Numerical constraints on XML data

Boundaries occur naturally in everyday life. This paper introduces numerical constraints into the framework of XML to take advantage of the benefits that result from the explicit specification of such boundaries. Roughly speaking, numerical constraints restrict the number of elements in an XML data fragment based on the data values of selected subelements. Efficient reasoning about numerical constraints provides effective means for predicting the number of answers to XQuery and XPath queries, the number of updates when using the XQuery update facility, and the number of encryptions or decryptions when using XML encryption. Moreover, numerical constraints can help to optimise XQuery and XPath queries, to exclude certain choices of indices from the index selection problem, and to generate views for efficient processing of common queries and updates.We investigate decision problems associated with numerical constraints in order to capitalise on the range of applications in XML data processing. To begin with we demonstrate that the implication problem is strongly coNP-hard for several classes of numerical constraints. These sources of potential intractability direct our attention towards the class of numerical keys that permit the specification of positive upper bounds. Numerical keys are of interest as they are reminiscent of cardinality constraints that are widely used in conceptual data modelling. At the same time, they form a natural generalisation of XML keys that are popular in XML theory and practice. We show that numerical keys are finitely satisfiable and establish a finite axiomatisation for their implication problem. Finally, we propose an algorithm that decides numerical key implication in quadratic time using shortest path methods.