The impact of disjunction on reasoning under existential rules - Morak, M (2014)
Ontological database management systems are a powerful tool that combine traditional database techniques with ontological reasoning methods. In this setting, a classical extensional database is enriched with an ontology, or a set of logical assertions, that describe how new, intensional knowledge can be derived from the extensional data. Conjunctive queries are therefore answered against this combined knowledge base of extensional and intensional data. Many languages that represent ontologies have been introduced in the literature. In this thesis we will focus on existential rules (also called tuple-generating dependencies or Datalog± rules), and three established languages in this area, namely guarded-based rules, sticky rules and weakly-acyclic rules. The main goal of the thesis is to enrich these languages with non-deterministic constructs (i.e. disjunctions) and investigate the complexity of the answering conjunctive queries under these extended languages. As is common in the literature, we will distinguish between combined complexity, where the database, the ontology and the query are considered as input, and data complexity, where only the database is considered as input. The latter case is relevant in practice, as usually the ontology and the query can be considered as fixed, and are usually much smaller than the database itself. After giving appropriate definitions to extend the considered languages to disjunctive existential rules, we establish a series of complexity results, completing the complexity picture for each of the above languages, and four different query languages: arbitrary conjunctive queries, bounded (hyper-)treewidth queries, acyclic queries and atomic queries. For the guarded-based languages, we show a strong 2EXPTIME lower bound for general queries that holds even for fixed ontologies, and establishes 2EXPTIME-completeness of the query answering problem in this case. For acyclic queries, the complexity can be reduced to EXPTIME, if the predicate arity is bounded, and the problem even becomes tractable for certain restricted languages, if only atomic queries are used. For ontologies represented by sticky disjunctive rules, we show that the problem becomes undecidable, even in the case of data complexity and atomic queries. Finally, for weakly-acyclic rules, we show that the complexity increases from 2EXPTIME to coN2EXPTIME in general, and from tractable to coNP in case of the data complexity, independent of which query language is used. After answering the open complexity questions, we investigate applications and relevant consequences of our results for description logics and give two generic complexity statements, respectively, for acyclic and general conjunctive query answering over description logic knowledge bases. These generic results allow for an easy determination of the complexity of this reasoning task, based on the expressivity of the considered description logic.