Limiting until in ordered tree query languages

Journal article

Marx and de Rijke have shown that the navigational core of the w3c XML query language XPath is not first-order complete – that is it cannot express every query definable in first-order logic over the naviga- tional predicates. How can one extend XPath to get a first-order complete language? Marx has shown that Conditional XPath – an extension of XPath with an “Until” operator – is first order complete. The com- pleteness argument makes essential use of the presence of upward axes in Conditional XPath. We examine whether it is possible to get “forward-only” languages that are first-order complete for Boolean queries on ordered trees. It is easy to see that a variant of the temporal logic CTL ∗ is first-order complete; the vari- ant has path quantifiers for downward, leftward and rightward paths, while along a path one can check arbitrary formulas of linear temporal logic (LTL). This language has two major disadvantages: it requires path quantification in both horizontal directions (in particular, it requires looking backward at the prior siblings of a node), and it requires the consideration of formulas of LTL of arbitrary complexity on vertical paths. This last is in contrast with Marx’s Conditional XPath, which requires only the checking of a single Until operator on a path. We investigate whether either of these restrictions can be eliminated. Our main results are negative ones. We show that if we restrict our CTL ∗ language by having an until operator in only one horizontal direction, then we lose completeness. We also show that no restriction to a “small” subset of LTL along vertical paths is sufficient for first order completeness. Smallness here means of bounded “Until Depth”, a measure of complexity of LTL formulas defined by Etessami and Wilke. In particular, it follows from our work that Conditional XPath with only forward axes is not expressively complete; this extends results proved by Rabinovich and Maoz in the context of infinite unordered trees.

Authors: Benedikt, M; Ley, C

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Association for Computing Machinery
• Journal: ACM Transaction on Computational Logic
• Volume: 17
• Issue: 2
• Article number: 14
• Publication date: 2016-03-01
• DOI: 10.1145/2856104
• EISSN: 1557-945X
• ISSN: 1529-3785