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