Rough Sets Based on Possible Indiscernibility Relations in Incomplete Information Tables with Continuous Values

Rough sets under incomplete information with continuous domains are examined on the basis of possible world semantics. We show an approach under possible indiscernibility relations, although the traditional approaches are done under possible tables. This is because the number of possible indiscernibility relations is finite, even if the number of possible tables is infinite. First, lower and upper approximations are described using the indiscernibility relation on an attribute in a complete information table. Second, these are addressed in an incomplete information table under possible world semantics. Two types of indiscernibility relations; namely, certain and possible ones, are obtained on an attribute in an information table. The actual indiscernibility relation is one of possible ones. The family of indiscernibility relations is a lattice for inclusion. The minimal element is the certain indiscernibility relation while the maximal one is the maximal possible indiscernibility relation. By using certain and possible indiscernibility relations, we obtain four types of approximations: certain lower, certain upper, possible lower, and possible upper approximations. The approach based on possible world semantics gives the same approximations as ones obtained from our extended approach, which is proposed in the previous work directly using indiscernibility relations.