Validity of Positive XPath Queries with Wildcard in the Presence of DTDs

This paper discusses the validity problem for positive XPath queries with wildcard in the presence of DTDs. A given XPath query p is valid under a DTD D if, for every XML document T conforming to D, the answer to p on T is nonempty. The validity problem is one of the basic static analyses of queries, together with the satisfiability and the containment problems. Although the validity problem is the dual of the satisfiability problem, the complexity of validity for positive XPath classes is not obvious because the XPath class does not contain the negation operator. In this paper, first, it is shown that the path union operator in XPath queries easily makes the validity problem intractable. Then, we focus on wildcard, which is a special case of path union and more popular than path union in the real world. Interestingly, wildcard together with child and descendant-orself axes and qualifier causes intractability while the validity problem becomes tractable for XPath classes defined by any combination of three of child axis, descendant-or-self axis, qualifier, and wildcard.