Rough Sets

Rough set theory due to Zdzisław Pawlak (1926-2006) [106, 108, 109, 110], is a mathematical approach to imperfect knowledge. The problem of imperfect knowledge has been tackled for a long time by philosophers, logicians and mathematicians. Recently it has become a crucial issue for computer scientists as well, particularly in the area of computational intelligence [129], [99]. There are many approaches to the problem of how to understand and manipulate imperfect knowledge. The most successful one is, no doubt, the fuzzy set theory proposed by Lotfi A. Zadeh [226]. Rough set theory presents still another attempt to solve this problem. It is based on an assumption that objects are perceived by partial information about them. Due to this some objects can be indiscernible. Indiscernible objects form elementary granules. From this fact it follows that some sets can not be exactly described by available information about objects. They are rough not crisp. Any rough set is characterized by its (lower and upper) approximations.