Min-max attribute-object bireducts: On unifying models of reducts in rough set theory

A decision table describes a finite set of objects OB by using a finite set of condition attributes C and a finite set of decision attributes D. Pawlak defines attribute reducts by considering the entire decision table. As a generalization, we introduce the notion of min-max attribute-object bireducts of a sub-table restricted by a pair (B, X) of a subset of condition attributes B and a subset of objects X. A pair (R, Z) is a min-max attribute-object bireduct of (B, X) if and only if R is a minimal subset of B such that R and B make the same correct decisions for objects in X and Z is a maximal subset of X for which B can make the correct decisions. We propose the notion of the decidability of objects and introduce the decidable region of a set of objects as a generalization of the positive region of the set. We define and interpret a min-max attribute-object bireduct based on the decidable region. Min-max attribute-object bireducts offer a general model and existing models of attribute reducts are special cases. The results lead to a unified framework for studying four types of attribute reducts.