Generalized rough approximations in ŁΠ12

AbstractIn this paper we will treat a generalization of inner and outer approximations of fuzzy sets, which we will call R-inner and R-outer approximations respectively (R being any finite set of rational numbers in [0,1]). In particular we will discuss the case of those fuzzy sets which are definable in the logic ŁΠ12 by means of step functions from the hypercube [0,1]k and taking value in an arbitrary (finite) subset of [0,1]∩Q. Then, we will show that if a fuzzy set is definable as truth table of a formula of ŁΠ12, then both its R-inner and R-outer approximation are definable as truth table of formulas of ŁΠ12. Finally, we will introduce a generalization of abstract approximation spaces and compare our approach with the notion of fuzzy rough set