Constructing the Optimal Approximation Sets of Rough Sets in Multi-granularity Spaces

Rough set theory is an important tool to solve the uncertain problems. How to use the existing knowledge granules to approximately describe an uncertain target concept X has been a key issue. However, current research on theories and methods is still not comprehensive enough. R-0.5(X), a kind of approximation sets of an uncertain concept, was proposed and analyzed in detail in our previous research work. However, whether R-0.5(X) is the optimal approximation set of an uncertain concept X is still unable to determine. As a result, in this paper, based on the approximation of an uncertain concept, the existence of the optimal approximation set is explored. Then an optimal approximation set R-Best(X) is proposed and discussed. At first, the definition of R-Best(X) is defined. Then several comparative analysis between R-Best(X) and other approximation sets is carried out. Next, operation properties of R-Best(X) are presented and proved respectively. Finally, with changing knowledge granularity spaces, the change rules of the similarity between an uncertain set X and its R-Best(X) are revealed.