Rough Sets Based on Possible World Semantics in Incomplete and Continuous Information Tables

Abstract

Rough sets under incomplete information with continuous domains are examined based on possible world se- mantics. We focus on possible indiscernibility relations, although the traditional approaches are done under possible tables. This is because we only obtain a finite number of possible indiscernibility relations even if an infinite number of possible tables are derived from an incomplete information table. A possible indiscernible class for an object is derived from every possible indiscernibility relation. The family of possibly indiscernible classes for the object is a lattice for inclusion. Lower and upper approximations are derived from using the minimal and the maximal elements in the lattice. Therefore, there is no computational complexity for the number of objects with incomplete information. Furthermore, 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 indiscernible classes. Therefore, the approach developed in this paper justifies our extended approach.