Structures of Association Rule Set

This paper shows a mathematical foundation for almost important features in the problem of discovering knowledge by association rules. The class of frequent itemsets and the association rule set are partitioned into disjoint classes by two equivalence relations based on closures. Thanks to these partitions, efficient parallel algorithms for mining frequent itemsets and association rules can be obtained. Practically, one can mine frequent itemsets as well as association rules just in the classes that users take care of. Then, we obtain structures of each rule class using corresponding order relations. For a given relation, each rule class splits into two subsets of basic and consequence. The basic one contains minimal rules and the consequence one includes in the rules that can be deducted from those minimal rules. In the rest, we consider association rule mining based on order relation min. The explicit form of minimal rules according to that relation is shown. Due to unique representations of frequent itemsets through their generators and corresponding eliminable itemsets, operators for deducting all remaining rules are also suggested. Experimental results show that mining association rules based on relation min is better than the ones based on relations of minmin and minMax in terms of reduction in mining times as well as number of basic rules.