Expressiveness and Complexity of Order Dependencies

Dependencies play an important role in databases. We study order dependencies (ODs)—and unidirectional order dependencies (UODs), a proper sub-class of ODs—which describe the relationships among lexicographical orderings of sets of tuples. We consider lexicographical ordering, as by the order-by operator in SQL, because this is the notion of order used in SQL and within query optimization. Our main goal is to investigate the inference problem for ODs, both in theory and in practice. We show the usefulness of ODs in query optimization. We establish the following theoretical results: (i) a hierarchy of order dependency classes; (ii) a proof of coNP-completeness of the inference problem for the subclass of UODs (and ODs); (iii) a proof of co-NP-completeness of the inference problem of functional dependencies (FDs) from ODs in general, but demonstrate linear time complexity for the inference of FDs from UODs; (iv) a sound and complete elimination procedure for inference over ODs; and (v) a sound and complete polynomial inference algorithm for sets of UODs over restricted domains.