An Axiom System for Functional Dependency Implication in Complete XML Documents II - Completeness

With the growing use of XML as a format for the permanent storage of data, the study of functional dependencies in XML (XFDs) is of fundamental importance in understanding how to effectively design XML databases without redundancy or update problems. In this article we investigate the implication problem, that is determining when a set of integrity constraints logically implies another integrity constraint, for a new type of integrity constraint called a weak ‘closest node’ XFD in the context of complete XML documents. The implication problem is the most important problem in dependency theory, irrespective of the type of integrity constraint, and a solution to it lies at the heart of any automated tools for manipulating integrity constraints. In a companion article we presented an axiom system for XFD implication and showed that the system is sound. In this article we show that the axiom system is complete. In establishing this result, we present a quadratic time algorithm for the XFD implication problem and prove that it is sound and complete. We also investigate the XFD implication problem when a Document Type definition (DTD) is present, and prove that for a class of DTDs called simple DTDs, the implication problem can be solved by reducing it to the implication problem of an augmented set of XFDs without a DTD.