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 matroids based on relations
Information Sciences: an International Journal
Concept Formation: Rough Sets and Scott Systems
Fundamenta Informaticae - To Andrzej Skowron on His 70th Birthday
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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.