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Enhancing Datalog with existential quantification gives rise to Datalog^∃, a powerful knowledge representation language widely used in ontology-based query answering. In this setting, a conjunctive query is evaluated over a Datalog^∃ program consisting of extensional data paired with so-called “existential” rules. Owing to their high expressiveness, such rules make the evaluation of queries undecidable, even when the latter are atomic. Decidable generalizations of Datalog by existential rules have been proposed in the literature (such as weakly acyclic and weakly guarded); but they pay the price of higher computational complexity, hindering the implementation of effective systems. Conversely, the results in this article demonstrate that it is definitely possible to enable fast yet powerful query answering over existential rules that strictly generalize Datalog by ensuring decidability without any complexity overhead …