As an essential way of knowledge representation and knowledge acquisition in uncertainty theory, variable precision rough set (VPRS) has been widely considered by researchers. Firstly, the corresponding CVPRS model via boundary region is investigated based on an existing covering-based rough set model. Its algebraic structures and properties are systematically studied. This model mainly depicts a pair of boundary operators and a pair of approximation operators with a certain threshold from the perspective of single knowledge granularity. And it not only relaxes the complete inclusion relation but also retains the basic properties of the classical rough set. Secondly, an attribute reduction approach is given for a covering-based decision information systemusing the proposed CVPRS model, which keeps the lower or upper approximation unchanged. The different performances of boundary operators and their related three indices in these reduction methods are compared. The results show that boundary operators and distribution functions have outstanding advantages in these applications. Finally, an extraction method of necessity rules and possibility rules corresponding to decision classes are established. The validity and security of the above methods are verified theoretically. Numerical experiments further show that the proposed models and methods have a certain application value.

作为不确定性理论中知识表示和知识获取的重要方式，变精度粗糙集（VPRS）已被研究人员广泛考虑。首先，在现有的基于覆盖的粗糙集模型的基础上，研究了通过边界区域对应的CVPRS模型。其代数结构和性质进行了系统的研究。该模型主要从单一知识粒度的角度描述了一对边界算子和一对具有一定阈值的逼近算子。它不仅放宽了完全包含关系，而且保留了经典粗糙集的基本性质。其次，使用所提出的CVPRS模型为基于覆盖的决策信息系统提供了一种属性约简方法，该方法保持下近似值或上近似值不变。比较了这些归约方法中边界算子的不同性能及其相关的三个指标。结果表明，边界算子和分布函数在这些应用中具有突出的优势。最后，建立了决策类对应的必要性规则和可能性规则的提取方法。从理论上验证了上述方法的有效性和安全性。数值实验进一步表明所提出的模型和方法具有一定的应用价值。