Datalog Rewritings Research Articles

Datalog rewritability and data complexity of [formula omitted] with closed predicates

We study the relative expressiveness of ontology-mediated queries (OMQs) formulated in the expressive Description Logic ALCHOIQ extended with closed predicates. In particular, we present a polynomial time translation from OMQs into Datalog with negation under the stable model semantics, the formalism that underlies Answer Set Programming. This is a novel and non-trivial result: the considered OMQs are not only non-monotonic, but also feature a tricky combination of nominals, inverse roles, and counting. We start with atomic queries and then lift our approach to a large class of first-order queries where quantification is “guarded” by closed predicates. Our translation is based on a characterization of the query answering problem via integer programming, and a specially crafted program in Datalog with negation that finds solutions to dynamically generated systems of integer inequalities. As an important by-product of our translation we get that the query answering problem is co-NP-complete in data complexity for the considered class of OMQs. Thus, answering these OMQs in the presence of closed predicates is not harder than answering them in the standard setting. This is not obvious as closed predicates are known to increase data complexity for some existing ontology languages.

On Monotonic Determinacy and Rewritability for Recursive Queries and Views

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.

A tetrachotomy of ontology-mediated queries with a covering axiom

Our concern is the problem of efficiently determining the data complexity of answering queries mediated by description logic ontologies and constructing their optimal rewritings to standard database queries. Originated in ontology-based data access and datalog optimisation, this problem is known to be computationally very complex in general, with no explicit syntactic characterisations available. In this article, aiming to understand the fundamental roots of this difficulty, we strip the problem to the bare bones and focus on Boolean conjunctive queries mediated by a simple covering axiom stating that one class is covered by the union of two other classes. We show that, on the one hand, these rudimentary ontology-mediated queries, called disjunctive sirups (or d-sirups), capture many features and difficulties of the general case. For example, answering d-sirups is Π2p-complete for combined complexity and can be in ▪ or L-, NL-, P-, or coNP-complete for data complexity (with the problem of recognising FO-rewritability of d-sirups being 2ExpTime-hard); some d-sirups only have exponential-size resolution proofs, some only double-exponential-size positive existential FO-rewritings and single-exponential-size nonrecursive datalog rewritings. On the other hand, we prove a few partial sufficient and necessary conditions of FO- and (symmetric/linear-) datalog rewritability of d-sirups. Our main technical result is a complete and transparent syntactic ▪/NL/P/coNP tetrachotomy of d-sirups with disjoint covering classes and a path-shaped Boolean conjunctive query. To obtain this tetrachotomy, we develop new techniques for establishing P- and coNP-hardness of answering non-Horn ontology-mediated queries as well as showing that they can be answered in NL.

A complete classification of the complexity and rewritability of ontology-mediated queries based on the description logic [formula omitted

We provide a fine-grained analysis of the data complexity and rewritability of ontology-mediated queries (OMQs) based on an EL ontology and a conjunctive query (CQ). Our main results are that every such OMQ is in ▪, ▪-complete, or ▪-complete and that containment in ▪ coincides with rewritability into linear Datalog (whereas containment in ▪ coincides with rewritability into first-order logic). We establish natural characterizations of the three cases in terms of bounded depth and (un)bounded pathwidth of certain minimal ABoxes on which the OMQ yields an answer. We also show that each of the associated meta problems such as deciding whether a given OMQ is rewritable into linear Datalog is ExpTime-complete. We also give a way to construct linear Datalog rewritings when they exist and prove that there is no constant bound on the arity of IDB relations in linear Datalog rewritings.

Dichotomies in Ontology-Mediated Querying with the Guarded Fragment

We study ontology-mediated querying in the case where ontologies are formulated in the guarded fragment of first-order logic (GF) or extensions thereof with counting and where the actual queries are (unions of) conjunctive queries. Our aim is to classify the data complexity and Datalog rewritability of query evaluation depending on the ontology O , where query evaluation w.r.t. O is in PT ime (resp. Datalog rewritable) if all queries can be evaluated in PT ime w.r.t. O (resp. rewritten into Datalog under O ), and co NP-hard if at least one query is co NP-hard w.r.t. O . We identify several fragments of GF that enjoy a dichotomy between Datalog-rewritability (which implies PT ime ) and co NP-hardness as well as several other fragments that enjoy a dichotomy between PT ime and co NP-hardness, but for which PT ime does not imply Datalog-rewritability. For the latter, we establish and exploit a connection to constraint satisfaction problems. We also identify fragments for which there is no dichotomy between PT ime and co NP. To prove this, we establish a non-trivial variation of Ladner’s theorem on the existence of NP-intermediate problems. Finally, we study the decidability of whether a given ontology enjoys PT ime query evaluation, presenting both positive and negative results, depending on the fragment.

Ontology-Mediated Queries

We give solutions to two fundamental computational problems in ontology-based data access with the W3C standard ontology language OWL 2 QL : the succinctness problem for first-order rewritings of ontology-mediated queries (OMQs) and the complexity problem for OMQ answering. We classify OMQs according to the shape of their conjunctive queries (treewidth, the number of leaves) and the existential depth of their ontologies. For each of these classes, we determine the combined complexity of OMQ answering and whether all OMQs in the class have polynomial-size first-order, positive existential, and nonrecursive datalog rewritings. We obtain the succinctness results using hypergraph programs, a new computational model for Boolean functions, which makes it possible to connect the size of OMQ rewritings and circuit complexity.

Datalog rewritability of Disjunctive Datalog programs and non-Horn ontologies

We study the problem of rewriting a Disjunctive Datalog program into an equivalent plain Datalog program (i.e., one that entails the same facts for every dataset). We show that a Disjunctive Datalog program is Datalog rewritable if and only if it can be rewritten into a linear program (i.e., having at most one IDB body atom in each rule), thus providing a novel characterisation of Datalog rewritability in terms of linearisability. Motivated by this result, we propose the class of markable programs, which extends both Datalog and linear Disjunctive Datalog and admits Datalog rewritings of polynomial size. We show that our results can be seamlessly applied to ontological reasoning and identify two classes of non-Horn ontologies that admit Datalog rewritings of polynomial and exponential size, respectively. Finally, we shift our attention to conjunctive query answering and extend our results to the problem of computing a rewriting of a Disjunctive Datalog program that yields the same answers to a given query w.r.t. arbitrary data. Our empirical results suggest that a fair number of non-Horn ontologies are Datalog rewritable and that query answering over such ontologies becomes feasible using a Datalog engine.

Datalog Rewritings of Regular Path Queries using Views

We consider query answering using views on graph databases, i.e. databases structured as edge-labeled graphs. We mainly consider views and queries specified by Regular Path Queries (RPQ). These are queries selecting pairs of nodes in a graph database that are connected via a path whose sequence of edge labels belongs to some regular language. We say that a view V determines a query Q if for all graph databases D, the view image V(D) always contains enough information to answer Q on D. In other words, there is a well defined function from V(D) to Q(D). Our main result shows that when this function is monotone, there exists a rewriting of Q as a Datalog query over the view instance V(D). In particular the rewriting query can be evaluated in time polynomial in the size of V(D). Moreover this implies that it is decidable whether an RPQ query can be rewritten in Datalog using RPQ views.

Ontology-Based Data Access

Ontology-based data access is concerned with querying incomplete data sources in the presence of domain-specific knowledge provided by an ontology. A central notion in this setting is that of an ontology-mediated query , which is a database query coupled with an ontology. In this article, we study several classes of ontology-mediated queries, where the database queries are given as some form of conjunctive query and the ontologies are formulated in description logics or other relevant fragments of first-order logic, such as the guarded fragment and the unary negation fragment. The contributions of the article are threefold. First, we show that popular ontology-mediated query languages have the same expressive power as natural fragments of disjunctive datalog, and we study the relative succinctness of ontology-mediated queries and disjunctive datalog queries. Second, we establish intimate connections between ontology-mediated queries and constraint satisfaction problems (CSPs) and their logical generalization, MMSNP formulas. Third, we exploit these connections to obtain new results regarding: (i) first-order rewritability and datalog rewritability of ontology-mediated queries; (ii) P/NP dichotomies for ontology-mediated queries; and (iii) the query containment problem for ontology-mediated queries.

Datalog Rewritability of Disjunctive Datalog Programs and its Applications to Ontology Reasoning

We study the problem of rewriting a disjunctive datalog program into plain datalog. We show that a disjunctive program is rewritable if and only if it is equivalent to a linear disjunctive program, thus providing a novel characterisation of datalog rewritability. Motivated by this result, we propose weakly linear disjunctive datalog -- a novel rule-based KR language that extends both datalog and linear disjunctive datalog and for which reasoning is tractable in data complexity. We then explore applications of weakly linear programs to ontology reasoning and propose a tractable extension of OWL 2 RL with disjunctive axioms. Our empirical results suggest that many non-Horn ontologies can be reduced to weakly linear programs and that query answering over such ontologies using a datalog engine is feasible in practice.

The price of query rewriting in ontology-based data access

We give a solution to the succinctness problem for the size of first-order rewritings of conjunctive queries in ontology-based data access with ontology languages such as OWL 2 QL, linear Datalog± and sticky Datalog±. We show that positive existential and nonrecursive datalog rewritings, which do not use extra non-logical symbols (except for intensional predicates in the case of datalog rewritings), suffer an exponential blowup in the worst case, while first-order rewritings can grow superpolynomially unless NP⊆P/poly. We also prove that nonrecursive datalog rewritings are in general exponentially more succinct than positive existential rewritings, while first-order rewritings can be superpolynomially more succinct than positive existential rewritings. On the other hand, we construct polynomial-size positive existential and nonrecursive datalog rewritings under the assumption that any data instance contains two fixed constants.