  Based on rectangle theory of formal concept and set covering theory, the
concept reduction preserving binary relations is investigated in this paper. It
is known that there are three types of formal concepts: core concepts, relative
necessary concepts and unnecessary concepts. First, we present the new judgment
results for relative necessary concepts and unnecessary concepts. Second, we
derive the bounds for both the maximum number of relative necessary concepts
and the maximum number of unnecessary concepts and it is a difficult problem as
either in concept reduction preserving binary relations or attribute reduction
of decision formal contexts, the computation of formal contexts from formal
concepts is a challenging problem. Third, based on rectangle theory of formal
concept, a fast algorithm for reducing attributes while preserving the
extensions for a set of formal concepts is proposed using the extension
bit-array technique, which allows multiple context cells to be processed by a
single 32-bit or 64-bit operator. Technically, the new algorithm could store
both formal context and extent of a concept as bit-arrays, and we can use
bit-operations to process set operations "or" as well as "and". One more merit
is that the new algorithm does not need to consider other concepts in the
concept lattice, thus the algorithm is explicit to understand and fast.
Experiments demonstrate that the new algorithm is effective in the computation
of attribute reductions.
