Introduction

Covering rough set theory, pioneered by Zakowski [62] in 1983, has become a useful mathematical tool for dealing with uncertain and imprecise information in practical situations. As a substantial con- stituent of granular computing, covering-based rough set theory has been applied to many fields such as feature selection and data mining without any prior knowledge. Especially, covering rough set theory is being attracting more and more attention in the era of artificial intelligence, which provides powerful supports for the development of data processing technique. Many researchers [1, 5, 8–۱۱, ۱۵, ۱۷–۲۲, ۲۵, ۲۷, ۲۸, ۳۱, ۳۴, ۴۱, ۴۲, ۴۸–۵۰, ۵۲–۵۶, ۵۸–۶۱, ۶۴–۶۹] have studied covering-based rough set theory. For example, Hu et al. [8] proposed a matrix representation of multigranulation approximations in optimistic and pessimistic multigranulation rough sets and matrixbased dynamic approaches for updating approximations in multigranulation rough sets when a single granular structure evolves over time. Lang et al. [15] presented incremental approaches to computing the second and sixth lower and upper approximations of sets in dynamic covering approximation spaces. Luo et al. [29] investigated the updating properties for dynamic maintenance of approximations when the criteria values in the set-valued decision system evolve with time and proposed two incremental algorithms for computing rough approximations with the addition and removal of criteria values. Wang et al. [49] transformed the set approximation computation into products of the type-1 and type-2 characteristic matrices and the characteristic function of the set in covering approximation spaces. Yang et al. [54] provided a new type of fuzzy covering-based rough set model by introducing the notion of fuzzy β-minimal description and generalized the model to L-fuzzy covering-based rough set which is defined over fuzzy lattices. Yang et al. [56] provided related family-based methods for computing attribute reducts and relative attribute reducts for covering rough sets, which remove superfluous attributes while keeping the approximation space of covering information system unchanged. Yao et al. [61] classified all approximation operators into element-based approximation operators, granule-based approximation operators, and subsystem-based approximation operators. Knowledge reduction of dynamic information systems [2–۴,۶,۷,۱۲–۱۴,۱۶,۱۹,۲۳,۲۴,۲۶,۲۹,۳۰,۳۲,۳۳, ۳۵, ۳۶, ۳۸–۴۰, ۴۳–۴۷, ۵۱, ۵۷, ۶۳, ۶۶] has attracted more attention. For example, Chen et al. [3] employed an incremental manner to update minimal elements in the discernibility matrices at the arrival of an incremental sample. Huang et al. [12] presented the extended variable precision rough set model based on the λ-tolerance relation in terms of Bhattacharyya distance and incremental mechanisms by the utilization of previously learned approximation results and region relation matrices for updating rough approximations in set-valued information systems. Lang et al. [14] focused on knowledge reduction of dynamic covering information systems with variations of objects using the type-1 and type-2 characteristic matrices. Li et al. [19] discussed the principles of updating P-dominating sets and P-dominated sets when some attributes are added into or deleted from the attribute set P. Luo et. al [30] analyzed the dynamic characteristics of conditional partition and decision classification on the universe when the insertion or deletion of objects occurs and presented incremental algorithms for updating probabilistic approximations, which are proficient to efficiently classify the incremental objects into decision regions by avoiding re-computation efforts. Qian et al.