Traditional rough set theory is mainly used to reduce attributes and extract rules in databases in which attributes are characterised by partitions, which the covering rough set theory, a generalisation of traditional rough set theory, covers. In this article, we posit a method to reduce the attributes of covering decision systems, which are databases incarnated in the form of covers. First, we define different covering decision systems and their attributes’ reductions. Further, we describe the necessity and sufficiency for reductions. Thereafter, we construct a discernible matrix to design algorithms that compute all the reductions of covering decision systems. Finally, the above methods are illustrated using a practical example and the obtained results are contrasted with other results.

#### Keywords

- discernible matrix
- information entropy
- decision system
- attribute

The Rough Set theory was proposed by Polish mathematician Zdzisław Pawlak in 1982 [1, 3, 17]. It can effectively handle uncertain, inaccurate and incomp Supposinge information. Recently, rough set theory has successfully been applied in many fields, including machine learning, pattern recognition, decision analysis, process control and data mining [4, 5, 10, 11, 12, 13, 14]. Therefore, this theory has received great attention from the international information science community, computer science and mathematics. A large number of studies in the literature have witnessed the development of rough sets in theory and application [26, 27, 28, 29]. Therefore, some scholars have extended the partition to encompass aspects explained in the above reasons, and the scope of rough set theory research has been greatly expanded. Although much attention is paid to the set approximation of cover, sparse research has been carried out in relation to the attribute reduction of covering rough sets. The attribute reduction under the algebraic point of view is generalised on the basis of the attribute reduction of conditional information entropy.

At present, certain scholars have studied the rough set theory based on their conclusions drawn from the information theory, and have proposed the information theory about rough set theory. Wang et al. [2] posited the reduction of the decision table and the common properties and different characteristics of information. Yang [6] proposed an approximate reduction method on basis of conditional information entropy in a decision table. On this basis, an approximate reduction method was proposed for vertical multi distribution decision tables [8]. Wang Yan et al also used a reduction algorithm on information entropy and identifiable Matrix, and presented a new combination algorithm [9]. After Hudan et al. added a probability measure to the rough set theory [7], some concepts and properties of information theory and rough set theory were compared, and a new method of rule extraction was obtained. Hu et al. [15] proposed a rough entropy method based on generalised rough set coverage reduction. Chen et al. [16] proposed an optimal section for reducing the superfluous cover. Yang [18] performed research on rough set methods, from the attribute reduction problem on inconsistent decision systems to the attribute reduction problem on consistent decision systems. Guo [19] studied knowledge reduction based on rough set theory for the inconsistent decision systems, such as generalised decision table, relative resolution and knowledge reduction. Shi et al. [20] proposed attribute reduction based on the Boolean matrix. Li and Yin [21] proposed a reduction algorithm of covering system on information theory. Ma [22] constructed a decision tree based on the covering rough set theory. Chen et al. [23] got a multi-label attribute reduction algorithm on neighbourhood rough set. Zhang et al. [24] developed the belief and plausibility functions from the evidence theory and these are employed to characterise attribute reductions in the covering decision information system. Zhang et al. [25] posited confidence-preserved attribute reduction and algorithms of rule acquisition in covering decision systems. Jiang et al. [30] presented an accelerator for multi-granularity attribute reduction knowledge-based systems from another angle. However, we resolve and analyse the problem in consistent and inconsistent covering decision systems based on conditional information entropy in this article.

In this article, we propose a method to reduce the attributes of covering decision systems, which are databases characterised by covers. First, we define two scenarios of covering decision systems and their attributes’ reductions. Second, we state the necessity and sufficiency for reductions. Third, we construct a discernible matrix to design algorithms that compute all the reductions of different covering decision systems. Finally, the above methods are illustrated using a practical example and the obtained results are in contrast to other results.

We go over the basic concepts related to covering rough sets which can be found in the literature [1, 4, 11, 16, 17, 19, 21, 22, 24, 25].

The ordered pair (

Supposing that (

Supposing that _{1}, _{2}, …, _{n}_{x}_{j}_{j}_{j}_{x}

Supposing that _{i}_{x}_{ix}_{ix}_{i}_{ix}_{x}

In this section, we focus on investigating the basic concepts and key results of consistent covering decision systems [1, 4, 16, 19, 21].

Supposing that {_{i}_{j}_{x}_{j}

The positive region of

For ∀

If every cover in _{d}_{d}_{d}_{d}_{i}, x_{j}_{xi}_{xj}_{d}_{i}_{d}_{i}_{d}) = _{d}_{xi}_{d}_{xj}_{d}_{j}_{d}_{j}_{d}).

If _{d}_{xi}_{d}_{xj}_{xi}_{xj}_{xi}_{xj}_{xj}_{xi}_{xi}_{xj}_{xj}_{xi}_{d}_{xi}_{d}_{xj}_{d}_{xi}_{d}_{xj}

On the other hand, if _{xi}_{xj}_{d}_{xi}_{d}_{xj}_{d}_{xi}_{d}_{xj}_{xi}_{xj}_{xj}_{xi}_{xi}_{xj}_{xi}_{xj}

Further, we define the relative reduction of a consistent covering decision system.

Supposing that _{i}_{i}_{i}_{i}_{i}_{i}_{d}

The relative reduction of a consistent covering decision system is the minimal set of conditional covers (attributes) which ensure that every decision rule is still consistent. For a single cover _{i}

_{y}, then x_{d.}

Since for every _{y}_{D}. If for every _{y}_{d}. On the other hand, for every _{y}_{d}, hence _{y}_{d}. Therefore, for every

_{π}

⇒ For every _{π}_{y}_{d}, therefore _{d}, and so we have [_{d} = [_{d}, i.e. _{y}_{d} ≠ ∅.

For every _{y}_{d} and by the Theorem 1, we get

⇐ Since
_{π}_{π}_{y}_{d}. By Theorem 1, _{y}_{d}; hence _{d}; then by _{y}_{d} and [_{d} ⊆ [_{d}, we have _{d}; hence _{π}_{π}

By the above two theorems of discussions, we obtain the following two corollaries.

_{i}_{i}_{x}_{i} is indispensable, i.e. H_{i}_{x}_{D} is not true.

If there exists _{x}_{D} is not true, by _{x}_{x}_{D} is not true. So _{i}

If _{i}_{x}_{D} is not true. Especially, _{x}_{D} is also not true.

It should be indicated that _{x}_{D} not being true means that (_{i}_{i}_{i}

_{i}_{i}_{x}_{i}_{i}_{i} is absolutely indispensable.

_{i}_{i}_{i} is called superfluous in B.

We assume that _{i}_{x}_{i}_{ix}_{i}_{x}_{i}_{x}_{i}_{x}_{ix}

If _{i}_{i}_{x}_{ix}_{i}_{i}_{i}_{i}

Theorem 4 implies that the superfluous knowledge in question could not supply new and useful information to the concerned information system. However, the necessary knowledge could give helpful information for information systems.

_{i}_{i} is superfluous relative tod in π if and only if H_{i}

_{i}_{i}_{i}

_{i}_{i}_{i}

By Theorem 4, _{i}_{i}_{i}_{i}

Based on the discussion of the above theorems, we could consider indicial form of information entropy as being equivalent to expressions of algebra for attribute reduction.

Assume that _{i}_{i}_{i}_{i}, x_{j}_{d}_{xi}_{d}_{xj}_{i}

⇒ We note that if _{i}_{x}_{i}_{0}, _{0} ∈ _{0} ∈ _{x}_{0} and _{0} ∉ [_{0}]_{d}, which implies _{y}_{0} ⊆ _{x}_{0} and _{x}_{0} ⊄ _{y}_{0},_{y}_{0} ⊄ _{x}_{0}, respectively, and _{0}, _{0} satisfying _{d}_{x}_{0}) ≠ _{d}_{y}_{0}), so the factor that has the foremost relation of _{0}, _{0} with regard to _{i}

⇐ Suppose _{0}, _{0} ∈ _{d}_{xi}_{d}_{xj}_{0}]_{d} ∩ [_{0}]_{d} = ∅, and _{x}_{0} ⊄ _{y}_{0},_{y}_{0} ⊄ _{x}_{0}, if their foremost relation _{0}, _{0} with regard to _{i}_{x}_{0} ⊄ _{y}_{0} or _{y}_{0} ⊄ _{x}_{0} is not effective. Thus if _{y}_{0} ⊄ _{x}_{0}, then _{y}_{0} ⊆ [_{0}]_{d} is not effective, otherwise it points out _{0} ∈ [_{0}]_{d} which is a contradiction; if _{x}_{0} ⊇ _{y}_{0}, then _{x}_{0} ⊆ [_{0}]_{d} is not virtual; if _{x}_{0} = _{y}_{0} then _{y}_{0} ⊆ [_{0}]_{d} and _{x}_{0} ⊆ [_{0}]_{d} are not effective. Hence _{i}

Theorem 6 implies that an indispensable cover can be characterised by the foremost relation between two elements in the universe. Thus, we have the following theorem to characterise a consistent decision system.

_{i}, x_{j}_{d}_{x}_{0}) ≠ _{d}_{y}_{0}), _{i}, x_{j} with regard to π is equivalent to their relation with regard to B, i.e. π_{xi}_{xj}, π_{xj}_{xi}_{xi}_{xj}, B_{xj}_{xi}.

Since the proof is similar, here there is no need to repeat it.

The intention of relative reduction of covering attribute

Supposing (_{1}, _{2}, … _{n}_{ij}_{xi} ≠ A_{xj}_{xi}_{xj}_{xj}_{xi}_{d}_{xi}_{d}_{xj}_{ij} π_{d}_{xi}_{d}_{xj}_{i}, x_{j}

Supposing (

(1) For every _{ij}

(2) _{d}_{ij}_{n}_{×n}(_{ij}

Assume _{d}_{xi}_{d}_{xj}_{ij}_{ij}_{d}_{xi}_{d}_{xj}_{xi}_{xi}_{i}_{d}, _{xj}_{xj}_{j}_{d}, then, we have _{d}_{xi}_{d}_{xj}_{xi}_{xj}, π_{xj}_{xi}_{xi}_{xj}, B_{xj}_{xi}_{i}_{0} ∈ _{i}_{0x} ≠ _{j}_{0y}, i.e. _{i}_{0x} ⊄ _{j}_{0y},_{j0y} ⊄ _{i}_{0x} and so we have _{ij}

On the contrary, if _{ij}_{d}_{x}_{d}_{y}

If every, then _{i}_{d}_{x}_{d}_{y}_{xi}_{xj}_{xj}_{xi}_{ij}

Hence we have _{D}_{ij}_{n}_{×n}(_{ij}_{ij}_{D}

The value of core in information system is exclusive, which is the most important part of knowledge category in the information system.

By the Theorem 8(2), a method is used which can directly obtain _{d}_{d}_{ij}_{ij}_{d}

_{1} ⊆ _{d}_{i}_{ij}_{1} _{ij},

⇐ Since _{1} ⊆ _{d}_{ij}_{1} for all _{ij}_{i}_{1} are reduced.

⇒ If the attributes on _{1} could be reduced at the same time, and there also exists a _{ij}_{ij}_{1}, then we obtain _{ij}_{1} are reduced at the same time, which point out the attributes on _{1} could not be reduced at the same time.

In many of practical problems, we always have inconsistent covering decision system. In this section, we propose attribute reductions for inconsistent covering decision systems. We understand that some rules extracted from inconsistent decision systems may not be consistent. As to covering decision system, experts can still give the decision-making in the case of inconsistent information, so it can be assumed that the decision-making property is not empty. So we have the following definition of attribute reduction. We always suppose _{i}_{x}_{d} : _{π}

In this subsection, we discuss the key definition of inconsistent covering decision systems which can be found in the literature [7, 16, 17, 19].

Suppose _{i}_{i}_{d}_{d}_{d}_{d}, then, _{π}

Suppose _{x}_{d}}.

Suppose _{i}_{i}_{i}

Suppose is an inconsistent covering decision system. For every

If ∀_{i}_{i}

Then _{π}

_{W}_{B}

Suppose _{x}_{x}_{W}_{x}_{x}_{1}, …, _{t}

Obviously, we can get _{1} ⊆ _{t}_{x}_{x} such that _{x}_{tx}_{(t−1)x} ⊆_{1x} ⊆ _{x}_{x}_{x}_{B}_{W}_{B}

_{π}_{B}_{π}_{B}

Here we will not prove it.

_{B}_{B}_{−{b}}(

If _{B}_{π}_{B}_{−{a}}(_{π}_{B}_{−{a}}(_{1} ∉ _{B}_{−{a}}(_{1} ∉ _{x1} ⊄ _{x1} ⊄ _{x1} ⊆ ((_{x1}, then ((_{x1} ((_{x1} ⊄ _{1} ∉ _{(B−{b})−{a}}(_{1} ≠ _{(B−{b})−{a}}(_{(B−{b})−a}(_{B}_{−{a}}(_{B}_{−{b}}(_{π}_{B}_{−{b}}(_{B}_{B}_{−{b}}(

_{B}_{Q}

For every _{i}_{y}_{i}_{d}⇔ _{y}_{i}_{d}. Since _{y}_{y}_{y}_{y}_{B}_{Q}

_{B}_{Q}

The same is true; here we will not prove it.

_{B}_{Q}

_{i}_{i} is a superfluous element relative to d in π, if and only if LH_{i}

If _{i}_{π}_{π}_{−{Ai}}(_{i}_{i}

If _{i}_{i}_{i}_{π}_{π}_{−{Ai}}. Therefore, _{i}

_{i}_{i} is dispensable relative to d in π, if and only if LH_{i}

_{i}_{i} is independent relative to d in π, if and only if LH_{i}

Suppose _{x}_{x}_{x}_{x}

Since

If (1) met _{x}_{d}, _{x}_{d} and _{x}_{x}

If (1) only met _{x}_{x}

Therefore,

Here we will not prove it.

Suppose _{x}_{x}_{x}_{x}_{x}_{x}_{x}_{x}_{x}_{x}_{x}_{x}_{x}_{1}(_{1} ∈ _{y}_{1}(_{1} ∈ _{x}_{1} ⊄ _{y}_{1}, if _{x}_{x}_{x}_{x}

Similarly, we can prove the following theorem:

From the description of Theorem 13 through Theorem 16, we know that there exists corresponding relation for one to one between rough of knowledge and information entropy in inconsistent covering decision system.

Well-known scholars in Poland initially proposed discernibility matrix [11], or discernibility function can be used to calculate all attribute reduction in the decision table. Although the resolution matrix and its approach have been widely used, but due to the definition of resolution matrix, the data regarding the degree of inconsistency and its effects are not fully taken into account, so there are limitations. Literature [4, 20, 21] improved methods and discussed the case of inconsistent decision table, so that the former method can obtain the correct (all) attribute reduction results. Hence, such research is of great significance and, ultimately, a new application used in inconsistent decision tables to distinguish Matrices. Further, a method is proposed to distinguish matrix in the literature [31] based on the past, that is, split-based strategies and to distinguish Matrices decision table attribute reduction. Literature [19, 30] presents rough set theory, algorithms and applications, but also specifically pointed out that the resulting matrix method to distinguish demand for inconsistent decision tables is relatively simple errors in the nuclear, and also presented the results of a detailed analysis. Please refer to literature [19, 30].

Supposing _{X inU/d}

Suppose _{1} → _{1} and inconsistent covering sub-table _{2} → _{2}. We denote a _{ij}_{u}, x_{j}

Supposing _{ij}_{ij}_{d}_{ij}_{ij}_{d}

We denote _{i}_{x}_{i}_{ij}_{i}_{j}_{i}_{yj}_{xj}

Suppose _{j}_{s}_{d})(_{d}_{j}_{d}_{j}_{d}_{j}_{j}_{y}_{d}) and _{xj}_{t}_{d}) ≠ ∅.

By Definition 12, _{i}_{j}_{xj}_{i}_{d}

On the other hand, if _{i}_{d}_{s}_{i}_{d})(

(1) there exists _{xs}_{j}_{d} ≠ ∅;

(2) _{xs}_{x}_{∈}_{U} ([_{d})) ≠ ∅

If Condition (1) is true, then there exists _{t}_{j}_{d} and _{j}_{i}_{xsj}_{xij}_{xsi}_{xti}_{s}_{i}_{d}), _{t}_{t}_{d}), then we can obtain _{d}_{s}_{d}_{t}_{s}_{t}_{st}_{i}

If Condition (2) is true, then there exists _{r}_{j}_{i}_{xsj}_{xrj}_{xsi}_{xri}_{s}_{y}_{sr}_{i}_{d}

_{1} ⊆ _{π}_{π−{E1}(d1)} _{ij}_{ij}_{1}

If there exists _{1} ⊆ _{1} does not belong to _{d}_{1} can be reduced. If for every _{ij}_{ij}_{1}, which still ensure _{π}_{π}_{−{E1}}(_{1}) after reduction of properties on _{1} at the same time.

If there exists _{ij}_{1} contemporarily such that _{ij}_{ij}_{1} cannot be reduced at the same time.

Theorem 18 implies that as long as there is a simple observation and treatment for discernibility matrix, we will have cores and reducts in the inconsistent covering decision system. The following Corollary 7 can be founded in the literature [21, 25].

Suppose _{ij}_{ij}

Here is a car which is to be considered for analysis. Suppose _{1}, … , _{10}} to be a set of ten cars, and

For attribute

For attribute

For attribute

For attribute

We think that the evaluation of every index is has the same importance. Therefore, we get a cover rather than a partition for every car attribute, which implies a certain uncertainty caused by the interpretation of the data.

Final decision _{1}, _{2}, _{4}, _{6}}, further evaluation for sale= {_{4}, _{5}, _{7}}, against sale= {_{1}, _{2}, _{4}, _{6}}. Suppose _{i}_{xi}_{i}_{i}_{xi}

The positive domain of

Supposing _{1}, then _{1} = {_{1}, _{2}, _{3}, _{6}}, _{2} = _{3} = {_{2}, _{3}, _{6}}, _{4} = _{5} = {_{2}, _{3}, _{4}, _{5}, _{6}, _{7}}, _{6} = {_{6}}, _{7} = {_{6}, _{7}}, _{8} = {_{6}, _{8}, _{9}}, _{9} = {_{5}, _{9}}, _{10} = {_{10}}, the positive domain of

As to (1) and (2), we could obtain _{B}_{π}_{i}

The uppermost relation of _{2}, _{4} to _{i}_{1} is a superfluous element of

Suppose _{1}, _{2}, _{3}, _{6}, _{10}} and _{4}, _{5}, _{7}, _{8}, _{9}}. By the Definition 12, we have the discernibility matrix of inconsistent covering decision system (_{i}_{1}, _{2}, _{3}, _{1}) = ∧{∨_{ij}_{ij}_{1} ∨ _{3} ∨ _{4}) ∧ (_{2} ∨ _{4}) ∧ (_{3} ∨ _{4}) ∧ (_{2} ∨ _{4}) ∧ (_{1} ∨ _{3} ∨ _{4}) ∧ ((_{1} ∨ _{3}) ∨ (_{2} ∨ _{3}) ∨ _{3} ∨ _{4}) ∧ (_{1} ∨ _{2} ∨ _{4}) ∧ ((_{2} ∧ _{3}) ∨ _{3} ∨ _{4}) ∧ (_{1} ∨ _{3} ∨ _{4}) ∧ (_{2} ∨ _{3} ∨ _{4}) ∧ (_{3} ∨ _{4}) ∧ (_{1} ∨ _{2} ∨ _{3} ∨ _{4}) ∧{_{2}} = (_{2} ∧_{4}) ∨ (_{2} ∧_{3}), so

It should be pointed out that if the covering decision system is consistent, then the method proposed in this section is equivalent to the one in Section 4. If is a partition, then the method adopted in this section is just the method for computing relative reducts of traditional rough sets in the literature [32] to ensure that we find the smallest reduction.

If these ten cars are trial samples, then we have two different kinds of evaluation references for other input samples: {colour; oil-consumption}, {colour; quality}. Clearly, the attribute is the key attribute for the evaluation of cars.

To illustrate the methods of space and computational complexity in the section, we will compare our methods with the methods of literature [21, 23, 24], such that if _{1}| + |_{2}| satisfies,

(1) The space complexity can be compared: without considering compression and storage of symmetric matrix, the elements of discernibility matrix in the section are |_{1}| ×

(2) The computational complexity can be compared: the computational complexity in literature [21, 23, 24] is ^{n}) + ^{2}), while the method in this section is ^{n}) + _{1}| ×

It is evident that the space and computational complexity in the section are lower than the literature [21]. Therefore, the methods in this section could be used effectively not only to reduce the computational cost, but also in providing a new framework to certain extent based on the covering rough sets theory.

According to classical rough sets theory, attributes of decision systems consist of two parts namely conditional attributes and decision attributes. Every conditional attribute decide a partition in a complete decision system. The abstract information systems which come from reality problems are mostly incomplete decision system. Every conditional attribute in this decision system determines a cover of U. This paper mainly studies theories and methods of systems and discusses about related attribute reduction for covering decision information reduction algorithms on the basis of conditional information entropy. Moreover, attribute reduction of covering decision systems also have wide applications in the three-way, which indicates the importance of the direction of research currently.

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Beijing–Tianjin–Hebei regional government guidance fund based on the entropy method The transfer of stylised artistic images in eye movement experiments based on fuzzy differential equations Research on behavioural differences in the processing of tenant listing information: An eye-movement experiment A review of the treatment techniques of VOC Some classes of complete permutation polynomials in the form of ( x ^{pm}−x +δ )^{s}+ax ^{pm}+bx overF _{p2m}The consistency method of linguistic information and other four preference information in group decision-making Research on the willingness of Forest Land’s Management Rights transfer under the Beijing Forestry Development A mathematical model of the fractional differential method for structural design dynamics simulation of lower limb force movement step structure based on Sanda movement Fractal structure of magnetic island in tokamak plasma Numerical calculation and study of differential equations of muscle movement velocity based on martial articulation body ligament tension Study on the maximum value of flight distance based on the fractional differential equation for calculating the best path of shot put Sports intensity and energy consumption based on fractional linear regression equation Analysis of the properties of matrix rank and the relationship between matrix rank and matrix operations Study on Establishment and Improvement Strategy of Aviation Equipment Research on Financial Risk Early Warning of Listed Companies Based on Stochastic Effect Mode Characteristics of Mathematical Statistics Model of Student Emotion in College Physical Education Mathematical Calculus Modeling in Improving the Teaching Performance of Shot Put Application of Nonlinear Differential Equation in Electric Automation Control System Nonlinear strategic human resource management based on organisational mathematical model Higher Mathematics Teaching Curriculum Model Based on Lagrangian Mathematical Model Optimization of Color Matching Technology in Cultural Industry by Fractional Differential Equations The Marketing of Cross-border E-commerce Enterprises in Foreign Trade Based on the Statistics of Mathematical Probability Theory The Evolution Model of Regional Tourism Economic Development Difference Based on Spatial Variation Function The Inner Relationship between Students' Psychological Factors and Physical Exercise Based on Structural Equation Model (SEM) Fractional Differential Equations in Sports Training in Universities Higher Education Agglomeration Promoting Innovation and Entrepreneurship Based on Spatial Dubin Model