Attribute reduction is a popular topic in research on rough sets. In the classical model, much progress has been made in the study of the attribute reduction of indiscernibility and discernibility relations. To enhance the fault tolerance of the model, concepts of both indiscernibility and discernibility relations involving uncertain or imprecise information are proposed in this paper. The attribute reductions of the relative -indiscernibility relation and relative -discernibility relation and their algorithms are proposed. When the precision satisfies certain conditions, the reduction of two relation concepts can be converted into a positive region reduction. Therefore, the discernibility matrix is used to construct the reductions of the two relation concepts and the positive region. Furthermore, the corresponding algorithm of the relative -indiscernibility (discernibility) relation reduction can be optimized when the precision is greater than 0.5, and this is used to develop an optimization algorithm that constructs the discernibility matrix more efficiently. Experiments show the feasibility of the two relation reduction algorithms. More importantly, the reduction algorithms of the two relations and the optimization algorithm are compared to demonstrate the feasibility of the optimization algorithm proposed in this paper.

Variable precision reduction (VPR) and positive region reduction (PRR) are common definitions in attribute reduction. The compacted decision table is an extension of a decision table. In this paper, we propose another extension, called the weighted decision table. In both types of decision tables, VPR is defined, and the corresponding discernibility matrices for the PRR are proposed. Then, algorithms for obtaining the PRR from the discernibility matrices are presented. In both types of decision tables, the relationship between VPR and PRR is established by comparing the corresponding discernibility matrices. If the precision of the VPR meets the given conditions, then the PRR algorithms can be used to obtain the results after modifying some decision values in the decision tables. An analysis of the modification process of the decision values and the compression process of decision tables is used to propose a new algorithm for VPR in decision tables that ensures credibility. The effectiveness of the proposed algorithm was evaluated by an experimental comparison with existing VPR algorithms.

Attribute reduction comes from machine learning and is an important component of rough set theory. Research on attribute reduction has produced many important achievements. The aim of attribute reduction is to reduce the complexity of data while retaining its original characteristics to the greatest extent. The concept of attribute reduction is of great significance in machine learning research. In previous studies, a variety of attribute reduction definitions have been proposed according to different rules. Based on the binary relations among objects and local decision rules, this paper describes a local indiscernibility relation reduction for information tables. The discernibility matrix for the proposed reduction is established, and examples for single- and multi-decision classes are presented to illustrate that the proposed local indiscernibility relation reduction can be applied to decision tables. According to the reduction concept developed in this paper, and considering a heuristic algorithm for calculating the significance of attributes and a binary integer programming algorithm based on the discernibility matrix, three reduction algorithms are proposed. Experiments are conducted using four classifiers and a number of publicly available datasets. A comparison of the experimental results presented in this paper demonstrates the feasibility of the proposed algorithms.

Dynamic reduction algorithms have become an important part of attribute reduction research because of their ability to perform dynamic updates without the need to retrain the original model. To enhance the efficiency of variable precision reduction algorithms in processing dynamic data, research has been conducted from the perspective of the construction process of the discernibility matrix. By modifying the decision values of some samples through an absolute majority voting strategy, a connection between variable precision reduction and positive region reduction has been established. Considering the increase and decrease of samples, dynamic variable precision reduction algorithms have been proposed. For four cases of sample increase, four corresponding scenarios have been discussed, and judgment conditions for the construction of the discernibility matrix have been proposed, which has led to the development of a dynamic variable precision reduction algorithm for sample increasing (DVPRA-SI). Simultaneously, for the scenario of sample deletion, three corresponding scenarios have been proposed, and the judgment conditions for the construction of the discernibility matrix have been discussed, which has resulted in the development of a dynamic variable precision reduction algorithm for sample deletion (DVPRA-SD). Finally, the proposed two algorithms and existing dynamic variable precision reduction algorithms were compared in terms of the running time and classification precision, and the experiments demonstrated that both algorithms are feasible and effective.

Variable precision reduction (VPR) and positive region reduction (PRR) are common definitions in attribute reduction. The compacted decision table is an extension of a decision table. In this paper, we propose another extension, called the weighted decision table. In both types of decision tables, VPR is defined, and the corresponding discernibility matrices for the PRR are proposed. Then, algorithms for obtaining the PRR from the discernibility matrices are presented. In both types of decision tables, the relationship between VPR and PRR is established by comparing the corresponding discernibility matrices. If the precision of the VPR meets the given conditions, then the PRR algorithms can be used to obtain the results after modifying some decision values in the decision tables. An analysis of the modification process of the decision values and the compression process of decision tables is used to propose a new algorithm for VPR in decision tables that ensures credibility. The effectiveness of the proposed algorithm was evaluated by an experimental comparison with existing VPR algorithms.

Since Shannon put forward information entropy and used it to measure the amount of information in the information system, people began to explore various new methods to measure the uncertainty in the information system. Rough set is a method to solve uncertainty problems, and the measurement of knowledge uncertainty is an important content in the research of rough set theory. Many scholars have explored this from different perspectives. This paper analyzes the uncertainty of knowledge from a new perspective based on Pawlak's definition of the degree of knowledge uncertainty contained in approximate sets, and provides a new measure of knowledge uncertainty. Compared to existing similar measures, this measure not only better reflects the connotation of knowledge uncertainty in the approximation space described by Pawlak, but also is computationally feasible. The research helps people better understand the causes of uncertainty in approximation spaces, and expands and enhances the applicability of rough set theory.