Minimal representation of directed hypergraphs
SIAM Journal on Computing
Minimum Covers in Relational Database Model
Journal of the ACM (JACM)
Quasi-Acyclic Propositional Horn Knowledge Bases: Optimal Compression
IEEE Transactions on Knowledge and Data Engineering
Discrete Applied Mathematics - Special issue: The 1998 conference on ordinal and symbolic data analysis (OSDA '98)
Exclusive and essential sets of implicates of Boolean functions
Discrete Applied Mathematics
Fundamental study: The multiple facets of the canonical direct unit implicational basis
Theoretical Computer Science
Ordered direct implicational basis of a finite closure system
Discrete Applied Mathematics
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We present results inspired by the study of closure systems with unique critical sets. Many of these results, however, are of a more general nature. Among those is the statement that every optimum basis of a finite closure system, in D. Maier's sense, is also right-side optimum. New parameters for the size of the binary part of a closure system are established. We introduce the K-basis of a closure system, which is a refinement of the canonical basis of V. Duquenne and J.L. Guigues, and discuss a polynomial algorithm to obtain it. The main part of the paper is devoted to closure systems with unique critical sets, and some subclasses of these 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. Consequently, closure systems without D-cycles can be effectively recognized. While the E-basis achieves an optimum in one of its parts, the optimization of the others is an NP-complete problem.