Handbook of logic in computer science (vol. 3)
Contributions to the theory of rough sets
Fundamenta Informaticae
Rough Sets: Mathematical Foundations
Rough Sets: Mathematical Foundations
Implicit programming and the logic of constructible duality
Implicit programming and the logic of constructible duality
Rough Set Algebras as Description Domains
Fundamenta Informaticae
Rough Set Algebras as Description Domains
Fundamenta Informaticae
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The present article deals with the problem whether and how the bilattice orderings of knowledge ⩽$_k$ and truth ⩽$_t$ might enrich the theory of rough sets. Passing to the chief idea of the paper, we develop a bilattice-theoretic generalisation of the concept of rough set to be called A-approximation. It is proved that A-approximations (induced by a topological approximation space) together with the knowledge ordering ⩽$_k$ constitute a complete partial order (CPO) and that the meet and join operations induced by the truth ordering ⩽$_t$ are continuous functions with respect to ⩽$_k$. Crisp sets are then obtained as maximal elements of this CPO. The second part of this article deals with the categorical and algebraic properties of A-approximations induced by an Alexandroff topological space. We build a *-autonomous category of A-approximations by means of the Chu construction applied to the Heyting algebra of open sets of Alexandroff topological space. From the algebraic point of view A-approximations under ⩽$_t$ ordering constitute a special Nelson lattice and, as a result, provide a semantics for constructive logic with strong negation. Such lattice may be obtained by means of the twist construction over a Heyting algebra which resembles very much the Chu construction. Thus A-approximations may be retrived from very elementary structures in elegant and intuitive ways.