The property of boundedness in Datalog formalizes whether a set of rules can be equivalently expressed by a non-recursive set of rules. Existential rules extend Datalog to the presence of existential variables in rule heads. In this paper, we introduce and study notions of boundedness for existential rules. We provide a notion of weak boundedness and a notion of strong boundedness for a rule set, and show that they correspond, respectively, to the notions of first-order rewritability of atomic queries and first-order rewritability of conjunctive queries over the set. While weak and strong boundedness are in general not equivalent, we show that, for some notable subclasses of existential rules, i.e., Datalog, single-head binary rules, and frontier-guarded rules, the two notions coincide

Civili, Cristina

Rosati, Riccardo

English

CIVILI, CRISTINA

ROSATI, Riccardo

Archivio della ricerca- Università di Roma La Sapienza

Bounded Implication for Existential Rules(Extended Abstract)Cristina Civili1 and Riccardo Rosati21 School of Informatics, University of Edinburgh, UK2 DIAG, Sapienza Universita` di Roma, ItalyThe problem This paper deals with the property of boundedness in rule languages.Boundedness is an important notion that formalizes the fact that a rule set Σ can beunfolded into a finite set Σ′ of acyclic (i.e., non-recursive) rules such that Σ and Σ′are equivalent on every database: it is therefore a crucial property for optimizing theprocessing of rules. Such a property has been extensively studied, especially for theDatalog rule language [9,4], and, recently, for Answer Set Programming [16].In Datalog, the (uniform) boundedness of a program P can be defined as the exis-tence of an integer k such that, for every databaseD, the number of iterated applications(in a forward chaining manner) of P to D that are necessary to compute the minimalmodel of P and D is bounded by k. This definition of boundedness is equivalent tothe existence of a finite, non-recursive program that is equivalent to P . Also, it is well-known that a Datalog query is bounded if and only if it is equivalent to a first-ordersentence [1,12].More recently, rule-based languages have been used in the context of ontology-based data access [15]. In this framework, the main focus is on the problem of answer-ing conjunctive queries over an ontology expressed by a set of rules, and one of themost studied properties is the first-order rewritability of conjunctive queries (CQFO-rewritability) over an ontology, which corresponds to the above mentioned first-orderexpressibility in Datalog: an ontologyO is CQFO-rewritable if every conjunctive queryq over the ontology can be equivalently rewritten into a first-order query q′, i.e., q′ issuch that, for every database instance D, the evaluation of q over O and D coincideswith the evaluation of q′ over D. Notably, in the case when the ontology is expressedas a set P of Datalog rules, the CQFO-rewritability of P and the boundedness of P areequivalent properties.Existential rules, which extend Datalog rules to the presence of existentially quan-tified variables and multiple atoms in rule heads, have been proposed and studied in thelast years as a specification language for ontology-based data access [5,2,14]. An exis-tential rule (or simply rule) σ over a relational schema S is an expression of the form∀x∀y(Φ(x,y, a¯)→ ∃zΨ(x, z, b¯)), where Φ(x,y, a¯) (the body of σ) and Ψ(x, z, b¯) (thehead of σ) are conjunctions of atoms over S. We call x the frontier variables (F(σ)),and z the existential head variables of σ (EH(σ)), while a¯, b¯ are the constants occur-ring in σ (Const(σ)). Several recent studies have focused on the first-order rewritabilityproperty for existential rules (e.g., [7,2,8]). On the other hand, the notion of bound-edness for existential rules has not been deeply investigated. To our knowledge, oneof the most relevant recent approaches to this problem is presented in [2], where thenotion of acyclic graph of rule dependencies (aGRD) is defined, which correspondsto a form of boundedness for existential rules. Instead, we start our study from a no-tion of boundedness for existential rules that generalizes the definition of boundednessprovided for Datalog to existential rules in a simpler way: we call such a notion strictboundedness. More precisely, we say that a set Σ of existential rules is strictly boundedif it is logically equivalent to a finite and acyclic set of rules.However, it can be immediately verified that, for arbitrary sets of existential rules,the above notion of boundedness is much stronger than the first-order rewritability ofconjunctive queries. That is, while strict boundedness of a rule set implies its CQFO-rewritability, there exist rule sets that are CQFO-rewritable but are not strictly bounded.Notice that the same property holds for the above mentioned notion of aGRD.The main goal of this paper is to answer the following question: is it possible togeneralize the notion of boundedness for Datalog to existential rules, in such a way thatthe correspondence with the notion of first-order rewritability of conjunctive queriesis preserved? Actually, from the forward chaining perspective, such a generalizationhas been provided by the bounded derivation depth property of the chase of existentialrules [5,10]. However, we would like to characterize this property in terms of equivalentrepresentations of the set of rules, and see how the alternative notion of boundedness asexistence of a finite and non-recursive equivalent rule set has to be extended to capturefirst-order rewritable rule sets.Our contribution Our approach to the study of boundedness for existential rules isinspired by the work in query rewriting for existential rules [2,13,6,11]. In particular,we extend the techniques presented in [2,13] to address the problem of computing anunfolding of a set of existential rules, and the problem of defining an appropriate notionof redundancy between rules.First, we define a notion of boundedness for existential rules that weakens strictboundedness by giving up the acyclicity condition: we call such a notion weakimplication-boundedness. It is based on the idea of looking for a finite representationof all the single-head rules (i.e., rules with one atom in the head) that are logically im-plied by the initial rule set, and on a notion of equivalence between rule sets, calledR-equivalence, that is different from (and stronger than) the standard logical equiva-lence. R-equivalence is based on a notion of redundancy between two rules.Given two rules σ : Φ(x,y, a¯) → ∃z Ψ(x, z, b¯), and σ′ : Φ′(x′,y′, c¯) →∃z′ Ψ(x′, z′, d¯), we say that σ is redundant with respect to σ′ if there exists a spe-cialization of σ′ η(σ′) = σ′s (where η : F(σ′) → F(σ′) ∪ Const(σ)) and a bijectivefunction  : F(σ′s)→ F(σ) such that the following first-order sentences are valid:∀x∀y body(σ)(x,y, a¯)→ ∃y′ (body(σ′s))(x,y′, e¯)∀x′∀z′ head(σ′s)(x′, z′, d¯)→ ∃z −(head(σ))(x′, z, f¯)where e¯ = c¯ ∪ a¯ ∪ b¯ and f¯ = d¯ ∪ a¯ ∪ b¯.Given two sets of rules Σ, Σ′, we say that Σ′ R-entails Σ if, for each non-tautological rule σ ∈ Σ there exists a rule σ′ ∈ Σ′ such that σ is redundant w.r.t.σ′. Moreover, we say that Σ and Σ′ are R-equivalent if both Σ′ R-entails Σ and ΣR-entails Σ′.Let Σ be a set of rules over a signature S. We define the SH-closure of Σ as the setΣ?s = {σ | σ is a single-head rule over S and Const(Σ), and Σ |= σ}.We say that a set Σ of rules is weakly implication-bounded if Σ?s is R-equivalentto a finite set of rules.It is immediate to verify that strict boundedness always implies weak implication-boundedness, while the converse does not always hold. Furthermore, it turns out thatweak implication-boundedness is not equivalent to CQFO-rewritability. More precisely,it is possible to show that weak implication-boundedness does not imply CQFO-rewritability for single-head ternary set of rules (i.e., rules over relations of arity ≤ 3).Similarly, it can be shown that weak implication-boundedness does not imply CQFO-rewritability for binary set of rules (i.e., over relations of arity ≤ 2). On the other hand,we are able to prove the correspondence between weak implication-boundedness andthe notion of AFO-rewritability, that is, first-order rewritability of all atomic queries(i.e., conjunctive queries consisting of a single atom).To arrive at a notion of boundedness that corresponds to CQFO-rewritability, we de-fine a second notion of strong implication-boundedness for existential rules. Roughlyspeaking, such a notion is obtained from weak implication-boundedness by discard-ing the restriction to single-head rules in the deductive closure of the rule set, and byconsidering projections of such a deductive closure.Let Σ be a set of rules over a signature S. We call closure of Σ the set Σ? ={σ | σ is a rule over S and Const(Σ), and Σ |= σ}. Moreover, let σ, σ′ betwo rules. We say that σ′ is head-unifiable w.r.t. σ if there exists a homomorphismµ : F(σ) → F(σ′) ∪ Const(σ′) and an isomorphism  : EH(σ) → EH(σ′) such thathead(µ((σ))) = head(σ′). Then, we call projection of a rule set Σ with respect to arule σ the set Πσ(Σ) = {σ′ ∈ Σ | σ′ is head-unifiable with σ}.We say that a set Σ of rules over a schema S is strongly implication-bounded if, foreach rule σ over S, Πσ(Σ?) is R-equivalent to a finite set of rules.The notion of weak implication-boundedness has the desired correspondence withthe CQFO-rewritability. Namely, every set of rules Σ is strongly implication-boundedif and only if Σ is CQFO-rewritable. This in turn implies the correspondence betweenstrong boundedness and the bounded derivation depth property [10].Moreover, the equivalence between weak and strong implication-boundedness ac-tually extends to two broad classes of existential rules: single-head binary rules, thatis, single-head rules over relations of arity not greater than 2, and frontier-guarded [2]rules.We believe that the equivalence between weak and strong implication-boundednessis a very important property for a set of existential rules. In particular, the above corre-spondence could be exploited in the optimization of query answering over ontologiesexpressed by rule sets belonging to the above classes.Finally, it is possible to show that checking strong (or, equivalently, weak)implication-boundedness is undecidable for single-head binary rules, and (using re-sults from [3]) decidable for frontier-guarded rules. These results complement the well-known undecidability of (strict) boundedness for Datalog [9].Acknowledgments This research is partially supported by the UK EPSRC ProjectVADA (grant n. M025268) and the EU Project Optique (grant n. FP7-318338).References1. Miklo´s Ajtai and Yuri Gurevich. Datalog vs first-order logic. J. Comput. Syst. Sci.,49(3):562–588, 1994.2. Jean-Franc¸ois Baget, Michel Lecle`re, Marie-Laure Mugnier, and Eric Salvat. On rules withexistential variables: Walking the decidability line. Artificial Intelligence, 175(9–10):1620–1654, 2011.3. Vince Ba´ra´ny, Michael Benedikt, and Balder ten Cate. Rewriting guarded negation queries.In Mathematical Foundations of Computer Science 2013 - 38th International Symposium,MFCS 2013, pages 98–110, 2013.4. Michael Benedikt, Balder ten Cate, Thomas Colcombet, and Michael Vanden Boom. Thecomplexity of boundedness for guarded logics. In 30th Annual ACM/IEEE Symposium onLogic in Computer Science, LICS 2015, Kyoto, Japan, July 6-10, 2015, pages 293–304, 2015.5. Andrea Calı`, Georg Gottlob, and Thomas Lukasiewicz. A general datalog-based frameworkfor tractable query answering over ontologies. Semantic Web J., 14:57–83, 2012.6. Andrea Calı`, Georg Gottlob, and Andreas Pieris. Advanced processing for ontologicalqueries. Proc. of the VLDB Endowment, 3(1):554–565, 2010.7. Andrea Calı`, Georg Gottlob, and Andreas Pieris. Towards more expressive ontology lan-guages: The query answering problem. Artificial Intelligence, 193:87–128, 2012.8. Cristina Civili and Riccardo Rosati. A broad class of first-order rewritable tuple-generatingdependencies. In Proc. of the 2nd Datalog 2.0 Workshop, 2012.9. Haim Gaifman, Harry G. Mairson, Yehoshua Sagiv, and Moshe Y. Vardi. Undecidable opti-mization problems for database logic programs. J. of the ACM, 40(3):683–713, 1993.10. Georg Gottlob, Stanislav Kikot, Roman Kontchakov, Vladimir V. Podolskii, ThomasSchwentick, and Michael Zakharyaschev. The price of query rewriting in ontology-baseddata access. Artif. Intell., 213:42–59, 2014.11. Georg Gottlob, Giorgio Orsi, and Andreas Pieris. Ontological queries: Rewriting and op-timization. In Proc. of the 27th IEEE Int. Conf. on Data Engineering (ICDE 2011), pages2–13, 2011.12. Phokion G. Kolaitis and Christos H. Papadimitriou. Some computational aspects of circum-scription. J. ACM, 37(1):1–14, 1990.13. Me´lanie Ko¨nig, Michel Lecle`re, Marie-Laure Mugnier, and Michae¨l Thomazo. Sound, com-plete and minimal ucq-rewriting for existential rules. Semantic Web, 6(5):451–475, 2015.14. Markus Kro¨tzsch and Sebastian Rudolph. Extending decidable existential rules by joiningacyclicity and guardedness. In Proc. of the 22nd Int. Joint Conf. on Artificial Intelligence(IJCAI 2011), pages 963–968, 2011.15. Antonella Poggi, Domenico Lembo, Diego Calvanese, Giuseppe De Giacomo, MaurizioLenzerini, and Riccardo Rosati. Linking data to ontologies. J. on Data Semantics, X:133–173, 2008.16. Heng Zhang and Yan Zhang. First-order expressibility and boundedness of disjunctive logicprograms. In IJCAI 2013, Proceedings of the 23rd International Joint Conference on Artifi-cial Intelligence, Beijing, China, August 3-9, 2013, 2013.

Bounded Implication for Existential Rules

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Bounded Implication for Existential Rules

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