Mining interesting rules without support requirement: a general universal existential upward closure property

International audience; Many studies have shown the limits of support/confidence framework used in Apriori-like algorithms to mine association rules. There are a lot of efficient implementations based on the antimonotony property of the support. But candidate set generation is still costly and many rules are uninteresting or redundant. In addition one can miss interesting rules like nuggets. We are thus facing a complexity issue and a quality issue. One solution is to get rid of frequent itemset mining and to focus as soon as possible on interesting rules. For that purpose algorithmic properties were first studied, especially for the confidence. They allow to find all confident rules without a preliminary support pruning. Recently, in the case of class association rules, the Universal Existential Upward Closure property of confidence has been exploited in an efficient manner. Indeed, it allows to use a pruning strategy for an Apriori-like but top down associative classification rules algorithm. We present a new formal framework which allows us to make the link between analytic and algorithmic properties of the measures. We then apply this framework to propose a General Universal Existential Upward Closure. We demonstrate a necessary and a sufficient condition of existence for this property. These results are then applied to 32 measures and we show that 13 of them do have the GUEUC property.