Bounded Implication for Existential Rules

The property of boundedness in Datalog formalizes whether a set of rules can be equivalently expressed by a non-recursive set of rules. Existential rules extend Datalog to the presence of existential variables in rule heads. In this paper, we introduce and study notions of boundedness for existential rules. We provide a notion of weak boundedness and a notion of strong boundedness for a rule set, and show that they correspond, respectively, to the notions of first-order rewritability of atomic queries and first-order rewritability of conjunctive queries over the set. While weak and strong boundedness are in general not equivalent, we show that, for some notable subclasses of existential rules, i.e., Datalog, single-head binary rules, and frontier-guarded rules, the two notions coincide.