On implicational bases of closure systems with unique critical sets
release_dunrrolubfd7xl333b2ugcwqj4
by
Kira Adaricheva, J.B. Nation
2013
Abstract
We show that every optimum basis of a finite closure system, in D.Maier's
sense, is also right-side optimum, which is a parameter of a minimum CNF
representation of a Horn Boolean function. New parameters for the size of the
binary part are also established. We introduce a K-basis of a general closure
system, which is a refinement of the canonical basis of Duquenne and Guigues,
and discuss a polynomial algorithm to obtain it. We study closure systems with
the unique criticals and some of its subclasses, where the K-basis is unique. A
further refinement in the form of the E-basis is possible for closure systems
without D-cycles. There is a polynomial algorithm to recognize the D-relation
from a K-basis. Thus, closure systems without D-cycles can be effectively
recognized. While E-basis achieves an optimum in one of its parts, the
optimization of the others is an NP-complete problem.
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1205.2881v2
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