ABSTRACT XML document markup is highly repetitive and therefore well compressible using grammar-based compression. Downward, navigational XPath can be executed over grammar-compressed trees in PTIME: the query is translated into an automaton which is executed in one pass over the grammar. This result is well-known and has been mentioned before. Here we present precise bounds on the time complexity of this problem, in terms of big-O notation. For a given grammar and XPath query, we consider three different tasks: (1) to count the number of nodes selected by the query (2) to materialize the pre-order numbers of the selected nodes, and (3) to serialize the subtrees at the selected nodes.