A Granular Reduction Algorithm Based on Covering Rough Sets

The granular reduction is to delete dispensable elements from a covering. It is an efficient method to 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 granular reducts of covering rough sets. Moreover, a discernibility matrix is simplified to the minimal format. In addition, a heuristic algorithm is proposed as well such that a granular reduct is generated rapidly.


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 set theory was raised by Pawlak 1, 2 to deal with uncertainty and vagueness, and it has been applied 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 of equivalence 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 rough sets 10,11,15,23,27,28 plays an important role in practice.A partition is no longer a partition if any of its elements is deleted, while a covering may still be a covering with invariant set approximations after dropping some elements.Therefore, there are two types

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 x generated 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 are called the lower and upper approximations of X, respectively.If R * X R * X , X is an R-exact set.Otherwise, it is an R-rough set.
Let R be a family of equivalence relations, and let 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. The discernibility matrix method is proposed to compute all reducts of information systems and relative reducts of decision systems 29 .
C is called a covering of U, where U is a nonempty domain of discourse, and C is a family of nonempty subsets of U and ∪C U.
It is clear that a partition of U is certainly a covering of U, so the concept of a covering is an extension of the concept of a partition.Definition 2.1 minimal description 9 .Let C be a covering of U, is called the minimal description of x.When there is no confusion, we omit the C from the subscript.
Definition 2.2 neighborhood 9, 19 .Let C be a covering of U, and Generally, we omit the subscript C when there is no confusion.
Minimal description and neighborhood are regarded as related information granules to describe x, which are used as approximation elements in rough sets as shown in Definition 2.3 .It shows that N x ∩{C ∈ C | x ∈ C} ∩Md x .The neighborhood of x can be seen as the minimum description of x, and it is the most precise description more details are referred to 9 .Definition 2.3 covering lower and upper approximation operations 19 .Let C be a covering of U. The operations CL C : P U → P U and CL C : P U → P U are defined as follows: for all X ∈ P U ,

2.3
We call CL C the first, the second, the third, or the fourth covering lower approximation operations and CL C the fifth, the sixth, or the seventh covering lower approximation operations, with respect to the covering C. The operations FH, SH, TH, RH, IH, XH, and V H : P U → P U are defined as follows: for all X ∈ P U , , and V H C are called the first, the second, the third, the fourth, the fifth, the sixth, and the seventh covering upper approximation operations with respect 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 applied in certain circumstance.We choose the suitable approximation operation according to the specific situation.So it is important to design the granular reduction algorithms for all of these models.
More precise approximation spaces are proposed in 30 .As a further result, a reasonable granular reduction of coverings is also introduced.Let is the approximation space of the first and the third types of covering rough sets, U, C is the approximation space of the second and the fourth types of covering rough sets, and U, N C is the approximation space of the fifth, the sixth, and the seventh types of covering rough sets referred to 30 for the details .In this paper, we design the algorithm of granular reduction for the fifth, the sixth, and the seventh type of covering rough sets.
Let C be a covering of U, denoting a covering approximation space.M C denotes an Mapproximation space.N C represents an N-approximation space.We omit C at the subscript when there is no confusion referred to 30 for the details .

Discernibility Matrixes Based on Covering Granular Reduction
In the original Pawlak's rough sets, a family of equivalence classes induced by equivalence relations 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 of redundant information in databases.Therefore, granular reduction is not applicable to the original 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 set approximations are invariant.The purpose of covering granular reduction is to find minimal subsets keeping the same set approximations.It is meaningful and necessary to develop the algorithm for covering granular reduction.
The quintuple U, C, CL, CH is called a covering rough set system CRSS , where C is a covering of U, CL and CH are the lower and upper approximation operations with respect to the covering C, and U, A C is the approximation space.According to the categories of covering approximation operations in 30 , there are two kinds of situations as follows.
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.
Consequently, an algorithm is needed to compute all granular reducts of C for the fifth, the sixth, and the seventh type of covering rough set models.
Next we examine the algorithm of granular reduction for the fifth, the sixth, and the seventh type of covering rough sets.Let C be a covering of U, since N C {N x | x ∈ U}, and N C is the collection of all approximation elements of the fifth, the sixth, or the seventh type of lower/upper approximation operations.N C is called the N-approximation space of 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 seventh type of covering rough set models, the purpose of granular reduction is to find the minimal subsets C of C such that N C N C .The granular reducts based on the N-approximation spaces are called the N-reducts.Nred C is the set of all N-reducts of C, and NI C is the set of 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 equivalence class containing x, we say that x and y are indiscernible.Otherwise, they are discernible.Let R {R 1 , R 2 , . . ., R n } be a family of equivalence relation on U, R i ∈ R. R i 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 after deleting R i from R. The attribute reduction of Pawlak's rough sets is to find minimal subsets of R which keep the relations invariant for any x, y ∈ U. Based on this statement, the method of discernibility matrix to compute all reducts of Pawlak's rough sets was proposed in 29 .In covering rough sets, however, the discernibility relation between x, y ∈ U is different from that in Pawlak's rough sets.
Let C be a covering on U, x, y ∈ U × U. Then we call x, y indiscernible if y ∈ 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's discernibility relation.In Pawlak's rough sets, x, y is indiscernible if and only if y, x is indiscernible.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 call these relations the relations of x, y with respect to C. The following theorem characterizes these relations.

, n} be a covering on U, and let
Suppose that there is x, y ∈ U × U whose discernibility relation with respect to C is changed after deleting C i from C. Put differently, x, y is discernible with respect to C, while x, y is indiscernible with respect to C − {C i }.Then we have y The purpose of granular reducts of a covering C is to find the minimal subsets of C which keep the same classification ability as C or, put differently, keep N C invariant.In Theorem 3.2, N C 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 with discernibility matrix.

Journal of Applied Mathematics
Definition 3.3.Let U {x 1 , x 2 , . . ., x n }, C be a covering on U. M U, C is an n×n matrix c ij n×n called a discernibility matrix of U, C , where 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 the characterizations of N-reduction.Proof.For every k 1, 2, . . ., l, ∧C k ≤ ∨c ij for any For any Step 1: M U, C c ij n×n , for each c ij , let c ij ∅.
Step 2: for each Step Step . ., l, and every element in C k only appears once.
The following example is used to illustrate our idea.
Example 3.10.Suppose that U {x 1 , x 2 , . . ., x 6 }, where x i , i 1, 2, . . ., 6 denote six objects, and let C i , i 1, 2, . . ., 7 denote seven properties; the information is presented in Table 1, that is, the ith object possesses the jth attribute is indicated by a * in the ij-position of the table.
{x 1 , x 2 , x 3 } is the set of all objects possessing the attribute C 1 , and it is denoted by  The discernibility matrix of U, C is exhibited as follows: 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.

The Simplification of Discernibility Matrixes
For the purpose of finding the set of all granular reducts, we have proposed the method by discernibility matrix.Unfortunately, it is at least an NP problem, since the discernibility matrix in this paper is more complex than the one in 33 .Accordingly, we simplify the discernibility matrixes in this section.In addition, a heuristic algorithm is presented to avoid the NP hard problem. where In summary, and it is easy to prove that C ∈ Nred C .However, C ∩ d i0j0 ∅, that is, M 0 U, C cannot compute all granular reducts of C. Thus, if M 0 U, C can compute all granular reducts of C, then d ij c ij for 1 ≤ i, j ≤ n.
From the above propositions, we know that the simplified discernibility matrix is the minimal discernibility matrix which can compute the same reducts as the original one.Hereafter, we only examine simplified discernibility matrixes instead of general discernibility matrixes.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: From the above example, it is easy to see that simplified discernibility matrix can simplify the computing processes remarkably.Especially when C is a consistent covering proposed in 30 , that is, Unfortunately, although the simplified discernibility matrixes are more simple, the processes of computing reducts by discernibility function are still NP hard.Accordingly, we develop a heuristic algorithm to obtain a reduct from a discernibility matrix directly.
Let M U, C c ij n×n be a discernibility matrix.We denote the number of the elements in c ij by then the elements in c ij may either be deleted from C or be preserved.Suppose that called the maximal element with respect to the simplified discernibility matrix SIM U, C .The heuristic algorithm to get a reduct from a discernibility matrix directly proceeds as follows.Step 1: M U, C c ij n×n , for each c ij , let c ij ∅.
Step 2: for each ∈ C}//get the discernibility matrix. Step For a maximal element C 4 of SIM U, C , let c 1 53 {C 4 }, then we get M 1 U, C as follows: For a maximal element C 7 of SIM U, C , let c 1 53 {C 7 }, then we get M 2 U, C as follows: } is also a granular reduct of C.
From the above example, we show that the heuristic algorithm can avoid the NP hard problem and generate a granular reduct from the simplified discernability matrix directly.With the heuristic algorithm, the granular reduction theory based on discernability matrix is no longer limited to the theoretic level but applicable in practical usage.

Conclusion
In this paper, we develop an algorithm by discernability matrixes to compute all the granular reducts with covering rough sets initially.A simplification of discernibility matrix is proposed for the first time.Moreover, a heuristic algorithm to compute a granular reduct is presented to avoid the NP hard problem in granular reduction such that a granular reduct is generated rapidly.

Theorem 3 . 8 .
Let C be a family of covering on U, let f U, C be the discernibility function, and let g U, C be the reduced disjunctive form of f U, C by applying the multiplication and absorption laws.If g U, C ∧C 1 ∨ • • • ∨ ∧C l , where C k ⊆ C, k 1, 2, . . ., l and every element in C k only appears once, then Nred C {C 1 , C 2 , . . ., C l }.

Definition 4 . 1 .
Let M U, C c ij n×n be the discernibility matrix of U, C .For any

Theorem 4 . 2 .
then we get a new discernibility matrix SIM U, C c ij n×n , which called the simplification discernibility matrix of U, C .Let M U, C be the discernibility matrix of U, C , and SIM U, C is the simplification discernibility matrix, C ⊆ C. Then C ∩ c ij / ∅ for any nonempty element c ij ∈ M U, C if and only

Proposition 4 . 5 .
Let SIM U, C c ij n×nbe the simplified discernibility matrix of U, C , then SIM U, C is the minimal matrix to compute all granular reducts of C, that is, for any matrixM 0 U, C d ij n×n where d ij ⊆ c ij , M 0 U, C can compute all granular reducts of C if and only if d ij c ij for 1 ≤ i, j ≤ n.Proof.If d ij c ij for 1 ≤ i, j ≤ n, then M 0 U, C SIM U, C ,and M 0 U, C can compute all granular reducts of C. Suppose that there is a nonempty c i0j0 ∈ SIM U, C such that d i0j0 ⊂ c i0j0 .If |c i0j0 | 1, suppose that c i0j0 {C 0 }, then d i0j0 ∅.From the definition of the simplification discernibility matrix, we know that C 0 / ∈ c ij for any c ij ∈ SIM U, C − {c i0j0 }, then C 0 / ∈ d ij for any d ij ∈ M 0 U, C .So M 0 U, C cannot compute any granular reducts of C. If |c i0j0 | ≥ 2, we suppose that d i0j0 / ∅.Then there is a C ∈ c i0j0 − d i0j0 , and let c 1 i0j0 {C}.For any

3 :Example 4 . 8 .
for each c ij ∈ M U, C , if there is a nonempty element c i0j0 ∈ M U, C − {c ij } such that c i0j0 ⊆ c ij , let c ij ∅ //get the simplified discernibility matrix.Step 4: for eachC i ∈ ∪M U, C , compute ||C i || and select the maximal element C 0 of SIM U, C .For each c ij ∈ M U, C , if C 0 ∈ c ij , let c ij {C 0 }.Step 5: if there is c ij ∈ M U, C such that |c ij | 2,return to Step 3; else output red ∪M U, C .Step 5: end.The simplified discernibility matrix of U, C in Example 3.10 is as follows: and only if there is C ∈ C such that x i ∈ C and x Boolean variables, C 1 , C 2 , . . ., C m , corresponding to the covering elements C 1 , C 2 , . . ., C m , respectively, defined as j / ∈ C , ⇔ for any c ij / ∅, C / ∅.Proposition 3.6.Suppose that C ⊆ C, then C ∈ Nred C if and only if C is a minimal set satisfying C ∩ c ij / ∅ for every c ij / ∅.Definition 3.7.Let U {x 1 , x 2 , . . ., x n }, let C {C 1 , C 2 , . . ., C m } be a covering of U, and let M U, C c ij n×n be the discernibility matrix of U, C .A discernibility function f U, C is a Boolean function of m

Table 1
since both C and C k 0 are reducts, and it is evident that C C k 0 .Consequently, Red C {C 1 , C 2 , . . ., C l }.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 Nirreducible elements.