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

Balbiani P., Iliev P., Vakarelov D.

Fundamenta Informaticae

2008

Vol. 83, nr 4

451-464

The natural modal logic corresponding to Pawlak's approximation spaces is S5, based on the box modality [R]A (and the diamond modality áRn~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 ROUGHn corresponding to the class of approximation spaces with rough cardinality n. We prove that the satisfiability problem for ROUGHn is NP-complete.