Adding Weight to DL-Lite

Adding Weight to DL-Lite

A. Artale,¹ D. Calvanese,¹R. Kontchakov,² and M. Zakharyaschev²

1 KRDB Research Centre Free University of Bozen-Bolzano

I-39100 Bolzano, Italy [email protected]

2 School of Comp. Science and Inf. Sys.

Birkbeck College London WC1E 7HX, UK {roman,michael}@dcs.bbk.ac.uk

1 Introduction

Description logics (DLs) have recently been used to provide access to large amounts of data through a high-level conceptual interface, which is of relevance in several application contexts, notably data integration and ontology-based data access. Besides the traditional reasoning services of knowledge base satisfiability and instance checking, a further important service in that context is that of an- swering complex database-like queries by fully taking into account the axioms in the TBox and the data stored in the ABox. The key property for such an approach to be viable in practice is the efficiency of query evaluation, in partic- ular for conjunctive queries and, more generally, for positive existential queries (this class of queries includes unions of conjunctive queries) [1]. To address these needs, the DL-Lite family of description logics has been proposed and investi- gated in [6–8, 16], with the aim of identifying a class of DLs that could capture typical conceptual modeling formalisms, such as UML class diagrams and ER models, and for which query answering could be performed efficiently in terms of data complexity. The data complexity measure presupposes that only the size of the ABox is considered as variable while the sizes of the TBox and the query are regarded as fixed. Such a measure is important since in the typical application contexts we are interested in here, the size of the data stored in the ABox largely dominates that of the TBox and the query. As shown in [6–8, 16], for the logics of the DL-Lite family, a (union of) conjunctive queries posed over a TBox can be answered by rewriting it into a new union of conjunctive queries that has

‘compiled in’ the assertions in the TBox, and that can simply be evaluated (by a relational engine) over the ABox to produce the correct answer to the original query. In other words, it was shown that such logics enjoy FO rewritability [7, 8], and so belong to the complexity class FO in terms of descriptive complexity theory, and to the classAC⁰in terms of circuit complexity [10].

Successive work [3] has shown that some of the nice computational properties ofDL-Litelogics can be preserved, even when they are extended with additional constructs used in conceptual modeling. In particular, it was proved in [3] that the data complexity of answering positive existential queries stays in AC⁰ for the logicDL-Lite^N_hornwhich allows conjunctions on the left-hand side of concept inclusions as well as arbitrary number restrictions. Moreover, the same data complexity bound holds also for satisfiability and instance checking in the logic

Language Combined complexity Data complexity

Satisfiability Instance checking Query answering DL-Lite^(RN)_core NLogSpace inAC⁰ inAC⁰ DL-Lite^(RN)_horn P ^≤[Th.1] inAC⁰ inAC^{0 ≤[Th.3]}

DL-Lite^(RN)_krom NLogSpace^≤[Th.1] inAC⁰ coNP^≥[18]

DL-Lite^(RN)_bool NP ^≤[Th.1] inAC^0≤[Th.2] coNP≤[14, 13, 9]

Table 1.Combined and data complexity.

DL-Lite^N_bool which allows full Booleans as concept constructs. (Note that these results hold only under the unique name assumption. DL-Lite logics without this assumption are investigated in [4].)

One aim of this paper is to extend DL-Lite^N_horn, DL-Lite^N_bool and their fragments with a number of new constructs without spoiling their computa- tional properties. The resulting logic is called DL-Lite^(RN)_horn. Another aim is to present explicit (exponential) FO rewritings of positive existential queries over DL-Lite^(RN)_horn KBs. The constructs we add to our logics are as follows: (i) role inclusions, (ii) qualified number restrictions, and (iii) role disjointness, symme- try, asymmetry, reflexivity, and irreflexivity constraints. Needless to say that when adding (i) and (ii), we have to restrict the interaction of these constructs with number restrictions (otherwise, even the logicDL-Lite^R,F_core with extremely primitive concept inclusions, but with unrestricted role inclusions and global functionality constraints isExpTime-complete for combined complexity andP- complete for data complexity [11].) This will be done by generalizing the ideas of [16]. Our main tool for dealing withDL-Litelogics is embedding into the one- variable fragmentQL¹of first-order logic without equality and function symbols, which seems to be a natural logic-based characterization of theDL-Lite logics.

The complexity results obtained in this paper are summarized in Table 1.

2 DL-Lite

and its fragments

We start by defining the description logicDL-Lite^(RN_bool⁾, the most expressive of our logics, which subsumes, in particular, all members of theDL-Litefamily [6–8].

The language ofDL-Lite^(RN_bool⁾containsobject names a₀, a₁, . . .,concept names A₀, A₁, . . ., and role names P₀, P₁, . . .. Complex concepts C and roles R are defined as follows:

B ::= ⊥ | Ai | ≥q R, R ::= Pi | P_i⁻, C ::= B | ¬C | ≥q R.C | C1uC2,

whereqis a positive integer. The concepts of the formB will be calledbasic. A DL-Lite^(RN)_bool TBox, T, is a finite set ofconcept inclusions (CIs, for short), role inclusions, androle constraints of the form:

C1 v C2, R1 v R2, Dis(R1, R2), Irr(Pk), and Ref(Pk).

We write inv(R) for P_k⁻ if R =Pk, and forPk ifR =P_k⁻. Denote byv^∗_T the reflexive and transitive closure of{(R, R⁰),(inv(R),inv(R⁰))|RvR⁰∈ T }. Say

thatR⁰ is aproper sub-role of R inT ifR⁰v^∗_T RandR6v^∗_T R⁰. We impose the following syntactic conditions onDL-Lite^(RN_bool⁾ TBoxesT (cf. DL-Lite_A [16]):

(inter) ifRhas a proper sub-role inT thenT contains nonegative occurrences¹ of number restrictions≥q Ror ≥qinv(R) withq≥2;

(exists) T may contain only positive occurrences of ≥q R.C, and if ≥q R.C occurs in T then T does not contain negative occurrences of ≥q⁰R or

≥q⁰inv(R), forq⁰ ≥2.

It follows that no TBox can contain both a functionality constraint≥2Rv ⊥ and an occurrence of≥q R.C, for some q≥1 and some roleR.

AnABox,A, is a finite set of assertions of the form: Ak(ai),Pk(ai, aj) and

¬Pk(ai, aj). Taken together,T andAconstitute theDL-Lite^(RN_bool⁾knowledge base (KB, for short)K= (T,A).

As usual in description logic, an interpretation, I = (∆^I,·^I), consists of a nonemptydomain ∆^I and an interpretation function ·^I that assigns to each object namea_ian elementa^I_i ∈∆^I, to each concept nameA_ia subsetA^I_i ⊆∆^I, and to each role name P_i a binary relation P_i^I ⊆ ∆^I×∆^I. In this paper, we adopt theunique name assumption(UNA): a^I_i 6=a^I_j, for alli6=j, and refer the reader to [4] for results on theDL-Litelogics without UNA.

The role and concept constructs are interpreted inI in the standard way.

We will also use standard abbreviations such as > = ¬⊥, ∃R = (≥1R) and

≤q R = ¬(≥q+ 1R). The satisfaction relation |= is also standard; we only mention here that I |=Dis(R1, R2) iff R^I₁ ∩R^I₂ =∅ (R1 andR2 are disjoint), I |= Irr(Pk) iff (x, x) ∈/ P_k^I for all x∈ ∆^I (Pk is irreflexive), I |=Ref(Pk) iff (x, x)∈P_k^Ifor allx∈∆^I(Pkisreflexive). Note that symmetric and asymmetric role constraints can be regarded as syntactic sugar in this language: Sym(Pk) and Asym(Pk) can be equivalently replaced with P_k⁻ v Pk and Dis(Pk, P_k⁻), respectively (extending a TBox with P_k⁻ v Pk cannot violate (inter) as P_k⁻ is not a proper sub-role of Pk). A KB K = (T,A) is said to be satisfiable (or consistent) if there is an interpretation,I, satisfying all the members ofT and A. In this case we writeI |=K (as well asI |=T andI |=A) and say thatI is a model of K.

It is to be emphasized that such constructs as role constraints and qual- ified number restrictions are used in conceptual modeling and also belong to the OWL 2 proposal; moreover, as we show, adding them does not affect the computational complexity of our logics.

Similarly to classical logic, we adopt the following definitions: a TBoxT is – aDL-Lite^(RN_horn⁾ TBox if its CIs are of the formB₁u · · · uB_k v B (theB_i

andB are basic concepts and, by definition, the empty conjunction is>);

– a DL-Lite^(RN_krom⁾ TBox² if its CIs are of the formB1 v B2, B1 v ¬B2 or

¬B1 v B2;

1 An occurrence of a concept on the right-hand (resp., left-hand) side of a concept inclusion is called negative if it is in the scope of an odd (resp., even) number of negations¬; otherwise the occurrence is calledpositive.

2 The Krom fragment of first-order logic consists of all formulas in prenex normal form whose quantifier-free part is a conjunction of binary clauses.

– aDL-Lite^(RN)_core TBox if its CIs are of the formB₁ v B₂ orB₁ v ¬B₂. As B1 v ¬B2 is equivalent to B1uB2 v ⊥, core TBoxes can be regarded as both Krom and Horn TBoxes. We note here that a conceptC occurring inT in some ≥q R.C can be a conjunction of any concepts allowed on the right-hand side of concept inclusions in the respective language.

3 DL-Lite in the Light of First-Order Logic

Our main aim in this section is to prove the upper combined complexity bounds for reasoning in DL-Lite^(RN_bool⁾ and its fragments and develop the technical tools we need to investigate the data complexity of query answering inDL-Lite^(RN)_horn.

For aDL-Lite^(RN)_bool KBK= (T,A), denote byrole^±(K) the set of role names occurring inKand their inverses and byob(A) the set of object names occurring in A. LetQ^R_T be the set of natural numbers containing 1 and all the numerical parameters qsuch that ≥q R or≥q R.C occurs inT. Note that|Q^R_T| ≥2 if T contains a functionality constraint forR. Our main result in this section is:

Theorem 1. (i)Satisfiability of DL-Lite^(RN_bool⁾ KBs is NP-complete; (ii) satisfi- ability ofDL-Lite^(RN_horn⁾ KBs is P-complete;and (iii) satisfiability ofDL-Lite^(RN)_krom andDL-Lite^(RN)_core KBs is NLogSpace-complete.

Let us consider first the sub-language ofDL-Lite^(RN)_bool without qualified num- ber restrictions and role constraints, which will be required for purely technical reasons; we denote it byDL-Lite^(RN)_bool ⁻. In Section 4, we will also useDL-Lite^(RN)_horn⁻.

Let K = (T,A) be a DL-Lite^(RN)_bool ⁻ KB and let Id be a distinguished role name, which will be used to simulate theidentity relationrequired for encoding the role constraints. We assume that either K does not contain Id at all or satisfies the following conditions:

(Id1) Id(ai, aj)∈ A iff i=j, for allai, aj∈ob(A), (Id2)

> v ∃Id, Id⁻vId ⊆ T, andQ^Id_T =Q^Id_T⁻ ={1},

(Id3) Idis only allowed in role inclusions of the formId⁻ vId andIdvR.

We assume, without loss of generality, thatQ^R_T ⊆Q^R_T⁰ wheneverRv^∗_T R⁰ (for if this is not the case we can always introduce the missing numbers inQ^R_T⁰, e.g., by adding⊥ v ≥q R⁰ to the TBox).

We present now a reduction of the satisfiability problem for DL-Lite^(RN)_bool ⁻ KBs to satisfiability of first-order formulas with one variable, or QL¹-formulas.

With every object name a_i ∈ ob(A) we associate the individual constant a_i of QL¹ and with every concept name A_i the unary predicate A_i(x) from the signature of QL¹. For each role R ∈ role^±(K), we introduce |Q^R_T|-many fresh unary predicatesE_qR(x),forq∈Q^R_T. The intended meaning of these predicates is as follows: for a role name P_k, E₁P_k(x) and E₁P_k⁻(x) represent the domain and range of Pk, respectively; more generally, for each q ∈ Q^R_T, EqPk(x) and EqP_k⁻(x) represent the sets of points withat leastqdistinctPk-successors andat

leastqdistinctP_k-predecessors, respectively. We writeinv(E_qR)(x) forE_qP_k⁻(x) if R = P_k, and for E_qP_k(x) if R = P_k⁻. Additionally, for every pair of roles P_k, P_k⁻ ∈role^±(K), we take two fresh individual constantsdp_k anddp⁻_k ofQL¹, which will serve as ‘representatives’ of the points from the domain and range of Pk, respectively (provided that it is not empty). Denote the set of all thosedpk

anddp⁻_k bydr(K) and write inv(dr) fordp⁻_k ifR=Pk, and for dpk ifR=P_k⁻. By induction on the construction of conceptC we define theQL¹-formulaC^∗:

⊥^∗ = ⊥, (Ai)^∗ = Ai(x), (≥q R)^∗ = EqR(x), (¬C)^∗ = ¬C^∗(x), (C₁uC₂)^∗ = C₁^∗(x)∧C₂^∗(x).

For every role R∈role^±(K), we need twoQL¹-formulas:

εR(x) = E1R(x)→inv(E1R)(inv(dr)), δR(x) = ^

q,q⁰∈Q^R_T, q⁰>q q⁰>q⁰⁰>qfor noq⁰⁰∈Q^R_T

Eq⁰R(x)→EqR(x) .

FormulaεR(x) says that if the domain ofR is not empty then its range is not empty either: it contains the constantinv(dr), the ‘representative’ of the domain ofinv(R). The meaning of δR(x) should be obvious. For a KBK, we define

K^‡e = ∀xh

T^∗R(x) ∧ ^

R∈role^±(K)

ε_R(x)∧δ_R(x)i

∧ A^‡e, where

T^∗R(x) = ^

C1vC2∈T

C₁^∗(x)→C₂^∗(x)

∧ ^

RvR⁰∈T or inv(R)vinv(R⁰)∈T

^

q∈Q^R_T

EqR(x)→EqR⁰(x) ,

A^‡e = ^

Ak(ai)∈A

A_k(a_i) ∧ ^

R(a,a⁰)∈Cl^e_T(A)

E_q^e

R,aR(a) ∧ ^

¬Pk(ai,aj)∈A

(¬P_k(a_i, a_j))^⊥e,

Cl^e_T(A) ={R⁰(ai, aj)|R(ai, aj)∈ A, Rv^∗_T R⁰},³ q_R,a^e is the maximum number in Q^R_T such that there are q^e_R,a many distinct ai with R(a, ai)∈ Cl^e_T(A), and (¬P_k(a_i, a_j))^⊥e =⊥ifP_k(a_i, a_j)∈Cl^e_T(A) and>otherwise. Note that the size ofK^‡e is linear in the size ofK,no matter whether the numerical parameters are coded in unary or in binary. The following lemma is an analogue of [3, Theorem 1]

(for the proof see [4]):

Lemma 1. A DL-Lite^(RN_bool⁾⁻ KB K is satisfiable iff the QL¹-sentence K^‡e is satisfiable.

It should be clear that the translation·^‡e can be computed inNLogSpacefor combined complexity. Indeed, this is trivial for the first conjunct of K^‡e. To computeA^‡e, we first need to be able to check, given a roleRand a pair of ob- jectsa_i,a_j, whetherR(a_i, a_j)∈Cl^e_T(A) and second, givenR(a, a⁰)∈Cl^e_T(A), to

3 We slightly abuse notation and writeR(ai, aj)∈ Ato indicate thatPk(ai, aj)∈ A ifR=Pk, orPk(aj, ai)∈ AifR=P_k⁻.

computeq^e_R,a. TheR(a_i, a_j)∈Cl^e_T(A) test can be done by anon-deterministic algorithm using space logarithmic in |role^±(K)| (see, e.g., the NLogSpacedi- rected graph reachability problem [12]). The following algorithm computesq^e_R,a: setq= 0 and then enumerate all object namesaiinAincrementingqeach time R(a, ai) ∈ Cl^e_T(A); stop if q = maxQ^R_T or the end of the object name list is reached. The resultingq^e_R,ais the maximum number in Q^R_T not exceedingq.

As follows from the proof of Lemma 1, for aDL-Lite^(RN_bool⁾⁻ KB K= (T,A), every modelMofK^‡e induces a model IM ofKwith the following properties:

(abox) For allai, aj∈ob(A), (a^I_i^M, a^I_j^M)∈R^IM iffR(ai, aj)∈Cl^e_T(A).

(uniq) The object namesa∈ob(A) induce a partitioning of∆^IM into disjoint labeled trees Ta = (Ta, Ea, `a) with nodes Ta, edges Ea, root a<sup>IM</sup>, and a labeling function`_a:E_a →role^±(K)\ {Id,Id⁻}.

(cp) There is a function cp:∆^IM →ob(A)∪dr(K) such that cp(a^IM) =afor a∈ob(A), andcp(w) =dr, for roleR such that w⁰ ∈Ta, (w⁰, w)∈Ea and

`a(w⁰, w) =inv(R), for somea∈ob(A).

(iso) For each R∈role^±(K), all labeled subtrees generated by w∈∆^IM with cp(w) =drare isomorphic.

(con) For all basic conceptsBinKandw∈∆^IM,w∈B^IMiffM|=B^∗[cp(w)].

(role) For every role name P_k, includingId, P_k^IM=

(a^I_i^M, a^I_j^M)|R(ai, aj)∈ A, Rv^∗_T Pk ∪

(w, w)|Idv^∗_T Pk ∪ (w, w⁰)∈Ea|a∈ob(A), `a(w, w⁰) =R, Rv^∗_T Pk . Such a model will be called anuntangled model of K (the untangled model ofK induced by M, to be more precise). It should be pointed out that there are two main distinguishing features of untangled models forDL-Lite^(RN)_bool KBs: (i) there are at most |ob(A)|+|role^±(K)| different types of points in them, and (ii) al- though two points may be connected by a set of rolesΩ, one can always select R∈Ωsuch thatΩis an upward closure of{R}underv^∗_T, provided that one of the points is not fromob(A).

The following lemma reduces satisfiability ofDL-Lite^(RN_bool⁾KBs to satisfiability ofDL-Lite^(RN)_bool ⁻ KBs (for the proof see [4, Lemma 5.17]):

Lemma 2. For every DL-Lite^(RN)_bool KB K⁰ = (T⁰,A⁰), one can construct (in linear time and logarithmic space)aDL-Lite^(RN)_bool ⁻ KB K= (T,A) such that

– every untangled modelIM ofK is a model of K⁰, provided that

there are noR₁(a_i, a_j), R₂(a_i, a_j)∈Cl^e_T(A)withDis(R₁, R₂)∈ T⁰, and there is no R(a_i, a_i)∈Cl^e_T(A) withIrr(R)∈ T⁰;

– every modelI⁰ of K⁰ gives rise to a modelI ofK based on the same domain asI⁰ and such that I agrees withI⁰ on all symbols from K⁰.

Theorem 1 follows from Lemmas 2 and 1, the observation thatK^‡e is aQL¹- formula, for aDL-Lite^(RN)_bool KB, a universal HornQL¹-formula, for aDL-Lite^(RN)_horn KB, and a universal Krom QL¹-formula, for a DL-Lite^(RN)_krom KB, and the com- plexity results for the respective fragments ofQL¹[15, 5].

For the data complexity the following result is proved in [4, Section 6]:

Theorem 2. The satisfiability and instance checking problems forDL-Lite^(RN)_bool KBs are inAC⁰ for data complexity.

4 FO Rewritability of Query Answering

In this section we study the data complexity of query answering overDL-Lite^(RN)_horn KBs. We assume that all concept and role names of a KB occur in its TBox and write role^±(T) and dr(T) instead of role^±(K) and dr(K), respectively. Denote byBcon(T) the set of basic concepts occurring inT (i.e., concepts of the form Aand≥q R, for a concept nameAoccurring in T,R∈role^±(T) andq∈Q^R_T).

A positive existential query q(x) is a first-order formula ϕ(x) constructed by means of conjunction, disjunction and existential quantification from atoms of the from A(t₁) and P(t₁, t₂), where t₁, t₂ are terms taken from the list of variablesy0, y1, . . . and object namesa0, a1, . . .. The free variables ofϕare called distinguished variablesofq. Anassignmentain∆^Iis a function associating with each variableyan elementa(y) of∆^I. We writea^I,a_i =a^I_i andy^I,a=a(y). For K = (T,A), say that a tuplea of object names from Ais a certain answer to q(x) w.r.t.K, and write K |= q(a), if I |=q(a) whenever I |= K. The query answering problem is, given K, a query q(x) and a ⊆ ob(A), decide whether K |=q(a). Our main result in this section is the following:

Theorem 3. The positive existential query answering problem forDL-Lite^(RN)_horn KBs is in AC⁰ for data complexity.

Proof. Suppose that we are given aconsistentDL-Lite^(RN_horn⁾KBK⁰= (T⁰,A⁰) and a positive existential query in prenex form q(x) = ∃yϕ(x,y), y = y₁, . . . , y_k, in the signature of K⁰. Consider the DL-Lite^(RN_horn⁾⁻ KBK = (T,A) provided by Lemma 2. The untangled models ofK produce exactly the same answers asK⁰: Lemma 3. For every tuple a of object names in K⁰, we have K⁰ |= q(a) iff I |=q(a)for all untangled modelsI of K.

Next we show that, as K^‡e is a universal Horn sentence, it is enough to consider just one special untangled model I0 of K. Let M0 be the minimal Herbrand model ofK^‡e. We remind the reader (see, e.g., [2, 17]) thatM0can be constructed by taking the intersection of all Herbrand models for K^‡e, that is, of all models based on the domainΛ=ob(A)∪dr(T). It follows that

M0|=B^∗[c] iff K^‡e |=B^∗(c), for allc∈Λ andB∈Bcon(T).

Denote byI₀the untangled model ofKinduced byM₀and its domain by∆^I0. By Lemma 1 with(con) and(cp),

a^I_i⁰ ∈B^I0 iff K |=B(a_i), for alla_i∈ob(A) andB∈Bcon(T). (1) For eachR∈role^±(T), by Lemma 1, ifR^I0 6=∅thenM0|= (∃R)^∗[dr] and thus, (T ∪ {∃R v ⊥},A) is not satisfiable, whence R^I 6= ∅, for all models I of K.

Moreover, ifR^I0 6=∅ then

w∈B^I0 iff K |=∃RvB, for allw∈∆^I0 withcp(w) =dr, (2)

wherecp:∆^I0 →Λis the function provided by(cp).

Lemma 4. If I0|=q(a)thenI |=q(a)for all untangled models I of K.

Proof. SupposeI |=K. Asq(a) is a positive existential sentence, it is enough to construct a homomorphism h: I0 → I. By (uniq), ∆^I0 is partitioned into trees Ta, for a ∈ ob(A). Define the depth of w ∈ ∆^I0 to be the length of the path in the respective tree from its root tow. Denote byWm the set of points of depth ≤m; in particular, W₀ = {a^I0 | a ∈ ob(A)}. We construct h as the union of homomorphisms h_m:W_m → I, m≥0, such thath_m+1(w) =h_m(w), for all w ∈ W_m. For the basis of induction, set h₀(a^I_i⁰) = a^I_i, for a_i ∈ ob(A);

h₀ is a homomorphism by (1) and(abox). For the induction step, supposeh_m has been defined. Ifv∈W_m, seth_m+1(v) =h_m(v). Otherwise,v∈W_m+1\W_m. By (uniq), there is a uniqueu∈Wmwith (u, v)∈Ea, for somea∈ob(A). Let

`a(u, v) = S. By(cp),cp(v) =inv(ds) and, by (role), u∈(∃S)^I0. As hm is a homomorphism,hm(u)∈(∃S)^I, whence there isw∈∆^I with (hm(u), w)∈S^I. Set hm+1(v) = w. As (∃inv(S))^I0 6= ∅, cp(v) = inv(ds) and w ∈ (∃inv(S))^I, by (2), if v ∈B^I0 thenw∈B^I, for allB ∈Bcon(T). It remains to show that (w, v)∈R^I0implies (hm+1(w), hm+1(v))∈R^I. By(role), we have (w, v)∈R^I0, forw∈Wm+1 andv∈Wm+1\Wm, just in two cases: eitherw∈Wm+1\Wm, and thenw=vwithIdv^∗_T R, orw∈Wm, and thenw=uwithSv^∗_T R. In the former case, (hm+1(v), hm+1(v))∈ Id^I ⊆ R^I. In the latter case, (u, v)∈ S^I0;

hence (hm+1(u), hm+1(v))∈S^I⊆R^I. ut

Our next lemma shows that to check whetherI₀|=q(a) it suffices to consider only the set of points Wm₀ of depth ≤ m0 in ∆^I, for somem0 that does not depend on|A|(see [4, Lemma 7.4] for the proof):

Lemma 5. Letm₀=k+|role^±(T)|. IfI0|=∃yϕ(a,y)then there is an assign- ment a₀ in W_m0 (i.e., a₀(y_i)∈W_m0 for alli)such that I0|=^a0 ϕ(a,y).

To complete the proof of Theorem 3, we encode the problem ‘K |= q(a)?’

as a model checking problem for first-order formulas. We fix a signature that contains a unary predicate Ak(x) for each concept nameAk and a binary pred- icatePk(x, y) for each role namePk, and then represent the ABox AofK as a first-order model AA with domainob(A): for eachai, aj∈ob(A),

A_A|=A_k[a_i] iff A_k(a_i)∈ A and A_A|=P_k[a_i, a_j] iff P_k(a_i, a_j)∈ A.

Now we define a first-order formulaϕ_T,q(x) in the above signature such that (i) ϕ_T,q(x) depends onT andqbut not onA, and (ii)A_A|=ϕ_T,q(a) iffI0|=q(a).

To simplify the presentation, we denote bye(T) the extension ofT with:

– ≥q⁰Rv ≥q R, for allR∈role^±(T) andq, q⁰∈Q^R_T withq⁰ > q, and – ≥q Rv ≥q R⁰, for allq∈Q^R_T andRvR⁰∈ T orinv(R)vinv(R⁰)∈ T. It follows from the definition of·^‡e and Lemma 1 that, for a Horn concept inclu- sionC vB, we have T |=C vB iff (C^∗(x)→B^∗(x)) is a logical consequence of{(C_i^∗(x)→B_i^∗(x))|CivBi∈e(T)}.

We begin by defining formulasψ_B(x),B∈Bcon(T), that describe the types of the elements ofob(A) in the model I₀in the following sense (cf. (1)):

AA|=ψB[ai] iff a^I_i⁰ ∈B^I0, forB ∈Bcon(T) andai∈ob(A). (3) These formulas are defined as the ‘fixed-points’ of sequences ψ_B⁰(x), ψ¹_B(x), . . . of formulas with one free variable, where

ψ⁰_B(x) =







A(x), ifB =A,

∃y1. . .∃yq

h ^

1≤i<j≤q

(yi6=yj)∧ ^

1≤i≤q

R^T(x, yi)i

, ifB =≥q R,

ψⁱ_B(x) =ψ⁰_B(x) ∨ _

B₁u···uBkvB∈e(T)

ψ_Bⁱ⁻¹

1 (x)∧ · · · ∧ψⁱ⁻¹_B

k(x)

, fori≥1,

and R^T(x, y) = W

P_kv^∗_TRPk(x, y) ∨W

P_k⁻v^∗_TRPk(y, x). Clearly, if there is an i such that, for allB∈Bcon(T),ψ_Bⁱ(x)≡ψ_Bⁱ⁺¹(x), i.e., everyψ_Bⁱ (x) is equivalent to ψ_Bⁱ⁺¹(x) in first-order logic, thenψ_Bⁱ(x)≡ψ^j_B(x) for allB∈Bcon(T),j ≥i.

The minimum suchidoes not exceedN =|Bcon(T)|, so we setψB(x) =ψ^N_B(x).

Next we introduce sentencesθB,dr, for B ∈Bcon(T) and dr∈ dr(T), that describe the types of the elements ofdr(T) inI0 in the following sense (cf. (2)):

A_A|=θB,dr iff w∈B^I0, for each (some)w∈∆^I0 with cp(w) =dr. (4) These sentences are defined similarly to the ψB(x): namely, for each B ∈ Bcon(T) anddr∈dr(T), we consider a sequenceθ⁰_B,dr, θ_B,dr¹ , . . . by taking

θ⁰_B,dr=ρ⁰_B,dr, θⁱ_B,dr=ρⁱ_B,dr ∨ _

B₁u···uBkvB∈e(T)

θ_Bⁱ⁻¹

1,dr∧ · · · ∧θ_Bⁱ⁻¹

k,dr

, fori≥1,

whereρⁱ_B,dr=⊥, for allB 6=∃Randi≥0, and ρ⁰_∃R,dr =∃x ψ_∃inv(R)(x) and ρⁱ_∃R,dr= _

ds∈dr(T)

θ_∃inv(R),dsⁱ⁻¹ , fori≥1.

We haveθ_B,drⁱ ≡θ_B,drⁱ⁺¹ for somei≤M =N· |role^±(T)|. So, letθB,dr =θ_B,dr^M . Now we consider the directed graphGT = (VT, ET), where VT is the set of all equivalence classes [R], [R] ={R⁰|Rv^∗_T R⁰, R⁰ v^∗_T R}, such that∃Ris not empty insome model of T, andE_T is the set of all pairs ([R_i],[R_j]) such that (path) T |=∃inv(Ri)v ≥q Rj and either inv(Ri)6v^∗_T Rj or q≥2, andRj has no proper sub-role satisfying(path). We have ([Ri],[Rj])∈E_T iff, for any ABoxA⁰, whenever the minimal untangled modelI0 of (T,A⁰) contains a copywofinv(dr_i⁰), forR⁰_i∈[Ri], thenwis connected to a copy ofinv(dr_j⁰), for R_j⁰ ∈[Rj], by all relationsS withRjv^∗_T S. LetΣT,m₀ be the set of all paths in GT of length≤m0(as in Lemma 5):

Σ_T,m₀= ε ∪

([R1], . . . ,[Rn])|1≤n≤m0& ([Rj],[Rj+1])∈E_T, forj < n .

Forσ, σ⁰∈Σ_T,m0 andR∈role^±(T), we writeσ→^R σ⁰ if (i)σ=σ⁰ andIdv^∗_T R or (ii)σ.[S] =σ⁰ or (iii) σ=σ⁰.[inv(S)], for someS withS v^∗_T R.

LetΣ_T^k_,m

0be the set of allk-tuples of the formσ= (σ₁, . . . , σ_k),σ_i∈Σ_T,m0. Intuitively, when evaluating the query∃yϕ(x,y) over I0, each bound, or non- distinguished, variable y_i is mapped to a point w in W_m0. However, the first- order model A_A does not contain the points fromW_m0\W₀, and to represent them, we use the following ‘trick.’ By(uniq), every pointwinW_m0 is uniquely determined by the pair (a, σ), wherea^I0 is the root of the treeTacontainingw, andσis the sequence of labels`a(u, v) on the path froma^I0 tow. It follows from the unraveling procedure and (path) that σ∈Σ_T,m₀. So, in the formula ϕ_T,q

we are about to define we assume that theyi range over W0 and represent the first component of the pairs (a, σ), whereas the second component is encoded in the ith member of σ (these yi should not be confused with the yi in the original queryq, which range over all ofWm₀). In order to treat arbitrary terms toccurring inϕ(x,y) in a uniform way, we sett^σ=ε, ift=a∈ob(A) ort=xi, andt^σ =σi, ift=yi(the distinguished variablesxiand the object namesaare mapped toW₀ and do not require the second component of the pairs).

Given an assignmenta0 inWm0 we denote bysplit(a0) the pair (a,σ), where a is an assignment inA_Aandσ= (σ₁, . . . , σ_k)∈Σ^k_T,m

0 are such that – for each distinguished variablexi,a(xi) =awitha^I0=a0(xi);

– for each bound variable yi, a(yi) = a and σi = ([R1], . . . ,[Rn]), n ≤ m0, witha^I0 being the root of the tree containinga0(yi) and R1, . . . , Rn being the sequence of labels`a(u, v) on the path from a^I0 toa0(yi).

Not every pair (a,σ), however, corresponds to an assignment in Wm₀ because some paths in σ may not exist in our I0: G_T represents possible paths in all models for the fixed TBoxT and varying ABox. As follows from the unraveling procedure, a point in Wm₀ \W0 corresponds to a ∈ ob(A) and σ ∈ ΣT,m₀, σ= ([R], . . .), iffahas not enoughR-witnesses inA:AA|=¬ψ⁰_{≥q R}[a]∧ψ≥q R[a], for some q ∈ Q^R_T. Thus, for every (a,σ) with σ = (σ1, . . . , σk), there is an assignmenta0in Wm₀ withsplit(a0) = (a,σ) iffA_A|=^aη^σ(y), where

η^σ(y) = ^

1≤i≤k σi6=ε

_

q∈Q^Ri_T

¬ψ⁰_{≥q Ri}(yi) ∧ ψ_{≥q Ri}(yi)

and eachRi, for 1≤i≤kwithσi 6=ε, is such that σi= ([Ri], . . .).

We define now, for everyσ∈Σ_T^k_,m

0, concept nameAand role nameR, A^σ(t) =

(ψA(t), ift^σ =ε,

θ_A,inv(ds), ift^σ =σ⁰.[S], for someσ⁰ ∈Σ_T,m₀,

R^σ(t₁, t₂) =







R^T(t₁, t₂), ift^σ₁ =t^σ₂ =ε,

(t1=t2), ift^σ₁ →^R t^σ₂ and eithert^σ₁ 6=εort^σ₂ 6=ε,

⊥, otherwise.

We claim that, for each assignmenta₀ inW_m0, (a, σ) =split(a₀) and termt, I₀|=^a0A(t) iff A_A|=^aA^σ(t), for all concept namesA, (5) I0|=^a0 R(t1, t2) iff AA|=^aR^σ(t1, t2), for all rolesR. (6) For A(a), A(x_i) or A(y_i) with σ_i = ε the claim follows from (3). For A(y_i) with σ_i = σ⁰.[S], by (cp), we have cp(a(y_i)) =inv(dr), for some R ∈ [S]; the claim then follows from (4). ForR(y_i1, y_i2) withσ_i1 =σ_i2=ε, the claim follows from (abox). Let us consider the case of R(y_i1, y_i2) with σ_i2 6= ε: we have a₀(y_i2)∈/W₀ and thus, by(role),I₀|=^a0R(y_i1, y_i2) iff

– a0(yi₁),a0(yi₂) are in the same treeTa, fora∈ob(A), i.e.,A_A|=^a(yi₁ =yi₂), – and either (a0(yi₁),a0(yi₂))∈Ea and then `a(a0(yi<sub>1</sub>),a0(yi<sub>2</sub>)) =S for some Sv<sup>∗</sup><sub>T</sub> R, or (a0(yi<sub>2</sub>),a0(yi<sub>1</sub>))∈Ea and then`a(a0(yi₂),a0(yi₁)) =Sfor some inv(S)v^∗_T R, ora0(yi₁) =a0(yi₂) and thenIdv^∗_T R, i.e.,σi₁

→R σi₂. Other cases are similar and left to the reader.

Finally, letϕ^σ(x,y) be the result of attaching the superscript^σto each atom ofϕand

ϕ_T,q(x) = ∃y _

σ∈Σ^k_T,m

0

ϕ^σ(x,y) ∧η^σ(y) .

As follows from (5)–(6), for every assignmenta0inW_m0, we haveI0|=^a0 ϕ(x,y) iffA_A|=^aϕ^σ(x,y) for (a, σ) =split(a₀). For the converse direction notice that, ifA_A|=^a η^σ(y) then there is an assignmenta₀ inW_m0 withsplit(a₀) = (a,σ).

Clearly,A_A|=ϕ_T,q(a) iffI₀ |=q(a), for every tuple a. We also note that, for every pair of tuplesaandbof object names inob(A),ϕ^σ(a,b) is a positive existential sentence with inequalities, and so domain-independent.⁴ It is also easily seen that, for each b, η^σ(b) is domain-independent. It follows from the minimality ofI0thatϕ_T,q(a) is domain-independent, for each tupleaof object names inob(A).

Finally, we note that the resulting query contains ≤ m^k·(k+m) disjuncts, wherem=|role^±(T)|andkis the number of bound variables inq. ut We also remark that although extending the DL-Lite^(RN_α ⁾ languages with transitive roles does not change the combined complexity of reasoning, it does change the data complexity: instance checking and satisfiability inDL-Lite^(RN_α ⁾, forα∈ {core,krom,horn,bool}, areNLogSpace-complete (rather than inAC⁰) and query answering over DL-Lite^(RN)_horn and DL-Lite^(RN_core⁾ KBs is NLogSpace- complete for data complexity (see Section 5.4 of [4]).

4 A queryq(x) is said to be domain-independent in caseAA|=^aq(x) iffA|=^aq(x), for eachAsuch that the domain ofAcontainsob(A), the active domain ofAA, and A^A=A^AA andP^A=P^AA, for all concept and role namesAandP.

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