A Granular Reduction Algorithm Based on Covering Rough Sets

Abstract

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 discernability matrixes 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.

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 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 of reduction on covering rough sets. One is to reduce redundant coverings from a family of coverings, referred to as the attribute reduction. The other is to reduce redundant elements from a covering, noted as the granular reduction. It is to find the minimal subsets of a covering which generate the same set approximations with the original covering. Employed to reduce granular structures and databases as well as interactive with the attribute reduction, we think the granular reduction should be ignored by no means. In this paper, we devote to investigate granular reduction of covering rough sets.

In order to compute all attribute reducts for Pawlak's rough sets, discernibility matrix is initially presented [29]. Tsang et al. [15] develop an algorithm of discernibility 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 covering rough set models initially. In addition, Yang et al. systematically examine the granular reduction 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 discernibility matrixes which is applicable to all granular reducts of covering rough sets. This algorithm can reduce granular structures and get rid of the redundant information from information systems. Then a discernibility matrix is simplified to the minimal format. Meanwhile, based on a simplification of discernibility matrix, a heuristic algorithm is proposed as well.

The remainder of this paper proceeds as follows. Section 2 reviews the relevant background knowledge about the granular reduction. Section 3 constructs the algorithm based on discernibility matrix. Section 4 simplifies the discernibility matrix and proposes a heuristic 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 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 𝑅_∗=∪[𝑥]_𝑅∣[𝑥]_𝑅⊆𝑋,𝑅^∗=∪[𝑥]_𝑅∣[𝑥]_𝑅∩𝑋≠Ø(2.1) 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 𝐴∈ℝ, denoted as IND(ℝ)=∩{𝑅∶𝑅∈ℝ}. 𝐴 is dispensable in ℝ if and only if IND(ℝ)=IND(ℝ−𝐴). Otherwise, 𝐴 is 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. 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, 𝑀𝑑_𝒞(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 the subscript.

Definition 2.2 (neighborhood [9, 19]). Let 𝒞 be a covering of 𝑈, and 𝑁_𝒞(𝑥)=∩{𝐶∈𝒞∣𝑥∈𝐶} is called the neighborhood of 𝑥. 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), 𝐶𝐿_𝒞(𝑋)=𝐶∪𝐿{𝐾_𝒞∈𝒞∣𝐾⊆𝑋}=∪{𝐾∣∃𝑥,s.t.(𝐾∈𝑀𝑑(𝑥))∧(𝐾⊆𝑋)},(𝑋)={𝑥∣𝑁(𝑥)⊆𝑋}=∪{𝑁(𝑥)∣𝑁(𝑥)⊆𝑋}.(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 𝑉𝐻∶𝑃(𝑈)→𝑃(𝑈) are defined as follows: for all 𝑋∈𝑃(𝑈), 𝐹𝐻_𝒞(𝑋𝑆)𝐻=_𝒞𝐶𝐿(𝑋)∪(∪{𝑀𝑑(𝑥)∣𝑥∈𝑋−𝐶𝐿(𝑋)}),𝑇𝐻(𝑋_𝒞)=∪𝐾∣𝐾∈𝒞,𝐾∩𝑋≠Ø,𝑅𝐻_𝒞(𝑋)=∪{𝑀𝑑(𝑥)∣𝑥∈𝑋},𝐼𝐻_𝒞(𝑋)=𝐶𝐿(𝑋)∪∪𝐾∣𝐾∩(𝑋−𝐶𝐿(𝑋))≠Ø,(𝑋)=𝐶𝐿(𝑋)∪𝑋𝐻(∪_𝒞{𝑁(𝑥)∣𝑥∈𝑋−𝐶𝐿(𝑋)}=∪{𝑁(𝑥)∣𝑥∈𝑋}),𝑉𝐻_𝒞(𝑋)=𝑥∣𝑁(𝑥)∩𝑋≠Ø,(𝑋)=∪𝑁(𝑥)∣𝑁(𝑥)∩𝑋≠Ø.(2.4)𝐹𝐻_𝒞, 𝑆𝐻_𝒞, 𝑇𝐻_𝒞, 𝑅𝐻_𝒞, 𝐼𝐻_𝒞, 𝑋𝐻_𝒞, and 𝑉𝐻_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 ℳ_𝒞=∪{𝑀𝑑(𝑥)∣𝑥∈𝑈}, 𝒩_𝒞={𝑁(𝑥)∣𝑥∈𝑈}. ⟨𝑈,ℳ_𝒞⟩ 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 ℳ-approximation 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).

3. 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.(1)If 𝒜_𝒞=𝒞 or 𝒜_𝒞=ℳ_𝒞, then 𝒜_𝒞⊆𝒞: thus; 𝒜_𝒞 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)If 𝒜_𝒞=𝒩_𝒞, generally, 𝒜_𝒞 is not a subset of 𝒞. 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={𝑅₁,𝑅₂,…,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.

Proposition 3.1. Let 𝒞={𝐶_𝑖∣𝑖=1,2,3,…,𝑛} be a covering on 𝑈, and let 𝒞_𝑥={𝐶_𝑖∈𝒞∣𝑥∈𝐶_𝑖}.(1)𝑦∈𝑁(𝑥) if and only if 𝒞_𝑥⊆𝒞_𝑦.(2)𝑦∉𝑁(𝑥) if and only if there is 𝐶_𝑖∈𝒞 such that 𝑥∈𝐶_𝑖 and 𝑦∉𝐶_𝑖.

Proof. (1)𝑦∈𝑁(𝑥)=∩𝒞_𝑥⇔ for any 𝐶_𝑖∈𝒞_𝑥, 𝑦∈𝐶_𝑖⇔ for any 𝐶_𝑖∈𝒞_𝑥, 𝐶_𝑖∈𝒞_𝑦⇔𝒞_𝑥⊆𝒞_𝑦.
(2) It is evident from (1).

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

Proof. Suppose that 𝒩_𝒞≠𝒩_{𝒞−{𝐶𝑖}}, then there is at least one element 𝑥∈𝑈 such that 𝑁_𝒞(𝑥)≠𝑁_{𝒞−{𝐶𝑖}}(𝑥), that is, 𝑁_𝒞(𝑥)⊂𝑁_{𝒞−{𝐶𝑖}}(𝑥). Since 𝑁_{𝒞−{𝐶𝑖}}(𝑥)−𝑁_𝒞(𝑥)≠Ø, suppose that 𝑦∈𝑁_{𝒞−{𝐶𝑖}}(𝑥)−𝑁_𝒞(𝑥), then 𝑦∈𝑁_{𝒞−{𝐶𝑖}}(𝑥) and 𝑦∉𝑁_𝒞(𝑥). Namely, (𝑥,𝑦) is discernible with respect to 𝒞, while (𝑥,𝑦) is indiscernible with respect to 𝒞−{𝐶_𝑖}.
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∈𝑁_{𝒞−{𝐶𝑖}}(𝑥) and 𝑦∉𝑁_𝒞(𝑥), so 𝑦∈𝑁_{𝒞−{𝐶𝑖}}(𝑥)−𝑁_𝒞(𝑥). Thus, 𝑁_𝒞(𝑥)≠𝑁_{𝒞−{𝐶𝑖}}(𝑥). It implies 𝒩_𝒞≠𝒩_{𝒞−{𝐶𝑖}}.

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.

Definition 3.3. Let 𝑈={𝑥₁,𝑥₂,…,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(1)𝑐_𝑖𝑗=Ø, 𝑥_𝑗∈𝑁(𝑥_𝑖),(2)𝑐_𝑖𝑗={𝐶∈𝒞∣𝑥_𝑖∈𝐶,𝑥_𝑗∉𝐶}, 𝑥_𝑗∉𝑁(𝑥_𝑖).

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.

Proposition 3.4. Consider that 𝑁𝐼(𝒞)={𝐶∣𝑐_𝑖𝑗={𝐶} for some 𝑐_𝑖𝑗∈𝑀(𝑈,𝒞)}.

Proof. For any 𝐶∈𝑁𝐼(𝒞), 𝒩_𝒞≠𝒩_{𝒞−{𝐶}}, then there is (𝑥_𝑖,𝑥_𝑗)∈𝑈×𝑈 such that 𝑥_𝑗∈𝑁_{𝒞−{𝐶}}(𝑥_𝑖) and 𝑥_𝑗∉𝑁_𝒞(𝑥_𝑖). It implies that 𝑥_𝑖∈𝐶 and 𝑥_𝑗∉𝐶. Moreover, for any 𝐶′∈𝒞−{𝐶}, since 𝑥_𝑗∈𝑁_{𝒞−{𝐶}}(𝑥_𝑖), we have 𝑥_𝑖∈𝐶′ if 𝑥_𝑖∈𝐶′. Thus, 𝑐_𝑖𝑗={𝐶}.
If 𝑐_𝑖𝑗={𝐶} for some 𝑐_𝑖𝑗∈𝑀(𝑈,𝒞), then 𝑥_𝑖∈𝐶 and 𝑥_𝑗∉𝐶. And for any 𝐶′∈𝒞−{𝐶}, if 𝑥_𝑖∈𝐶′, then 𝑥_𝑖∈𝐶′, that is, 𝑥_𝑗∈𝑁_{𝒞−{𝐶}}(𝑥_𝑖) and 𝑥_𝑗∉𝑁_𝒞(𝑥_𝑖), then 𝑁_{𝒞−{𝐶}}(𝑥_𝑖)≠𝑁_𝒞(𝑥_𝑖). Namely, 𝒩_𝒞≠𝒩_{𝒞−{𝐶}}, which implies 𝐶∈𝑁𝐼(𝒞).

Proposition 3.5. Suppose that 𝒞′⊆𝒞, then 𝒩_𝒞=𝒩_𝒞 if and only if 𝒞′∩𝑐_𝑖𝑗≠Ø for every 𝑐_𝑖𝑗≠Ø.

Proof. 𝒩_𝒞=𝒩_𝒞 ⇔ for any (𝑥_𝑖,𝑥_𝑗)∈𝑈×𝑈, 𝑥_𝑗∉𝑁_𝒞(𝑥_𝑖) if and only if 𝑥_𝑗∉𝑁_𝒞(𝑥_𝑖), ⇔ for any (𝑥_𝑖,𝑥_𝑗)∈𝑈×𝑈, there is 𝐶∈𝒞 such that 𝑥_𝑖∈𝐶 and 𝑥_𝑗∉𝐶 if and only if there is 𝐶′∈𝒞′ such that 𝑥_𝑖∈𝐶′ and 𝑥_𝑗∉𝐶′, ⇔ for any 𝑐_𝑖𝑗≠Ø, 𝒞′≠Ø.

Proposition 3.6. Suppose that 𝒞′⊆𝒞, then 𝒞^∈𝑁red(𝒞) if and only if 𝒞′ is a minimal set satisfying 𝒞′∩𝑐_𝑖𝑗≠Ø for every 𝑐_𝑖𝑗≠Ø.

Definition 3.7. Let 𝑈={𝑥₁,𝑥₂,…,𝑥_𝑛}, let 𝒞={𝐶₁,𝐶₂,…,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 Boolean variables, 𝐶₁,𝐶₂,…,𝐶_𝑚, corresponding to the covering elements 𝐶₁,𝐶₂,…,𝐶_𝑚, respectively, defined as 𝑓(𝑈,𝒞)(𝐶₁,𝐶₂,…,𝐶_𝑚)=∧{∨(𝑐_𝑖𝑗)∣𝑐_𝑖𝑗∈𝑀(𝑈,𝒞),𝑐_𝑖𝑗≠Ø}.

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)=(∧𝒞₁)∨⋯∨(∧𝒞_𝑙), where 𝒞_𝑘⊆𝒞, 𝑘=1,2,…,𝑙 and every element in 𝒞_𝑘 only appears once, then 𝑁red(𝒞)={𝒞₁,𝒞₂,…,𝒞_𝑙}.

Proof. For every 𝑘=1,2,…,𝑙, ∧𝒞_𝑘≤∨𝑐_𝑖𝑗 for any 𝑐_𝑖𝑗∈𝑀(𝑈,𝒞), so 𝒞_𝑘∩𝑐_𝑖𝑗≠Ø. Let 𝒞^𝑘=𝒞_𝑘−{𝐶} for any 𝐶∈𝒞_𝑘, then 𝑔(𝑈,𝒞)≨∨^{𝑘−1𝑡=1}(∧𝒞_𝑡)∨(∧𝒞^𝑘)∨(∨^{𝑙𝑡=𝑘+1}(∧𝒞_𝑡)). If for every 𝑐_𝑖𝑗∈𝑀(𝑈,𝒞), we have 𝒞^𝑘∩𝑐_𝑖𝑗≠Ø, then ∧𝒞^𝑘≤∨𝑐_𝑖𝑗 for every 𝑐_𝑖𝑗∈𝑀(𝑈,𝒞), that is, 𝑔(𝑈,𝒞)≥∨^{𝑘−1𝑡=1}(∧𝒞_𝑡)∨(∧𝒞^𝑘)∨(∨^{𝑙𝑡=𝑘+1}(∧𝒞_𝑡)), which is a contradiction. It implies that there is 𝑐_𝑖0𝑗0∈𝑀(𝑈,𝒞) such that 𝒞^𝑘∩𝑐_𝑖0𝑗0=Ø. Thus, 𝒞_𝑘 is a reduct of 𝒞.
For any 𝒞′∈Red(𝒞), we have 𝒞′∩𝑐_𝑖𝑗≠Ø for every 𝑐_𝑖𝑗∈𝑀(𝑈,𝒞), so 𝑓(𝑈,𝒞)∧(∧𝒞′)=∧(∨𝑐_𝑖𝑗)∧(∧𝒞′)=∧𝒞′, which implies ∧𝒞′≤𝑓(𝑈,𝒞)=𝑔(𝑈,𝒞). Suppose that, for every 𝑘=1,2,…,𝑙, we have 𝒞_𝑘−𝒞′≠Ø, then for every 𝑘, there is 𝒞_𝑘∈𝒞_𝑘−𝒞′. By rewriting 𝑔(𝑈,𝒞)=(∨^𝑙𝑘=1𝒞_𝑘)∧Φ, ∧𝒞′≤∨^𝑙𝑘=1𝒞_𝑘. Thus, there is 𝐶_𝑘0 such that ∧𝒞′≤𝒞_𝑘0, that is, 𝒞_𝑘0∈𝒞′, which is a contradiction. So 𝒞_𝑘0⊆𝒞′ for some 𝑘₀, since both 𝒞′ and 𝒞_𝑘0 are reducts, and it is evident that 𝒞′=𝒞_𝑘0. Consequently, Red(𝒞)={𝒞₁,𝒞₂,…,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 𝒩-irreducible elements. Step 1: 𝑀(𝑈,𝒞)=(𝑐_𝑖𝑗)_𝑛×𝑛, for each 𝑐_𝑖𝑗, let 𝑐_𝑖𝑗=Ø.  Step 2: for each 𝑥_𝑖∈𝑈, compute 𝑁(𝑥_𝑖)=∩{𝐶∈𝒞∣𝑥_𝑖∈𝐶}.    If𝑥_𝑗∉𝑁(𝑥_𝑖),𝑐_𝑖𝑗={𝐶∈𝒞∣𝑥_𝑖∈𝐶,𝑥_𝑗∉𝐶}.  Step 3: 𝑓(𝑈,𝒞)(𝐶₁,𝐶₂,…,𝐶_𝑚)=∧{∨(𝑐_𝑖𝑗)∣𝑐_𝑖𝑗∈𝑀(𝑈,𝒞),𝑐_𝑖𝑗≠Ø}.  Step 4: compute 𝑓(𝑈,𝒞) to 𝑔(𝑈,𝒞)=(∧𝒞₁)∨⋯∨(∧𝒞_𝑙)// where 𝒞_𝑘⊆𝒞,     𝑘=1,2,…,𝑙, and every element in 𝒞_𝑘 only appears once. Step 4: output 𝑁red(𝒞)={𝒞₁,𝒞₂,…,𝒞_𝑙}, 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 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.

Table 1 

{𝑥₁,𝑥₂,𝑥₃} is the set of all objects possessing the attribute 𝐶₁, and it is denoted by 𝐶₁={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 covering on 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}.

The discernibility matrix of (𝑈,𝒞) is exhibited as follows: ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ØØ𝐶₂,𝐶₄Ø,𝐶₅,𝐶₇𝐶₂,𝐶₄Ø,𝐶₅,𝐶₇𝐶𝐶₁₁𝐶₁,𝐶₂,𝐶₅𝐶₁,𝐶₂,𝐶₄𝐶₃𝐶₁𝐶₁𝐶₃,𝐶₆𝐶₃,𝐶₆Ø𝐶₁𝐶₁,𝐶₆𝐶₁,𝐶₃,𝐶₆𝐶,₃𝐶₆𝐶₂,𝐶₃,𝐶₄,𝐶₅,𝐶₆,𝐶₇𝐶₂,𝐶₄,𝐶₅,𝐶₇Ø𝐶₂,𝐶₅,𝐶₆𝐶₂,𝐶₃,𝐶₄,𝐶₆Ø𝐶₃𝐶,₅𝐶₄,𝐶₇𝐶₄,𝐶₇ØØ𝐶₃,𝐶₄𝑓(𝑈,Δ),𝐶₇𝐶₅,𝐶₇Ø𝐶₅Ø⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,(3.1)=∧𝐶∨¹,𝑐𝐶_2𝑖𝑗,𝐶₃,𝐶₄,𝐶₅,𝐶₆,𝐶₇=𝐶∣𝑖,𝑗=1,2,…,6,𝑐_𝑖𝑗≠Ø₂∧𝐶₁∨𝐶₄∨𝐶₅∨𝐶₇∧𝐶₂∨𝐶₄∨𝐶₅∨𝐶₇∧𝐶₁∧𝐶₁∨𝐶₂∨𝐶₅∧𝐶₁∨𝐶₂∨𝐶₄∧𝐶₁∧𝐶₁∧𝐶₁∧𝐶₃∨𝐶₆∧𝐶₃∨𝐶₆∧𝐶₁∧𝐶₁∨𝐶₆∧𝐶₂∨𝐶₃∨𝐶₆∧𝐶₃∨𝐶₆∧𝐶₂∨𝐶₃∨𝐶₄∨𝐶₅∨𝐶₆∨𝐶₇∧𝐶₂∨𝐶₄∨𝐶₅∨𝐶₇∧𝐶₃∨𝐶₅∨𝐶₆∧𝐶₂∨𝐶₃∨𝐶₄∨𝐶₆∧𝐶₃∧𝐶₃∨𝐶₄∨𝐶₇∧𝐶₄∨𝐶₇∨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.

Table 2 

Table 3 

4. 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.

Definition 4.1. Let 𝑀(𝑈,𝒞)=(𝑐_𝑖𝑗)_𝑛×𝑛 be the discernibility matrix of (𝑈,𝒞). For any 𝑐_𝑖𝑗∈𝑀(𝑈,𝒞), if there is an nonempty element 𝑐_𝑖0𝑗0∈𝑀(𝑈,𝒞)−{𝑐_𝑖𝑗} such that 𝑐_𝑖0𝑗0⊆𝑐_𝑖𝑗, let 𝑐^𝑖𝑗=Ø; otherwise, 𝑐^𝑖𝑗=𝑐_𝑖𝑗, 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. 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≠Ø for any nonempty element 𝑐_𝑖𝑗∈𝑀(𝑈,𝒞) if and only if 𝒞′∩𝑐^𝑖𝑗≠Ø for any nonempty element 𝑐^𝑖𝑗∈SIM(𝑈,𝒞).

Proof. If 𝒞′∩𝑐_𝑖𝑗≠Ø for every 𝑐_𝑖𝑗≠Ø and 𝑐_𝑖𝑗∈𝑀(𝑈,𝒞), it is evident that 𝒞′∩𝑐^𝑖𝑗≠Ø for every 𝑐^𝑖𝑗≠Ø and 𝑐^𝑖𝑗∈SIM(𝑈,𝒞).
Suppose that 𝒞′∩𝑐^𝑖𝑗≠Ø for every 𝑐^𝑖𝑗≠Ø and 𝑐^𝑖𝑗∈SIM(𝑈,𝒞). For any nonempty 𝑐_𝑖𝑗∈𝑀(𝑈,𝒞), if there is an nonempty element 𝑐_𝑖0𝑗0∈𝑀(𝑈,𝒞)−{𝑐_𝑖𝑗} such that 𝑐_𝑖0𝑗0⊆𝑐_𝑖𝑗, and for any nonempty element 𝑐_𝑖1𝑗1∈𝑀(𝑈,𝒞)−{𝑐_𝑖𝑗,𝑐_𝑖0𝑗0}, 𝑐_𝑖1𝑗1⊈𝑐_𝑖0𝑗0, then 𝑐^{𝑖0𝑗0}=𝑐_𝑖0𝑗0≠Ø. Since 𝒞′∩𝑐^{𝑖0𝑗0}≠Ø, then 𝒞′∩𝑐_𝑖0𝑗0≠Ø; thus, 𝒞′∩𝑐_𝑖𝑗≠Ø. If 𝑐_𝑖0𝑗0⊈𝑐_𝑖𝑗 for any nonempty element 𝑐_𝑖0𝑗0∈𝑀(𝑈,𝒞)−{𝑐_𝑖𝑗}, then 𝑐^𝑖𝑗=𝑐_𝑖𝑗. Since 𝒞′∩𝑐^𝑖𝑗≠Ø, then 𝒞′∩𝑐_𝑖𝑗≠Ø. Thus, 𝒞′∩𝑐_𝑖𝑗≠Ø for every nonempty 𝑐_𝑖𝑗∈𝑀(𝑈,𝒞).

Proposition 4.3. Suppose that 𝒞′⊆𝒞, then 𝒞^∈𝑁red(𝒞) if and only if 𝒞′ is a minimal set satisfying 𝒞′∩𝑐^𝑖𝑗≠Ø for every 𝑐^𝑖𝑗≠Ø and 𝑐^𝑖𝑗∈<?brm?>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^𝑖𝑗 and 𝑐^𝑖𝑗∩𝑁𝐼(𝒞)=Ø. For any 𝑐^𝑖𝑗∈SIM(𝑈,𝒞), if 𝐶∈𝑐^𝑖𝑗, let 𝑐^1𝑖𝑗={𝐶}. Otherwise, 𝑐^1𝑖𝑗={𝐶_𝑖𝑗}, where 𝐶_𝑖𝑗∈𝑐^𝑖𝑗. Suppose that 𝑀₁(𝑈,𝒞)=(𝑐^1𝑖𝑗)_𝑛×𝑛; it is easy to prove that 𝐶∈∪{𝑐^1𝑖𝑗∣𝑐^1𝑖𝑗∈𝑀₁(𝑈,𝒞)}∈𝑁red(𝒞). Thus, 𝐶∈∪𝑁red(𝒞).
Suppose that 𝐶∈∪𝑁red(𝒞), then there is 𝒞_𝑘∈𝑁red(𝒞) such that 𝐶∈𝒞_𝑘. From Proposition 4.3, we know that 𝒞_𝑘 is a minimal set satisfying 𝒞_𝑘∩𝑐′_𝑖𝑗≠Ø for every 𝑐^𝑖𝑗≠Ø and 𝑐^𝑖𝑗∈SIM(𝑈,𝒞). So there is a 𝑐^𝑖𝑗∈SIM(𝑈,𝒞) such that 𝐶∈𝑐^𝑖𝑗, or else 𝐶 is redundant in 𝒞_𝑘. 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), then SIM(𝑈,𝒞) is the minimal matrix to compute all granular reducts of 𝒞, that is, for any matrix 𝑀₀(𝑈,𝒞)=(𝑑_𝑖𝑗)_𝑛×𝑛 where 𝑑_𝑖𝑗⊆𝑐^𝑖𝑗, 𝑀₀(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 𝑀₀(𝑈,𝒞)=SIM(𝑈,𝒞), and 𝑀₀(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}={𝐶₀}, then 𝑑_𝑖0𝑗0=Ø. From the definition of the simplification discernibility matrix, we know that 𝐶₀∉𝑐^𝑖𝑗 for any 𝑐^𝑖𝑗∈SIM(𝑈,𝒞)−{𝑐^{𝑖0𝑗0}}, then 𝐶₀∉𝑑_𝑖𝑗 for any 𝑑_𝑖𝑗∈𝑀₀(𝑈,𝒞). So 𝑀₀(𝑈,𝒞) cannot compute any granular reducts of 𝒞. If |𝑐^{𝑖0𝑗0}|≥2, we suppose that 𝑑_𝑖0𝑗0≠Ø. Then there is a 𝐶∈(𝑐^{𝑖0𝑗0}−𝑑_𝑖0𝑗0), and let 𝑐^1𝑖0𝑗0={𝐶}. For any 𝑐^𝑖𝑗∈SIM(𝑈,𝒞)−{𝑐^{𝑖0𝑗0}}, if 𝐶∈𝑐^𝑖𝑗, let 𝑐^1𝑖𝑗=Ø. Otherwise, let 𝑐^1𝑖𝑗={𝐶_𝑖𝑗} where 𝐶_𝑖𝑗∈𝑐^𝑖𝑗−𝑐^{𝑖0𝑗0}. Let 𝑀₁(𝑈,𝒞)=(𝑐^1𝑖𝑗)_𝑛×𝑛 and 𝒞′=∪{𝑐^1𝑖𝑗∣𝑐^1𝑖𝑗∈𝑀₁(𝑈,𝒞)}, and it is easy to prove that 𝒞′∈𝑁red(𝒞). However, 𝒞′∩𝑑_𝑖0𝑗0=Ø, that is, 𝑀₀(𝑈,𝒞) cannot compute all granular reducts of 𝒞. Thus, if 𝑀₀(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: ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝𝐶ØØ𝐶Ø₅ØØ𝐶Ø₁ØØØ₃ØØØØØØ𝐶₄ØØØØØØØØ𝑓(𝑈,Δ)ØØØ,𝐶₇ØØØØØØ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,𝐶₁,𝐶₂,𝐶₃,𝐶₄,𝐶₅,𝐶₆,𝐶₇==𝐶∧₅∨𝑐^𝑖𝑗∣𝑖,𝑗=1,2,…,6,𝑐_𝑖𝑗≠Ø=𝐶∧₅𝐶₁∧𝐶₃∧𝐶₄∨𝐶₇∧𝐶₁∧𝐶₃∧𝐶₄∨𝐶₅∧𝐶₁∧𝐶₃∧𝐶₇.(4.1) So 𝑁red(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 can simplify the computing processes remarkably. Especially when 𝒞 is a consistent covering proposed 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, 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 |𝑐_𝑖𝑗|. For any 𝐶∈𝒞, ||𝐶|| denotes the number of 𝑐_𝑖𝑗 which contain 𝐶. Let 𝑐_𝑖𝑗∈𝑀(𝑈,𝒞), if for any 𝐶∈𝑁𝐼(𝒞), 𝐶∉𝑐_𝑖𝑗, then 𝑐^𝑖𝑗=𝑐_𝑖𝑗. 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. Suppose that 𝐶₀∈∪{𝑐^𝑖𝑗∣|𝑐^𝑖𝑗|≥2}, if ||𝐶₀||≥||𝐶|| for any 𝐶∈∪{𝑐^𝑖𝑗∣|𝑐^𝑖𝑗|≥2}, 𝐶₀ is 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.

Algorithm 4.7. Consider the following:
input: ⟨𝑈,𝒞⟩,
output: granular reducts red Step 1: 𝑀(𝑈,𝒞)=(𝑐_𝑖𝑗)_𝑛×𝑛, for each 𝑐_𝑖𝑗, let 𝑐_𝑖𝑗=Ø. Step 2: for each 𝑥_𝑖∈𝑈, compute 𝑁(𝑥_𝑖)=∩{𝐶∈𝒞𝑥_𝑖∈𝐶}.    If𝑥_𝑗∉𝑁(𝑥_𝑖),     𝑐_𝑖𝑗={𝐶∈𝒞∣𝑥_𝑖∈𝐶,𝑥_𝑗∉𝐶}//getthediscernibilitymatrix. Step 3: 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=Ø // get the simplified discernibility matrix. Step 4: for each 𝐶_𝑖∈∪𝑀(𝑈,𝒞), compute ||𝐶_𝑖|| and select the maximal      element 𝐶₀ of SIM(𝑈,𝒞).    For each 𝑐_𝑖𝑗∈𝑀(𝑈,𝒞),    if 𝐶₀∈𝑐_𝑖𝑗,     let 𝑐_𝑖𝑗={𝐶₀}.  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.

Example 4.8. The simplified discernibility matrix of (𝑈,𝒞) in Example 3.10 is as follows: ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝𝐶ØØ𝐶Ø₅ØØ𝐶Ø₁ØØØ₃ØØØØØØ𝐶₄ØØØØØØØØØØØ,𝐶₇ØØØØØØ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠.(4.2)
For a maximal element 𝐶₄ of SIM(𝑈,𝒞), let 𝑐¹⁵³={𝐶₄}, then we get 𝑀₁(𝑈,𝒞) as follows: ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝𝐶ØØ𝐶Ø₅ØØ𝐶Ø₁ØØØØ₃ØØØØØ𝐶ØØ₄ØØØØØØØØØØØØØØØ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠.(4.3)
Thus, {𝐶₁,𝐶₃,𝐶₄,𝐶₅}=∪{𝑐^1𝑖𝑗∣𝑐^1𝑖𝑗∈𝑀₁(𝑈,𝒞)} is a granular reduct of 𝒞.
For a maximal element 𝐶₇ of SIM(𝑈,𝒞), let 𝑐¹⁵³={𝐶₇}, then we get 𝑀₂(𝑈,𝒞) as follows: ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝𝐶ØØ𝐶Ø₅ØØ𝐶Ø₁ØØØØ₃ØØØØØ𝐶ØØ₇ØØØØØØØØØØØØØØØ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠.(4.4) Thus, {𝐶₁,𝐶₃,𝐶₅,𝐶₇}=∪{𝑐^2𝑖𝑗∣𝑐^2𝑖𝑗∈𝑀₂(𝑈,𝒞)} 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.

5. 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.

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.