Chase Termination Research Articles

Chase Termination Beyond Polynomial Time

The chase is a widely implemented approach to reason with tuple-generating dependencies (tgds), used in data exchange, data integration, and ontology-based query answering. However, it is merely a semi-decision procedure, which may fail to terminate. Many decidable conditions have been proposed for tgds to ensure chase termination, typically by forbidding some kind of "cycle'' in the chase process. We propose a new criterion that explicitly allows some such cycles, and yet ensures termination of the standard chase under reasonable conditions. This leads to new decidable fragments of tgds that are not only syntactically more general but also strictly more expressive than the fragments defined by prior acyclicity conditions. Indeed, while known terminating fragments are restricted to PTime data complexity, our conditions yield decidable languages for any k- ExpTime. We further refine our syntactic conditions to obtain fragments of tgds for which an optimised chase procedure decides query entailment in PSpace or k- ExpSpace, respectively.

Semi-Oblivious Chase Termination for Linear Existential Rules: An Experimental Study

The chase procedure is a fundamental algorithmic tool in databases that allows us to reason with constraints, such as existential rules, with a plethora of applications. It takes as input a database and a set of constraints, and iteratively completes the database as dictated by the constraints. A key challenge, though, is the fact that it may not terminate, which leads to the problem of checking whether it terminates given a database and a set of constraints. In this work, we focus on the semi-oblivious version of the chase, which is well-suited for practical implementations, and linear existential rules, a central class of constraints with several applications. In this setting, there is a mature body of theoretical work that provides syntactic characterizations of when the chase terminates, algorithms for checking chase termination, and precise complexity results. Our main objective is to experimentally evaluate the existing chase termination algorithms with the aim of understanding which input parameters affect their performance, clarifying whether they can be used in practice, and revealing their performance limitations.

General Acyclicity and Cyclicity Notions for the Disjunctive Skolem Chase

The disjunctive skolem chase is a sound, complete, and potentially non-terminating procedure for solving boolean conjunctive query entailment over knowledge bases of disjunctive existential rules. We develop novel acyclicity and cyclicity notions for this procedure; that is, we develop sufficient conditions to determine chase termination and non-termination. Our empirical evaluation shows that our novel notions are significantly more general than existing criteria.

Uniform Restricted Chase Termination

The chase procedure is a fundamental algorithmic tool in database theory with a variety of applications. A central problem concerning the chase procedure is uniform (a.k.a. all-instances) chase 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, in general, undecidable, it is natural to ask whether known well-behaved classes of TGDs ensure decidability. We focus on the main paradigms that led to robust TGD-based formalisms, namely guardedness and stickiness, that have been introduced in the context of knowledge-enriched databases. Although uniform chase termination is well understood for the oblivious version of the chase (2EXPTIME-complete for guarded, and PSPACE-complete for sticky TGDs), the more subtle case of the restricted (a.k.a. the standard) chase is rather unexplored. We show that uniform restricted chase termination under guarded single-head TGDs and sticky single-head TGDs is decidable in elementary time. In the case of guardedness, we provide a reduction to the satisfiability problem of monadic second-order logic over infinite trees of bounded degree, while for stickiness we provide a reduction to the emptiness problem of deterministic Büchi automata. Those reductions build on a series of technical results of independent interest related to the notion of fairness of the restricted chase, and the existence of critical databases that characterize nontermination of the restricted chase via databases of a certain form.

All-Instances Restricted Chase Termination for Linear TGDs

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 chase 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, in general, undecidable, it is natural to ask whether well-behaved classes of TGDs, introduced in different contexts, ensure decidability. It has been recently shown that the problem is decidable for the restricted (a.k.a. standard) version of the chase, and linear TGDs, a prominent class of TGDs that has been introduced in the context of ontological query answering, under the assumption that only one atom appears in TGD-heads. We provide an alternative proof for this result based on Monadic Second-Order Logic, which we believe is simpler that the ones obtained from the literature.

Semi-Oblivious Chase Termination: The Sticky Case

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, introduced in different contexts such as ontological reasoning, ensure decidability. We consider a prominent paradigm that led to a robust TGD-based formalism, called stickiness. We show that for sticky sets of TGDs, all-instances 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 a graph-based syntactic characterization of chase termination that is of independent interest.

Checking Chase Termination over Ontologies of Existential Rules with Equality

The chase is a sound and complete algorithm for conjunctive query answering over ontologies of existential rules with equality. To enable its effective use, we can apply acyclicity notions; that is, sufficient conditions that guarantee chase termination. Unfortunately, most of these notions have only been defined for existential rule sets without equality. A proposed solution to circumvent this issue is to treat equality as an ordinary predicate with an explicit axiomatisation. We empirically show that this solution is not efficient in practice and propose an alternative approach. More precisely, we show that, if the chase terminates for any equality axiomatisation of an ontology, then it terminates for the original ontology (which may contain equality). Therefore, one can apply existing acyclicity notions to check chase termination over an axiomatisation of an ontology and then use the original ontology for reasoning. We show that, in practice, doing so results in a more efficient reasoning procedure. Furthermore, we present equality model-faithful acyclicity, a general acyclicity notion that can be directly applied to ontologies with equality.

Checking Chase Termination: Cyclicity Analysis and Rewriting Techniques

The aim of this paper is to present more general criteria and techniques for chase termination. We first present extensions of the well-known stratification criterion and introduce a new criterion, called local stratification , which generalizes both super-weak acyclicity and stratification -based criteria (including the class of constraints which are inductively restricted ). Next, the paper presents a rewriting algorithm transforming the original set of constraints $\Sigma$ into an “equivalent” set $\Sigma ^\alpha$ and verifying the structural properties for chase termination on $\Sigma ^\alpha$ . The rewriting of constraints allows us to recognize larger classes of constraints for which chase termination is guaranteed. In particular, we show that if $\Sigma$ satisfies chase termination conditions $\cal T$ , then the rewritten set $\Sigma ^\alpha$ satisfies $\cal T$ as well, but the vice versa is not true, that is there are significant classes of constraints for which $\Sigma ^\alpha$ satisfies $\cal T$ and $\Sigma$ does not. A more general rewriting algorithm producing as output an equivalent set of dependencies and a Boolean value stating whether a sort of cyclicity has been detected is also proposed. The new rewriting technique and the checking of acyclicity allow us to introduce the class of acyclic constraints, which generalizes local stratification and guarantees that all chase sequences are finite with a length polynomial in the size of the input database.

Existential Rule Languages with Finite Chase: Complexity and Expressiveness

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 complexity can be arbitrarily high. Furthermore, we prove that all the rule languages with finite chase that extend the weakly acyclic language are of the same expressiveness as the weakly acyclic one, while rule languages with higher combined complexity are in general more succinct than those with lower combined complexity.

Acyclicity Notions for Existential Rules and Their Application to Query Answering in Ontologies

Answering conjunctive queries (CQs) over a set of facts extended with existential rules is a prominent problem in knowledge representation and databases. This problem can be solved using the chase algorithm, which extends the given set of facts with fresh facts in order to satisfy the rules. If the chase terminates, then CQs can be evaluated directly in the resulting set of facts. The chase, however, does not terminate necessarily, and checking whether the chase terminates on a given set of rules and facts is undecidable. Numerous acyclicity notions were proposed as sufficient conditions for chase termination. In this paper, we present two new acyclicity notions called model-faithful acyclicity (MFA) and model-summarising acyclicity (MSA). Furthermore, we investigate the landscape of the known acyclicity notions and establish a complete taxonomy of all notions known to us. Finally, we show that MFA and MSA generalise most of these notions. Existential rules are closely related to the Horn fragments of the OWL 2 ontology language; furthermore, several prominent OWL 2 reasoners implement CQ answering by using the chase to materialise all relevant facts. In order to avoid termination problems, many of these systems handle only the OWL 2 RL profile of OWL 2; furthermore, some systems go beyond OWL 2 RL, but without any termination guarantees. In this paper we also investigate whether various acyclicity notions can provide a principled and practical solution to these problems. On the theoretical side, we show that query answering for acyclic ontologies is of lower complexity than for general ontologies. On the practical side, we show that many of the commonly used OWL 2 ontologies are MSA, and that the number of facts obtained by materialisation is not too large. Our results thus suggest that principled development of materialisation-based OWL 2 reasoners is practically feasible.

Stratification criteria and rewriting techniques for checking chase termination

The Chase is a fixpoint algorithm enforcing satisfaction of data dependencies in databases. Its execution involves the insertion of tuples with possible null values and the changing of null values which can be made equal to constants or other null values. Since the chase fixpoint evaluation could be non-terminating, in recent years the problem know as chase termination has been investigated. It consists in the detection of sufficient conditions, derived from the structural analysis of dependencies, guaranteeing that the chase fix-point terminates independently from the database instance. Several criteria introducing sufficient conditions for chase termination have been recently proposed [9, 8, 13, 12]. The aim of this paper is to present more general criteria and techniques for chase termination. We first present extensions of the well-known stratification conditions and introduce a new criterion, called local stratification ( LS ), which generalizes both super-weak acyclicity and stratification-based criteria (including the class of constraints which are inductively restricted). Next the paper presents a rewriting algorithm, whose structure is similar to the one presented in [10]; the algorithm takes as input a set of tuple generating dependencies and produces as output an equivalent set of dependencies and a boolean value stating whether a sort of cyclicity has been detected. The output set, obtained by adorning the input set of constraints, allows us to perform a more accurate analysis of the structural properties of constraints and to further enlarge the class of tuple generating dependencies for which chase termination is guaranteed, whereas the checking of acyclicity allows us to introduce the class of acyclic constraints ( AC ), which generalizes LS and guarantees chase termination.

Chase termination

Several database areas such as data exchange and integration share the problem of fixing database instance violations with respect to a set of constraints. The chase algorithm solves such violations by inserting tuples and setting the value of nulls. Unfortunately, the chase algorithm may not terminate and the problem of deciding whether the chase process terminates is undecidable. Recently there has been an increasing interest in the identification of sufficient structural properties of constraints which guarantee that the chase algorithm terminates [8, 10, 14, 15]. In this paper we propose an original technique which allows to improve current conditions detecting chase termination. Our proposal consists in rewriting the original set of constraints Σ into an 'equivalent' set Σ α and verifying the structural properties for chase termination on Σ α . The rewriting of constraints allows to recognize larger classes of constraints for which chase termination is guaranteed. In particular, we show that if Σ satisfies chase termination conditions T, then the rewritten set Σ α satisfies T as well, but the vice versa is not true, that is there are significant classes of constraints for which Σ α satisfies T and Σ does not.