Three-layer granular structures and three-way informational measures of a decision table

Attribute reduction in rough set theory serves as a fundamental topic for information processing, and its basis is usually a decision table (D-Table). D-Table attribute reduction concerns three hierarchical types, and only classification-based reduction is related to information-theoretic representation. Aiming at inducing comprehensive D-Table attribute reduction with hierarchies and information, this paper concretely constructs a D-Table's three-layer granular structures and three-way informational measures via granular computing and Bayes' theorem. With regard to the D-Table, the micro-bottom, meso-middle, and macro-top are hierarchically organized according to the formal structure and systematic granularity. Then, different layers produce different three-way informational measures by developing Bayes' theorem. Thus, three-way weighted entropies originate from three-way probabilities at the micro-bottom and further evolve from the meso-middle to the macro-top, and their granulation monotonicity and evolution systematicness are acquired. Furthermore, three-way informational measures are analyzed by three-layer granular structures to achieve their hierarchical evolution, superiority, and algorithms. Finally, structural and informational results are effectively illustrated by a D-Table example. This study establishes D-Table's hierarchical structures to reveal constructional mechanisms and systematic relationships of informational measures. The obtained results underlie the D-Table's hierarchical, systematic, and informational attribute reduction, and they also enrich the three-way decisions theory.