Modal logic
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Fundamenta Informaticae - New Frontiers in Scientific Discovery - Commemorating the Life and Work of Zdzislaw Pawlak
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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.