Rough approximation operators on R0 R 0 -algebras (nilpotent minimum algebras) with an application in formal logic L? L ?
An abstract axiomatization to Pawlak rough set theory in the context of R0 R 0 -algebras (equivalently, NM -algebras) has been proposed in the present paper. More precisely, by employing the conjunction operator ? ? and the disjunction operator ? ? in R0 R 0 -algebras, the notions of rough upper approximation operator U and rough lower approximation operator L  on R0 R 0 -algebras are proposed, respectively. Owing to the logical properties of ? ? and ? ? , any R0 R 0 -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 R 0 -algebras can naturally induce a pair of rough (upper, lower) truth degrees for formulae in L? 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.
rough approximation approximate reasoning r0 r 0 -algebra r0 r 0 r0 r 0 r 0 rough truth degree rough similarity degree