In ontology-based data access (OBDA) [17], an ontology provides an abstract and formal representation of the domain of interest, which is used as a virtual schema when formulating queries over the data. Current research in OBDA mostly focuses on ontology specification languages for which conjunctive query answering is first-order (FO) rewritable. In a nutshell, FO-rewritability means that query answering can be performed by rewriting the input query into a first-order query which encodes the relevant knowledge from the ontology, and then evaluating the resulting query over the data. Among FO-rewritable ontology languages, description logics (DLs) of the DL-Lite family [8, 2] have played an especially prominent role and notably served as the inspiration for the OWL 2 QL profile 3 of the latest version of the OWL web ontology language. In real-world applications involving large amounts of data and/or multiple data sources, chances are that the data will be inconsistent with the ontology. Standard OBDA querying algorithms are next to useless in such circumstances, since first-order logic semantics (upon which DLs and standard ontology languages are based) dictates that everything can be derived from a contradiction. Appropriate mechanisms for handling inconsistent data are thus critical to the success of OBDA in practice. Clearly, the best solution is to restore consistency by removing the pieces of data that are responsible for the inconsistencies. However, this strategy cannot always be applied, since the system may not have enough information to localize the errors, or may lack the authorization to modify the data (as is often the case in information integration applications). Thus, a robust OBDA system must be capable of providing meaningful answers to user queries in the presence of inconsistent data. Recently, several approaches have pursued the idea of adopting an inconsistencytolerant semantics for OBDA, taking inspiration from the work on consistent query answering in databases [1, 4]. The most well-known and intuitive among such semantics, which we will call the CQA semantics, considers as a repair of a knowledge base (KB) consisting of an ontology T and a dataset A, a maximal subset of A that is consistent with T . Query answering under the CQA semantics then amounts to computing those answers that hold in every repair of the KB. Unfortunately, conjunctive query answering (as well as simpler forms of reasoning) under CQA semantics is computationally 3 http://www.w3.org/TR/owl2-profiles/
hard, even for extremely simple ontology languages for which reasoning under classical
semantics is tractable [
To overcome this computational problem, approximations of the CQA semantics
have been recently proposed. In particular, [
In this paper, we address the above issues and provide the following contributions: (i) we propose two new families of inconsistency-tolerant semantics, called k-defeater and k-support semantics, that approximate the CQA semantics from above (complete approximations) and from below (sound approximations), respectively, and converge to the CQA semantics in the limit;
(ii) we study the data complexity of conjunctive query answering under the new
semantics, and show a general tractability result for a broad class of ontology languages
that includes all known first-order rewritable languages, in particular almost all DLs of
the DL-Lite family and several rule-based languages of the Datalog+/- family [
The k-support and k-defeater semantics proposed in this paper provide the basis for a semantically grounded and computationally tractable approximation of the CQA semantics in OBDA systems. In particular, we envision a flexible, iterated execution of query q under both k-support and k-defeater semantics with increasing values of k, which stops as soon as the answers to q under both semantics coincide, or when the user is not interested in (or does not want to pay further computational cost for) an exact classification of the tuples that are answers to q under the CQA semantics. 2
Ontologies and KBs An ontology T is a finite set of first-order logic sentences, and an ontology (specification) language L is a (typically infinite) set of first-order logic sentences. If T L, then T is called an L ontology. A knowledge base (KB) is a pair consisting of an ontology T and a finite set A of ground facts. A KB hT ; Ai is said to be consistent if the first-order theory T [ A has a model. Otherwise, it is inconsistent, which we denote by hT ; Ai j= ?.
We are interested in the problem of answering instance queries and conjunctive queries over KBs. Without loss of generality, and for ease of exposition, we only consider Boolean queries (i.e. queries without free variables). A first-order (FO) query, or simply query, is a first-order sentence. An instance query (IQ) is a FO query consisting of a single ground fact. A conjunctive query (CQ) is a FO query of the form 9x( 1 ^ : : : ^ n) where every i is an atom whose arguments are either constants or variables from x. A query q is entailed by a KB K under classical semantics (denoted by K j= q) if q is satisfied in every model of K. The instance checking problem consists in deciding, for a KB K and IQ q, whether K j= q. The conjunctive query entailment problem is defined analogously, but with q a CQ.
We introduce some terminology for referring to sets of facts which are responsible for inconsistency or query entailment. A set S of ground facts is called T -consistent if hT ; Si 6j= ?. A minimal T -inconsistent subset of A is any S A such that hT ; Si j= ? and every S0 ( S is T -consistent. A set of facts S A is said to be a T -support for query q in A if S is T -consistent and hT ; Si j= q, and it is called a minimal T -support for q in A if no proper subset of S is a T -support for q in A. We sometimes omit “for q” or “in A”, when these are understood.
Given a set of ground facts A, we define IA as the interpretation isomorphic to A,
i.e., the interpretation defined over the domain of constants occurring in A and such that
the interpretation of every relation R in IA is equal to the set fa j R(a) 2 Ag.
DL-Lite ontology languages We focus on DLs of the DL-Lite family [
B ! A j 9R
C ! B j :B
R ! P j P with A a concept name (unary relation) and P a role name (binary relation). In a DL-LiteHorn ontology, inclusions take the form B1 u : : : u Bn v C, with B1; : : : ; Bn and C as above.
The classical semantics of DL-Lite and DL-LiteHorn ontologies is obtained by translating inclusions into first-order sentences using the following function : (A(x)) = A(x) (9P (x)) = 9y(P (x; y)) (9P (x)) = 9y(P (y; x))
(:B(x)) = : (B(x)) (B1 u B2(x)) = (B1(x)) ^ (B2(x))
(C v D) = 8x( (C(x)) ! (D(x)) The classical semantics of a DL-LiteHorn KB hT ; Ai (and in particular, the notions of model, consistency, and entailment) corresponds to the semantics of the first-order KB h (T ); Ai. Note that when considering DL KBs, we assume as is standard that the dataset A uses only unary and binary relations.
First-order rewritability We say that an ontology T is first-order (FO) rewritable
(for CQ answering) under semantics S if, for every CQ q, there exists an effectively
computable FO query q0 such that, for every set of ground facts A, hT ; Ai entails q
4 This DL is referred to as DL-Litecore in [
In this section, we formally introduce the consistent query answering (CQA) semantics and other relevant inconsistency-tolerant semantics. All of the semantics considered in this paper rely on the notion of a repair, defined as follows: Definition 1. A repair of a KB K = hT ; Ai is an inclusion-maximal subset of A that is T -consistent. We use Rep(K) to denote the set of repairs of K.
The repairs of a KB correspond to the different ways of achieving consistency while retaining as much of the original data as possible. Hence, if we consider that the data is mostly reliable, then it is reasonable to assume that one of the repairs accurately reflects the correct portion of the data.
The consistent query answering semantics (also known as the AR semantics [
Example 1. Consider the DL-Lite ontology Tuniv:
which states that professors, lecturers, and research fellows are disjoint classes of faculty, that professors and lecturers must teach something, and that whatever is taught is not faculty. Now let Asam be as follows: fProf(sam); Lect(sam); Fellow(sam)g. It is easy to see that KB hTuniv; Asami is inconsistent and has 3 repairs: R1 = fProf(sam)g, R2 = fLect(sam)g and R3 = fFellow(sam)g. Observe that from each of the repairs, we can infer q1 = Faculty(sam), so hTuniv; Asami j=CQA q1. However, q2 = hTuniv; Asami 6j=CQA q2.
Unfortunately, while the CQA semantics is intuitively appealing, it is well-known
that answering queries under this semantics is usually intractable w.r.t. data complexity
[
To overcome the computational problems of the CQA semantics, a sound
approximation of it, called the IAR semantics, was proposed in [
Definition 3. A query q is entailed by a KB K = hT ; Ai under the IAR semantics, written hT ; Ai j=IAR q, if hT ; Di j= q where D = TB2Rep(K) B.
The IAR semantics is more conservative than the CQA semantics, as it only uses those facts which are not involved in any contradiction. This has the advantage of yielding query results which are almost surely correct, but also the drawback that some plausible inferences may be missed, as demonstrated by the following example. Example 2. Reconsider the KB hTuniv; Asami and CQ q1 from Example 1. The intersection of the repairs R1 \ R2 \ R3 is the empty set, so hTuniv; Asami 6j=IAR q1, despite the fact that all the information in Asam supports q1 being true.
From the computational perspective, the IAR semantics can be much better-behaved
than the CQA semantics. Indeed, it was shown in [
Finally, to obtain a natural overapproximation of the CQA semantics, we introduce its brave version.
Definition 4. A query q is entailed by a KB K = hT ; Ai under the brave semantics, written hT ; Ai j=brave q, if hT ; Bi j= q for some repair B 2 Rep(K).
Example 3. As q2 is entailed by hTuniv; R1i, we have hTuniv; Asami j=brave q2. Also note that every fact in Asam appear in some repair, hence, all facts in Asam are entailed under the brave semantics.
As Example 3 demonstrates, the brave semantics has the undesirable feature of allowing contradictory statements to be entailed. Nonetheless, this semantics can still serve a useful role by providing a means of showing that a query is not entailed under the CQA semantics. 4
In this section, we propose two new families of inconsistency-tolerant semantics, which provide increasingly fine-grained under- and over-approximations of the CQA semantics. As these semantics will be shown in Section 5 to enjoy the same nice computational properties as the IAR semantics, our new approach allows us to marry the advantages of the IAR and CQA semantics.
We begin by presenting our new family of sound approximations of the CQA semantics. The intuition is as follows: if a query q is entailed under the CQA semantics, then this is because there is a set fS1; : : : ; Sng of T -supports for q such that every repair contains some Si. The k-support semantics we propose is obtained by allowing a maximum of k different supports to be used.
Definition 5. A query q is entailed by K = hT ; Ai under the k-support semantics, written K j=k-supp q, if there exist (not necessarily distinct) subsets S1; : : : ; Sk of A satisfying the following conditions: – each Si is a T -support for q in A – for every R 2 Rep(K), there is some Si with Si R have hTuniv; Asami j=3-supp q1, but hTuniv; Asami 6j=2-supp q1.
Example 4. The three repairs of hTuniv; Asami all use different supports for q1. We thus
The following theorem resumes the important properties of the family of k-support semantics, showing that they interpolate between the IAR and CQA semantics. Theorem 1. Let K = hT ; Ai be a KB and q a query. Then: 1. K j=IAR q if and only if K j=1-supp q 2. K j=CQA q if and only if K j=k-supp q for some k 3. for every k 0, if K j=k-supp q, then K j=k+1-supp q
The k-support semantics allows us to approximate more and more closely the set of queries entailed under the CQA semantics, but provides no way of showing that a particular query is not entailed under this semantics. This motivates the study of complete approximations of the CQA semantics.
The observation underlying our new family of complete approximations is the following: if a query q is not entailed under the CQA semantics, this is because there is a T -consistent set of facts which contradicts all of the T -supports of q. The k-defeater semantics corresponds to there being no way to construct such a “defeating” set using at most k facts.
Definition 6. A query q is entailed by K = hT ; Ai under the k-defeater semantics, written K j=k-def q, if there does not exist a T -consistent subset S of A with jSj k such that hT ; S [ Ci j= ? for every minimal T -support C A of q.
Note that if q has no T -support, then it is not entailed under 0-defeater semantics since one can simply take S = ;.
Example 5. We have hTuniv; Asami 6j=1-def q2, since by choosing S = fFellow(sam)g, we can invalidate the minimal T -supports of q2, which are fProf(sam)g and fLect(sam)g.
The next theorem shows that the family of k-defeater semantics provides increasingly closer over-approximations of the CQA semantics, starting from the brave semantics presented in Section 3.
Theorem 2. Let K = hT ; Ai be a KB and q a query. Then: 1. K j=brave q if and only if K j=0-def q 2. K j=CQA q if and only if K j=k-def q for every k 3. for every k 1, if K j=k+1-def q, then K j=k-def q
In this section, we study the data complexity of conjunctive query answering under the k-support and k-defeater semantics. Our main result is the following theorem which shows that, for a broad class of ontology languages, conjunctive query answering under these semantics can be done using FO-rewriting, and hence is in AC0 w.r.t. data complexity.
Theorem 3. Let T be an ontology that is FO-rewritable for CQ answering under classical semantics and such that for every CQ q, there exist `; m such that for every A, every minimal T -support for q relative to A has cardinality at most `, and every minimal T -inconsistent subset of A has cardinality at most m. Then: (i) for every k 1, T is FO-rewritable for conjunctive query answering under the k-support semantics; (ii) for every k 0, T is FO-rewritable for conjunctive query answering under the k-defeater semantics.
Proof (sketch). Let T be as stated, and let q be a CQ. By assumption, we can find ` and m such that for every A, the minimal T -supports for q relative to A have cardinality at most `, and the minimal T -inconsistent subsets of A have cardinality bounded by m. For point (i), a FO-rewriting of q relative to T for the k-support semantics can be obtained by considering the first-order query 'q = q1 _ : : : _ qn, where the disjuncts qi correspond to the different possible choices of k T -supports for q of cardinality at most `, and each qi asserts that the chosen supports are present in A and that there is no T -consistent subset of A of cardinality at most km which conflicts with each of the supports. For point (ii), the desired FO-rewriting of q takes the form :(q1 _ : : : _ qn), where every qi asserts the existence of a T -consistent set of facts of cardinality at most k which conflicts with every minimal T -support for q. Here we again utilize the fact that the size of minimal T -supports is bounded by `, and hence there are only finitely many types of supports to consider.
Theorem 3 significantly strengthens earlier positive results for the IAR semantics
[
The following examples illustrate the construction of FO-rewritings for the k-support and k-defeater semantics.
Example 6. We consider how to rewrite the CQ q1 under the k-support semantics. When k = 1, we can take as our FO-rewriting the disjunction of the formulas:
Note that each disjunct expresses that one of the four possible T -supports is present and is not contradicted by other facts. To obtain the rewriting for k = 2, we must introduce additional disjuncts which assert that a pair of T -supports is present and cannot be simultaneously contradicted. We obtain three new disjuncts (the other combinations being subsumed by one of the other disjuncts):
Finally, for k = 3, we must add further disjuncts to check for the existence of a triple of T -supports which are present and cannot be defeated. In our case, this leads to one new (non-subsumed) disjunct:
Note that this last disjunct is satisfied in IAsam , witnessing hTuniv; Asami j=3-supp q1. Notice also that in this particular example, the CQA and 3-support semantics coincide, and so the FO-rewriting for k = 3 is also a FO-rewriting under the CQA semantics. Example 7. We now consider how to rewrite the query q2 under the k-defeater semantics. When k = 0, the construction yields the following FO-rewriting: : :(9x Faculty(sam) ^ teaches(sam; x)) ^ :Prof(sam)
^ :Lect(sam) ^ :(9x Fellow(sam) ^ teaches(sam; x)) Inside the negation, there is a single disjunct which asserts that the empty set conflicts with every T -support, or equivalently, that there are no T -supports. When k = 1, we must add further disjuncts inside the negation to capture single facts which conflict with all T -supports. In our case, we must add two new disjuncts:
The first disjunct is needed since any fact of the form teaches(x; sam) contradicts Faculty(sam), and hence, every T -support for q2. The second disjunct treats the case where there is no atom teaches(x; sam), in which case the only possible T -supports for q2 are Prof(sam) and Lect(sam), both of which are contradicted by Fellow(sam). Notice that this last disjunct holds in IAsam , which proves that hTuniv; Asami 6j= q2.
We briefly remark that polynomial data complexity is not preserved under the new
semantics. Indeed, in the lightweight DL E L?, CQ answering and unsatisfiability are
P-complete w.r.t. data complexity, but it was shown in [
To gain further insight into the computational properties of the different inconsistencytolerant semantics considered in this paper, we study the combined complexity of instance checking and CQ entailment for DL-Lite and DL-LiteHorn KBs under these semantics. The results of our analysis are reported in Figure 1. IC
DL-Lite NLSPACE
DL-LiteHorn coNP CQ DL-Lite
DL-LiteHorn
NP NP 2p[O(log n)] 2p[O(log n)] k-supp (k > 1) NLSPACE coNP 2p[O(log n)] coNP coNP p 2 p 2 k-def (k > 0) brave NLSPACE NP NP NP
NLSPACE NP NP NP
Before presenting the results in more detail, let us begin with some general observations. First, it is interesting to note that for DL-Lite KBs, the complexities obtained for the IAR, k-support, brave, k-defeater, and classical semantics all coincide, and are strictly lower than the complexity w.r.t. the CQA semantics. By contrast, for DL-LiteHorn KBs, instance checking under any of the considered inconsistency-tolerant semantics is of higher complexity than under classical semantics. Moreover, we lose the symmetry between the sound and complete approximations. Indeed, for CQ entailment, the complexities of the sound approximations (IAR and k-support) is higher than for the complete approximations (brave and k-defeater semantics).
Finally, we remark that in several cases, and in particular, for the k-support semantics, the complexity for DL-LiteHorn is higher than for DL-Lite. This can be explained by the fact that for DL-Lite KBs, the size of a minimal T -support of a query is linear in the size of the query and independent of T , whereas for DL-LiteHorn KBs, the bound on minimal T -supports depends also on the size of T . Overall, these results suggest that while the k-support and k-defeater semantics are tractable w.r.t. data complexity for both DL-Lite and DL-LiteHorn , it will likely be much easier to obtain practical algorithms for DL-Lite KBs.
We now present our different complexity results and some brief ideas concerning the proofs. We start by showing that for DL-Lite, instance checking under the proposed semantics has the same low complexity as under classical semantics.
Theorem 4. In DL-Lite, instance checking under the k-support semantics is NLSPACEcomplete w.r.t. combined complexity, for every k 1. The same holds for the k-defeater semantics, for every k 0.
Proof (idea). The proof exploits the fact that when T is a DL-Lite ontology, minimal T -supports for IQs consist of single facts, and minimal T -inconsistent subsets contain at most two facts. This means in particular that every k-tuple of minimal T -supports contains at most k facts, and at most k facts are needed to contradict all k supports. This enables a NLSPACE procedure which guesses k facts and verifies that each fact is a T -support, and that there is no set with at most k facts which contradicts all of the guessed facts. The upper bound for the k-defeater semantics uses similar ideas.
In DL-LiteHorn , instance checking is intractable already for the IAR and brave semantics, and the lower bounds can be used to show intractability also for the k-support and k-defeater semantics. For the k-defeater semantics, Theorem 6 provides a matching upper bound, while the precise complexity for the k-support semantics remains open. Theorem 5. Instance checking in DL-LiteHorn is coNP-complete w.r.t. combined complexity under the IAR semantics, coNP-hard w.r.t. combined complexity under k-support semantics, and NP-complete w.r.t. combined complexity under both the brave semantics and k-defeater semantics.
Proof (idea). We sketch the coNP lower bound for the IAR semantics, which is by reduction from UNSAT. Let ' = c1 ^ : : : ^ cn be a propositional CNF over variables x1; : : : ; xm. Consider the DL-LiteHorn KB with
fTi u Fi v ? j 1 T =fTi v Cj j xi 2 cj g [ fFi v Cj j :xi 2 cj g[ i
mg[fA u C1 u : : : u Cn v ?g and A = fA(a)g[fTi(a); Fi(a) j 1 i mg. Then it can be shown that hT ; Ai j=IAR A(a) if and only if the formula ' is unsatisfiable.
We next consider the complexity of CQ entailment under our proposed semantics. For DL-Lite, we obtain precisely the same complexity as under the classical semantics. Theorem 6. In DL-Lite, CQ entailment under the k-support semantics is NP-complete w.r.t. combined complexity, for every k 1. For both DL-Lite and DL-LiteHorn , CQ entailment under the k-defeater semantics is NP-complete w.r.t. combined complexity, for every k 1.
Proof (idea). We sketch the upper bound for the k-defeater semantics. Fix a DL-LiteHorn KB hT ; Ai and a CQ q. Let S1; : : : ; Sm be the T -consistent subsets of A with cardinality at most k. Guess a sequence C1; : : : ; Cm of subsets of A of cardinality at most c = 2 jqj jT j, together with polynomial certificates that hT ; Cii j= q, for each Ci. Output yes if for every 1 i m, the certificate is valid and Si [ Ci is T -consistent. As m is polynomial in jAj (since k is fixed), and both conditions can be verified in polynomial time for DL-LiteHorn KBs, we obtain an NP procedure. Correctness relies on the fact that because T is a DL-LiteHorn ontology, every minimal T -support for q has cardinality at most c.
For DL-LiteHorn , CQ entailment under the IAR and k-support semantics rises to 2p[O(log n)]-complete.
Theorem 7. In DL-LiteHorn , CQ entailment under k-support semantics is
complete w.r.t. combined complexity, for every k 1.
2p[O(log
n)]Proof (idea). The lower bound is by a non-trivial reduction from the Parity(SAT)
problem [
Every minimal T -support for q contains at most c facts. It follows that the algorithm
returns yes if hT ; Ai j=k-supp q. Conversely, if the output is yes, with (C1; : : : ; Cn) the
k-tuple from Step 2, then by (i), every Ci is a T -support for q. Moreover, (ii) ensures
that every repair contains some Ci, for it not, we could find some k-tuple ( 1; : : : ; k) 2
C1 : : : Cn which does not belong to S, contradicting (ii). Note that the algorithm
runs in polynomial time with an NP oracle, since there are only polynomially many
k-tuples of facts to consider, for fixed k. As the oracle calls can be organized into a tree,
membership in 2p[O(log n)] follows by a result from [
Finally, we determine the combined complexity of instance checking and CQ entailment for the CQA semantics (prior results only considered data complexity). Theorem 8. For DL-Lite and DL-LiteHorn , instance checking (resp. CQ entailment) under CQA semantics is coNP-complete (resp. 2p-complete) for combined complexity. Proof (idea). The upper bounds are easy: guess a repair and show that it does not entail the query. The coNP-lower bound for instance checking follows from the coNPhardness of this problem w.r.t. data complexity. The 2p-hardness result involves a nontrivial reduction from 2-QBF validity.
We should point out that our proofs are quite generic and can be directly used (or trivially extended) to obtain results for a whole rangle DL-Lite dialects (as well as other ontology languages). 7
The present work can be extended in several directions. First, we believe that our
approach can have a practical impact on OBDA systems, so we aim to implement and
experiment with the approach. It would also be very interesting to investigate the
connections between our approach and approximate knowledge compilation [
Acknowledgments. The first author has been supported by a Universite´ Paris-Sud Attractivite´ grant and ANR project PAGODA (ANR-12-JS02-007-01). The second author has been partially supported by EU FP7 project Optique – Scalable End-user Access to Big Data (grant n. FP7-318338).