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Rough sets through algebraic logic. (English) Zbl 0864.03041
Summary: While studying rough equality within the framework of the modal system $S_5$, an algebraic structure called rough algebra [{\it M. Banerjee} and {\it M. K. Chakraborty}, Bull. Pol. Acad. Sci., Math. 41, No. 4, 293-297 (1993; Zbl 0795.03035)], came up. Its features were abstracted to yield a topological quasi-Boolean algebra (tqBa). In this paper, it is observed that rough algebra is more structured than a tqBa. Thus, enriching the tqBa with additional axioms, two more structures, viz. pre-rough algebra and rough algebra, are defined. Representation theorems of these algebras are also obtained. Further, the corresponding logical systems ${\cal L}_1$, ${\cal L}_2$ are proposed and eventually, ${\cal L}_2$ is proved to be sound and complete with respect to a rough set semantics.

03G25Other algebras related to logic
03B45Modal logic, etc.
68T27Logic in artificial intelligence
68T30Knowledge representation