Rough Set Theory and Logic-Algebraic Structures

Abstract

Any Rough Set System induced by an Approximation Space can be given several logic-algebraic interpretations related to the intuitive reading of the notion of Rough Set. In this paper Rough Set Systems are investigated, first, within the framework of Nelson algebras and the structure of the resulting subclass is inherently described using the properties of Approximation Spaces. In particular, the logic-algebraic structure given to a Rough Set System, understood as a Nelson algebra is equipped with a weak negation and a strong negation and, since it is a finite distributive lattice, it can also be regarded as a Heyting algebra equipped with its own pseudo-complementation. The double weak negation and the double pseudo-complementation are shown to be projection operations connected to the notion of definability in Approximation Spaces. From this analysis we obtain an interpretation of Rough Sets Systems connected to three-valued Lukasiewicz algebras where the roles of projections operators are played by the two endomorphisms of these algebras. Finally, continuing to explore the point of view of Multi-Valued Logics suggested by the latter interpretation we achieve in a quite "natural" wayan interpretation based on the notion of Chain Based Lattice. Here the projection operators are provided by the pseudo-supplementation and dual pseudo-supplementation.

Algebraic models of deviant modal operators based on de Morgan and Kleene lattices

“…As a particular remark, the implications given by equations (7) are a generalization of Pagliani's work Pagliani (1998).…”

Algebras from Rough Sets

“…In the following section, we discuss in some detail the work of Iwinski [1S], Obtulowicz [25], Pomykala [30,31], Comer [12,13], Bonikowski [S], Chuchro [10], Pagliani [27], Banerjee and Chakraborty [3,4], Wasilewska [37] and Iturrioz [17]. The results that are reported here have been chosen in compliance with the angle of presentation of this chapter, and constitute only a fragment of the work done by these authors.…”

A Framework for Reasoning with Rough Sets

“…Most of the research in the rough sets field has been focused on the following issues: algebraic characterization and interpretation of rough sets [Pag97]; relations of rough set theory with other theories to represent knowledge, like modal logics [YL96]; integration of rough sets with other techniques like inductive logic programming [MK00] or fuzzy sets [Wyg89,DP92]; extensions to the basic rough set formalism using different types of indiscernibility relations and more general definitions of upper and lower approximations [YL96,SS96,Zia93]; construction of software tools for data mining based on rough sets methods [ØK97]; application of rough set techniques to real problems [KØ99,MK02,ZF02].…”

“…Different definitions for the concept of rough set have been proposed in the literature [Pag97]. For instance, a rough set could be defined as the pair S = (S, S).…”