  Existential rules are a very popular ontology-mediated query language for
which the chase represents a generic computational approach for query
answering. It is straightforward that existential rule queries exhibiting chase
termination are decidable and can only recognize properties that are preserved
under homomorphisms. In this paper, we show the converse: every decidable query
that is closed under homomorphism can be expressed by an existential rule set
for which the standard chase universally terminates. Membership in this
fragment is not decidable, but we show via a diagonalisation argument that this
is unavoidable.
