Algorithms and analyses of logical difference of graded lower and upper approximation operators

Grade serves as a fundamental quantitative index, and the graded rough set model is a basic model. Thus, this paper aims to investigate an original logical difference in the graded rough set model. According to the specific logical requirement on the grade index, the logical difference is put forward based on the graded lower and upper approximation operators; the new concept has the practical logical quantitative meaning. Then, both the macroscopic essence and microscopic fundamental structure are obtained for the new concept. For calculation, two algorithms, the conventional and microscopic algorithms are proposed and analyzed emphatically, and we conclude that the microscopic algorithm has more advantages in time and space complexity. Finally, the logical difference and its algorithms are both illustrated by an example.