Abstract: Adopting Zakowski-s upper approximation operator
C and lower approximation operator C, this paper investigates
granularity-wise separations in covering approximation spaces. Some
characterizations of granularity-wise separations are obtained by
means of Pawlak rough sets and some relations among granularitywise
separations are established, which makes it possible to research
covering approximation spaces by logical methods and mathematical
methods in computer science. Results of this paper give further
applications of Pawlak rough set theory in pattern recognition and
artificial intelligence.
Abstract: Covering approximation spaces is a class of important
generalization of approximation spaces. For a subset X of a covering
approximation space (U, C), is X definable or rough? The
answer of this question is uncertain, which depends on covering
approximation operators endowed on (U, C). Note that there are many
various covering approximation operators, which can be endowed
on covering approximation spaces. This paper investigates covering
approximation spaces endowed ten covering approximation operators
respectively, and establishes some relations among definable subsets,
inner definable subsets and outer definable subsets in covering approximation
spaces, which deepens some results on definable subsets
in approximation spaces.
Abstract: Let (U;D) be a Gr-covering approximation space
(U; C) with covering lower approximation operator D and covering
upper approximation operator D. For a subset X of U, this paper
investigates the following three conditions: (1) X is a definable subset
of (U;D); (2) X is an inner definable subset of (U;D); (3) X is an
outer definable subset of (U;D). It is proved that if one of the above
three conditions holds, then the others hold. These results give a
positive answer of an open problem for definable subsets of covering
approximation spaces.