We present a method for evaluating positive unary queries expressed in terms of Core XPath axes on any XML document t, which may be presented in a fully or partially compressed form. To each Core XPath query of a certain basic type, we associate a word automaton; these automata run on the graph of dependency between the non-terminals of the minimal straightline regular tree grammar associated to the given document, or along complete sibling chains in this grammar. Any given Core XPath query can be decomposed into queries of the basic type, and the answer to the query can then be expressed as a sub-document of t, suitably labeled under the runs of such automata.