Relative expressive power of downward fragments of navigational query languages on trees and chains

Motivated by the continuing interest in the tree data model, we study the expressive power of downward fragments of navigational query languages on trees. The basic navigational query language we consider expresses queries by building binary relations from the edge relations and the identity relation, using composition and union. We study the effects on the expressive power when we add transitive closure, projections, coprojections, intersection, and difference. We study expressiveness at the level of boolean queries and path queries, on labeled and unlabeled trees, and on labeled and unlabeled chains. In all these cases, we are able to present the complete Hasse diagram of relative expressiveness. In particular, we were able to decide, for each fragment of the navigational query languages that we study, whether it is closed under difference and intersection when applied on trees.