  The chase is a sound and complete algorithm for conjunctive query answering
over ontologies of existential rules with equality. To enable its effective
use, we can apply acyclicity notions; that is, sufficient conditions that
guarantee chase termination. Unfortunately, most of these notions have only
been defined for existential rule sets without equality. A proposed solution to
circumvent this issue is to treat equality as an ordinary predicate with an
explicit axiomatisation. We empirically show that this solution is not
efficient in practice and propose an alternative approach. More precisely, we
show that, if the chase terminates for any equality axiomatisation of an
ontology, then it terminates for the original ontology (which may contain
equality). Therefore, one can apply existing acyclicity notions to check chase
termination over an axiomatisation of an ontology and then use the original
ontology for reasoning. We show that, in practice, doing so results in a more
efficient reasoning procedure. Furthermore, we present equality model-faithful
acyclicity, a general acyclicity notion that can be directly applied to
ontologies with equality.
