About the Lossless Reduction of the Minimal Generator Family of a Context

Minimal generators (MGs), aka minimal keys, play an important role in many theoretical and practical problem settings involving closure systems that originate in graph theory, relational database design, data mining, etc. As minima of the equivalence classes associated to closures, MGs underlie many compressed representations: For instance, they form premises in canonical implication/ association rules - with closures as conclusions - that losslessly represent the entire rule family of a closure system. However, MGs often show an intra-class combinatorial redundancy that makes an exhaustive storage and use impractical. In this respect, the succinct system of minimal generators (SSMG) recently introduced by Dong et al. is a first step towards a lossless reduction of this redundancy. However, as shown elsewhere, some of the claims about SSMG, e.g., its invariant size and lossless nature, do not hold. As a remedy, we propose here a new succinct family which restores the losslessness by adding few further elements to the SSMG core, while theoretically grounding the whole. Computing means for the new family are presented together with the empirical evidences about its relative size w.r.t. the entire MG family and similar structures from the literature.