A formal study of a rough set model integrating relational and neighbourhood system approaches

This article introduces a rough set model that integrates two approximation operators - one induced by a relation and the other by a neighbourhood system - within a unified framework. The proposed model formalizes concept approximation in heterogeneous settings where multiple granulation mechanisms contribute to information processing. The framework effectively combines the strengths of relation-based and neighbourhood-based rough set models by employing conjunctive and disjunctive fusion rules. To provide a rigorous foundation, we develop a logical study of the resulting approximation operators using the modal language with two unary modal operators. We introduce semantics that fuses Kripke and neighbourhood interpretations of modal operators, establish sound and complete deductive systems, and investigate definability properties. This study contributes to both rough set theory and modal logic by offering a formal perspective on the fusion of approximation operators.