Reduction target structure-based hierarchical attribute reduction for two-category decision-theoretic rough sets

Attribute reduction is an essential subject in rough set theory, but because of quantitative extension, it becomes a problem when considering probabilistic rough set (PRS) approaches. The decision-theoretic rough set (DTRS) has a threshold semantics and decision feature and thus becomes a typical and fundamental PRS. Based on reduction target structures, this paper investigates hierarchical attribute reduction for a two-category DTRS and is divided into five parts. (1) The knowledge-preservation property and reduct are explored by knowledge coarsening. (2) The consistency-preservation principle and reduct are constructed by a consistency mechanism. (3) Region preservation is analyzed, and the separability between consistency preservation and region preservation is concluded; thus, the double-preservation principle and reduct are studied. (4) Structure targets are proposed by knowledge structures, and an attribute reduction is further described and simulated; thus, general reducts are defined to preserve the structure targets or optimal measures. (5) The hierarchical relationships of the relevant four targets and reducts are analyzed, and a decision table example is provided for illustration. This study offers promotion, rationality, structure, hierarchy and generalization, and it establishes a fundamental reduction framework for two-category DTRS. The relevant results also provide some new insights into the attribute reduction problem for PRS.