Rough Sets and Rule Induction from Indiscernibility Relations Based on Possible World Semantics in Incomplete Information Systems with Continuous Domains

Rough sets and rule induction in an incomplete and continuous information table are investigated under possible world semantics. We show an approach using possible indiscernibility relations, whereas the traditional approaches use possible tables. This is because the number of possible indiscernibility relations is finite, although we have the infinite number of possible tables in an incomplete and continuous information table. First, lower and upper approximations are derived directly using the indiscernibility relation on a set of attributes in a complete and continuous information table. Second, how these approximations are derived are described applying possible world semantics to an incomplete and continuous information table. Lots of possible indiscernibility relations are obtained. The actual indiscernibility relation is one of possible ones. The family of possible indiscernibility relations is a lattice for inclusion with the minimum and the maximum indiscernibility relations. Under the minimum and the maximum indiscernibility relations, we obtain four kinds of approximations: certain lower, certain upper, possible lower, and possible upper approximations. Therefore, there is no computational complexity for the number of values with incomplete information. The approximations in possible world semantics are the same as ones in our extended approach directly using indiscernibility relations. We obtain four kinds of single rules: certain and consistent, certain and inconsistent, possible and consistent, and possible and inconsistent ones from certain lower, certain upper, possible lower, and possible upper approximations, respectively. Individual objects in an approximation support single rules. Serial single rules from the approximation are brought into one combined rule. The combined rule has greater applicability than single rules that individual objects support.