Extensive research in the field of ontology-based query answering has led to the identification of numerous fragments of existential rules (also known as tuple-generating dependencies) that exhibit decidable answering of atomic and conjunctive queries. Motivated by the increased theoretical and practical interest in navigational queries, this paper considers the question for which of these fragments decidability of querying extends to regular path queries (RPQs). In fact, decidability of RPQs has recently been shown to generally hold for the comprehensive family of all fragments that come with the guarantee of universal models being reasonably well-shaped (that is, being of finite cliquewidth). Yet, for the second major family of fragments, known as finite unification sets (short: fus), which are based on first-order-rewritability, corresponding results have been largely elusive so far. We complete the picture by showing that RPQ answering over arbitrary fus rulesets is undecidable. On the positive side, we establish that the problem is decidable for the prominent fus subclass of sticky rulesets, with the caveat that a very mild extension of the RPQ formalism turns the problem undecidable again.

We study consistent query answering over knowledge bases expressed by existential rules. Specifically, we establish the data complexity of consistent query answering and repair checking under tuple-deletion semantics for a general class of disjunctive existential rules and for several subclasses thereof (acyclic, linear, full, guarded, and sticky). In particular, we identify several cases in which the above problems are tractable or even first-order rewritable, and present new query rewriting techniques that can be the basis for practical inconsistency-tolerant query answering systems.

We propose a generic framework for establishing the decidability of a wide range of logical entailment problems (briefly called querying), based on the existence of countermodels that are structurally simple, gauged by certain types of width measures (with treewidth and cliquewidth as popular examples). As an important special case of our framework, we identify logics exhibiting width-finite finitely universal model sets, warranting decidable entailment for a wide range of homomorphism-closed queries, subsuming a diverse set of practically relevant query languages. As a particularly powerful width measure, we propose Blumensath's partitionwidth, which subsumes various other commonly considered width measures and exhibits highly favorable computational and structural properties. Focusing on the formalism of existential rules as a popular showcase, we explain how finite-partitionwidth sets of rules subsume other known abstract decidable classes but - leveraging existing notions of stratification - also cover a wide range of new rulesets. We expose natural limitations for fitting the class of finite-unification sets into our picture and provide several options for remedy.

Notation3 Logic (\nthree) is an extension of RDF that allows the user to write rules introducing new blank nodes to RDF graphs. Many applications (e.g., ontology mapping) rely on this feature as blank nodes -- used directly or in auxiliary constructs -- are omnipresent on the Web. However, the number of fast \nthree reasoners covering this very important feature of the logic is rather limited. On the other hand, there are engines like VLog or Nemo which do not directly support Semantic Web rule formats but which are developed and optimized for very similar constructs: existential rules. In this paper, we investigate the relation between \nthree rules with blank nodes in their heads and existential rules. We identify a subset of \nthree which can be mapped directly to existential rules and define such a mapping preserving the equivalence of \nthree formulae. In order to also illustrate that in some cases \nthree reasoning could benefit from our translation, we then employ this mapping in an implementation to compare the performance of the \nthree reasoners EYE and cwm to VLog and Nemo on \nthree rules and their mapped counterparts. Our tests show that the existential rule reasoners perform particularly well for use cases containing many facts while especially the EYE reasoner is very fast when dealing with a high number of dependent rules. We thus provide a tool enabling the Semantic Web community to directly use existing and future existential rule reasoners and benefit from the findings of this active community.

This paper establishes alternative characterizations of very expressive classes of existential rule sets with decidable query entailment. We consider the notable class of greedy bounded-treewidth sets (gbts) and a new, generalized variant, called weakly gbts (wgbts). Revisiting and building on the notion of derivation graphs, we define (weakly) cycle-free derivation graph sets ((w)cdgs) and employ elaborate proof-theoretic arguments to obtain that gbts and cdgs coincide, as do wgbts and wcdgs. These novel characterizations advance our analytic proof-theoretic understanding of existential rules and will likely be instrumental in practice.

We consider the issue of answering unions of conjunctive queries (UCQs) with disjunctive existential rules and mappings. While this issue has already been well studied from a chase perspective, query rewriting within UCQs has hardly been addressed yet. We first propose a sound and complete query rewriting operator, which has the advantage of establishing a tight relationship between a chase step and a rewriting step. The associated breadth-first query rewriting algorithm outputs a minimal UCQ-rewriting when one exists. Second, we show that for any ``truly disjunctive'' nonrecursive rule, there exists a conjunctive query that has no UCQ-rewriting. It follows that the notion of finite unification sets (fus), which denotes sets of existential rules such that any UCQ admits a UCQ-rewriting, seems to have little relevance in this setting. Finally, turning our attention to mappings, we show that the problem of determining whether a UCQ admits a UCQ-rewriting through a disjunctive mapping is undecidable. We conclude with a number of open problems.

Chase algorithms are indispensable in the domain of knowledge base querying, which enable the extraction of implicit knowledge from a given database via applications of rules from a given ontology. Such algorithms have proved beneficial in identifying logical languages which admit decidable query entailment. Within the discipline of proof theory, sequent calculi have been used to write and design proof-search algorithms to identify decidable classes of logics. In this paper, we show that the chase mechanism in the context of existential rules is in essence the same as proof-search in an extension of Gentzen's sequent calculus for first-order logic. Moreover, we show that proof-search generates universal models of knowledge bases, a feature also exhibited by the chase. Thus, we formally connect a central tool for establishing decidability proof-theoretically with a central decidability tool in the context of knowledge representation.

Several types of dependencies have been proposed for the static analysis of existential rule ontologies, promising insights about computational properties and possible practical uses of a given set of rules, e.g., in ontology-based query answering. Unfortunately, these dependencies are rarely implemented, so their potential is hardly realised in practice. We focus on two kinds of rule dependencies -- positive reliances and restraints -- and design and implement optimised algorithms for their efficient computation. Experiments on real-world ontologies of up to more than 100,000 rules show the scalability of our approach, which lets us realise several previously proposed applications as practical case studies. In particular, we can analyse to what extent rule-based bottom-up approaches of reasoning can be guaranteed to yield redundancy-free "lean" knowledge graphs (so-called cores) on practical ontologies.

Answering conjunctive queries (CQs) over a set of facts extended with existential rules is a key problem in knowledge representation and databases. This problem can be solved using the chase (aka materialisation) algorithm; however, CQ answering is undecidable for general existential rules, so the chase is not guaranteed to terminate. Several acyclicity conditions provide sufficient conditions for chase termination. In this paper, we present two novel such conditions--model-faithful acyclicity (MFA) and model-summarising acyclicity (MSA)--that generalise many of the acyclicity conditions known so far in the literature. Materialisation provides the basis for several widely-used OWL 2 DL reasoners.

Answering queries in information systems that allow for ex- pressive inferencing is currently a field of intense research. This problem is often referred to as ontology-based data ac- cess (OBDA). We focus on conjunctive query entailment un- der logical rules known as tuple-generating dependencies, existential rules or Datalog /-. One of the most expressive decidable classes of existential rules known today is that of greedy bounded treewidth sets (gbts). We propose an algo- rithm for this class, which is worst-case optimal for data and combined complexities, with or without bound on the pred- icate arity. A beneficial feature of this algorithm is that it allows for separation between offline and online processing steps: the knowledge base can be compiled independently from queries, which are evaluated against the compiled form. Moreover, very simple adaptations of the algorithm lead to worst-case-optimal complexities for specific subclasses of gbts which have lower complexities, such as guarded rules.