible if and only if (y, x) isindiscernible. However, for a general covering, if N(y) ⊆ N(x) and N(y)/=N(x), that is,y ∈ N(x) and x /∈ N(y), (y, x) is discernible while (x, y) is indiscernible. Thereafter, we callthese relations the relations of (x, y) with respect to C. The following theorem characterizesthese relations.

Proposition 3.1. Let C = {Ci | i = 1, 2, 3, . . . , n} be a covering onU, and let Cx = {Ci ∈ C | x ∈ Ci}.(1) y ∈ N(x) if and only if Cx ⊆ Cy.

(2) y /∈ N(x) if and only if there is Ci ∈ C such that x ∈ Ci and y /∈ Ci.

Proof. (1) y ∈ N(x) = ∩Cx ⇔ for any Ci ∈ Cx, y ∈ Ci ⇔ for any Ci ∈ Cx, Ci ∈ Cy ⇔ Cx ⊆ Cy.(2) It is evident from (1).

Theorem 3.2. Let C be a covering on U, Ci ∈ C. Then NC /=NC−{Ci} if and only if there is (x, y) ∈U ×U whose discernibility relation with respect to C is changed after deleting Ci from C.

Proof. Suppose that NC /=NC−{Ci}, then there is at least one element x ∈ U such thatNC(x)/=NC−{Ci}(x), that is, NC(x) ⊂ NC−{Ci}(x). Since NC−{Ci}(x) − NC(x)/=∅, suppose thaty ∈ NC−{Ci}(x) − NC(x), then y ∈ NC−{Ci}(x) and y /∈ NC(x). Namely, (x, y) is discerniblewith respect to C, while (x, y) is indiscernible with respect to C − {Ci}.

Suppose that there is (x, y) ∈ U ×U whose discernibility relation with respect to C ischanged after deleting Ci from C. Put differently, (x, y) is discernible with respect to C, while(x, y) is indiscernible with respect to C − {Ci}. Then we have y ∈ NC−{Ci}(x) and y /∈ NC(x),so y ∈ NC−{Ci}(x) −NC(x). Thus, NC(x)/=NC−{Ci}(x). It implies NC /=NC−{Ci}.

The purpose of granular reducts of a covering C is to find the minimal subsets ofC which keep the same classification ability as C or, put differently, keep NC invariant.In Theorem 3.2, NC is kept unchanged to make the discernibility relations of any (x, y) ∈U × U invariant. Based on this statement, we are able to compute granular reducts withdiscernibility matrix.


Definition 3.3. LetU = {x1, x2, . . . , xn}, C be a covering onU.M(U,C) is an n×nmatrix (cij)n×ncalled a discernibility matrix of (U,C), where

(1) cij = ∅, xj ∈ N(xi),

(2) cij = {C ∈ C | xi ∈ C, xj /∈ C}, xj /∈ N(xi).

This definition of discernibility matrix is more concise than the one in [11, 15]due to the reasonable statement of the discernibility relations. Likewise, we restate thecharacterizations of N-reduction.

Proposition 3.4. Consider thatNI(C) = {C | cij = {C} for some cij ∈ M(U,C)}.

Proof. For anyC ∈ NI(C),NC /=NC−{C}, then there is (xi, xj) ∈ U×U such that xj ∈ NC−{C}(xi)and xj /∈ NC(xi). It implies that xi ∈ C and xj /∈ C. Moreover, for any C′ ∈ C − {C}, sincexj ∈ NC−{C}(xi), we have xi ∈ C′ if xi ∈ C′. Thus, cij = {C}.

If cij = {C} for some cij ∈ M(U,C), then xi ∈ C and xj /∈ C. And for any C′ ∈ C − {C},if xi ∈ C′, then xi ∈ C′, that is, xj ∈ NC−{C}(xi) and xj /∈ NC(xi), then NC−{C}(xi)/=NC(xi).Namely, NC /=NC−{C}, which implies C ∈ NI(C).

Proposition 3.5. Suppose that C′ ⊆ C, thenNC = NC′ if and only if C′ ∩ cij /=∅ for every cij /=∅.

Proof. NC = NC′

⇔ for any (xi, xj) ∈ U ×U, xj /∈ NC(xi) if and only if xj /∈ NC′(xi),

⇔ for any (xi, xj) ∈ U ×U, there is C ∈ C such that xi ∈ C and xj /∈ C if and only ifthere is C′ ∈ C′ such that xi ∈ C′ and xj /∈ C′,

⇔ for any cij /=∅, C′ /=∅.

Proposition 3.6. Suppose that C′ ⊆ C, then C′ ∈ Nred(C) if and only if C′ is a minimal set satisfyingC′ ∩ cij /=∅ for every cij /=∅.

Definition 3.7. Let U = {x1, x2, . . . , xn}, let C = {C1, C2, . . . , Cm} be a covering of U, and letM(U,C) = (cij)n×n be the discernibility matrix of (U,C). A discernibility function f(U,C)is a Boolean function of m Boolean variables, C1, C2, . . . , Cm, corresponding to the coveringelements C1, C2, . . . , Cm, respectively, defined as f(U,C)(C1, C2, . . . , Cm) = ∧{∨(cij) | cij ∈M(U,C), cij /=∅}.

Theorem 3.8. Let C be a family of covering on U, let f(U,C) be the discernibility function, andlet g(U,C) be the reduced disjunctive form of f(U,C) by applying the multiplication and absorptionlaws. If g(U,C) = (∧C1) ∨ · · · ∨ (∧Cl), where Ck ⊆ C, k = 1, 2, . . . , l and every element in Ck onlyappears once, thenNred(C) = {C1,C2, . . . ,Cl}.

Proof. For every k = 1, 2, . . . , l, ∧Ck ≤ ∨cij for any cij ∈ M(U,C), so Ck ∩ cij /=∅. Let C′k = Ck −

{C} for anyC ∈ Ck, then g(U,C) � ∨k−1t=1 (∧Ct)∨(∧C′

k)∨(∨lt=k+1(∧Ct)). If for every cij ∈ M(U,C),

we have C′k ∩ cij /=∅, then ∧C′

k ≤ ∨cij for every cij ∈ M(U,C), that is, g(U,C) ≥ ∨k−1t=1 (∧Ct) ∨

(∧C′k)∨(∨l

t=k+1(∧Ct)), which is a contradiction. It implies that there is ci0j0 ∈ M(U,C) such thatC′k∩ ci0j0 = ∅. Thus, Ck is a reduct of C.


Table 1

Objects C1 C2 C3 C4 C5 C6 C7

x1 ∗ ∗ ∗ ∗ ∗x2 ∗x3 ∗ ∗ ∗x4 ∗ ∗ ∗ ∗ ∗ ∗x5 ∗ ∗ ∗x6 ∗ ∗

For any C′ ∈ Red(C), we have C′ ∩ cij /=∅ for every cij ∈ M(U,C), so f(U,C) ∧ (∧C′) =∧(∨cij) ∧ (∧C′) = ∧C′, which implies ∧C′ ≤ f(U,C) = g(U,C). Suppose that, for every k =1, 2, . . . , l, we have Ck − C′ /=∅, then for every k, there is Ck ∈ Ck − C′. By rewriting g(U,C) =(∨l

k=1Ck) ∧Φ, ∧C′ ≤ ∨lk=1Ck. Thus, there is Ck0 such that ∧C′ ≤ Ck0 , that is, Ck0 ∈ C′, which is a

contradiction. So Ck0 ⊆ C′ for some k0, since both C′ and Ck0 are reducts, and it is evident thatC′ = Ck0 . Consequently, Red(C) = {C1,C2, . . . ,Cl}.

Algorithm 3.9. Consider the following:input: 〈U,C〉,output: Nred(C) and NI(C)// The set of all granular reducts and the set of all N-

irreducible elements.

Step 1: M(U,C)=(cij)n×n, for each cij , let cij = ∅.

Step 2: for each xi ∈ U, compute N(xi) = ∩{C ∈ C | xi ∈ C}.

If xj /∈ N(xi), cij = {C ∈ C | xi ∈ C, xj /∈ C}.

Step 3: f(U,C)(C1, C2, . . . , Cm) = ∧{∨(cij) | cij ∈ M(U,C), cij /=∅}.

Step 4: compute f(U,C) to g(U,C) = (∧C1) ∨ · · · ∨ (∧Cl)// where Ck ⊆ C,

k = 1, 2, . . . , l, and every element in Ck only appears once.

Step 4: output Nred(C) = {C1,C2, . . . ,Cl}, NI(C) = ∩Nred(C).

Step 5: end.

The following example is used to illustrate our idea.

Example 3.10. Suppose that U = {x1, x2, . . . , x6}, where xi, i = 1, 2, . . . , 6 denote six objects,and let Ci, i = 1, 2, . . . , 7 denote seven properties; the information is presented in Table 1, thatis, the ith object possesses the jth attribute is indicated by a ∗ in the ij-position of the table.

{x1, x2, x3} is the set of all objects possessing the attribute C1, and it is denoted byC1 = {x1, x2, x3}. Similarly, C2 = {x1, x4}, C3 = {x3, x4, x5}, C4 = {x1, x4, x5}, C5 = {x1, x4, x6},C6 = {x3, x4}, and C7 = {x1, x4, x5, x6}. Evidently, C = {C1, C2, C3, C4, C5, C6, C7} is a coveringon U.

Then, N(x1) = {x1}, N(x2) = {x1, x2, x3}, N(x3) = {x3}, N(x4) = {x4}, N(x5) ={x4, x5}, and N(x6) = {x4, x6}.


Table 2

Objects C1 C3 C4 C5

x1 ∗ ∗ ∗x2 ∗x3 ∗ ∗x4 ∗ ∗ ∗x5 ∗ ∗x6 ∗

Table 3

Objects C1 C3 C5 C7

x1 ∗ ∗ ∗x2 ∗x3 ∗ ∗x4 ∗ ∗ ∗x5 ∗ ∗x6 ∗ ∗

The discernibility matrix of (U,C) is exhibited as follows:

⎛⎜⎜⎜⎜⎜⎜⎜⎝

∅ {C2, C4, C5, C7} {C2, C4, C5, C7} {C1} {C1, C2, C5} {C1, C2, C4}∅ ∅ ∅ {C1} {C1} {C1}

{C3, C6} {C3, C6} ∅ {C1} {C1, C6} {C1, C3, C6}{C3, C6} {C2, C3, C4, C5, C6, C7} {C2, C4, C5, C7} ∅ {C2, C5, C6} {C2, C3, C4, C6}{C3} {C3, C4, C7} {C4, C7} ∅ ∅ {C3, C4}

∅ {C5, C7} {C5, C7} ∅ {C5} ∅

⎞⎟⎟⎟⎟⎟⎟⎟⎠

,

(3.1)

f(U,Δ)(C1, C2, C3, C4, C5, C6, C7

)

= ∧{∨(cij) | i, j = 1, 2, . . . , 6, cij /=∅

}

= (C2 ∨ C4 ∨ C5 ∨ C7) ∧ (C2 ∨ C4 ∨ C5 ∨ C7) ∧ C1 ∧ (C1 ∨ C2 ∨ C5)

∧ (C1 ∨ C2 ∨ C4) ∧ C1 ∧ C1 ∧ C1 ∧ (C3 ∨ C6) ∧ (C3 ∨ C6) ∧ (C1) ∧ (C1 ∨ C6)

∧ (C1 ∨ C3 ∨ C6) ∧ (C3 ∨ C6) ∧ (C2 ∨ C3 ∨ C4 ∨ C5 ∨ C6 ∨ C7) ∧ (C2 ∨ C4 ∨ C5 ∨ C7)

∧ (C2 ∨ C5 ∨ C6) ∧ (C2 ∨ C3 ∨ C4 ∨ C6) ∧ C3 ∧ (C3 ∨ C4 ∨ C7) ∧ (C4 ∨ C7)

∧ (C3 ∨ C4) ∧ (C5 ∨ C7) ∧ (C5 ∨ C7) ∧ C5

= (C5 ∧ C1 ∧ C3 ∧ C4) ∨ (C5 ∧ C1 ∧ C3 ∧ C7).(3.2)

So Nred(C) = {{C1, C3, C4, C5}, {C1, C3, C5, C7}}, NI(C) = {C1, C3, C5}. As a result,Table 1 can be simplified into Table 2 or Table 3, and the ability of classification is invariant.Obviously, the granular reduction algorithm can reduce data sets as shown.


4. The Simplification of Discernibility Matrixes

For the purpose of finding the set of all granular reducts, we have proposed the methodby discernibility matrix. Unfortunately, it is at least an NP problem, since the discernibilitymatrix in this paper is more complex than the one in [33]. Accordingly, we simplify thediscernibility matrixes in this section. In addition, a heuristic algorithm is presented to avoidthe NP hard problem.

Definition 4.1. Let M(U,C) = (cij)n×n be the discernibility matrix of (U,C). For any cij ∈M(U,C), if there is an nonempty element ci0j0 ∈ M(U,C) − {cij} such that ci0j0 ⊆ cij , letc′ij = ∅; otherwise, c′ij = cij , then we get a new discernibility matrix SIM(U,C) = (c′ij)n×n,which called the simplification discernibility matrix of (U,C).

Theorem 4.2. LetM(U,C) be the discernibility matrix of (U,C), and SIM(U,C) is the simplificationdiscernibility matrix, C′ ⊆ C. Then C′ ∩ cij /=∅ for any nonempty element cij ∈ M(U,C) if and onlyif C′ ∩ c′ij /=∅ for any nonempty element c′ij ∈ SIM(U,C).

Proof. If C′ ∩ cij /=∅ for every cij /=∅ and cij ∈ M(U,C), it is evident that C′ ∩ c′ij /=∅ for everyc′ij /=∅ and c′ij ∈ SIM(U,C).

Suppose that C′ ∩ c′ij /=∅ for every c′ij /=∅ and c′ij ∈ SIM(U,C). For any nonempty cij ∈M(U,C), if there is an nonempty element ci0j0 ∈ M(U,C) − {cij} such that ci0j0 ⊆ cij , andfor any nonempty element ci1j1 ∈ M(U,C) − {cij , ci0j0}, ci1j1/⊆ci0j0, then c′i0j0 = ci0j0 /=∅. SinceC′ ∩ c′i0j0 /=∅, then C′ ∩ ci0j0 /=∅; thus, C′ ∩ cij /=∅. If ci0j0/⊆cij for any nonempty element ci0j0 ∈M(U,C) − {cij}, then c′ij = cij . Since C′ ∩ c′ij /=∅, then C′ ∩ cij /=∅. Thus, C′ ∩ cij /=∅ for everynonempty cij ∈ M(U,C).

Proposition 4.3. Suppose that C′ ⊆ C, then C′ ∈ Nred(C) if and only if C′ is a minimal set satisfyingC′ ∩ c′ij /=∅ for every c′ij /=∅ and c′ij ∈ SIM(U,C).

Proposition 4.4. Consider that ∪{c′ij | c′ij ∈ SIM(U,C)} = ∪Nred(C).

Proof. Suppose that C ∈ ∪{c′ij | c′ij ∈ SIM(U,C)}, then there is c′ij ∈ SIM(U,C) such that C ∈ c′ijand c′ij ∩NI(C) = ∅. For any c′ij ∈ SIM(U,C), if C ∈ c′ij , let c

1ij = {C}. Otherwise, c1ij = {Cij},

where Cij ∈ c′ij . Suppose that M1(U,C) = (c1ij)n×n; it is easy to prove that C ∈ ∪{c1ij | c1ij ∈M1(U,C)} ∈ Nred(C). Thus, C ∈ ∪Nred(C).

Suppose that C ∈ ∪Nred(C), then there is Ck ∈ Nred(C) such that C ∈ Ck. FromProposition 4.3, we know that Ck is a minimal set satisfying Ck ∩ c′ij /=∅ for every c′ij /=∅ andc′ij ∈ SIM(U,C). So there is a c′ij ∈ SIM(U,C) such that C ∈ c′ij , or else C is redundant in Ck.Thus, C ∈ ∪{c′ij | c′ij ∈ SIM(U,C)}.

In summary, ∪{c′ij | c′ij ∈ SIM(U,C)} = ∪Nred(C).

Proposition 4.5. Let SIM(U,C) = (c′ij)n×n be the simplified discernibility matrix of (U,C), thenSIM(U,C) is the minimal matrix to compute all granular reducts of C, that is, for any matrixM0(U,C) = (dij)n×n where dij ⊆ c′ij , M0(U,C) can compute all granular reducts of C if and only ifdij = c′ij for 1 ≤ i, j ≤ n.

Proof. If dij = c′ij for 1 ≤ i, j ≤ n, then M0(U,C) = SIM(U,C), and M0(U,C) can compute allgranular reducts of C.


Suppose that there is a nonempty c′i0j0 ∈ SIM(U,C) such that di0j0 ⊂ c′i0j0. If |c′i0j0| =1, suppose that c′i0j0 = {C0}, then di0j0 = ∅. From the definition of the simplificationdiscernibility matrix, we know that C0 /∈ c′ij for any c′ij ∈ SIM(U,C) − {c′i0j0}, then C0 /∈ dij

for any dij ∈ M0(U,C). So M0(U,C) cannot compute any granular reducts of C. If |c′i0j0| ≥ 2,we suppose that di0j0 /=∅. Then there is a C ∈ (c′i0j0 − di0j0), and let c1i0j0 = {C}. For anyc′ij ∈ SIM(U,C)−{c′i0j0}, ifC ∈ c′ij , let c

1ij = ∅. Otherwise, let c1ij = {Cij}whereCij ∈ c′ij−c′i0j0. Let

M1(U,C) = (c1ij)n×n and C′ = ∪{c1ij | c1ij ∈ M1(U,C)}, and it is easy to prove that C′ ∈ Nred(C).However, C′ ∩ di0j0 = ∅, that is, M0(U,C) cannot compute all granular reducts of C. Thus, ifM0(U,C) can compute all granular reducts of C, then dij = c′ij for 1 ≤ i, j ≤ n.

From the above propositions, we know that the simplified discernibility matrix isthe minimal discernibility matrix which can compute the same reducts as the original one.Hereafter, we only examine simplified discernibility matrixes instead of general discernibilitymatrixes. The following example is used to illustrate our idea.

Example 4.6. The discernibility matrix of (U,C) in Example 3.10 is as follows:

⎛⎜⎜⎜⎜⎜⎜⎜⎝

∅ {C5} ∅ {C1} ∅ ∅

∅ ∅ ∅ ∅ ∅ ∅

{C3} ∅ ∅ ∅ ∅ ∅

∅ ∅ ∅ ∅ ∅ ∅

∅ ∅ {C4, C7} ∅ ∅ ∅

∅ ∅ ∅ ∅ ∅ ∅

⎞⎟⎟⎟⎟⎟⎟⎟⎠

,

f(U,Δ)(C1, C2, C3, C4, C5, C6, C7

)= ∧

{∨(c′ij

)| i, j = 1, 2, . . . , 6, cij /=∅

}

= C5 ∧ C1 ∧ C3 ∧ (C4 ∨ C7)

= (C5 ∧ C1 ∧ C3 ∧ C4) ∨ (C5 ∧ C1 ∧ C3 ∧ C7).

(4.1)

SoNred(C) = {{C1, C3, C4, C5}, {C1, C3, C5, C7}}, NI(C) = {C1, C3, C5}.

From the above example, it is easy to see that simplified discernibility matrix cansimplify the computing processes remarkably. Especially when C is a consistent coveringproposed in [30], that is, Nred(C) = {NI(C)}, the unique reduct Nred(C) = {∪{c′ij | c′ij ∈SIM(U,C)}}.

Unfortunately, although the simplified discernibility matrixes are more simple, theprocesses of computing reducts by discernibility function are still NP hard. Accordingly, wedevelop a heuristic algorithm to obtain a reduct from a discernibility matrix directly.

LetM(U,C) = (cij)n×n be a discernibilitymatrix.We denote the number of the elementsin cij by |cij |. For anyC ∈ C, ||C|| denotes the number of cij which containC. Let cij ∈ M(U,C),if for any C ∈ NI(C), C /∈ cij , then c′ij = cij . Since ∪{c′ij | |c′ij | ≥ 2} = ∪Nred(C) − NI(C), if|c′ij | ≥ 2, then the elements in c′ij may either be deleted from C or be preserved. Supposethat C0 ∈ ∪{c′ij | |c′ij | ≥ 2}, if ||C0|| ≥ ||C|| for any C ∈ ∪{c′ij | |c′ij | ≥ 2}, C0 is called themaximal element with respect to the simplified discernibility matrix SIM(U,C). The heuristicalgorithm to get a reduct from a discernibility matrix directly proceeds as follows.


Algorithm 4.7. Consider the following:input: 〈U,C〉,output: granular reducts red

Step 1: M(U,C)=(cij)n×n, for each cij , let cij = ∅.

Step 2: for each xi ∈ U, compute N(xi) = ∩{C ∈ Cxi ∈ C}.If xj /∈ N(xi),

cij = {C ∈ C | xi ∈ C, xj /∈ C}//get the discernibility matrix.

Step 3: for each cij ∈ M(U,C),if there is a nonempty element ci0j0 ∈ M(U,C) − {cij} such that

ci0j0 ⊆ cij , let cij = ∅ // get the simplified discernibility matrix.

Step 4: for each Ci ∈ ∪M(U,C), compute ||Ci|| and select the maximal

element C0 of SIM(U,C).For each cij ∈ M(U,C),if C0 ∈ cij ,

let cij = {C0}.Step 5: if there is cij ∈ M(U,C) such that |cij | � 2,

return to Step 3;

else

output red = ∪M(U,C).Step 5: end.

Example 4.8. The simplified discernibility matrix of (U,C) in Example 3.10 is as follows:

⎛⎜⎜⎜⎜⎜⎜⎜⎝

∅ {C5} ∅ {C1} ∅ ∅

∅ ∅ ∅ ∅ ∅ ∅

{C3} ∅ ∅ ∅ ∅ ∅

∅ ∅ ∅ ∅ ∅ ∅

∅ ∅ {C4, C7} ∅ ∅ ∅

∅ ∅ ∅ ∅ ∅ ∅

⎞⎟⎟⎟⎟⎟⎟⎟⎠

. (4.2)

For amaximal elementC4 of SIM(U,C), let c153 = {C4}, thenwe getM1(U,C) as follows:

⎛⎜⎜⎜⎜⎜⎜⎜⎝

∅ {C5} ∅ {C1} ∅ ∅

∅ ∅ ∅ ∅ ∅ ∅

{C3} ∅ ∅ ∅ ∅ ∅

∅ ∅ ∅ ∅ ∅ ∅

∅ ∅ {C4} ∅ ∅ ∅

∅ ∅ ∅ ∅ ∅ ∅

⎞⎟⎟⎟⎟⎟⎟⎟⎠

. (4.3)

Thus, {C1, C3, C4, C5} = ∪{c1ij | c1ij ∈ M1(U,C)} is a granular reduct of C.


For amaximal elementC7 of SIM(U,C), let c153 = {C7}, thenwe getM2(U,C) as follows:

⎛⎜⎜⎜⎜⎜⎜⎜⎝

∅ {C5} ∅ {C1} ∅ ∅

∅ ∅ ∅ ∅ ∅ ∅

{C3} ∅ ∅ ∅ ∅ ∅

∅ ∅ ∅ ∅ ∅ ∅

∅ ∅ {C7} ∅ ∅ ∅

∅ ∅ ∅ ∅ ∅ ∅

⎞⎟⎟⎟⎟⎟⎟⎟⎠

. (4.4)

Thus, {C1, C3, C5, C7} = ∪{c2ij | c2ij ∈ M2(U,C)} is also a granular reduct of C.

From the above example, we show that the heuristic algorithm can avoid the NP hardproblem and generate a granular reduct from the simplified discernability matrix directly.With the heuristic algorithm, the granular reduction theory based on discernability matrix isno longer limited to the theoretic level but applicable in practical usage.

5. Conclusion

In this paper, we develop an algorithm by discernability matrixes to compute all the granularreducts with covering rough sets initially. A simplification of discernibility matrix is proposedfor the first time. Moreover, a heuristic algorithm to compute a granular reduct is presentedto avoid the NP hard problem in granular reduction such that a granular reduct is generatedrapidly.

Acknowledgments

This work is supported by National Natural Science Foundation of China under Grant no.11201490 and no. 11061004, Science and Technology Plan Projects of Hunan Province no.2011FJ3152.

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A Granular Reduction Algorithm Based on Covering Rough Setsdownloads.hindawi.com/journals/jam/2012/970576.pdf · algorithm for covering granular reduction. The quintuple U, C,CL,CH