Approach to approximate distribution reduct in incomplete ordered decision system

The original rough set model cannot be used to deal with the incomplete information systems. Nevertheless, by relaxing the indiscernibility relation to more general binary relations, many improved rough set models have been successfully applied into the incomplete information systems for knowledge acquisition. This article presents an explorative research focusing on the transition from incomplete decision system to a more complex system---the incomplete ordered decision system. In such incomplete decision system, all attributes have preference-ordered domains. With introduction of the concept of approximate distribution reduct into the incomplete ordered decision system, four new notions of approximate distribution reduct are proposed. They are the minimal subsets of condition attributes, which preserve lower and upper approximations of all upward unions and downward unions of the decision classes respectively. The judgment theorems and discernibility matrices associated with these approximate distribution reducts are also obtained. For further illustration, an example is analyzed. The research is meaningful both in the theory and in applications for the acquisition of rules in complex information systems.