Optimizations of approximation operators in covering rough set theory

Classical rough set theory fundamentally requires upper and lower approximations to be definite sets for precise knowledge representation. However, a significant problem arises as many widely used approximation operators inherently produce rough approximations (with non-empty boundaries), contradicting this core theoretical intent and undermining practical applicability. To resolve this core discrepancy, we introduce stable approximation operators and stable sets, and develop an optimization method that transforms unstable operators into stable ones, ensuring definite approximations. This method includes detailing the optimization process with algorithmic implementation, analyzing the topological structure of resulting approximation spaces and connections between optimized operators, and enhancing computational efficiency via matrix-based computation. This work may strengthen rough set theory's foundation by bridging the gap between theory and practice while enhancing its scope for practical applications.