1 Rough Sets In Data Analysis: Foundations and Applications Lech Polkowski 1,2 and Piotr Artiemjew 2 1 Polish-Japanese Institute of Information Technology, Koszykowa 86, 02008 Warszawa, Poland polkow@pjwstk.edu.pl 2 Department of Mathematics and Computer Science, University of Warmia and Mazury, ˙ Zo lnierska 14, Olsztyn, Poland artem@matman.uwm.edu.pl Summary. Rough sets is a paradigm introduced in order to deal with uncertainty due to ambiguity of classification caused by incompleteness of knowledge. The idea proposed by Z. Pawlak in 1982 goes back to classical idea of representing uncertain and/or inexact notions due to the founder of modern logic, Gottlob Frege: uncertain notions should possess around them a region of uncertainty consisting of objects that can be qualified with certainty neither into the notion nor to its complement. The central tool in realizing this idea in rough sets is the relation of uncertainty based on the classical notion of indiscernibility due to Gottfried W. Leibniz: objects are indiscernible when no operator applied to each of them yields distinct values. In applications, knowledge comes in the form of data; those data in rough sets are organized into an information system: a pair of the form (U, A) where U is a set of objects and A is a set of attributes, each of them a mapping a : U → Va, the value set of a. Each attribute a does produce the a-indiscernibility relation IND(a)= {(u, v): a(u)= a(v)}. Each set of attributes B does induce the B-indiscernibility relation IND(B)=  IND(a): a ∈ B. Objects u, v that are in the relation IND(B) are B-indiscernible. Classes [u]B of the relation IND(B) form B–elementary granules of knowledge. Rough sets allow for establishing dependencies among groups of attributes: a group B depends functionally on group C when IND(C) ⊆ IND(B): in that case values of attributes in B are functions of values of attributes in C. An important case is when data are organized into a decision system: a triple (U, A, d) where d is a new attribute called the decision. The decision gives a clas- sification of object due to an expert, an external oracle; establishing dependencies between groups B of attributes in A and the decision is one of tasks of rough set theory. The language for expressing dependencies is the descriptor logic. A descriptor is a formula (a = v) where v ∈ Va, interpreted in the set U as [a = v]= {u : a(u)= v}. Descriptor formulas are obtained from descriptors by means of connectives ∨, ∧, ¬, ⇒