Intuitionistic fuzzy \(\beta \)-covering-based rough sets

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.