A Modal Logic for Pawlak's Approximation Spaces with Rough Cardinality n

Abstract

The natural modal logic corresponding to Pawlak's approximation spaces is S5, based on the box modality [R]A (and the diamond modality 〈R〉A=¬[R]¬A), where R is the corresponding indiscernibility relation of the approximation space S=(W,R). However the expressive power of S5 is too weak and, for instance, we cannot express that the space S has exactly n equivalence classes (we say that S is roughly-finite and n is the rough cardinality of S). For this reason we extend the modal logic S5 with a new box modality [S]A, where S is the complement of R i.e. the discernibility relation of W. We propose a complete axiomatization, in this new language, of the logic ROUGH ^n corresponding to the class of approximation spaces with rough cardinality n. We prove that the satisfiability problem for ROUGH ^n is NP-complete.