Extending Acyclicity Notions for Existential Rules (\emph{long version})

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