XPath satisfiability in the presence of DTDs

Abstract

We study the satisfiability problem associated with XPath in the presence of DTDs. This is the problem of determining, given a query p in an XPath fragment and a DTD D, whether or not there exists an XML document T such that T conforms to D and the answer of p on T is nonempty. We consider a variety of XPath fragments widely used in practice, and investigate the impact of different XPath operators on satisfiability analysis. We first study the problem for negation-free XPath fragments with and without upward axes, recursion and data-value joins, identifying which factors lead to tractability and which to NP-completeness. We then turn to fragments with negation but without data values, establishing lower and upper bounds in the absence and in the presence of upward modalities and recursion. We show that with negation the complexity ranges from PSPACE to EXPTIME. Moreover, when both data values and negation are in place, we find that the complexity ranges from NEXPTIME to undecidable. Finally, we give a finer analysis of the problem for particular classes of DTDs, exploring the impact of various DTD constructs, identifying tractable cases, as well as providing the complexity in the query size alone.

Alternation-free modal mu-calculus for data trees

“…Following [2,4], we assume without loss of generality that document type definitions (DTDs) [6] are given as regular tree languages. More precisely, we consider a DTD to be a nondeterministic top-down tree automaton T with alphabet (Σ∪Σ ) {d, r} and without ε-transitions.…”

“…The fragment of XPath [7] below contains all operators commonly found in practice and was considered in [2,17]. The grammars of queries p and qualifiers q are mutually recursive.…”

“…The most basic static analysis problem for XPath, with a variety of applications, is satisfiability in the presence of DTDs. The two recent surveys on its complexity by Benedikt, Fan and Geerts [2,17] convey that relatively little is known for fragments with negation and data (i.e., equality comparisons between attribute values). The only decidability result in [2,17] that allows negation and data does not allow axes which are recursive (such as "self or descendant") or between siblings.…”

“…The two recent surveys on its complexity by Benedikt, Fan and Geerts [2,17] convey that relatively little is known for fragments with negation and data (i.e., equality comparisons between attribute values). The only decidability result in [2,17] that allows negation and data does not allow axes which are recursive (such as "self or descendant") or between siblings. By representing XML documents as data trees and translating from XPath to FO 2 (+1, ∼), a decidable fragment with negation, data and all nonrecursive axes was obtained in [4].…”

“…Another fragment of XPath was considered in [18], but it lacks concatenation, recursive axes and sibling axes. Hence, decidability for the largest downward fragment, which was one of the main questions posed in [2], remains open.…”