Extension of rough set under incomplete information systems

The classical rough set theory is based on complete information systems. It classifies objects using upper-approximation and lower-approximation defined on an indiscernibility relation that is a kind of equivalent relation. In order to process incomplete information systems, the classical rough set theory needs to be extended, especially, the indiscernibility relation needs to be extended to some inequivalent relation. There are several extensions for the indiscernibility relation at present, such as tolerance relation, non-symmetric similarity relation, and valued tolerance relation. Unfortunately, these extensions have their own limitation. We develop a new extension of rough set theory that is based on a limited tolerance relation