In our pursuit of generic criteria for decidable ontology-based querying, we introduce 'finite-cliquewidth sets' (FCS) of existential rules, a model-theoretically defined class of rule sets, inspired by the cliquewidth measure from graph theory. By a generic argument, we show that FCS ensures decidability of entailment for a sizable class of queries (dubbed 'DaMSOQs') subsuming conjunctive queries (CQs). The FCS class properly generalizes the class of finite-expansion sets (FES), and for signatures of arity at most 2, the class of bounded-treewidth sets (BTS). For higher arities, BTS is only indirectly subsumed by FCS by means of reification. Despite the generality of FCS, we provide a rule set with decidable CQ entailment (by virtue of first-order-rewritability) that falls outside FCS, thus demonstrating the incomparability of FCS and the class of finite-unification sets (FUS). In spite of this, we show that if we restrict ourselves to single-headed rule sets over signatures of arity at most 2, then FCS subsumes FUS.
We consider existential rules (aka Datalog+) as a formalism for specifying ontologies. In recent years, many classes of existential rules have been exhibited for which conjunctive query (CQ) entailment is decidable. However, most of these classes cannot express transitivity of binary relations, a frequently used modelling construct. In this paper, we address the issue of whether transitivity can be safely combined with decidable classes of existential rules. First, we prove that transitivity is incompatible with one of the simplest decidable classes, namely aGRD (acyclic graph of rule dependen…
The Sticky Path to Expressive Querying: Decidability of Navigational Queries under Existential Rules
TU Dresden（德累斯顿工业大学）；University of Wrocław（弗罗茨瓦夫大学）；Center for Scalable Data Analytics and Artificial Intelligence Dresden/Leipzig（德累斯顿/莱比锡可扩展数据分析与人工智能中心）
Existential rules are a very popular ontology-mediated query language for which the chase represents a generic computational approach for query answering. It is straightforward that existential rule queries exhibiting chase termination are decidable and can only recognize properties that are preserved under homomorphisms. In this paper, we show the converse: every decidable query that is closed under homomorphism can be expressed by an existential rule set for which the standard chase universally terminates. Membership in this fragment is not decidable, but we show via a diagonalisation argume…
Datalog is one of the best-known rule-based languages, and extensions of it are used in a wide context of applications. An important Datalog extension is Disjunctive Datalog, which significantly increases the expressivity of the basic language. Disjunctive Datalog is useful in a wide range of applications, ranging from Databases (e.g., Data Integration) to Artificial Intelligence (e.g., diagnosis and planning under incomplete knowledge). However, in recent years an important shortcoming of Datalog-based languages became evident, e.g. in the context of data-integration (consistent query-answeri…
Warded Datalog+- extends the logic-based language Datalog with existential quantifiers in rule heads. Existential rules are needed for advanced reasoning tasks, e.g., ontological reasoning. The theoretical efficiency guarantees of Warded Datalog+- do not cover extensions crucial for data analytics, such as arithmetic. Moreover, despite the significance of arithmetic for common data analytic scenarios, no decidable fragment of any Datalog+- language extended with arithmetic has been identified. We close this gap by defining a new language that extends Warded Datalog+- with arithmetic and prove …
Diekert, Matiyasevich and Muscholl proved that the existential first-order theory of a trace monoid over a finite alphabet is decidable. We extend this result to a natural class of trace monoids with infinitely many generators. As an application, we prove that for every ordinal $λ$ less than $\varepsilon_0$, the existential theory of the set of successor ordinals less than $λ$ equipped with multiplication is decidable.
FBK-IRST（布鲁诺·凯斯勒基金会研究院）；University Of Trento（特伦托大学）；Otto-von-Guericke University of Magdeburg（马格德堡奥托·冯·格里克大学）
The proliferation of contextualized knowledge in the Semantic Web (SW) has led to the popularity of knowledge formats such as \emph{quads} in the SW community. A quad is an extension of an RDF triple with contextual information of the triple. In this paper, we study the problem of query answering over quads augmented with forall-existential bridge rules that enable interoperability of reasoning between triples in various contexts. We call a set of quads together with such expressive bridge rules, a quad-system. Query answering over quad-systems is undecidable, in general. We derive decidable c…
University of Hildesheim（希尔德斯海姆大学）；University of Bremen（不来梅大学）；University of Wrocław（弗罗茨瓦夫大学）
We study the problem to decide, given sets T1,T2 of tuple-generating dependencies (TGDs), also called existential rules, whether T2 is a conservative extension of T1. We consider two natural notions of conservative extension, one pertaining to answers to conjunctive queries over databases and one to homomorphisms between chased databases. Our main results are that these problems are undecidable for linear TGDs, undecidable for guarded TGDs even when T1 is empty, and decidable for frontier-one TGDs.
Existential rules, long known as tuple-generating dependencies in database theory, have been intensively studied in the last decade as a powerful formalism to represent ontological knowledge in the context of ontology-based query answering. A knowledge base is then composed of an instance that contains incomplete data and a set of existential rules, and answers to queries are logically entailed from the knowledge base. This brought again to light the fundamental chase tool, and its different variants that have been proposed in the literature. It is well-known that the problem of determining, g…
Simple conceptual graphs are considered as the kernel of most knowledge representation formalisms built upon Sowa's model. Reasoning in this model can be expressed by a graph homomorphism called projection, whose semantics is usually given in terms of positive, conjunctive, existential FOL. We present here a family of extensions of this model, based on rules and constraints, keeping graph homomorphism as the basic operation. We focus on the formal definitions of the different models obtained, including their operational semantics and relationships with FOL, and we analyze the decidability and …
Ontology-based query answering (OBQA) asks whether a Boolean conjunctive query is satisfied by all models of a logical theory consisting of a relational database paired with an ontology. The introduction of existential rules (i.e., Datalog rules extended with existential quantifiers in rule-heads) as a means to specify the ontology gave birth to Datalog+/-, a framework that has received increasing attention in the last decade, with focus also on decidability and finite controllability to support effective reasoning. Five basic decidable fragments have been singled out: linear, weakly-acyclic, …
University of Western Sydney（西悉尼大学）；University of Alberta（阿尔伯塔大学）
Finite chase, or alternatively chase termination, is an important condition to ensure the decidability of existential rule languages. In the past few years, a number of rule languages with finite chase have been studied. In this work, we propose a novel approach for classifying the rule languages with finite chase. Using this approach, a family of decidable rule languages, which extend the existing languages with the finite chase property, are naturally defined. We then study the complexity of these languages. Although all of them are tractable for data complexity, we show that their combined …
It is shown that order-invariance of two-variable first-logic is decidable in the finite. This is an immediate consequence of a decision procedure obtained for the finite satisfiability problem for existential second-order logic with two first-order variables ($\mathrm{ESO}^2$) on structures with two linear orders and one induced successor. We also show that finite satisfiability is decidable on structures with two successors and one induced linear order. In both cases, so far only decidability for monadic $\mathrm{ESO}^2$ has been known. In addition, the finite satisfiability problem for $\ma…
We define a class of Separation Logic formulae, whose entailment problem: given formulae $ϕ, ψ_1, \ldots, ψ_n$, is every model of $ϕ$ a model of some $ψ_i$? is 2EXPTIME-complete. The formulae in this class are existentially quantified separating conjunctions involving predicate atoms, interpreted by the least sets of store-heap structures that satisfy a set of inductive rules, which is also part of the input to the entailment problem. Previous work consider established sets of rules, meaning that every existentially quantified variable in a rule must eventually be bound to an allocated locatio…
Datalog+/- is a Datalog-based language family enhanced with existential quantification in rule heads, equalities and negative constraints. Query answering over databases with respect to a Datalog+/- theory is generally undecidable, however several syntactic restrictions have been proposed to remedy this fact. However, a useful and natural feature however is as of yet missing from Datalog+/-: The ability to express uncertain knowledge, or choices, using disjunction. It is the precise objective of the doctoral thesis herein discussed, to investigate the impact on the complexity of query answerin…
DKM, FBK-IRST（FBK-IRST 的 DKM 部门）；DISI, University Of Trento（特伦托大学 DISI 系）
The recent outburst of context-dependent knowledge on the Semantic Web (SW) has led to the realization of the importance of the quads in the SW community. Quads, which extend a standard RDF triple, by adding a new parameter of the `context' of an RDF triple, thus informs a reasoner to distinguish between the knowledge in various contexts. Although this distinction separates the triples in an RDF graph into various contexts, and allows the reasoning to be decoupled across various contexts, bridge rules need to be provided for inter-operating the knowledge across these contexts. We call a set of…
