Complete axiomatizations for XPath fragments

We provide complete axiomatizations for several fragments of XPath: sets of equivalences from which every other valid equivalence is derivable. Specifically, we axiomatize downward single axis fragments of Core XPath (that is, Core XPath(↓) and Core XPath(↓)) as well as the full Core XPath. We make use of techniques from modal logic. XPath is a language for navigating through XML documents. In this paper, we consider the problem of finding complete axiomatizations for fragments of XPath. By an axiomatization we mean a finite set of valid equivalences between XPath expressions. These equivalences can be thought of as (undirected) rewrite rules. Completeness then means that any two equivalent expressions can be rewritten to each other using the given equivalences. Completeness tells us, in a mathematically precise way, that the given set of equivalences captures everything there is to say about semantic equivalence of XPath expressions. We are aware of two complete axiomatizations for XPath fragments. The first is for the downward, positive and filter-free fragment of XPath [1], a rather limited fragment, and the second [5] concerns Core XPath 2.0, a very expressive language, with non-elementary complexity for query containment (see [4]). In this paper, we study Core XPath 1.0, which was introduced in [7, 8] to capture the navigational core of XPath 1.0. Our main results are: – A complete axiomatization for Core XPath(↓) and for Core XPath(↓), i.e., the fragments with only the descendant and only the child axis, respectively. The axiomatizations are complete both for node expressions and for path expressions. ? This technical report is the full version of a paper accepted for LID 2008 workshop. If you are kindly going to quote it in your work, please check http://www.dcs.bbk.ac.uk/~tadeusz/ for the most current version published. ?? Currently on leave visiting IBM Almaden and UC Santa Cruz. – An axiomatization for full Core XPath that is complete for node expressions. The completeness can be extended to path expressions as well, at the cost of introducing an additional rule of inference. Proofs utilize techniques and known results from modal logic. We will describe the connection in more detail in Section 4. Acknowledgements The first author is supported by the Netherlands Organization for Scientific Research (NWO) grant 639.021.508. The second author was supported by NWO grant 680–50–0613 during the period when most of the results were obtained. At present, he is supported by EPSRC grant EP/F002262/1. We would like to thank the anonymous referees for their comments.