Rough approximation operators on R0-algebras (nilpotent minimum algebras) with an application in formal logic L∗

An abstract axiomatization to Pawlak rough set theory in the context of R0-algebras (equivalently, NM-algebras) has been proposed in the present paper. More precisely, by employing the conjunction operator ⊗ and the disjunction operator ⊕ in R0-algebras, the notions of rough upper approximation operator U and rough lower approximation operator L on R0-algebras are proposed, respectively. Owing to the logical properties of ⊗ and ⊕, any R0-algebra, equipped with L and U, forms an abstract approximation space in the sense of G. Cattaneo. A duality relationship between the set of lower crisp elements and the set of upper crisp elements is established, and some important properties are examined. Moreover, its connection with Tarski closure-interior approximation space and Halmos closure-interior approximation is studied. Such a pair of rough approximations on R0-algebras can naturally induce a pair of rough (upper, lower) truth degrees for formulae in L∗. Some uncertainty measures such as roughness degree and accuracy degree are subsequently presented and two kinds of approximate reasoning methods merging rough approximation and fuzzy logic are eventually established.