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