An axiomatic approach to the roughness measure of rough sets

TL;DR

This paper introduces an axiomatic framework for roughness measures in rough set theory, unifying existing measures and proposing new ones with desirable properties like monotonicity.

Contribution

It provides an axiomatic foundation for roughness measures, unifies existing approaches, and introduces new strong Pawlak roughness measures with proven properties.

Findings

Existing roughness measures are special cases of the proposed framework.

The axiomatic approach simplifies deriving properties of roughness measures.

New strong Pawlak roughness measures are introduced with advantageous properties.

Abstract

In Pawlak's rough set theory, a set is approximated by a pair of lower and upper approximations. To measure numerically the roughness of an approximation, Pawlak introduced a quantitative measure of roughness by using the ratio of the cardinalities of the lower and upper approximations. Although the roughness measure is effective, it has the drawback of not being strictly monotonic with respect to the standard ordering on partitions. Recently, some improvements have been made by taking into account the granularity of partitions. In this paper, we approach the roughness measure in an axiomatic way. After axiomatically defining roughness measure and partition measure, we provide a unified construction of roughness measure, called strong Pawlak roughness measure, and then explore the properties of this measure. We show that the improved roughness measures in the literature are special instances of our strong Pawlak roughness measure and introduce three more strong Pawlak roughness measures as well. The advantage of our axiomatic approach is that some properties of a roughness measure follow immediately as soon as the measure satisfies the relevant axiomatic definition.