We investigate the complexity of the evaluation problem for ECRPQ: Conjunctive Regular Path Queries (CRPQ), extended with synchronous relations (aka regular or automatic). We give a characterization for the evaluation and parameterized evaluation problems of ECRPQ in terms of the underlying structure of queries. As we show, complexity can range between PSpace, NP and polynomial time for the evaluation problem, and between XNL, W[1], and FPT for parameterized evaluation.

The class of synchronous relations, also known as automatic or regular, is one of the most studied subclasses of rational relations. It enjoys many desirable closure properties and is known to be logically characterized: the synchronous relations are exactly those that are defined by a first-order formula on the structure of all finite words, with the prefix, equal-length and last-letter predicates. Here, we study the quantifier alternation hierarchy of this logic. We show that it collapses at level Sigma_3 and that all levels below admit decidable characterizations. Our results reveal the connections between this hierarchy and the well-known hierarchy of first-order defined languages of finite words.