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Answering conjunctive queries (CQs) over a set of facts extended withexistential rules is a prominent problem in knowledge representation anddatabases. This problem can be solved using the chase algorithm, which extendsthe given set of facts with fresh facts in order to satisfy the rules. If thechase terminates, then CQs can be evaluated directly in the resulting set offacts. The chase, however, does not terminate necessarily, and checking whetherthe chase terminates on a given set of rules and facts is undecidable. Numerousacyclicity notions were proposed as sufficient conditions for chasetermination. In this paper, we present two new acyclicity notions calledmodel-faithful acyclicity (MFA) and model-summarising acyclicity (MSA).Furthermore, we investigate the landscape of the known acyclicity notions andestablish a complete taxonomy of all notions known to us. Finally, we show thatMFA and MSA generalise most of these notions.  Existential rules are closely related to the Horn fragments of the OWL 2ontology language; furthermore, several prominent OWL 2 reasoners implement CQanswering by using the chase to materialise all relevant facts. In order toavoid termination problems, many of these systems handle only the OWL 2 RLprofile of OWL 2; furthermore, some systems go beyond OWL 2 RL, but without anytermination guarantees. In this paper we also investigate whether variousacyclicity notions can provide a principled and practical solution to theseproblems. On the theoretical side, we show that query answering for acyclicontologies is of lower complexity than for general ontologies. On the practicalside, we show that many of the commonly used OWL 2 ontologies are MSA, and thatthe number of facts obtained by materialisation is not too large. Our resultsthus suggest that principled development of materialisation-based OWL 2reasoners is practically feasible.

Acyclicity Notions for Existential Rules and Their Application to Query Answering in Ontologies