Rough Approximations from Indiscernibility Relations under Incomplete Information

Rough sets and rule induction are formulated by directly using indiscernibility relations in information tables. First, we describe them in information tables with complete information. Second, they are shown on the basis of possible world semantics from the viewpoint of certainty and possibility, as was done by Lipski in the field of databases, in order to examine the fundamentals of rough sets and rule induction under incomplete information. As a result, we have the four approximations: certain lower, certain upper, possible lower, and possible upper ones. The certain lower and upper approximations are the lower bounds of the actual ones and also the possible lower and upper ones the upper bounds. Using these four approximations, the lower and upper approximations are expressed by interval sets. Third, the approach based on indiscernibility relations is extended in the case where objects that are characterized by incomplete information approximate a set of objects with incomplete information. The extended approach derives the same rough sets and rule induction as ones obtained from possible world semantics. This justifies our extension.