Covering-based rough set is an important extended type of classical rough set model. In this model, concepts are approximated through substitution of a partition in classical rough set theory with a covering in covering-based rough set theory. Various generalized covering-based rough sets have been investigated, however, little work has been done on extending four classical covering-based rough set to intuitionistic fuzzy (IF) settings. In this study, four novel IF covering-based rough set models are developed by combining an IF $$\beta $$ -covering with four classical covering-based rough set models. First, we present the concept of IF $$\beta $$ -minimal description, and then construct four order relations on IF $$\beta $$ approximation space. Second, we propose four IF $$\beta $$ -covering-based rough set models and derive that they are generalizations of four existing covering-based rough sets in IF settings. We also discuss the properties of these IF $$\beta $$ -covering-based rough sets and reveal their relationships. We use the existing distance between two IF sets to characterize the uncertainty of the presented IF $$\beta $$ -covering-based rough sets. Third, we define the reducts of IF $$\beta $$ -covering decision systems and examine their discernibility-function-based reduction methods for these IF $$\beta $$ -covering-based rough sets. Fourth, we present four optimistic and pessimistic multi-granulation IF $$\beta $$ -covering-based rough sets and analyze their properties and uncertainty measures from multi-granulation perspective. Fifth, we study the discernibility-function-based reduction methods for the presented multi-granulation IF $$\beta $$ -covering-based rough sets. Finally, we discuss another two neighborhood-based IF covering-based rough sets. This study can provide a covering-based rough set method for acquiring knowledge from IF decision systems.

中文翻译：

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直觉模糊 $$\beta $$ β -covering-based 粗糙集

基于覆盖的粗糙集是经典粗糙集模型的重要扩展类型。在这个模型中，概念是通过用基于覆盖的粗糙集理论中的覆盖替代经典粗糙集理论中的分区来近似的。已经研究了各种基于覆盖的广义粗糙集，然而，在将四种经典的基于覆盖的粗糙集扩展到直觉模糊（IF）设置方面几乎没有做任何工作。在这项研究中，通过将 IF $$\beta $$ -covering 与四个经典的基于覆盖的粗糙集模型相结合，开发了四个新颖的​​基于 IF 覆盖的粗糙集模型。首先，我们提出IF $$\beta $$ - 极小描述的概念，然后在IF $$\beta $$ 逼近空间上构造四阶关系。第二，我们提出了四个基于 IF $$\beta $$ -covering-based 的粗糙集模型，并推导出它们是 IF 设置中四个现有的基于覆盖的粗糙集的推广。我们还讨论了这些基于 IF $$\beta $$ -covering 的粗糙集的性质并揭示了它们的关系。我们使用两个 IF 集之间的现有距离来表征所呈现的 IF $$\beta $$ -covering-based 粗糙集的不确定性。第三，我们定义了 IF $$\beta $$ -covering 决策系统的约简，并针对这些 IF $$\beta $$ -covering-based 粗糙集检查了它们基于可识别性函数的约简方法。第四，我们提出了四种乐观和悲观的多粒度 IF $$\beta $$ -covering-based 粗糙集，并从多粒度的角度分析了它们的性质和不确定性度量。第五，我们研究了所提出的多粒度 IF $$\beta $$ -covering-based 粗糙集的基于辨别函数的约简方法。最后，我们讨论另外两个基于邻域的 IF 覆盖粗糙集。本研究可为从 IF 决策系统获取知识提供一种基于覆盖的粗糙集方法。