Nonmonotonic Existential Rules for Non-Tree-Shaped Ontological Modelling

Description logic (DL) formalisms are extensively used for ontological modelling, e.g., in biology [16] and chemistry [19]. One such example is the ChEBI ontology [20] – a reference terminology adopted by various biological databases for chemical annotation [23,8,10]. Despite their wide range of expressive features, DLs are severely limited in their ability to represent structures that are not tree-shaped. This explains, e.g., why ChEBI does not capture molecular structures in its ontology, thus excluding its main content from logical reasoning. In order to overcome this deficiency, numerous rulebased extensions of DLs have been proposed that provide certain kinds of graph-based modelling, such as description graphs [27]. However, in order to retain decidability of description graphs several constraints are imposed, such as role separation restrictions and a cumbersome acyclicity condition, that restrict the range of structures that can be modelled and thus hinder practical usability [26]. Moreover, when performing structurebased classification a form of closed-world assumption is often needed to reason about the absence of structural features, e.g., to conclude that a molecule is inorganic if it does not contain carbon. Expressing completeness (closure) of finite structures in DLs is prohibitively inefficient, whereas nonmonotonic extensions of DLs remain largely unrealised in tools and applications [28]. This motivates the use of logical languages that draw upon rules enriched with nonmonotonic negation for the representation and classification of objects with a complex internal structure. Existential rules—function-free Horn rules with existential quantifiers in rule heads—have been proposed as a new expressive ontological language [6,2], and can be viewed as a restricted kind of logic programs with function symbols. Recent works have considered nonmonotonic rule-based ontology languages using stratified negation [5,26], stable model semantics [13], and well-founded semantics [17]. If we additionally remove the stratification requirement, then the resulting language allows for the accurate modelling of complex finite structures such as those found in ChEBI. Unfortunately, reasoning in these formalisms is computationally challenging. If negation is stratified, then all of these semantics agree, and programs have uniquely determined stable models; this is highly desirable and easy to check, but too restrictive for many applications. Moreover, even without negation, satisfiability, fact entailment,