Knowledge Dependency and Rule Induction on Tolerance Rough Sets

Classical rough set theory(RST) is based on equivalence relations. Tolerance relations are more generic than equivalence relations. We extend some concepts in classical RST to tolerance relations by proposing that the knowledge representation in rough set models based on tolerance relations, such as weak, strong and central dependency, as well as the relationships among them. A general complete theorem about knowledge representation is given. We give formal proofs of the theorem and verify its correctness with some examples. A case study is presented to show how to extract certain rules from an incomplete information table. It is more elaborate than the restriction of equivalence relations for the classical rough set theory. The proposed approach is indeed effective, and therefore of practical value to many real-world problems.