  In this paper, we consider existential rules, an expressive formalism well
suited to the representation of ontological knowledge and data-to-ontology
mappings in the context of ontology-based data integration. The chase is a
fundamental tool to do reasoning with existential rules as it computes all the
facts entailed by the rules from a database instance. We introduce
parallelisable sets of existential rules, for which the chase can be computed
in a single breadth-first step from any instance. The question we investigate
is the characterization of such rule sets. We show that parallelisable rule
sets are exactly those rule sets both bounded for the chase and belonging to a
novel class of rules, called pieceful. The pieceful class includes in
particular frontier-guarded existential rules and (plain) datalog. We also give
another characterization of parallelisable rule sets in terms of rule
composition based on rewriting.
