Inconsistency Handling in Ontology-Mediated Query Answering: A Progress Report

Abstract

This paper accompanies an invited talk on inconsistency handling in OMQA and presents a concise summary of the research that has been conducted in the area.

Introduction

It is widely acknowledged that real-world data is plagued with numerous data quality issues, among them the presence of erroneous facts. While already a serious issue for `plain' databases, the problem of handling imperfect data is even more critical in the setting of ontology-mediated query answering (OMQA), where even a single erroneous fact can provoke an inconsistency, thereby rendering classical OMQA semantics useless. This has motivated DL and KR researchers to study a variety of approaches for handling inconsistent data in OMQA, adapting and extending techniques initially proposed for databases. Now that there has been about a decade of research on inconsistency handling in OMQA, the time is ripe to take a step back and evaluate the progress that has been made and what remains to be done.

In this paper, we will try to summarize what is now quite a large collection of works related to inconsistency handling in OMQA. Our treatment will necessarily be incomplete. We will focus on the case of inconsistencies in the data (i.e., we assume the ontology has been properly debugged) and mainly discuss how inconsistency-tolerant semantics can be used to obtain meaningful information from inconsistent knowledge bases. The material we cover can be mostly found in the survey chapter [ 9 ], and we invite interested readers to consult that chapter for a much more detailed presentation, with complete references and further examples and discussion.

Given the target audience, we will assume that the reader is familiar with DLs and the basics of OMQA. Throughout the paper, we assume that K = hT ; Ai is a knowledge base (KB) with TBox T and ABox A, q is a conjunctive query, and a is a tuple of constants from A of the same arity as q. We say that A is T -consistent if the KB hT ; Ai is consistent, i.e. it has at least one model, and otherwise, it is T -inconsistent. A subset A0 A is called a minimal T inconsistent subset of A if (i) A0 is T -inconsistent, and (ii) every A00 ( A0 is T -consistent. A subset C A is a(consistent) T -support of q(a) if (i) C is T -consistent, and (ii) hT ; Ci j= q(a), i.e. a is a certain answer to q w.r.t. hT ; Ci.

Inconsistency-Tolerant Semantics

As mentioned in the introduction, the usual rst-order semantics of DLs does not provide any useful information in the presence of inconsistencies, as everything can be inferred from a contradiction. To address this limitation of classical semantics, several inconsistency-tolerant semantics have been proposed with the aim of returning meaningful answers to queries posed over inconsistent KBs.

A key notion that underlies many of the proposed semantics is that of a repair, which intuitively captures the di erent ways of restoring consistency while retaining as much of the original information as possible. If we use set inclusion to select the maximal ABoxes, as was proposed in [ 21 ] and many subsequent works, then repairs can be formalized as follows.

De nition 1. An (ABox) repair of an ABox A w.r.t. a TBox T is an inclusionmaximal subset of A that is T -consistent. We use Rep(A; T ) to denote the set of repairs of A w.r.t. T , which we abbreviate to Rep(K) when K = hT ; Ai.

Each repair is T -consistent, so it is possible to query a repair using classical semantics. The di culty, however, is that there are typically several di erent repairs for an inconsistent KB, so we need to decide how to combine the answers obtained from the di erent repairs. Arguably the most natural approach is to require that a tuple be a certain answer no matter which repair is considered. This idea is captured by the AR semantics, which was rst de ned in [ 21 ] and can be seen as the OMQA analog of the consistent query answering approach long studied in the database literature [ 1, 16, 5 ].

De nition 2 (AR semantics). A tuple a is an answer to q over K = hT ; Ai under the AR (ABox Repair) semantics, written K j=AR q(a), just in the case that hT ; Bi j= q(a) for every repair B 2 Rep(K).

A more conservative semantics, termed the IAR semantics [ 21 ], is obtained by querying the intersection of the repairs (or equivalently, the set of assertions not participating in any minimal inconsistent subset).

De nition 3 (IAR semantics). A tuple a is an answer to q over K under the IAR (Intersection of ABox Repairs) semantics, written K j=IAR q(a), just in the case that hT ; Di j= q(a) where D = TB2Rep(K) B.

The more adventurous brave semantics, rst explored in the OMQA setting in [ 14 ], merely requires that an answer hold w.r.t. at least some repair. De nition 4 (Brave semantics). A tuple a is an answer to q over K = hT ; Ai under the brave semantics, written K j=brave q(a), just in the case that hT ; Bi j= q(a) for some repair B 2 Rep(K).

Before proceeding further, let us illustrate the AR, IAR, and brave semantics on an example (borrowed and adapted from [ 9 ]): k-lazy 2-def.eater

.. k-defeater k-support

...

brave = 0-defeater non-objection

CAR AR

ICAR

IAR = 1-support = 0-lazy

ICR The KB hTuniv; Aunivi is inconsistent and can be shown to have twelve repairs. If we evaluate the query q(x) = Faculty(x) using the three semantics, we obtain: { 3 answers for AR semantics: sam, kim, alex { 1 answer for IAR semantics: alex { 4 answers for brave semantics: sam, kim, alex, jane The preceding three semantics can be related as follows (see also Fig. 1): K j=IAR q(a) )

K j=AR q(a) )

K j=brave q(a) In other words, the brave and IAR semantics provide respectively upper and lower bounds on the set of answers w.r.t. AR semantics.

We can also compare these (and later) semantics based upon the properties they satisfy. Three salient properties for an inconsistency-tolerant semantics are:

Consistent Support

Unique Base

IAR, ICR, ICAR

Semantics which satisfy the property

non-objection, CAR, ICAR, AR, ICR, k-support, k-lazy, IAR brave, k-defeater, non-objection, AR, ICR, k-support, k-lazy, IAR

The interest of Consistent Results is that it allows users to safely combine the results obtained when querying under semantics S. The Consistent Support property means that every answer can be backed up by exhibiting a consistent subset of the original ABox. Finally, the Unique Base property is a nice feature from the implementation point of view, since it means we compute in an o ine phase a consistent ABox and then employ any existing querying algorithms.

As seen in Fig. 2, the brave semantics satis es only Consistent Support, the AR semantics satis es both Consistent Support and Consistent Results, while the IAR semantics satis es all three properties.

Let us now continue on to other semantics that have been proposed in the OMQA literature, starting with the ICR semantics, de ned in [ 7 ]: De nition 5 (ICR semantics). A tuple a is an answer to q over K = hT ; Ai under the ICR (Intersection of Closed Repairs) semantics just in the case that hT ; Di j= q(a) where D = TB2Rep(K) closeT (B).

By closing the repairs before intersecting them, the ICR semantics provides a better lower approximation of the AR semantics than the IAR semantics, and it can shown to satisfy the three properties. This semantics coincides with the AR semantics on instance queries, which means that in our example, the ICR semantics would return sam, kim, alex as answers to q(x) = Faculty(x).

The idea of adding inferred assertions in order to keep more information is also at the heart of the CAR and ICAR semantics proposed in [ 21 ]. The key difference is that a modi ed closure operator is applied to the original inconsistent ABox, and the enriched ABox is then used to de ne closed ABox repairs: De nition 6 (Closed ABox repair). Let closeT (A) = f j 9S A such that S is T -consistent and hT ; Si j= g. A subset R closeT (A) is a closed ABox repair of A w.r.t. T if (i) it is T -consistent, and (ii) there is no T -consistent R0 closeT (A) such that R\A ( R0 \A or R\A = R0 \A and R ( R0. If K = hT ; Ai, the set of closed ABox repairs of A w.r.t. T is denoted ClosedRep(K).

Closed ABox repairs can be seen as maximally `completing' the (plain) ABox repairs with assertions from closeT (A) n A. Note that while a closed ABox repair of hT ; Ai is always a repair of the KB hT ; closeT (A)i, repairs of hT ; closeT (A)i need not be closed ABox repairs of hT ; Ai (see [ 9 ] for an example). De nition 7 (CAR and ICAR semantics). A tuple a is an answer to q over K = hT ; Ai under the CAR (Closed ABox Repair) semantics just in the case that hT ; Ri j= q(a) for every R 2 ClosedRep(K). It is an answer under the ICAR (Intersection of Closed ABox Repairs) semantics just in the case that hT ; Di j= q(a) where D is the intersection of the closed ABox repairs of A w.r.t.T .

On the KB from Example 1, the CAR semantics gives the same answers as the AR semantics, and the ICAR semantics coincides with ICR semantics. We provide another example (borrowed from [ 9 ]) to show how these semantics di er. Example 2. Reconsider Tuniv and Auniv from Example 1, and let Tu0niv be obtained by adding 9Teaches v Faculty to Tuniv. Applying the closure operator yields: closeTu0niv (Auniv) = Auniv [ fFaculty(sam); Faculty(kim); Faculty(alex); Course(cs48);

Faculty(jane); Course(jane); Faculty(cs34)g Since Faculty(cs34) is not involved in any contradictions, it appears in every closed ABox repair, so cs34 is an answer to q(x) = Faculty(x) under ICAR and CAR semantics. Note however that cs34 is not an answer under AR semantics since some (standard) repairs do not contain Teaches(cs34; jane), which is required to infer Faculty(cs34).

As displayed in Fig. 2, the CAR semantics satis es Consistent Results, and ICAR semantics further satis es Unique Base, but neither satis es Consistent Support (here again we refer to [ 9 ] for an example).

We next consider a parameterized family of semantics, called the k-support semantics, that were introduced in [ 15 ] in order to provide increasingly more ne-grained lower approximations of the AR semantics (while enjoying certain desirable computational properties, see Section 3).

De nition 8 (k-support semantics). Tuple a is an answer to q over K = hT ; Ai under the k-support semantics, written hT ; Ai j=k-supp q(a), if there exist (not necessarily distinct) subsets S1; : : : ; Sk of A that satisfy the following: { each Si is a T -support for q(a) in A { for every R 2 Rep(K), there is some Si with Si R

The intuition for the k-support semantics is to restrict the number of distinct supports of the query that can be used to `cover' all of the repairs. When k = 1, the same support must be present in every repair, so the 1-support semantics coincides with the IAR semantics. By increasing k and allowing larger and larger supports, the set of answers will increase until it coincides with the AR-answers. Like the AR semantics, the k-support semantics satisfy both Consistent Results and Consistent Support.

Example 3. Returning to Example 1, we evaluate Faculty(x) using the k-support semantics. When k = 1, the semantics coincides with IAR, so we only get alex. For k = 2, we gain an additional answer, kim, by considering the pair of supports fProf(kim)g and fLect(kim)g. Finally, for k 3, we have one further answer, sam, by considering the supports fProf(sam)g, fLect(sam)g, fFellow(sam)g.

A second parameterized class of semantics, the k-defeater semantics, was introduced in the same work [ 15 ] in order to provide increasingly tighter upper approximations of the AR semantics.

De nition 9 (k-defeater semantics). A tuple a is an answer to q over K = hT ; Ai under the k-defeater semantics, written K j=k-def q(a), if there does not exist a T -consistent subset S of A with jSj k such that hT ; S [ Ci j= ? for every inclusion-minimal T -support C A of q(a).

It can be shown that 0-defeater semantics coincides with brave semantics and that the set of answers under k-defeater semantics decreases as the value of k increases, until the set of AR-answers is reached.

We should also mention another parameterized family of inconsistency-tolerant semantics, called k-lazy [ 24 ], whose de nition involves another notion of repair and will be omitted for lack of space. By taking k large enough, the k-lazy semantics coincides with the AR semantics. However, in contrast to the k-support semantics, the convergence is not monotone, meaning that a tuple might be an answer for k = ` but no longer an answer when k = ` + 1. Due to this behaviour, the k-lazy semantics are not always under-approximations of AR semantics, though they do satisfy Consistent Results and Consistent Support.

More recently, a new upper approximation of the AR semantics has been proposed, called non-objection inference [ 4 ]: De nition 10. A tuple a is an answer to q over K = hT ; Ai under the nonobjection (no) semantics if (i) there is some B 2 Rep(K) with hT ; Bi j= q(a), and (ii) for every B 2 Rep(K), there is a model of hT ; Bi where q(a) is satis ed.

The non-objection semantics lies between the brave and AR semantics, and it satis es both Consistent Results and Consistent Support. Two further variants are considered in [ 4 ], by focusing on cardinality-maximal repairs.

As noted at the beginning of the section, repairs are usually de ned using set inclusion. However, in some cases, it can be more appropriate to work with cardinality-maximal repairs, or to select only the most preferred repairs according to some criteria. Several di erent notions of preferred repair, based cardinality, priority levels or weighted assertions, were explored in [ 10 ] and used to de ne variants of the IAR and AR semantics.

We further remark that the preceding works focused on repairing data given in the form of an ABox, and the de nitions need to be adapted to handle the setting of ontology-based data access (OBDA), where existing data sources are linked to a TBox via mappings. This issue is explored in a recent paper [ 8 ], where two di erent approaches (repair-at-source and map-then-repair) are contrasted.

Finally, we should emphasize that there is no single `best' semantics, and the choice of which to use needs to be based upon the acceptable level of risk and performance requirements. Moreover, it can be fruitful to utilize multiple semantics in combination, either for computational bene t or to identify answers with di erent levels of reliability. 3

Complexity of Inconsistency-Tolerant Querying

The complexity of query answering under inconsistency-tolerant semantics has been the subject of numerous works. We brie y present what is known and refer to [ 9 ] for further details and full references.

Let us start by the most well-studied case, namely, DL-Lite KBs1. Figure 3 displays the complexity landscape for querying DL-Lite KBs under a range of inconsistency-tolerant semantics, considering both data and combined complexity and both conjunctive queries (CQs) and instance queries (IQs). We observe that there are several semantics for which query answering is in AC0 in data complexity. These upper bounds are shown by means of rst-order query rewriting. For the IAR semantics, the rough idea is to modify a usual rewriting by adding negated atoms that forbid the use of ABox assertions that do not belong to the intersection of repairs [ 22, 6 ]. Subsequent work [ 15 ] established general rewritability results that apply to the families of k-support and k-defeater semantics and arbitrary FO-rewritable ontology languages. We note that the AC0 result for non-objection semantics2 has not been stated in the literature but can be shown by adapting query rewriting techniques for the brave and IAR semantics. For the AR semantics, which is arguably the most natural, query answering is intractable in data complexity, even in simpler settings, like IQs [ 21 ] or very simple TBoxes (a single axiom T v F su ces [ 7 ]). Looking now to combined complexity, we observe that the semantics that are well behaved for data complexity remain so for combined complexity (i.e. their complexity matches that of classical semantics), while the semantics with intractable data complexity exhibit higher combined complexities than classical semantics. Finally, we note that the complexity of querying with variants of AR and IAR based upon 1 The presented results apply to the most common DL-Lite dialects, such as DL

Litecore, DL-LiteR, and DL-LiteA, see [ 9 ] for details. 2 In [ 4 ], only polynomial data complexity is proven, which we improve to AC0. It is also not too hard to show that the combined complexity matches classical semantics. Semantics classical AR IAR brave CAR ICAR ICR k-support (k k-defeater (k k-lazy (k 1) non-objection DL EL? ALC preferred repairs (cardinality, weights, priorities) has also been studied [ 10 ], and the general message is that incorporating preferences leads to higher complexity.

We now brie y consider the situation for DLs beyond DL-Lite. Figure 4 displays complexity results for two representative DLs (E L? and ALC) and three prominent semantics (AR, IAR, brave). The main observation with regards to E L? (and other Horn DLs) is that the IAR and brave semantics are no longer tractable in data complexity. Essentially, the reason is that in constrast to DL-Lite and other FO-rewritable languages, it is not possible in general to bound the size of minimal T -inconsistent subsets nor minimal T -supports. For ALC, the adoption of inconsistency-tolerant semantics leads to a rise in data complexity, but leaves the combined complexity unchanged (since the repairs can be enumerated in exponential time).

CQs NP p 2 NP NP p 2

NP 2p[O(log n)]

NP NP p 2 NP

IQs NL coNP NL NL NL NL coNP NL NL in P

NL CQs NP p 2 2p[O(log n)] NP Exp Exp Exp Exp

IQs P coNP coNP NP Exp Exp Exp Exp

4 Implementations of Inconsistency-Tolerant Querying

We provide a brief overview of systems that have been implemented and tested for inconsistency-tolerant query answering over DL knowledge bases. QuID system [ 26, 23 ] This system3 performs conjunctive query answering under the IAR semantics in an extension of DL-LiteA with denial and identi cation constraints. Three di erent approaches have been implemented and compared: rst-order query rewriting, ABox annotation (in which assertions are marked as safe or problematic depending on whether they belong to the intersection of repairs, and the query is modi ed to only use safe assertions), and ABox cleaning (in which assertions not belonging to the intersection of repairs are removed, and the resulting dataset is queried as usual). The latter two approaches generally proved to be more e cient than the rewriting approach, but they have the downside of involving data modi cations.

CQAPri system [ 10, 13 ] This system4 computes answers to CQs over DL-LiteR KBs under the IAR, brave, and AR semantics (as well as prioritized versions of AR and IAR). Answers are rst computed for the IAR and brave semantics, by evaluating a UCQ-rewriting and ltering the results using a pre-computed set of minimal T -inconsistent subsets. To identify the AR-answers among the remaining tuples (i.e. those holding under brave semantics but not under IAR semantics), CQAPri constructs a (usually quite small) instance of UNSAT for every such tuple, which is passed to an o -the-shelf SAT solver. In addition to using the IAR and brave semantics to reduce the number of calls to the SAT solver, the three semantics are used to partition query answers into three levels of reliability: (Almost) Sure (those answers holding under IAR semantics), Likely (answers holding under AR but not IAR semantics), and Possible (answers only holding under brave semantics). Experiments conducted on the modi ed LUBM benchmark (which was further augmented with negative inclusions and con icting assertions) showed that despite its intractable data complexity, it is feasible to compute query answers under the AR semantics, thanks in part to the fact that many AR-answers can be identifed using the tractable IAR semantics. SaQAI system [ 28 ] This system5 implements the IAR and ICAR semantics for DL-LiteR KBs and CQs. For the IAR semantics, the authors follow the ABox cleaning approach from QuID, using query rewriting to identify then remove the assertions that do not appear in the intersection of repairs. For the ICAR semantics, a combination of saturation and query rewriting is employed, together with some optimizations. The experiments conducted using the CQAPri benchmark show a better performance than QuID and CQAPri for the IAR semantics. 3 QuID system: http://www.dis.uniroma1.it/~ruzzi/quid/ 4 CQAPri system and benchmark: https://www.lri.fr/~bourgaux/CQAPri. 5 SaQAI system: http://www.image.ece.ntua.gr/~etsalap/SaQAI/ System from [ 27 ] This system targets the IAR semantics and currently supports the DL E LHd?r. It checks whether su cient conditions for producing an IARrewriting are ful lled (leveraging the FO-rewritability checker Grind [ 19 ]) and constructs such a rewriting when one exists by adding negated conjuncts to a classical rewriting. Experiments were conducted on seven existing ontologies (which sometimes needed to be enriched with negative inclusions) and for six of them, the su cient conditions were satis ed, suggesting that a rewriting-based approach to IAR may be feasible in practice for ontologies beyond DL-Lite. System from [ 4 ] This system implements the non-objection semantics for ground CQs (i.e. CQs without existentially quanti ed variables) for DL-LiteR KBs. Experiments on the CQAPri benchmark con rm that query answers can be e ciently computed (in accordance with the tractable data complexity). System from[ 18 ] This system can be used to query SHIQ KBs under a variant of the AR semantics with weight on ABox assertions and is restricted to ground CQs. Like CQAPri, it employs SAT solvers, and a form of reachability analysis to identify a query-relevant fragment of the KB. 5

Related Reasoning Services for Inconsistency Handling

We mention some related reasoning and analysis tasks. First, to render inconsistency-tolerant OMQA systems more usable, it is important to be able to explain the results to users. This issue has been taken up in [ 11 ], where a formal framework was presented for justifying why a given tuple appears as an answer under the considered inconsistency-tolerant semantics (AR, IAR, or brave) or why it is not part of the results. The approach has been implemented by exploiting di erent functionalities of SAT solvers and integrated into the CQAPri system. Closely related is a line of work [ 3, 2 ] on utilizing argumentation and dialogues with users to explain query answers under various inconsistency-tolerant semantics (ICR, IAR, brave, and AR).

Another important question is how to aid users in repairing their data, in order to improve the quality of the data. An interactive query-driven approach to this question has been presented in [ 12 ]. The idea is to allow users may provide feedback on which query results are missing or erroneous, and then interact with the user in order to identify a set of ABox modi cations (additions and deletions of assertions) that x the identi ed aws.

Finally, let us mention a formal study of the consistency properties of OBDA speci cations [ 17 ], in which a database schema is said to protect an OBDA speci cation if every legal data instance for the database constraints is consistent w.r.t. the ontology and mappings, and it said to be faithful to a speci cation if the database constraints do not exclude any instance that is allowed by the ontology and mappings. (Un)decidability and complexity results are shown for the analysis tasks of checking the protection and faithfulness properties.

Concluding Remarks

We hope to have showcased the large body of research that has been developed over the past decade or so around the issue of inconsistency handling in OMQA. Signi cant progress has been made on proposing di erent semantics for querying inconsistent KBs in a principled manner and exploring their computational properties: complexity, algorithms, and implemented prototypes. There remain several interesting challenges to tackle going forward, let us mention just three.

First, while we start to have a reasonable idea of how to approach the issue for DL-Lite KBs, there remains a need to develop practical algorithms for DLs beyond DL-Lite. Indeed, due to the prevalence of data quality issues, every OMQA system should be equipped with some sort of inconsistency handling mechanism (beyond simply reporting that the KB is inconsistent!), and the challenge is to nd ways of incorporating such features while limiting the impact on performance. Some rst steps towards this goal can be found in [ 28, 27 ].

Second, a very nice but extremely challenging theoretical question is to classify the complexity of inconsistency-tolerant query answering at the level of ontology-mediated queries (that is, ontology-query pairs). Some preliminary results in this direction have been presented in [ 6, 7 ]. We note that this problem is closely related to work on classifying the complexity of consistent query answering in the presence of functional dependencies, where signi cant progress has been made (see e.g. [ 20 ]), but a full classi cation has proven elusive.

Third, it would also be worthwhile to develop quantitative approaches to inconsistency-tolerant OMQA, both to be able to quantify the con dence in di erent results, and to be able to take advantage of numeric / probabilistic / statistical information when it is available. For instance, data that results from information extraction systems is often annotated with a con dence value, and mined data quality rules (see e.g. [ 25 ]) that act as soft constraints can prove useful in detecting inconsistencies and determining the most likely xes. Acknowledgements Let me take this opportunity to thank the steering committee and workshop chairs for asking me to be an invited speaker, and while I'm at it, a big thanks to my many co-authors and the DL community at large for providing a scienti cally stimulating and friendly research environment.

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