Rough Set Theory Zdzislaw Pawlak

After almost 20 years of pursuing rough set theory and its applications the theory has reached a certain degree of maturity. In recent years we witnessed a rapid grow of interest in rough set theory and its application world wide. The theory has attracted attention of many researchers and practitioners all over the world, who contributed essentially to its development and applications. Numerous non trivial real-life applications of the theory have shown its usefulness. Many international workshops, conferences and seminars on rough sets have been held in recent years. More than 2000 papers, and several books have been published on rough set theory and its applications till now. What is rough set theory? The theory has two faces – logical (or rather philosophical) and practical. Both are equally important. From logical point of view rough set theory is a new approach to vagueness. Vague concepts attracted attention of logicians and philosophers at least for one hundred years and in recent thirty years, or so, vagueness became also a hot topic in computer science. A concept is vague if it cannot be defined uniquely. For example, the concept of an “odd” (“even”) number is crisp, whereas the concept of a “beautiful woman” is vague. According to father of modern logic, Gottlob Frege, vague concepts are characterized by a “boundary region”, which consists of all elements which cannot be classified to the concept or its complement. This approach is known in a philosophical literature as boundary-line approach to vagueness. Rough set theory is a kind of implementation of Frege’s ideas and no doubt the theory contributed substantially to understand better the idea of vagueness. Rough set theory refers also to some ideas of Gottfried Leibniz (indiscernibility), George Boole (reasoning methods), Jan Lukasiewicz (multivalued logics), and Thomas Bayes (inductive reasoning). From practical point of view rough set theory seems to be of fundamental significance to AI and cognitive sciences, especially to machine learning, knowledge discovery, decision analysis, inductive reasoning and pattern recognition. It seems also important to decision support systems and data mining. In fact it is a new mathematical approach to data analysis. Rudiments of the theory can easily be understood and applied. Several software systems based on rough set theory have been implemented and many real life, nontrivial applications of this methodology have been reported, e.g., in medicine, pharmacology, engineering, banking, market analysis, conflict analysis, pattern recognition, environment, linguistics, gene expression and many more. Rough set theory is based on sound mathematical foundation. The theory is not competitive but complementary to other methods and can also be often used jointly with other approaches (e.g., statistical methods, neural networks, genetic algorithms, fuzzy sets, etc.). Rudiments of the rough set theory can be found in [5,6]. For recent developments see [7,8,11,13]. Various extension of the theory can be found in [1,3,11,13]. Some applications of rough set theory are discussed in [2,7,9,11]. For more information the reader is advised to consult the Internet. 2 Approximations – basic concepts of rough set theory