Rough Set Processing of Vague Information Using Fuzzy Similarity Relations

This work uses a similarity relation which is only reflexive, relaxing therefore the properties of symmetry and transitivity to approximate a given set represented by objects having the same description in terms of decision attributes by means of an indiscernibility binary relation.

Abstract

The rough sets theory has proved to be a very useful tool for analysis of information tables describing objects by means of disjoint subsets of condition and decision attributes. The key idea of rough sets is approximation of knowledge expressed by decision attributes using knowledge expressed by condition attributes. From a formal point of view, the rough sets theory was originally founded on the idea of approximating a given set represented by objects having the same description in terms of decision attributes, by means of an indiscernibility binary relation linking pairs of objects having the same description by condition attributes. The indiscernibility relation is an equivalence binary relation (reflexive, symmetric and transitive) and implies an impossibility to distinguish two objects having the same description in terms of the condition attributes. It produces crisp granules of knowledge that are used to build approximations. In reality, due to vagueness of the available information about objects, small differences are not considered significant. This situation may be formally modelled by similarity or tolerance relations instead of the indiscernibility relation. We are using a similarity relation which is only reflexive, relaxing therefore the properties of symmetry and transitivity. Moreover, the credibility of our similarity relation is gradual, i.e., it is a fuzzy similarity relation. As a consequence, the granules of knowledge produced by this relation are fuzzy and their credibility can change gradually from any finite degree to an infinitely small degree, thus giving different credibility of rough approximations.