On the k-Boundedness for Existential Rules

The chase is a fundamental tool for existential rules. Several chase variants are known, which differ on how they handle redundancies possibly caused by the introduction of nulls. Given a chase variant, the halting problem takes as input a set of existential rules and asks if this set of rules ensures the termination of the chase for any factbase. It is well-known that this problem is undecidable for all known chase variants. The related problem of boundedness asks if a given set of existential rules is bounded, i.e., whether there is a predefined upper bound on the number of (breadth-first) steps of the chase, independently from any factbase. This problem is already undecidable in the specific case of datalog rules. However, knowing that a set of rules is bounded for some chase variant does not help much in practice if the bound is unknown. Hence, in this paper, we investigate the decidability of the k-boundedness problem, which asks whether a given set of rules is bounded by an integer k. We prove that k-boundedness is decidable for three chase variants, namely the oblivious, semi-oblivious and restricted chase.Comment: 20 pages, revised version of the paper published at RuleML+RR 201

All-Instances Restricted Chase Termination

The chase procedure is a fundamental algorithmic tool in database theory with a variety of applications. A key problem concerning the chase procedure is all-instances termination: for a given set of tuple-generating dependencies (TGDs), is it the case that the chase terminates for every input database? In view of the fact that this problem is undecidable, it is natural to ask whether known well-behaved classes of TGDs ensure decidability. We consider here the main paradigms that led to robust TGD-based formalisms, that is, guardedness and stickiness. Although all-instances termination is well-understood for the oblivious version of the chase, the more subtle case of the restricted (a.k.a. the standard) chase is rather unexplored. We show that all-instances restricted chase termination for guarded and sticky single-head TGDs is decidable

Oblivious Chase Termination:The Sticky Case

The chase procedure is one of the most fundamental algorithmic tools in database theory. A key algorithmic task is uniform chase termination, i.e., given a set of tuple-generating dependencies (tgds), is it the case that the chase under this set of tgds terminates, for every input database? In view of the fact that this problem is undecidable, no matter which version of the chase we consider, it is natural to ask whether well-behaved classes of tgds, introduced in different contexts such as ontological reasoning, make our problem decidable. In this work, we consider a prominent decidability paradigm for tgds, called stickiness. We show that for sticky sets of tgds, uniform chase termination is decidable if we focus on the (semi-)oblivious chase, and we pinpoint its exact complexity: PSpace-complete in general, and NLogSpace-complete for predicates of bounded arity. These complexity results are obtained via graph-based syntactic characterizations of chase termination that are of independent interest

The data-exchange chase under the microscope

In this paper we take closer look at recent developments for the chase procedure, and provide additional results. Our analysis allows us create a taxonomy of the chase variations and the properties they satisfy. Two of the most central problems regarding the chase is termination, and discovery of restricted classes of sets of dependencies that guarantee termination of the chase. The search for the restricted classes has been motivated by a fairly recent result that shows that it is undecidable to determine whether the chase with a given dependency set will terminate on a given instance. There is a small dissonance here, since the quest has been for classes of sets of dependencies guaranteeing termination of the chase on all instances, even though the latter problem was not known to be undecidable. We resolve the dissonance in this paper by showing that determining whether the chase with a given set of dependencies terminates on all instances is coRE-complete. For the hardness proof we use a reduction from word rewriting systems, thereby also showing the close connection between the chase and word rewriting. The same reduction also gives us the aforementioned instance-dependent RE-completeness result as a byproduct. For one of the restricted classes guaranteeing termination on all instances, the stratified sets dependencies, we provide new complexity results for the problem of testing whether a given set of dependencies belongs to it. These results rectify some previous claims that have occurred in the literature.Comment: arXiv admin note: substantial text overlap with arXiv:1303.668

Revisiting Chase Termination for Existential Rules and their Extension to Nonmonotonic Negation

Existential rules have been proposed for representing ontological knowledge, specifically in the context of Ontology- Based Data Access. Entailment with existential rules is undecidable. We focus in this paper on conditions that ensure the termination of a breadth-first forward chaining algorithm known as the chase. Several variants of the chase have been proposed. In the first part of this paper, we propose a new tool that allows to extend existing acyclicity conditions ensuring chase termination, while keeping good complexity properties. In the second part, we study the extension to existential rules with nonmonotonic negation under stable model semantics, discuss the relevancy of the chase variants for these rules and further extend acyclicity results obtained in the positive case.Comment: This paper appears in the Proceedings of the 15th International Workshop on Non-Monotonic Reasoning (NMR 2014