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Association rules are among the most widely employed data analysis methods inthe field of Data Mining. An association rule is a form of partial implicationbetween two sets of binary variables. In the most common approach, associationrules are parameterized by a lower bound on their confidence, which is theempirical conditional probability of their consequent given the antecedent,and/or by some other parameter bounds such as "support" or deviation fromindependence. We study here notions of redundancy among association rules froma fundamental perspective. We see each transaction in a dataset as aninterpretation (or model) in the propositional logic sense, and considerexisting notions of redundancy, that is, of logical entailment, amongassociation rules, of the form "any dataset in which this first rule holds mustobey also that second rule, therefore the second is redundant". We discussseveral existing alternative definitions of redundancy between associationrules and provide new characterizations and relationships among them. We showthat the main alternatives we discuss correspond actually to just two variants,which differ in the treatment of full-confidence implications. For each ofthese two notions of redundancy, we provide a sound and complete deductioncalculus, and we show how to construct complete bases (that is,axiomatizations) of absolutely minimum size in terms of the number of rules. Weexplore finally an approach to redundancy with respect to several associationrules, and fully characterize its simplest case of two partial premises.

Redundancy, Deduction Schemes, and Minimum-Size Bases for Association Rules