DL-Lite
?Diego Calvanese1, Magdalena Ortiz2, Mantas Šimkus2, and Giorgio Stefanoni12
1 KRDB Research Centre for Knowledge and Data Free University of Bozen-Bolzano, Italy
calvanese@inf.unibz.it, giorgio.stefanoni@gmail.com
2 Institute of Information Systems Vienna University of Technology, Austria
ortiz@kr.tuwien.ac.at, simkus@dbai.tuwien.ac.at
Abstract. In order to meet usability requirements, most logic-based applications provide explanation facilities for reasoning services. This holds also for DLs, where research focused on the explanation of both TBox reasoning and, more recently, query answering. Besides explaining the presence of a tuple in a query answer, it is important to explain also why a given tuple is missing. We address this latter problem for (conjunctive) query answering over DL-Lite ontologies, by adopting abductive reasoning, that is, we look for additions to the ABox that force a given tuple to be in the result. As reasoning tasks, we consider existence and recognition of an explanation, and relevance and necessity of a certain asser- tion for an explanation. We characterize the computational complexity of these problems for subset minimal and cardinality minimal solutions.
1 Introduction
Query answering over ontologies formulated in Description Logics (DLs) has received considerable attention both in research and industry. Given an ontology, users typically pose queries over the conceptual schema and get answers that take into account the con- straints specified at the conceptual level. Many efforts have concentrated on lightweight description logics. For instance,DL-LiteAhas been tailored for query answering over large data sets [7]. For this reason, expressive power is traded in favor of a better com- putational behavior in terms of data-complexity. In fact, conjunctive query answering in DL-LiteAenjoys FOL-rewritability, i.e., it can be reduced to the problem of evaluating a suitably constructed FOL query over a database instance.
In order to meet usability requirements set by domain users, most logic-based ap- plications provide explanation algorithms for reasoning services. This holds also for DLs, where research focused on the explanation of both TBox reasoning [12,6,14,3]
and, more recently, query answering [5]. In addition, the latter paper advocates the importance of tackling the problem of explaining the absence of a tuple in the an- swers to a query over an ontology. This problem stems from the database community, where it has been solved in the context of databases extended with provenance infor- mation [9]. We address this problem by considering explanations for the absence of a
?This work was partially supported by the Austrian Science Fund (FWF) grant P20840.
tuple in the context of query answering overDL-LiteAontologies. We adoptabductive reasoning[10,11], that is, we consider which additions need to be made to the ABox to force the given tuple to be in the result. More precisely, given a TBoxT, an ABoxA, and a queryq, anexplanationfor a given tupletis a new ABoxUsuch that the answer toqoverhT,A ∪ U icontainst. An important aspect in explanations is to provide the user with solutions that are simple to understand and free of redundancy, hence as small as possible. To address this requirement, we study various restrictions on solutions, in particular, we focus on subset minimal and cardinality minimal ones. We consider stan- dard decision problems associated to logic-based abduction:(i) existenceof an expla- nation,(ii) recognitionof a given ABox as being an explanation, and(iii) relevanceand (iv) necessityof an ABox assertion, i.e., whether it occurs in some or all explanations.
After motivating such problems and formalizing them, we provide algorithms to solve them and a precise characterization of their computational complexity forDL-LiteA. The complexity results for the various reasoning tasks are summarized in Table 1.
2 Preliminaries
DL-LiteA. DL-LiteAis a member of theDL-Litefamily of DLs [7], which have been designed for dealing efficiently with large amounts of extensional information. InDL- LiteA, concept expressionsC, denoting sets of objects, and role expressionsR, denoting binary relations between objects, are formed as follows:
C −→ A | ∃R, R −→ P | P−.
whereAdenotes an atomic concept and P an atomic role3. In a DL-LiteA ontology O=hT,Ai, the TBoxT consists of axioms of the form
C1vC2, C1v ¬C2,
R1vR2,
R1v ¬R2, (functR),
and the ABoxAconsists of assertions of the formA(c)andR(c, c0), wherec,c0 are constants (or,individuals) from a countably infinite setC. An interpretation is a pair I =h∆I,·Ii, where∆I is a non-emptydomain, and theinterpretation function·I is defined as usual. We adopt here theunique name assumption(UNA), i.e.,cI1 6=cI2 for allc1, c2∈Cwithc16=c2. We refer to [7] for more details.
Conjunctive Queries. LetVbe a countably infinite set of variables. ExpressionsA(t) andP(t, t0) are called atoms, wheret, t0 ∈ V∪C. Aconjunctive query (CQ)q is an expressionq(x1, . . . , xn) ← a1, . . . , am,where eachai,1 ≤ i ≤ m, is an atom.
Let V(q)denote the set of variables occurring in q, C(q)the set of constants in q, and letat(q) =S
1≤i≤m{ai}. Amatchforqin an interpretationI is a mapping π : V(q)∪C(q) → ∆I such that πis the identity on constants, π(t) ∈ AI for each
3We ignore here the distinction between data values and objects, since it is immaterial for our results. As a consequence, we do not consider value domains and attributes, which are present inDL-LiteA.
A(t)∈at(q), andhπ(t), π(t0)i ∈PIfor eachP(t, t0)∈at(q). The tuplehx1, . . . , xni is the tuple ofanswer variablesofq. TheanswertoqoverI, denotedans(q,I), is the set of alln-tupleshd1, . . . , dni ∈Cnsuch thathdI1, . . . , dIni=hπ(x1), . . . , π(xn)ifor some matchπforqinI. Aunion of conjunctive queries(UCQ) is a set of CQs with the same answer variable tuple. For a UCQq, we letans(q,I) =S
q0∈qans(q0,I). The certain answerto a CQ or a UCQqoverOis defined ascert(q,O) ={c∈Cn |c∈ ans(q,I)for each modelIofO}.
3 Explaining Negative Query Answers
We now define the problem considered in this paper:
Definition 1. LetO=hT,Aibe an ontology,qa UCQ, andca tuple of constants. We callP =hO, q,ciaquery abduction problem(QAP). AsolutiontoP(or anexplana- tion forP) is any ABox U such that the ontologyO0 =hT,A ∪ U iis consistent and c∈cert(q,O0). The set of all explanations forPis denotedexpl(P).
Ifc∈/ cert(q,O), then we callcanegative answertoqoverO. Note that a query over the ontology can have a negative answer only if the ontology is satisfiable. Dif- ferently, if the ontology is unsatisfiable then the QAPP does not have any solution.
In the following, we will examine various restrictions toexpl(P)to reduce redundancy in explanations. This is achieved by the introduction of a preference relation among explanations. This relation is reflexive and transitive, i.e., we have a pre-order among solutions.
Definition 2. Assume a QAPP. Letdenote a pre-order on the setexpl(P)of solu- tions. We writeU ≺ U0ifU U0andU0U. The preferred explanationsexpl(P)of a QAPP under the pre-orderare defined as follows:expl(P) ={ U ∈expl(P)| there is noU0 ∈expl(P)s.t.U0 ≺ U }, i.e.,expl(P)contains all the-explanations that areminimalunder.
Two preference orders are considered here: thesubset-minimality order, denoted by
⊆, and theminimum explanation size order, denoted by≤. The latter order is defined byU ≤ U0iff|U | ≤ |U0|. Observe thatexpl≤(P)⊆expl⊆(P).
We define now four decision problems related to (minimal) explanations, which are parametric w.r.t. the chosen preference order. Given a QAPP:
– -EXISTENCE: Does there exist a-explanation forP?
– -RECOGNITION: Is a setU of ABox assertions a-explanation forP? – -RELEVANCE: Does an assertionαoccur in some-explanation forP? – -NECESSITY: Does an assertionαoccur in all-explanations forP?
In the following, whenever no preference is applied (i.e., whenis the identity) we omit to writein front of the problem’s names. We provide an example, in which we highlight the consequences of choosing among the various orderings.
Table 1: Summary of main complexity results (completeness) -EXISTENCE -RECOGNITION -RELEVANCE -NECESSITY
none PTIME(4.1) NP PTIME(4.3) PTIME(4.2)
≤ PTIME DP PNPk PNPk (4.2)
⊆ PTIME DP Σ2P(4.3) PTIME(4.2)
Example 1. LetO=hT,Aibe an ontology describing a university domain, whereT is
PostGrad vStudent, UnderGrad vStudent, UnderGrad v ¬PostGrad,
PartTime vUnderGrad.
Tutor vProfessor,
∃hasTutor vPartTime,
∃hasTutor− vTutor,
AdvancedvCourse,
∃teachesvProfessor,
∃teaches−vCourse,
That is, there are two different kinds of students,PostGradandUnderGrad. More- over,PartTime students are tutored byTutors, who are particular professors. Addi- tionally, the university offers someAdvanced courses. Let the ABoxAconsist of the assertionsteaches(rob,SWT),hasTutor(peter,rob). Now, assume that a user is in- terested in finding all those who both teach an advanced course and tutor a student.
Then, she would write the query
q(x)←teaches(x, y),Advanced(y),hasTutor(z, x).
Moreover, she may expect rob to be part of the result, i.e., rob ∈ cert(q,O), but this is not the case. Intuitively, rob satisfies all the constraints imposed by the query, except that the SWT course is not known to be Advanced. One can easily see that {teaches(rob,TOC),Advanced(TOC),hasTutor(john,rob)} is an expla- nation, {teaches(rob,ALG),Advanced(ALG)} is a ⊆-minimal explanation, while {Advanced(SWT)}is a≤-minimal explanation.
In the next section, the complexity of the four main problems is studied in the light of the different preference relations.
4 Complexity of Explanations
Table 1 provides an overview of our complexity results. Recall that the class Σ2P is a member of the Polynomial Hierarchy [13]; it is the class of all decision problems solvable in non-deterministic polynomial time using an NP oracle. Moreover, the class PNPk contains all the decision problems that can be solved in polynomial time with an NP oracle, where all oracle calls must be first prepared and then issued in parallel. The class DP contains all problems that, considered as languages, can be characterized as the intersection of a language in NP and a language inCONP [13]. Note that: PTIME⊆ NP ⊆DP⊆PNPk ⊆Σ2Pis believed to be a strict hierarchy of inclusions and here we make such an assumption.
Our results can be explained as follows. We show in the next section that EX-
ISTENCEcan be reduced to the PTIME-complete satisfiability problem forDL-LiteA without the UNA [1], which justifies our PTIME upper bound. This result can then be used to characterize the complexity of RELEVANCE,NECESSITY, and⊆-NECESSITY.
≤-RELEVANCEand≤-NECESSITYare harder. The reason being that in order to solve these problems one has to compute first the minimal size of a solution and, then, inspect all the solutions of that size. Additionally, there is another increase in complexity when dealing with⊆-RELEVANCE. The intuition is that there is an exponential number of candidate solutions to examine and for each of them one has to check that none of its subsets is itself a solution, which requires aCONP computation. Due to space limita- tions, the results on-RECOGNITIONare not detailed in this paper (see [8] for more details). The intuition for the NP bound forRECOGNITIONis that one needs simply to check consistency and perform query evaluation to solve the problem. In case a prefer- ence order is in place, one has to check minimality as well, which is aCONP check for
⊆- and≤-minimal explanations that leads to completeness for DP.
4.1 Complexity of-EXISTENCE
For -EXISTENCE note first that the existence of an explanation for P implies the existence of an explanation under the⊆and≤orderings. Thus, we only considerEX-
ISTENCE. Our first task is to show that we can restrict ourselves to explanations built from the original signature of the input QAP plus a small number of fresh constants.
Proposition 1. If P = hO, q,cihas a solution, thenP has a solutionU0 with con- cepts, roles, and attributes only fromOand at mostm=maxqi∈q|at(qi)|fresh ABox individuals.
Proof. Assume an arbitrary solutionUtoP. Given the consistency ofO0=hT,A ∪ U i, it follows that there exists a modelI ofO0 under the UNA. W.l.o.g. we assume that
∆I=CwithcI =cfor eachc∈C. Additionally, the interpretationIadmits a match πfor someqi(x)∈q, such thatπ(x) =c. LetU0={A(o)|A(t)∈at(qi)andπ(t) = o} ∪ {R(o, o0) | R(t, t0) ∈ at(qi)andπ(t) = oandπ(t0) = o0}. Observe that U0 has no more individuals thanqi has variables. It remains to see thatU0 is a solution.
Clearly the original matchπwitnesses alsoc∈ans(qi,A ∪ U0). It remains to see that O00 =hT,A ∪ U0iis consistent. But this follows from the fact thatIis a model ofO0
and that the atoms inU0hold inI ut
The above restriction allows us to considercanonical explanations, i.e., explana- tions resulting from suitable instantiations of the bodies of CQsqi ∈ q. Keeping in mind that CQs, seen as FOL formulae, are always satisfiable, an explanation does not exist only if the structure of the query is not compliant with the constraints expressed in the ontology. That is, for all the interpretationsJ ofqwithans(q,J)6=∅, there is no modelIofO, such thatI ∪ J |=O. To check whether a UCQ is compliant with the ontological constraints, a naïve method is to iteratively go through all the CQs inqand instantiate them in the ABox. If for none of the CQs we obtain a consistent ontology, then the query violates some of the constraints imposed at the conceptual level.
Proposition 2. For DL-LiteAontologies,EXISTENCEisPTIME-complete.
Proof. (MEMBERSHIP) Note thatP =hO, q,ci, withqa UCQ, has a solution iffPq0 = hO, q0,cihas a solution for someq0 ∈ q. Hence, it suffices to show the upper bound for CQs. To this end, we provide a logspace reduction fromEXISTENCEto consistency inDL-LiteAwithout UNA, which in turn is PTIME-complete [1]. Assume a QAPP = hO, q,ci, whereqis a CQ andO= hT,Ai. We argue thatP has a solution iffO0 = hT ∪ T0,A ∪ Uq∪ A0i is consistent, whereO0 is an ontology obtained fromO and q(c)as follows. The ABoxUq is obtained from at(q(c))by replacing each variable xwith a fresh individual nameax. The ABoxA0 consists of assertionsAo(o)for all constantso occurring inT,AandUq, where eachAo is a fresh concept name. The TBoxT0consists of axiomsAov ¬Ao0for all pairso6=o0of constants occurring inA andq(c). We now show thatPhas a solution iffO0is consistent.
Assume thatP has a solutionU. Then, due to consistency ofO00 = (T,A ∪ U), there is a modelIofO00under the UNA. Additionally,Iadmits a matchπforq(c). Let I0 be the extension ofIthat additionally interprets(i)constants inUq asaIx0 =π(x) for all variablesxinq, and(ii)AIo0 ={oI}for all freshly introducedAo. It remains to show thatI0is a model ofO0. Observe that sinceIis under the UNA, we have that AIo0∩AIo00 =∅for all constant pairso6=o0. ThusI0satisfies all the disjointness axioms Aov ¬Ao0 inT0. The assertions inA0are satisfied due to(ii), while the assertions in Uqdue to(i)above.
The other direction of the proof is obvious and we omit it here.
(HARDNESS) Let us reduce consistency inDL-LiteAwithout UNA toEXISTENCE. GivenO=hT,Ai, we create QAPP =hO0, q(),hiisimply by encoding the ABoxA in the CQqby replacing each constanta∈ Aby a distinct variable namexainq, while
the ontologyO0consists only of the TBoxT. ut
4.2 Complexity of (-)NECESSITY
The existence of an explanation is most of the times not very informative to the user.
In fact, given a negative answer to a query, it is important to delineate the fundamental reasons leading to the absence of the expected tuple. That is, users would like to know which assertions occur in all the solutions to a QAPP.
Proposition 3. For DL-LiteAontologies, theNECESSITYand⊆-NECESSITYproblems arePTIME-complete.
Proof. (MEMBERSHIP) We assume a QAPP =hO, q,ciwithO=hT,Ai, an asser- tionϕ(t)and considerNECESSITYfirst. This problem can be reduced to non-EXISTENCE, which was shown to be in PTIMEin the previous section. We buildO0 =hT0,A0isuch thatϕ(t)occurs in all explanations forP iffP0 = hO0, q,cihas no explanation. We defineO0 by settingA0 =A ∪ {ϕ(t)}¯ andT0 =T ∪ {ϕ¯v ¬ϕ}, whereϕ¯is a fresh predicate name. It is easy to see that ifP0has no explanation, then eitherP has no ex- planation as well, or, all the explanations forP must containϕ(t). For⊆-NECESSITY, observe thatϕ(t)occurs in all explanation forP iffϕ(t)occurs in all⊆-minimal ex- planations for P. Thus ⊆-NECESSITY can be decided in polynomial time using our algorithm forNECESSITY.
(HARDNESS) The lower-bound can be proved through a logspace reduction from
EXISTENCEto non-NECESSITY, that is, deciding whether there exists a solution to a QAP that does not contain the given assertion. LetP = hO, q,cibe a QAP with q a CQ, we buildhP, αisuch thatP has a solution iffhP, αiis a negative instance to
NECESSITY. Letα = A(o), for some fresh conceptA and constantonot occurring inP. Now, assumeP has a solutionU. By Proposition 1, we know that there exists a solutionU0toP containing only concept and role names fromO. Hence,A(o)6∈ U0, since concept nameAis not in the ontology. Therefore, one can conclude thatA(o)is not a necessary assertion toP.
The other direction is straightforward. ut
Let us now show the complexity of necessity under the≤preference order.
Theorem 1. For DL-LiteAontologies,≤-NECESSITYis PNPk -complete.
Proof. (MEMBERSHIPonly,HARDNESSin [8]) Let’s assume a QAPP =hO, q,ciand an assertionα. By the use of canonical explanations, we know that the size mof the largest solution toPcorresponds to the size of the largest CQ inq. Observe that(P, α) is a negative instance of≤-NECESSITYiff there is an0 ≤i ≤msuch that(a)P has an explanationU with|U | =iandα 6∈ U, and(b)U is≤-minimal. Thus, we use an auxiliary problemSIZE-OUT, which is to decide given a tuplehP0, α0, n0i, whereP0is a QAP,α0 is an assertion, andn0 is an integer, whether there exists an explanationU0 forP0 such that|U0|=n0andα0 6∈ U0. Furthermore, the problemNO-SMALLERis to decide given a tuplehP0, n0iof a QAP and an integer whether there is no explanation U0 forP0 such that|U0| < n0. Observe thatSIZE-OUTis in NP, whileNO-SMALLER
is in CONP. Take the tupleS = hA0, B0, . . . , Am, Bmi, where(a) Ai = (P, α, i), for all 0 ≤ i ≤ m, and (b) Bi = (P, i), for all 0 ≤ i ≤ m. Due to the above observation,αoccurs in all≤-minimal explanationsU forPiff for all0≤i≤m, one of the following holds:(i)Aiis a negative instance ofSIZE-OUT, or(ii)Biis a negative instance ofNO-SMALLER. Note thatScan be built in polynomial time in the size of the input, while the positivity of the instances inScan be decided by making2mparallel calls to an NP oracle. Thus we obtain membership in PNPk . ut
4.3 Complexity of-RELEVANCE
A domain user faced with a negative answer to a query may ask herself whether, the absence of a certain ABox assertionαin the ontology is related with the lack of the tuple in the results. That is, she would like to know whetherαoccurs in some explanation to QAPP.
Proposition 4. For DL-LiteAontologies,RELEVANCEisPTIME-complete.
Proof. (MEMBERSHIP) We assume a QAPP = hO, q,ciwithO = hT,Aiand an assertionφ(t). We now provide a reduction fromRELEVANCEtoEXISTENCE. We con- structO0=hT,A0i, whereA0=A∪φ(t). Then,Phas an explanationUwithφ(t)∈ U iff there exists an explanation toP0 = (O0, q,c). This is because, any explanation to P0can be extended by addingφ(t). It is simple to see that any such explanation is an
explanation toP, as well. Finally, ifP0 does not admit explanations, then φ(t)is a source of inconsistency inP.
(HARDNESS) Hardness can again be proved with a reduction fromEXISTENCEin a
similar way as done in Section 4.2. ut
Let us now tackle the problem of⊆-RELEVANCE, for which we recall the FOL- rewritability of query answering inDL-LiteA. Given an ABoxA, letDB(A)be the fol- lowing interpretation:(i)∆DB(A)is the set of constants occurring inA,(ii)ADB(A)= {o∈∆DB(A)|A(o)∈ A}, for each atomic conceptA, and(iii)PDB(A)={ho, o0i ∈
∆DB(A)|P(o, o0)∈ A}, for each atomic roleP.
Proposition 5 ([7]). Let PerfectRef(q,T)be the perfect reformulation ofq w.r.t. T, which is a UCQ obtained by applying the rewrite rules given in [7]. Then,cert(q,O) = S
r∈PerfectRef(q,T)ans(r, DB(A)).
In other words, the certain answers to a UCQ can be computed by rewriting the query into a FOL query to be evaluated over the ABox.
Theorem 2. For DL-LiteAontologies,⊆-RELEVANCEisΣ2P-complete.
Proof. (MEMBERSHIP) The membership inΣ2Pis clear from Algorithm 1, which works as follows. An explanation U containing φ(t) is non-deterministically computed by guessing an instantiation of a subquery inPerfectRef(q(c),T), whereAnonis a set of fresh ABox individuals (see Proposition 1). Let HAS-SUBEXPLsolve the problem of deciding whether a solution U has a subset which is itself an explanation. The prob- lem can be easily proved to be in NP. Then, the algorithm checks the complement of HAS-SUBEXPLin order to assure that none of the subsets of U is itself an expla- nation, from which it follows that φ(t) is ⊆-relevant. Checking the complement of
HAS-SUBEXPLrequires the power of aCONP machine. For this reason, the algorithm is solvable in non-deterministic polynomial time by a TM with an NP oracle.
(HARDNESS) We prove it by a reduction from theΣ2P-complete problem co-CERT3COL
[15] (see also [4]). An instance of co-CERT3COLis given by a graphG= (V, E)with vertices V = {0, . . . , n−1} such that every edge is labeled with a disjunction of two literals over the Boolean variables{p(i,j) | i, j < n}.Gis a positive instance if there is a truth value assignmentt to the Boolean variables such that the grapht(G)
Algorithm 1
INPUT: QAPP=hq,O,ciand ABox assertionφ(t) OUTPUT: yes iffφ(t)is relevant toP
1: Guessqi∈ {q1, . . . , qn}=q
2: Guess the derivation of one rewritingr(c)inPerfectRef(qi(c),T) 3: Guess a set of atomsU ⊆at(r)
4: Guess a mappingπfromV(q)to constants inDB(A)andAnon 5: Check that(T,A ∪ U)6|=⊥, whereUis the instantiation ofUthroughπ.
6: Check thatφ(t)∈ Uandπis a match forr(c)overDB(A ∪ U) 7: Check thatHAS-SUBEXPL(P,U) =no.
obtained fromGby including only those edges whose label evaluates to true undert is not 3-colorable. Assume an instanceGof co-CERT3COL. We show how to build in polynomial time a QAPPG = h(TG,AG), qG,cGiand an ABox assertion αG such that:Gis a positive instance of co-CERT3COL iffαG is⊆-relevant forPG. We use an empty TBox and a Boolean query, thusTG = ∅ andcG = hi. The query qG is a UCQ qG = qe1 ∪ · · · ∪qek ∪q0, where {e1, . . . , ek} = E, each qei is an atomic queryqei()← Wei(x, y), andq0is defined as follows. AssumeB ={t, f} to be the set of truth values. The queryq0has the following atoms for each edgee= (i, j)inE:
(a)B(xi),Re(xi, ye),Re(ye, xj),B(xj), and(b)P(ye, zpi),Api(zpi),Wpi(zpi, z0p
i), wherepi ∈ {p1, p2}andp1, p2are the first and the second proposition in the labeling ofe, respectively. The queryq0simply incorporatesGtogether with the disjunctions on the edges. Observe that if two edges have the same proposition in their label, then this will be reflected inq0by some shared variableszpi.
To buildAG we use individualscp andc¬p for the truth value of proposition p.
Intuitively, the truth value ofpwill be determined by eitherAp(cp)orAp(c¬p)being in the update. Assume a tuplet=he, v1, v2, a, bi, wheree∈E,{v1, v2} ⊆ B, anda, b are individuals. Letp1, p2be the first and the second propositions ofe. Fori∈ {1,2}
andvi=t, letli=piifpiis positive andli=¬piotherwise. Similarly, fori∈ {1,2}
andvi =f, letli =¬pi ifpi is positive andli =piotherwise. Then, the ABoxA(t) consists of the assertionsRe(a, dT),Re(dT, b),P(dT, cl1)andP(dT, cl2)depending on the boolean values in input.
The ABoxAGis the union of the following ABoxes:
(A1) A(he, v, v0, ai, aji)for alle∈E,v, v0∈ B,0≤i, j≤2, andi6=j;
(A2) A(he, f, f, ai, aii)for alle∈E, and0≤i≤2;
(A3) A(he, v, v0, b, bi)for alle∈E,v, v0 ∈ B;
(A4) The ABox{B(a0), B(a1), B(a2)};
(A5) The assertionsWp(cp, c¬p)andWp(c¬p, cp)for all propositions.
LetαG = B(b). It is not too difficult to see that Gis a positive instance of co-
CERT3COL iff there exists an ⊆-explanation U to P such that αG ∈ U. Basically, definitions (A1)-(A3) encode a triangular structureTin which edges inGthat evaluate to false according to a given truth assignment can be mapped on any edge ofT, reflexive edges included. If an edge ofGevaluates to true, then it must be mapped to one of the non-reflexive edges. This ensures that ifGcan be mapped toT under truth assignment t, thent(G)is 3-colorable. Instead, definitions (A4)-(A5) define a cyclic structureC into which any graphGcan be embedded. It has to be noted that the nodeb is not asserted to be a member ofB, henceqGcannot be mapped there directly with any truth assignment. We see this more formally next:
“⇒” Suppose there is a truth assignmenttsuch thatt(G)is not 3-colorable. Let U ={B(b)}∪U1, whereU1={Ap(cp)|t(p) =t}∪{Ap(c¬p)|t(p) =f}. It remains to argue thatU is a⊆-explanation toP. It is not hard to see thatU is an explanation.
IndeedqGmatches already in the ABox obtained by point (A3) (hint: sinceB(b)∈ U, we matchqGby mapping all variables ofqG to (interpretation of)b). Suppose there is a smaller updateU0 ⊂ U. Observe thatU1⊆ U0. This is because for all propositionsp, the symbolApdoes not occur inAGbut does occur inqG. Then,U \ {B(b)}must be an
update. If this is the case, thenqGcan be matched inAGwithout the ABox from (A3), i.e. in the triangle part. Thent(G)is 3-colorable, which contradicts the assumption.
“⇐” LetU be a⊆-minimal explanation containingB(b). Due to the presence of qe1 ∪. . . ∪qek inqG and the assumption, the role Wp cannot occur in U for any proposition p. SinceU is an explanation, by the definition of q0 and (A5) we have thatAp(cp) ∈ U orAp(c¬p) ∈ U for all propositionsp. Since for any propositionp we have thatApoccurs inqGwith one and only variablezp, we know that exactly one ofAp(cp) ∈ U andAp(c¬p) ∈ U holds. Due to the atomsWp(zp, zp0)inqG, we also have that individuals of the formcpandc¬pare the only ones that can get anAplabel.
Consider the assignmenttdefined as follows:t(e) = tifAp(cp) ∈ U, andt(e) = f ifAp(c¬p)∈ U. It is not difficult to argue thatt(G)is not 3-colorable and thusGis a positive instance of co-CERT3COL. Indeed, ift(G)was 3-colorable,Qshould be map- pable into the triangle part obtained in (A1)-(A3). ThenU \ {B(b)}would be a smaller
update, which would mean a contradiction. ut
Note that the above lower bound holds already for empty TBoxes.
Proposition 6. For DL-LiteAontologies,≤-RELEVANCEis PNPk -complete.
Proof. (MEMBERSHIPonly,HARDNESSin [8])≤-RELEVANCEcan be tackled in a way similar to≤-NECESSITY. In fact, the algorithm described in Theorem 1 can be modi- fied in order to solve this problem. LetSIZE-IN solve the following problem: given a tuplehP, α, ni, where P is a QAP,αan assertion, andnan integer, decide whether there exists an explanationU, with|U |=nandα∈ U. Then, we change the positivity condition of the≤-NECESSITYalgorithm as follows:αoccurs in some≤-minimal ex- planationsU forPiff for somei,0≤i≤m, it holds that: (i)Aiis a positive instance of SIZE-IN, and (ii)Bi is a positive instance of NO-SMALLER.It is easy to see that
SIZE-INis solvable in NP, hence the whole problem is again in PNPk . ut
5 Conclusions
In this paper we have provided the formalization of a new problem, namely the explana- tion of negative answers to user queries over ontologies. A tuple is said to be a negative answer, if the user expects it to be part of cert(q,O)but the tuple is actually not. In our framework, an explanation consists of an ABox that when added to the ontology leads the negative answer to be returned in the results of the query. We define various problems that help us in characterizing the complexity of finding explanations, such as the existence of explanations and relevance/necessity of assertions. We further con- sider a minimality criterion to be applied over explanations, such as subset-minimal and minimum-size preference orders. Within this framework, we provide a characterization of the computational complexity of the various problems for the DLDL-LiteA.
Future work includes studying the application of this framework to other lightweight description logics, starting with theEL-family. We would also like to investigate the problem in the case where the ontology signature and the explanation signature may be different. That is, the signature over which explanations can be constructed is restricted only to a subset of the ontology signature [2].
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