  Existential rules have been proposed for representing ontological knowledge,
specifically in the context of Ontology-Based Query Answering. Entailment with
existential rules is undecidable. We focus in this paper on conditions that
ensure the termination of a breadth-first forward chaining algorithm known as
the chase. First, we propose a new tool that allows to extend existing
acyclicity conditions ensuring chase termination, while keeping good complexity
properties. Second, we consider the extension to existential rules with
nonmonotonic negation under stable model semantics and further extend
acyclicity results obtained in the positive case.
