Succinct System of Minimal Generators: A Thorough Study, Limitations and New Definitions

Minimal generators (MGs) are the smallest ones (w.r.t. the number of items) among equivalent itemsets sharing a common set of objects, while their associated closed itemset (CI) is the largest one. The pairs composed by MGs and their associated CI divide the itemset lattice into distinct equivalence classes. Such pairs were at the origin of various works related to generic association rule bases, concise representations, arbitrary boolean expressions, etc. Furthermore, the MG set presents some important properties like the order ideal. The latter helped some level-wise bottom-up and even slightly modified depth-first algorithms to efficiently extract interesting knowledge. Nevertheless, the inherent absence of a unique MG associated to a given CI motivates an in-depth study of the possibility of discovering a kind of redundancy within the MG set. This study was started by Dong et al. who introduced the succinct system of minimal generators (SSMG) as an attempt to eliminate the redundancy within this set. In this paper, we give a thorough study of the SSMG as formerly defined by Dong et al. Then, we show that the latter suffers from some drawbacks. After that, we introduce new definitions allowing to overcome the limitations of their work. Finally, an experimental evaluation shows that the SSMG makes it possible to eliminate without information loss an important number of redundant MGs.