Barceló, Pablo

Berger, Gerald

Pieris, Andreas

Edinburgh Research Explorer

     Edinburgh Research Explorer                                      Guarded Ontology-Mediated Queries Distributing OverComponentsCitation for published version:Barceló, P, Berger, G & Pieris, A 2017, Guarded Ontology-Mediated Queries Distributing OverComponents. in Proceedings of the 11th Alberto Mendelzon International Workshop on Foundations of DataManagement and the Web.Link:Link to publication record in Edinburgh Research ExplorerDocument Version:Peer reviewed versionPublished In:Proceedings of the 11th Alberto Mendelzon International Workshop on Foundations of Data Management andthe WebGeneral rightsCopyright for the publications made accessible via the Edinburgh Research Explorer is retained by the author(s)and / or other copyright owners and it is a condition of accessing these publications that users recognise andabide by the legal requirements associated with these rights.Take down policyThe University of Edinburgh has made every reasonable effort to ensure that Edinburgh Research Explorercontent complies with UK legislation. If you believe that the public display of this file breaches copyright pleasecontact openaccess@ed.ac.uk providing details, and we will remove access to the work immediately andinvestigate your claim.Download date: 05. Jun. 2018Guarded Ontology-Mediated QueriesDistributing Over Components⋆Pablo Barcelo´1, Gerald Berger2, and Andreas Pieris31 Center for Semantic Web Research & DCC, University of Chilepbarcelo@dcc.uchile.cl2 Institute of Information Systems, TU Wien gberger@dbai.tuwien.ac.at3 School of Informatics, University of Edinburgh apieris@inf.ed.ac.uk1 IntroductionOntology-based data access (OBDA) has recently emerged as a promising applicationof knowledge representation and reasoning technologies in information managementsystems. The aim of OBDA is to facilitate access to data that is significantly heteroge-neous and incomplete. This is achieved via an ontology that provides a unified concep-tual view of the data, and makes it accessible via database queries formulated solely inthe vocabulary of the ontology. The actual database query and the ontology can be seenas two components of one composite query, called ontology-mediated query [8]. OBDAcan then be realized as the problem of answering ontology-mediated queries.An important topic for declarative database query languages, and in particular (ex-tensions of) Datalog, is coordination-free evaluation. This has its roots in declarativenetworking [15], an approach where distributed computations are programmed usingDatalog-based formalisms. In this setting, programs (or queries) are specified over aglobal schema, and are executed by multiple computing nodes over which the databaseis distributed. These nodes can perform local computations, and also communicateasynchronously with each other via messages. The model assumes that messages cannever be lost but can be arbitrarily delayed. An intrinsic source of inefficiency in suchsystems are the global barriers raised by the need for synchronization in computing theresult of queries. This has motivated a fruitful line of research for isolating classes ofqueries that can be evaluated in a coordination-free manner [1–4, 16].It is natural to ask whether the results on coordination-free evaluation for declarativedatabase query languages can be transferred to ontology-mediated queries. Undoubt-edly, a positive answer to this question, apart from possible applications to declarativenetworking, will significantly contribute towards more efficient procedures for OBDA,since it will enable distributed ontology-mediated query evaluation in a coordination-free way. The goal of the present work is to initiate the study of coordination-free eval-uation for ontology-mediated queries, and give strong indications that the answer to theabove challenging question is affirmative.Distribution Over Components. More concretely, we focus on the question whetherthe answer to an ontology-mediated query can be computed by parallelizing it over the⋆ This short paper is based on [6, 7].(maximally connected) components of the database, i.e., whether the query distributesover components. In other words, given an ontology-mediated query Q, the question iswhether for every database D the answer to Q over D, denoted Q(D), coincides with⋃1≤i≤nQ(Di), where D1, . . . , Dn are the components of D. In this case, Q(D) canbe computed without any communication over a network using a distribution whereevery computing node is assigned some of the connected components of the database,and every component is assigned to at least one computing node. The notion of dis-tribution over components has been introduced in [2], and explicitly considered forDatalog queries in [3]. It has been shown that connected Datalog, that is, the fragmentof Datalog where all rule-bodies are connected, provides an effective syntax for Data-log queries that distribute over components, while the problem of deciding whether aDatalog query distributes over components is undecidable.Aims and Objectives. As said, this work concentrates on ontology-mediated queriesand their distribution over components. We focus on guarded ontology-mediated querieswhere the database query is a conjunctive query and the ontology is formulated usingguarded existential rules (a.k.a. tuple-generating dependencies and Datalog± rules), thatis, sentences of the form ∀x¯∀y¯(ϕ(x¯, y¯) → ∃z¯ ψ(x¯, z¯)) where ϕ, ψ are conjunctions ofatoms with ϕ being guarded, i.e., it has an atom, called guard, that contains all the vari-ables (x¯ ∪ y¯) [10, 11]. It is widely accepted that the class of guarded existential rulesforms a natural and convenient way for modeling ontologies, which generalizes promi-nent description logics such as ELHI [5]. The goal of this work is to answer the fol-lowing questions: (1) Can we characterize the fragment of guarded ontology-mediatedqueries that distribute over components via the notion of connectedness, i.e., when therule-bodies and the conjunctive query are connected? (2) What is the complexity ofdeciding whether a guarded ontology-mediated query distributes over components?Our Results. Our results can be summarized as follows:– The fragment of guarded ontology-mediated queries that distribute over compo-nents has the same expressive power as the fragment of guarded ontology-mediatedqueries where the conjunctive query is connected. Notice that the body of a guardedexistential rule is always connected due to the existence of the guard atom. Thus,this result provides a positive answer to the first question above.– Regarding the second question, we show that deciding whether a guarded ontology-mediated query distributes over components is strongly related to a classical staticanalysis task, namely containment. We show that the problem in question is feasi-ble in polynomial time assuming access to an oracle that can solve containment forguarded ontology-mediated queries. We then show that the containment problemis feasible in 2EXPTIME, which immediately implies that distribution over compo-nents is also feasible in 2EXPTIME. A matching lower bound can be easily shownby exploiting the fact that evaluation of guarded OMQs is 2EXPTIME-hard [10].2 Distribution of Queries and ConnectednessLet us first recall the basics on ontology-mediated querying. An ontology-mediatedquery (OMQ) over a (relational) schema S is a triple (S, Σ, q), where Σ is a set ofexistential rules (the ontology), and q is a conjunctive query (CQ). A guarded OMQis an OMQ as the one above where Σ is a set of guarded existential rules; we write(G,CQ) for the OMQ language that consists of all guarded OMQs.4 The semantics ofan OMQ is given in terms of certain answers. The evaluation of Q = (S, Σ, q) over anS-database D, which is a finite set of atoms of the form R(t¯), where R ∈ S and t¯ is atuple of constants, denotedQ(D), is defined as the certain answers to q w.r.t. D and Σ.Equivalently, Q(D) is the evaluation of q over the canonical instance of D and Σ thatcan be constructed by the well-known chase procedure. Recall that the chase adds newatoms to D as dictated by Σ until the final result, written chase(D,Σ), satisfies all therules of Σ. For more details on ontology-mediated queries see, e.g., [7].The notion of distribution over components has been introduced in [2], and it statesthat the answer to a query can be computed by parallelizing it over the (maximallyconnected) components of the input database. But let us first make precise what a com-ponent is. A set A of atoms is connected if for all terms c, d occurring in A there existsa sequence α1, . . . , αn of atoms in A such that c occurs in α1, d occurs in αn, and αi,αi+1 have at least one term in common, for each i ∈ {1, . . . , n − 1}. We call B ⊆ Aa component of A if (i) B is connected, and (ii) for every α ∈ A \ B, B ∪ {α} isnot connected. Let co(A) be the set of components of A. We are now ready to intro-duce the notion of distribution over components. Consider an OMQ Q = (S, Σ, q).We say that Q distributes over components if Q(D) = Q(D1) ∪ · · · ∪ Q(Dn), whereco(D) = {D1, . . . , Dn}, for every S-database D. Roughly, the centralized answer toQ w.r.t. D is precisely obtained when we parallelize Q over the components of D. LetDIST be the class of OMQs that distribute over components.We proceed to characterize the expressive power of the fragment of (G,CQ) thatdistributes over components. This is done via connectedness of CQs. A CQ q is con-nected if the set of atoms occurring in q is connected; we write CCQ for the class ofconnected CQs. The main result of this section states that every guarded OMQ that dis-tributes over components is equivalent to a connected guarded OMQ. Given two OMQlanguages O1 and O2, we write O1 = O2 to state that they are equally expressive, i.e.,for every query Q1 ∈ O1 over S we can construct a query Q2 ∈ O2 over S such thatQ1(D) = Q2(D), for every S-database D, and vice versa. Then:Theorem 1. (G,CQ) ∩ DIST = (G,CCQ).For the (⊇) direction we show that connectedness ensures distribution over compo-nents. This is a consequence of the fact that, given a queryQ = (S, Σ, q) ∈ (G,CCQ),for every S-databaseD with co(D) = {D1, . . . , Dm}, chase(D,Σ) can be partitionedinto {I1, . . . , Im} such that, for each i ∈ {1, . . . ,m}, Ii depends solely onDi. Considernow a query Q = (S, Σ, q(x¯)) ∈ (G,CQ) ∩ DIST. Observe that if Q is unsatisfiable,i.e., there is no S-database D such that Q(D) 6= ∅, then the claim holds trivially;simply choose an arbitrary unsatisfiable query from (G,CCQ). The interesting caseis when Q is satisfiable. Assuming that {q1, . . . , qk} are the components of q, we canshow that there exists i ∈ {1, . . . , k} such that Qi = (S, Σ, qi(x¯)) is well-defined andequivalent to Q. Since Qi ∈ (G,CCQ) the claim follows.4 Notice that, in general, OMQs are defined for arbitrary ontology languages based on first-orderlogic (not only on existential rules), and arbitrary query languages (not only CQs).3 Deciding Distribution Over ComponentsIn this section we focus on the problem of deciding whether a guarded OMQ distributesover components. In fact, this problem can be defined for an OMQ language based onan arbitrary class C of existential rules:PROBLEM : Dist(C,CQ)INPUT : An OMQ Q ∈ (C,CQ).QUESTION : Does Q distribute over components?Our goal is to pinpoint the complexity of Dist(G,CQ). Towards this direction, wefirst show that it is closely related to the crucial task of containment: given two OMQsQ1, Q2 ∈ (G,CQ) over a schema S, decide whether Q1(D) ⊆ Q2(D), for every S-database D, written as Q1 ⊆ Q2. By exploiting Theorem 1 we can show the following:Proposition 1. Let Q = (S, Σ, q(x¯)) ∈ (G,CQ). The following are equivalent:1. Q distributes over components.2. Q is unsatisfiable or there exists qˆ(x¯) ∈ co(q)5 such that (S, Σ, qˆ(x¯)) ⊆ Q.Notice that checking unsatisfiability can be easily reduced to containment. There-fore, the above results essentially states that Dist(G,CQ) is feasible in polynomial timeassuming access to a C oracle, where C is a complexity class powerful enough for check-ing containment among guarded OMQs. The latter is a highly non-trivial problem, andwe have recently shown that is 2EXPTIME-complete [6]. To obtain the 2EXPTIME up-per bound, we make use of techniques based on two-way alternating parity automata ontrees (2WAPA) [12]. We first show that ifQ1 andQ2 are guarded OMQs such thatQ1 isnot contained in Q2, then this is witnessed over a class of “tree-like” databases that canbe represented as the set of trees accepted by a 2WAPA A. We then build a 2WAPA Bwith exponentially many states that recognizes those trees accepted by A that representwitnesses to non-containment of Q1 in Q2. Hence, Q1 is contained in Q2 iff B acceptsno tree. Since the emptiness problem for 2WAPA is feasible in exponential time in thenumber of states [12], we obtain that containment for guarded OMQs is in 2EXPTIME.A matching lower bound follows from [9]. Thus, Proposition 1 together with the factthat containment for guarded OMQs is in 2EXPTIME implies that Dist(G,CQ) is in2EXPTIME, while a matching lower bound can be shown by exploiting the fact thatevaluation of (G,CQ) queries is 2EXPTIME-hard [10]. Then:Theorem 2. Dist(G,CQ) is 2EXPTIME-complete.4 Next StepsHere is a couple of interesting open problems that we are planning to tackle: (i) distri-bution over components in the presence of equality and denial constraints is not well5 This denotes the set of components of the set of atoms in q, or simply, the components of q.understood, and (ii) we do not know how distribution over components behaves in thecase of guarded OMQs enriched with non-monotonic negation [13, 14].Acknowledgements. Barcelo´ is supported by the Millenium Nucleus Center for Se-mantic Web Research under grant NC120004. Berger is funded by the Austrian ScienceFund (FWF), project W1255-N23. Pieris is supported by the EPSRC Programme GrantEP/M025268/ “VADA: Value Added Data Systems – Principles and Architecture”.References1. Alvaro, P., Conway, N., Hellerstein, J.M., Maier, D.: Blazes: Coordination analysis for dis-tributed programs. In: ICDE. pp. 52–63 (2014)2. Ameloot, T.J., Ketsman, B., Neven, F., Zinn, D.: Weaker forms of monotonicity for declar-ative networking: A more fine-grained answer to the calm-conjecture. In: PODS. pp. 64–75(2014)3. Ameloot, T.J., Ketsman, B., Neven, F., Zinn, D.: Datalog queries distributing over compo-nents. In: ICDT. pp. 308–323 (2015)4. Ameloot, T.J., Neven, F., den Bussche, J.V.: Relational transducers for declarative network-ing. J. ACM 60(2), 15 (2013)5. Baader, F., Brandt, S., Lutz, C.: Pushing the el envelope. In: Proc. of IJCAI. pp. 364–369(2005)6. Barcelo´, P., Berger, G., Pieris, A.: Containment for rule-based ontology-mediated queries.CoRR abs/1703.07994 (2017), http://arxiv.org/abs/1703.079947. Berger, G., Pieris, A.: Ontology-mediated queries distributing over components. In: IJCAI.pp. 943–949 (2016)8. Bienvenu, M., ten Cate, B., Lutz, C., Wolter, F.: Ontology-based data access: A study throughdisjunctive datalog, csp, and MMSNP. ACM Trans. Database Syst. 39(4), 33:1–33:44 (2014)9. Bienvenu, M., Hansen, P., Lutz, C., Wolter, F.: First order-rewritability and containment ofconjunctive queries in horn description logics. In: IJCAI. pp. 965–971 (2016)10. Calı`, A., Gottlob, G., Kifer, M.: Taming the infinite chase: Query answering under expressiverelational constraints. J. Artif. Intell. Res. 48, 115–174 (2013)11. Calı`, A., Gottlob, G., Lukasiewicz, T.: A general Datalog-based framework for tractablequery answering over ontologies. J. Web Sem. 14, 57–83 (2012)12. Cosmadakis, S.S., Gaifman, H., Kanellakis, P.C., Vardi, M.Y.: Decidable optimization prob-lems for database logic programs (preliminary report). In: STOC. pp. 477–490 (1988)13. Gottlob, G., Hernich, A., Kupke, C., Lukasiewicz, T.: Stable model semantics for guardedexistential rules and description logics. In: KR (2014)14. Hernich, A., Kupke, C., Lukasiewicz, T., Gottlob, G.: Well-founded semantics for extendedDatalog and ontological reasoning. In: PODS. pp. 225–236 (2013)15. Loo, B.T., Condie, T., Garofalakis, M.N., Gay, D.E., Hellerstein, J.M., Maniatis, P., Ramakr-ishnan, R., Roscoe, T., Stoica, I.: Declarative networking. Commun. ACM 52(11), 87–95(2009)16. Zinn, D., Green, T.J., Luda¨scher, B.: Win-move is coordination-free (sometimes). In: ICDT.pp. 99–113 (2012)

Guarded Ontology-Mediated Queries Distributing Over Components

Abstract

Similar works

Full text

Available Versions

Edinburgh Research Explorer

Repositorio Académico de la Universidad de Chile