Decidability of Downward XPath

International audience; We investigate the satisfiability problem for downward-XPath, the fragment of XPath that includes the child and descendant axes, and tests for (in)equality of attributes' values. We prove that this problem is decidable, ExpTime-complete. These bounds also hold when path expressions allow closure under the Kleene star operator. To obtain these results, we introduce a Downward Data automata model (DD automata) over trees with data, which has a decidable emptiness problem. Satisfiability of downward-XPath can be reduced to the emptiness problem of DD automata and hence its decidability follows. Although downward-XPath does not include any horizontal axis, DD automata are more expressive and can perform some horizontal tests. Thus, we show that the satisfiability remains in ExpTime even in the presence of the regular constraints expressible by DD automata. However, the same problem in the presence of any regular constraint is known to have a non-primitive recursive complexity. Finally, we give the exact complexity of the satisfiability problem for several fragments of downward-XPath.