Bit-Vector Representation of Dominance-Based Approximation Space

Abstract

Dominance-based Rough Set Approach (DRSA) introduced by Greco et al. is an extension of Pawlak\'s classical rough set theory by using dominance relations in place of equivalence relations for approximating sets of preference ordered decision classes. The elementary granules in DRSA are P-dominating and P-dominated sets. Recently, Chan and Tzeng introduced the concept of indexed blocks for representing dominance-based approximation space with generalized dominance relations on evaluations of objects. This paper shows how to derive indexed blocks from P-dominating and P-dominated sets in DRSA. Approximations are generalized to any family of decision classes in terms of indexed blocks formulated as binary neighborhood systems. We present algorithms for generating indexed blocks from multi-criteria decision tables and for encoding indexed blocks as bit-vectors to facilitate the computation of approximations and rule generation. A new form of representing decision rules by using interval and set-difference operators is introduced, and we give a procedure of how to generate this type of rules that can be implemented as SQL queries.