Reasoning about attribute-relative approximations in multi-source environments: a modal framework with axiomatization

Abstract The study of rough set theory in multi-agent/multi-source contexts, and its extension to attribute-relative approximations, constitutes a significant line of research in the rough set literature. Motivated by these, we propose a unified modal framework based on multiple-source information structures, where accessibility relations are indexed by sources and parameterized by attribute sets. We introduce two semantics, W-semantics and S-semantics, interpreting modal formulas via weak and strong approximation operators from multi-granulation rough set models. These provide a formal system for reasoning about uncertainty that is both attribute-relative and source-sensitive. We establish sound and complete axiomatizations for the corresponding logics over classes of models satisfying structural properties like reflexivity, symmetry and transitivity. Our completeness proofs use a step-by-step construction adapted to the semantics, avoiding reliance on copying techniques. This work deepens the modal foundations of rough set theory and contributes to the logic of complex approximation systems.