Systematic attribute reductions based on double granulation structures and three-view uncertainty measures in interval-set decision systems

Attribute reductions eliminate redundant information to become valuable in data reasoning. In the data context of interval-set decision systems (ISDSs), attribute reductions rely on granulation structures and uncertainty measures; however, the current structures and measures exhibit the singleness limitations, so their enrichments imply corresponding improvements of attribute reductions. Aiming at ISDSs, a fuzzy-equivalent granulation structure is proposed to improve the existing similar granulation structure, dependency degrees are proposed to enrich the existing condition entropy by using algebra-information fusion, so $3\times 2$ attribute reductions are systematically formulated to contain both a basic reduction algorithm (called CAR) and five advanced reduction algorithms. At the granulation level, the similar granulation structure is improved to the fuzzy-equivalent granulation structure by removing the granular repeatability, and two knowledge structures emerge. At the measurement level, dependency degrees are proposed from the algebra perspective to supplement the condition entropy from the information perspective, and mixed measures are generated by fusing dependency degrees and condition entropies from the algebra-information viewpoint, so three-view and three-way uncertainty measures emerge to acquire granulation monotonicity/non-monotonicity. At the reduction level, the two granulation structures and three-view uncertainty measures two-dimensionally produce $3\times 2$ heuristic reduction algorithms based on attribute significances, and thus five new algorithms emerge to improve an old algorithm (i.e., CAR). As finally shown by data experiments, $3\times 2$-systematic construction measures and attribute reductions exhibit the effectiveness and development, comparative results validate the three-level improvements of granulation structures, uncertainty measures, and reduction algorithms on ISDSs. This study resorts to tri-level thinking to enrich the theory and application of three-way decision.