Construction of Basic Probability Assignment for D-S Rough Sets Based on Inclusion-Degree Gradient

Dempster–Shafer evidence theory and rough set theory are two important mathematical tools for dealing with uncertain information, and their integration provides a powerful theoretical framework for quantifying cognitive uncertainty. The classical D-S rough set model usually adopts the basic probability assignment BPA construction method based on the cardinality of equivalence classes, which fails to fully explore the correlation information between the internal structure of knowledge granularity and the target concept, thus limiting its representation and discrimination ability. To break through this limitation, this paper proposes a novel BPA construction theory based on the inclusion-degree gradient, aiming to significantly enhance the flexibility and semantic richness of the model by finely modeling the “partial inclusion” relationship. Following the progressive logic of “from classic to extended”, this study systematically constructs a hierarchical extended model system of D-S rough sets. First, within the classical framework, the equivalent relationships between the lower approximation and belief function, and the upper approximation and plausibility function are clarified. The core innovation lies in reconstructing the traditional fixed BPA assignment into a dynamic weighting process with inclusion degree as the core variable. Then, this innovative framework is systematically extended to variable precision rough sets and probabilistic rough sets: in variable precision rough sets, a continuous weight function with parameter β as the threshold is designed to realize adaptive reward for knowledge granularity with high inclusion degree, thus obtaining the β-weighted BPA; in probabilistic rough sets, a three-region BPA assignment model of positive region, boundary region, and negative region based on (α,β) threshold division is constructed, and the differentiated mass functions of evidence for each region are given. Through rigorous mathematical derivation, the proposed models are shown to satisfy the basic axioms of Dempster–Shafer evidence theory and to possess theoretical properties including monotonicity, continuity, and compatibility with classical constructions. The numerical example further illustrates how the introduction of inclusion-degree information redistributes evidence mass and changes the resulting belief–plausibility intervals. In particular, under the considered illustrative setting, the β-weighted BPA model produces a more concentrated belief assignment and a narrower belief–plausibility interval than the classical cardinality-based construction, while the three-region BPA model explicitly represents boundary-induced nonspecific evidence. These numerical results characterize changes in the representation and concentration of evidential uncertainty and should not be interpreted as evidence of improved predictive or diagnostic accuracy.