Horn Axiomatizations for Sequences

Abstract

This chapter focuses on the study of association rules for ordered data by using the system of closed sets of sequences, defined by operator $\triangle$. Our contribution is a notion of deterministic association rules with order where a set of sequences always implies another sequence in the data. The central advantage of dealing with deterministic rules is that they do not require to select, with little or no formal guidance, one single measure of strength of implication because they always hold. Moreover, since they are pure standard implications, they can be studied in purely logical terms. Indeed, in the second chapter we already mentioned that the set of deterministic association rules derived from classical lattice-theoretic methods axiomatize the minimal Horn upper bound of a binary relation [14]. On the basis of this formalization, the main result of this chapter is a similar characterization of the implications with order as the empirical Horn approximation of the input set of sequences. To allow for this characterization, we will require the definition of certain background Horn conditions to ensure the consistency of the theory. As a consequence of this main result, we can also prove the isomorphy of the lattice of closed sets of sequences and the classical binary lattice when the background Horn conditions hold. Finally, we discuss the computation of all these rules in practice.