The chase is a ubiquitous algorithm in database theory. However, for existential rules (aka tuple-generating dependencies), its termination is not guaranteed, and even undecidable in general. The problem of termination becomes particularly difficult for the restricted (or standard) chase, for which the order of rule application matters. Thus, decidability of restricted chase termination is still open for many well-behaved classes such as linear or guarded multi-headed rules. We make a step forward by showing that all-instances restricted chase termination is decidable in the linear multi-headed case.

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 to employ 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 suggest several options for remedy.

This paper addresses one of the fundamental open questions in the realm of existential rules: the conjecture on the finite controllability of bounded derivation depth rule sets (bdd⇒fc). We take a step toward a positive resolution of this conjecture by demonstrating that universal models generated by BDD rule sets cannot contain arbitrarily large tournaments (arbitrarily directed cliques) without entailing a loop query, ∃ E (x,x). This simple yet elegant result narrows the space of potential counterexamples to the (bdd⇒fc) conjecture.

Query Containment Problem (QCP) is a fundamental decision problem in query processing and optimization. While QCP has for a long time been completely understood for the case of set semantics, decidability of QCP for conjunctive queries under multi-set semantics (QCP CQ bag ) remains one of the most intriguing open problems in database theory. Certain effort has been put, in the last 30 years, to solve this problem and some decidable special cases of QCP CQ bag were identified, as well as some undecidable extensions, including QCP UCQ bag . In this paper we introduce a new technique which produces, for a given UCQ Φ, a CQ φ such that the application of φ to a database D is, in some sense, an approximation of the application of Φ to D . Using this technique we could analyze the status of QCP bag when one of the queries in question is a CQ and the other is a UCQ, and we reached conclusions which surprised us a little bit. We also tried to use this technique to translate the known undecidability proof for QCP UCQ bag into a proof of undecidability of QCP CQ bag . And, as you are going to see, we got stopped just one infinitely small ε before reaching this ultimate goal.

We study the interaction of views, queries, and background knowledge in the form of existential rules. The motivating questions concern monotonic determinacy of a query using views w.r.t. rules, which refers to the ability to recover the query answer from the views via a monotone function. We study the decidability of monotonic determinacy, and compare with variations that require the “recovery function” to be in a well-known monotone query language, such as conjunctive queries or Datalog. Surprisingly, we find that even in the presence of basic existential rules, the borderline between well-behaved and badly-behaved answerability differs radically from the unconstrained case. In order to understand this boundary, we require new results concerning entailment problems involving views and rules.

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.

The decidability of axiomatic extensions of the modal logic K with modal reduction principles, i.e. axioms of the form ⋄kp → ⋄np, has remained a long-standing open problem. In this paper, we make significant progress toward solving this problem and show that decidability holds for a large subclass of these logics, namely, for quasi-dense logics. Such logics are extensions of K with modal reduction axioms such that 0 < k < n (dubbed quasi-density axioms). To prove decidability, we define novel proof systems for quasi-dense logics consisting of disjunctive existential rules, which are first-order formulae typically used to specify ontologies in the context of database theory. We show that such proof systems can be used to generate proofs and models of modal formulae, and provide an intricate model-theoretic argument showing that such generated models can be encoded as finite objects called templates. By enumerating templates of bound size, we obtain an ExpSpace decision procedure as a consequence.

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 ≤ 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 ≤ 2, then fcs subsumes fus.

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 the main tool for establishing decidability proof-theoretically with a central decidability tool in the context of knowledge representation.

A query Q is monotonically determined over a set of views V if Q can be expressed as a monotonic function of the view image. In the case of relational algebra views and queries, monotonic determinacy coincides with rewritability as a union of conjunctive queries, and it is decidable in important special cases, such as for conjunctive query views and queries. We investigate the situation for views and queries in the recursive query language Datalog. We give both positive and negative results about the ability to decide monotonic determinacy, and also about the co-incidence of monotonic determinacy with Datalog rewritability.

In their classical 1993 paper Chaudhuri and Vardi notice that some fundamental database theory results and techniques fail to survive when we try to see query answers as bags (multisets) of tuples rather than as sets of tuples.

We consider (first-order) query rewritability in the context of theory-mediated query answering. The starting point of our journey is the FUS/FES conjecture, which states that any theory that is a finite expansion set (FES) and admits query rewriting (BDD, FUS) must be uniformly bounded. We show that this conjecture holds for a large class of BDD theories, which we call "local". Upon investigating how "non-local" BDD theories can actually get, we discover unexpected phenomena that, we think, are at odds with prevailing intuitions about BDD theories.

The chase is a famous algorithmic procedure in database theory with numerous applications in ontology-mediated query answering. We consider static analysis of the chase termination problem, which asks, given set of TGDs, whether the chase terminates on all input databases. The problem was recently shown to be undecidable by Gogacz et al. for sets of rules containing only ternary predicates. In this work, we show that undecidability occurs already for sets of single-head TGD over binary vocabularies. This question is relevant since many real-world ontologies, e.g., those from the Horn fragment of the popular OWL, are of this shape.

In our paper [Głuch, Marcinkowski, Ostropolski-Nalewaja, LICS ACM, 2018] we have solved an old problem stated in [Calvanese, De Giacomo, Lenzerini, Vardi, SPDS ACM, 2000] showing that query determinacy is undecidable for Regular Path Queries. Here a strong generalisation of this result is shown, and - we think - a very unexpected one. We prove that no regularity is needed: determinacy remains undecidable even for finite unions of conjunctive path queries.

For a given set of queries (which are expressions in some query language) Q = {Q1, Q2, ... Qk} and for another query Q0 we say that Q determines Q0 if -- informally speaking -- for every database D, the information contained in the views Q(D) is sufficient to compute Q0(D).

In the Maximum Duo-Preservation String Mapping problem we are given two strings and wish to map the letters of the former to the letters of the latter as to maximise the number of duos. A duo is a pair of consecutive letters that is mapped to a pair of consecutive letters in the same order. This is complementary to the well-studied Minimum Common String Partition problem, where the goal is to partition the former string into blocks that can be permuted and concatenated to obtain the latter string. Maximum Duo-Preservation String Mapping is APX-hard. After a series of improvements, Brubach [WABI 2016] showed a polynomial-time 3.25-approximation algorithm. Our main contribution is that, for any eps>0, there exists a polynomial-time (2+eps)-approximation algorithm. Similarly to a previous solution by Boria et al. [CPM 2016], our algorithm uses the local search technique. However, this is used only after a certain preliminary greedy procedure, which gives us more structure and makes a more general local search possible. We complement this with a specialised version of the algorithm that achieves 2.67-approximation in quadratic time.