Rough Sets in Incomplete Information Systems with Order Relations Under Lipski's Approach

Rough sets and rule induction based on them are described in incomplete information tables where attribute values are ordered. We apply possible world semantics to an incomplete information table, as Lipski did in incomplete databases. The set of possible tables on a set of attributes is derived from the original incomplete information table. Rough sets, a pair of lower and upper approximations, are obtained from every possible table. An object is certainly included in an approximation when it is in the approximation in all possible tables, while an object is possibly included in an approximation when it is in the approximation in some possible tables. From this, we obtain certain and possible approximations. The actual approximation is greater than the certain one and less than the possible one. Finally, we obtain the approximation in the form of interval sets. There exists a gap between rough sets and rule induction from them. To bridge rough sets and rule induction, we give expressions that correspond to certain and possible approximations. The expressions consist of a pair of an object and a rule that the object supports. Consequently, four types of rule supports: certain and consistent, certain and inconsistent, possible and consistent, and possible and inconsistent supports, are obtained from the expressions. The formulae can be applied to the case where not only attributes used to approximate but also attributes approximated have a value with incomplete information. The results give a correctness criterion of rough sets and rule induction based on them in incomplete ordered information systems, as the results of Lipski's work are so in incomplete databases.