In this extended abstract we discuss the relationship between confidentiality-preserving frameworks and inconsistency-tolerant repair and update semantics in Description Logics (DL). In particular, we consider the well-known problems of Consistent Query Answering, Controlled Query Evaluation, and Knowledge Base Update in DL and introduce a unifying framework that can be naturally instantiated to capture significant settings for the above problems, previously investigated in the literature

Lembo, Domenico

Rosati, Riccardo

Savo, Domenico Fabio

English

Domenico Lembo

Riccardo Rosati

Domenico Fabio Savo

Archivio della ricerca- Università di Roma La Sapienza

A comprehensive framework for Controlled Query Evaluation,Consistent Query Answering and KB Updates in Description LogicsDomenico Lembo, Riccardo Rosati, Domenico Fabio SavoSapienza University of Romelastname@diag.uniroma1.itIn this extended abstract we discuss the relation-ship between confidentiality-preserving frameworks andinconsistency-tolerant repair and update semantics in De-scription Logics (DL). In particular, we consider the well-known problems of Consistent Query Answering, Con-trolled Query Evaluation, and Knowledge Base Update inDL and introduce a unifying framework that can be natu-rally instantiated to capture significant settings for the aboveproblems, previously investigated in the literature.Consistent Query Answering (CQA). It is a declara-tive approach to manage inconsistency that has been ex-tensively studied in both databases and knowledge bases(KBs) (Bertossi 2011; Bienvenu and Bourgaux 2016). Thecrucial notions for CQA are those of repair and consistentanswers. For a DL KB K = 〈T ,A〉, where T is a consistentTBox and A is an ABox, a repair is an ABox A′ that “mini-mally” differs from A and that is consistent with T . Severalnotions of minimality can be adopted, which in general giverise to the existence of several repairs. For example, in theAR-semantics, a repair is a maximal subset of A that is con-sistent with T (Lembo et al. 2010). In the CAR-semantics,a repair is a maximal subset of the consistent closure of Kthat is consistent with T , where the consistent closure of Kis the set of facts implied by T and by any subset of A con-sistent with T (Lembo et al. 2010). The consistent answersto a query q specified over K are the answers to q that aretrue in every K′ = 〈T ,A′〉, where A′ is a repair of A.Controlled Query Evaluation (CQE). It is aconfidentiality-preserving framework, in which a cer-tain policy is declaratively specified on the schema levelof a database or knowledge base to regulate the access tothe underlying data. Users ask queries on the schema, butwhen these are answered no secrets have to be disclosed. Tothis aim, a so-called censor can alter the returned answersaccording to the specified policy. The main reasoning taskin CQE thus amounts to computing an optimal censor,given a KB or a database, a policy, and a query language.Intuitively, to be optimal a censor has to guarantee the policyby “minimally” modifying the answers to queries, so that ithides only the portion of data that is necessary to preservesecrets, but at the same time it returns as many answers asCopyright c© 2018, Association for the Advancement of ArtificialIntelligence (www.aaai.org). All rights reserved.possible among those that the system is allowed to return.Various ways to obtain a censor have been investigated,which at an abstract level correspond to various criteria ofminimal modification of the query answers. The notion ofcensor has been first studied in the context of databases(e.g., (Biskup and Bonatti 2004)), but recently has beeninvestigated in the setting of DL ontologies. In this paperwe mainly refer to the framework studied in (Cuenca Grauet al. 2015), where the policy is defined as a conjunctivequery over a DL KB, with the intended meaning that theanswers to this query should not be returned to the users.Knowledge Base Update (KBU). It is the problem of mod-ifying a KB in order to adhere to a change in the domain ofinterest. Typically such a change is represented by means ofa set of formulas that have to be entailed by the KB resultingfrom the update. If the new knowledge contradicts the oldone, a portion of the latter has to be discarded for restoringthe consistency. Similarly to CQA, the idea is to “minimizethe distance” between the original KB and the new one re-sulting from the application of an evolution operator imple-menting the update (see (Flouris et al. 2008) for a survey). Inour investigation, we focus on the so-called formula-basedapproaches, which define the result of the update in termsof a formula that differs minimally from the formula ex-pressing the original KB. Basically, solutions for this casecan be divided into two categories: foundational approachesand coherence approaches (Ga¨rdenfors 1990). The two dif-fer since in the foundational approaches the axioms assertedin the KB serve as a justification for all the knowledge ofthe KB, while in the coherence approaches axioms are seenonly as a means of expressing the knowledge. Intuitively,given two KBs K and KN , for updating K with KN in thefoundational approaches the maximal subsets of K that areconsistent with KN need to be computed. Instead, in the co-herence approaches, one intends to preserve as many asser-tions as possible of those entailed by K, and thus aims tocompute the maximal subsets of the logical consequences ofK that are consistent with KN .Motivations. The three problems we are considering canbe actually seen as belief revision problems. Belief revisionstudies the issue of integrating new information with previ-ous knowledge. Whereas the new knowledge has to be pre-served in the resulting theory, the old one should be revisedto guarantee consistency. In CQA in DL the TBox can beseen as the new information reflecting a modification aboutthe knowledge of the domain, whereas the previous knowl-edge is the ABox. In KBU the new knowledge is explicit inthe framework and reflects a change in the world. In CQE inDL the policy together with the TBox of the KB can be con-sidered as the new knowledge, whereas the underlying ABoxplays the role of the previous knowledge. Under this per-spective, for instance, the policy considered in (Cuenca Grauet al. 2015) is seen as a constraint over the KB obtained bynegating the policy query, and confidential data are thosefacts in the ABox that violate such constraint. As belief re-vision problems, CQA, CQE and KBU share the objectiveof modifying knowledge to ensure consistency by applyingminimal changes to the original theory. In CQE, for instance,this means to minimally alter query answers to enforce thepolicy.Framework. Based on the above common aspects shared byCQA, CQE and KBU, we introduce below the new notionof consistent projections of a KB. Intuitively, the consistentprojection of a KB K is a KB that maximally preserves theknowledge that K infers in a certain projection language Qand that is consistent with some new knowledge, which wedenote with Kc. In the following we indicate with Mod(K)the set of models of K, and with K |= φ the fact that K im-plies a first-order sentence φ. We also recall that K is con-sistent if Mod(K) 6= ∅, inconsistent otherwise.We give now the notion of consistent Q-entailment set ofK, denoted by CES (K,Q), which is the set of sentences{φ | φ ∈ Q and there exists K′ ⊆ K such thatMod(K′) 6= ∅ and K′ |= φ}.If K is a consistent KB, CES (K,Q) contains all the sen-tences in the language Q that are implied by K. If K is in-consistent, CES (K,Q) resorts to consistent subsets of K,and contains all the sentences in Q that are implied by atleast one of these subsets.We are now ready to formally define the notion of consis-tent projection of a KB.Definition 1 Given a KB K, a consistent KB Kc, and alanguage Q of first-order sentences, a Kc-consistent Q-projection of K is a set of sentences Φ such that:1. Φ ⊆ CES (K,Q)2. Mod(Kc ∪ Φ) 6= ∅3. there exists no Φ′ such that Φ ⊂ Φ′ ⊆ CES (K,Q) andMod(Kc ∪ Φ′) 6= ∅.The set of the Kc-consistent Q-projections of K is denotedby CPQ(K,Kc).Therefore, a Kc-consistent Q-projection of K is a max-imal subset of the consistent Q-entailment set of K that isconsistent with Kc. Intuitively, with the above notion weaim to capture all the logical consequences of all the con-sistent portions of K that are expressible in Q (i.e., the Q-projections of K) and that are consistent with Kc.The above framework can be instantiated to express rele-vant CQA, CQE, and KBU settings studied in the literature.For example, it can be shown that the set of repairs underthe AR-semantics for CQA over a DL KB K = 〈T ,A〉 cor-responds to the setCPQ(K,Kc), whenKc = T andQ = A,i.e., if the projection language includes only the facts of theABox. By suitably setting Kc and Q we can also show thatthe notion of consistent projection captures the one of repairunder the CAR-semantics studied in CQA for DL, the one ofoptimal censor proposed in (Cuenca Grau et al. 2015), andgeneral notions of update in the foundational and coherenceformula-based approaches and their instance-level versionsfor DL studied in (De Giacomo et al. 2016).In our framework we study a form of entailment that cap-tures and generalizes skeptical entailment in CQA, CQE andKBU under the correspondences mentioned above. A formaldefinition is given below.Definition 2 Given a KB K, a consistent KB Kc, a KBKn ⊆ Kc, a projection language Q and a first-order sen-tence φ, Consistent Projection Entailment is the problem ofdeciding whether Kn ∪Φ |= φ for every Φ ∈ CPQ(K,Kc).In the above definition, Kn is the portion of Kc that isrelevant for reasoning. To have an intuition on the role ofKn, consider for example CQE under the perspective of ourframework, where consistent projections are computed withrespect to the DL TBox and (the negation of) the policy,whereas query answering is on the TBox only (and the un-derlying data).We have some preliminary complexity results for thisgeneral form of entailment that straightforwardly apply toCQA, CQE and KBU, and we believe that our frameworkpaves the way for a unified approach to such problems.ReferencesBertossi, L. E. 2011. Database Repairing and ConsistentQuery Answering. Synthesis Lectures on Data Management.Morgan & Claypool Publishers.Bienvenu, M., and Bourgaux, C. 2016. Inconsistency-tolerant querying of description logic knowledge bases. InRW 2012 Tutorial Lectures, 156–202.Biskup, J., and Bonatti, P. A. 2004. Controlled query eval-uation for enforcing confidentiality in complete informationsystems. Int. J. Inf. Sec. 3(1):14–27.Cuenca Grau, B.; Kharlamov, E.; Kostylev, E. V.; andZheleznyakov, D. 2015. Controlled query evaluation for dat-alog and OWL 2 profile ontologies. In Proc. of IJCAI 2015,2883–2889.De Giacomo, G.; Oriol, X.; Rosati, R.; and Savo, D. F. 2016.Updating DL-Lite ontologies through first-order queries. InProc. of ISWC 2016, 167–183.Flouris, G.; Manakanatas, D.; Kondylakis, H.; Plexousakis,D.; and Antoniou, G. 2008. Ontology change: Classificationand survey. Knowledge Engineering Review 23(2):117–152.Ga¨rdenfors, P. 1990. The dynamics of belief systems:Foundations vs. coherence theories. Revue internationalede philosophie 24–46.Lembo, D.; Lenzerini, M.; Rosati, R.; Ruzzi, M.; and Savo,D. F. 2010. Inconsistency-tolerant semantics for descriptionlogics. In Proc. of RR 2010, 103–117.

A Comprehensive Framework for Controlled Query Evaluation, Consistent Query Answering and KB Updates in Description Logics

Savo, Domenico Fabio (ORCID:0000-0002-8391-8049)

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A Comprehensive Framework for Controlled Query Evaluation, Consistent Query Answering and KB Updates in Description Logics

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