DL-Lite with Attributes and Sub-Roles (Extended Abstract)

DL-Lite with Attributes and Sub-Roles (Extended Abstract)

A. Artale,¹ A. Ib´a˜nez Garc´ıa,¹ R. Kontchakov,²and V. Ryzhikov¹

1 KRDB Research Centre Free University of Bozen-Bolzano, Italy

{lastname}@inf.unibz.it

2 Dept. of Comp. Science and Inf. Sys.

Birkbeck College, London, UK roman@dcs.bbk.ac.uk

1 Introduction

TheDL-Litefamily of description logics has recently been proposed and investi- gated in [5–7] and later extended in [1, 8, 3]. The relevance of theDL-Litefamily is witnessed by the fact that it forms the basis of OWL 2 QL, one of the three profiles of OWL 2 (http://www.w3.org/TR/owl2-profiles/). According to the offi- cial W3C profiles document, the purpose of OWL 2 QL is to be the language of choice for applications that use very large amounts of data.

This paper extends theDL-Lite languages of [3] by relaxing the restriction on the interaction between cardinality constraints (N) and role inclusions (or hierarchies, H). We also introduce a new family of languages, DL-Lite^{HN A}_α , α∈ {core,krom,horn,bool}, extendingDL-Lite withattributes (A).

The notion of attributes, borrowed from conceptual modeling formalisms, introduces a distinction between (abstract) objects and application domain val- ues, and consequently, between concepts (sets of objects) and datatypes (sets of values), and between roles (relating objects to objects) and attributes (relating objects to values). The advantage of the presented languages overDL-Lite_A[8]

is that the range restrictions for attributes can belocal (and not onlyglobal as inDL-Lite_A). Indeed,DL-Lite^{HN A}_α has a possibility to express concept inclusion axioms of the form C v ∀U.T, for an attribute U and its datatypeT. In this way, we allow re-use of the very same attribute associated to different concepts with different range restrictions. For example, we can say that employees’ salary is of typeInteger, researchers’ salary is in the range 35,000–70,000 (enumeration type) and professors’ salary in the range 55,000–100,000—while both researchers and professors are employees. Note that local attributes are strictly more expres- sive than global ones. For example, concept disjointness (or unsatisfiability) can be inferred just from datatype disjointness for the same (existentially qualified) attribute. SinceDL-Litelanguages have been proved useful in capturing concep- tual data models [8, 4, 2], the extension with attributes, as presented here, will generalize their use even further.

We aim at establishing computational complexity of knowledge base satisfi- ability in these new DLs. In particular we prove the following results:

1. We can relax the restrictions presented in [3] limiting the interaction between sub-roles and number restrictions without increasing the complexity of rea- soning as far as the problem is limited to TBox satisfiability checking. As

for KB satisfiability, the presence of the ABox should be taken into account if we want to preserve the complexity results.

2. The introduction of local range restrictions for attributes is for free for the languagesDL-Lite^{N A}_bool,DL-Lite^{N A}_hornandDL-Lite^{N A}_core.

2 The Description Logic DL-Lite

The language of DL-Lite^{HN A}_bool contains object names a0, a1, . . ., value names v1, v2, . . ., concept names A0, A1, . . ., role names P0, P1, . . ., attribute names U0, U1, . . ., and datatype names T0, T1, . . .. Complex roles R and concepts C are defined below:

R ::= Pi | P_i⁻,

B ::= > | ⊥ | Ai | ≥q R | ≥q Ui

C ::= B | ¬C | C1uC2,

whereqis a positive integer. The concepts of the formBare calledbasic concepts.

A DL-Lite^{HN A}_bool TBox, T, is a finite set of concept, role, attribute and datatypeinclusion axioms of the form:

C₁vC₂, Cv ∀U. T, R₁vR₂, U vU⁰, T vT⁰, TuT⁰v ⊥, and anABox,A, is a finite set of assertions of the form:

Ak(ai), ¬Ak(ai), Pk(ai, aj), ¬Pk(ai, aj), Uk(ai, vj) and Tk(vj).

We standardly abbreviate≥1Rand≥1U by∃Rand∃U, respectively. Absence of an attribute (i.e.,Cv ∀U.⊥) can be expressed asCu ∃U v ⊥.

Together, a TBoxT and an ABoxAconstitute theDL-Lite^{HN A}_bool knowledge base (KB)K = (T,A). In the following, we denote byrole(K) andatt(K) the sets of role and attribute names occurring inK, respectively;role^±(K) denotes the set{Pk, P_k⁻|Pk ∈role(K)}.

Semantics.As usual in description logic, an interpretation, I = (∆^I,·^I), con- sists of a nonemptydomain ∆^Iand an interpretation function·^I. The interpre- tation domain ∆^I is the union of two non-empty disjoint sets: the domain of objects ∆^I_O and the domain of values ∆^I_V. We assume that all interpretations agree on the semantics assigned to each datatypeT_i, as well as on each constant v_i. In particular,T_i^I=val(T_i)⊆∆^I_V is the set of values of datatypeT_i, and each v_iis interpreted as one specific value, denotedval(v_i), i.e.,v_i^I=val(v_i)∈val(T_i).

Furthermore, ·^I assigns to each object name a_i an element a^I_i ∈ ∆^I_O, to each concept name A_k a subset A^I_k ⊆ ∆^I_O of the domain of objects, to each role name P_k a binary relation P_k^I ⊆∆^I_O×∆^I_O over the domain of objects, and to each attribute name Uk a binary relation U_k^I ⊆∆^I_O×∆^I_V. We adopt here the unique name assumption (UNA): a^I_i 6=a^I_j, for alli6=j. The role and concept

constructs are interpreted inI in the standard way:

(P_k⁻)^I = {(y, x)∈∆^I_O×∆^I_O|(x, y)∈P_k^I}, (inverse role)

>^I = ∆^I_O, (domain of objects)

⊥^I = ∅, (the empty set)

(≥q R)^I =

x∈∆^I_O |]{y|(x, y)∈R^I} ≥q , (at leastq R-successors) (≥q U)^I =

x∈∆^I_O |]{v|(x, v)∈U^I} ≥q , (at leastq U-attributes) (∀U. T)^I =

x∈∆^I_O | ∀v.(x, v)∈U^I →v∈T^I , (attribute value restriction)

(¬C)^I = ∆^I_O\C^I, (not inC)

(C1uC2)^I = C₁^I∩C₂^I, (both inC1and inC2) where]X is the cardinality ofX. Thesatisfaction relation |= is also standard:

I |=C₁vC₂ iff C₁^I ⊆C₂^I, I |=R₁vR₂ iff R^I₁ ⊆R^I₂, I |=T1vT2 iff T₁^I ⊆T₂^I, I |=U1vU2 iff U₁^I ⊆U₂^I, I |=T1uT2v ⊥ iff T₁^I∩T₂^I=∅, I |=Pk(ai, aj) iff (a^I_i, a^I_j)∈P_k^I,

I |=A_k(a_i) iff a^I_i ∈A^I_k, I |=¬P_k(a_i, a_j) iff (a^I_i, a^I_j)∈/P_k^I I |=¬Ak(ai) iff a^I_i ∈/ A^I_k, I |=Uk(ai, vj) iff (a^I_i, v_j^I)∈U_k^I,

I |=T_k(v_j) iff v^I_j ∈T_k^I.

A KBK= (T,A) is said to besatisfiable (orconsistent) if there is an interpre- tation, I, satisfying all the members of T and A. In this case we write I |=K (as well asI |=T andI |=A) and say thatI is amodel of K (ofT andA).

2.1 Fragments of DL-Lite^{HN A}_bool

We consider various syntactical restrictions on the language of DL-Lite^{HN A}_bool along two axes: (i) the Boolean operators (bool) on concepts, (ii) the role and attribute inclusions (H). Similarly to classical logic, we adopt the following def- initions. A TBox T will be called a Krom TBox¹ if its concept inclusions are restricted to:

B1 v B2, B1 v ¬B2 and ¬B1 v B2, (Krom) (here and below all the B_i andB are basic concepts). T will be called aHorn TBox if its concept inclusions are restricted to:

l

k

Bk v B. (Horn)

Finally, we callT a core TBox if its concept inclusions are restricted to:

B₁ v B₂ and B₁uB₂ v ⊥. (core)

1 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.

As B₁ v ¬B₂ is equivalent to B₁uB₂ v ⊥, core TBoxes can be regarded as sitting in the intersection of Krom and Horn TBoxes. In this paper we study the following logics, forα∈ {core,krom,horn,bool}:

DL-Lite^{HN A}_krom ,DL-Lite^{HN A}_horn ,DL-Lite^{HN A}_core are the fragments ofDL-Lite^{HN A}_bool with Krom, Horn, and core TBoxes, respectively;

DL-Lite^HN_α is the fragment ofDL-Lite^{HN A}_α without attributes and datatypes;

DL-Lite^{N A}_α is the fragment of DL-Lite^{HN A}_α without role and attribute inclu- sions.

As shown in [3], reasoning inDL-Lite^HN_α is already rather costly (ExpTime- complete) due to the interaction between role inclusions and number restrictions.

However, both of these constructs turn out to be useful for the purposes of conceptual modeling. By limiting their interplay one can get languages with a better computational properties [8, 3]. Before presenting such limitations we need to introduce some notation. For a roleR, letinv(R) =P_k⁻ ifR=P_k and inv(R) =Pk if R =P_k⁻. Given a TBoxT we denote byv^∗_T the reflexive and transitive closure of the relation{(R, R⁰),(inv(R), inv(R⁰))|RvR⁰ ∈ T }.We say that R ≡^∗_T R⁰ iff R v^∗_T R⁰ andR⁰ v^∗_T R. Say that R⁰ is a proper sub-role of R in T ifR⁰ v^∗_T R andR 6v^∗_T R⁰. A proper sub-role R⁰ of R is said to be a direct sub-role ofR if there is no other proper sub-roleR⁰⁰ ofR such thatR⁰ is a proper sub-role ofR⁰⁰; the set of direct sub-roles ofRis denoted asdsub_T(R).

The languageDL-Lite^(HN_α ⁾[3] is the result of imposing the following syntactic restriction onDL-Lite^HN_α TBoxesT:

(inter) ifRhas a proper sub-role inT thenT contains no negative occurrences of number restrictions≥q Ror≥qinv(R) withq≥2

(an occurrence of a concept on the right-hand (left-hand) side of a concept inclusion is called negative if it is in the scope of an odd (even) number of negations ¬; otherwise it is calledpositive). We will formulate two alternative versions of restriction(inter).

Definition 1. Given a TBoxT and a roleR∈role^±(T), we define the following parameters:

ubound(R,T) = min {∞} ∪ {q−1|q≥2 and ≥q R occurs negatively in T } , lbound(R,T) = max {0} ∪ {q| ≥q Roccurs positively inT }

, rank(R,T) = max lbound(R,T),P

R⁰∈dsubT(R)rank(R⁰,T) ,

rank(R,A) = max{0}∪{n|Ri(a, ai)∈ A, Riv^∗_T R, for distinct a1, . . . , an} . Consider the languages obtained from DL-Lite^HN_α by imposing one of the fol- lowing two restrictions:

(inter1) for every R ∈ role^±(T), if R has a proper sub-role in T then ubound(R,T)≥rank(R,T);

(inter2) for every R ∈ role^±(T), if R has a proper sub-role in T then ubound(R,T)≥rank(R,T) + max{1,rank(R,A)}.

language (inter)[3] (inter1) (inter2) non-restrict.

DL-Lite^HN_core NLogSpace[3] NLogSpace[Th.1]

DL-Lite^HN_horn PTime[3] PTime[Th.1]

DL-Lite^HN_krom NLogSpace[3] ≥NP[Th.2]

NLogSpace[Th.1] ExpTime[3]

DL-Lite^HN_bool NP[3] NP[Th.1]

DL-Lite^{HN A}_core NLogSpace[Th.3] NLogSpace[Th.3]

DL-Lite^{HN A}_horn PTime[Th.3] PTime[Th.3]

DL-Lite^{HN A}_krom NP[Th.4] ≥NP[Th.2]

NP[Th.4] ExpTime

DL-Lite^{HN A}_bool NP[Th.3] NP[Th.3]

DL-Lite^{N A}core NLogSpace[Th.3]

DL-Lite^{N A}horn PTime[Th.3]

DL-Lite^{N A}_krom NA NA NA

NP[Th. 4]

DL-Lite^{N A}_bool NP[Th.3]

Table 1: Complexity ofDL-Litelogics (NA = Non-Applicable).

These new restrictions are in some way weaker than (inter) and, for ex- ample, allow for the specialization of functional roles: KB K = (T,A) with T ={≥2Rv ⊥, R1vR2, R2vR}, and A={R(a, b), R1(a1, b1), R2(a2, b2)}

does not satisfy (inter), but it satisfies both (inter1) and (inter2). Finally, the above restrictions can also be applied to sub-attributes in the languages DL-Lite^{HN A}_α . Table 1 summarizes the obtained complexity results (with numer- ical parametersqcoded in binary).

3 Reasoning in DL-Lite

In this section, we investigate the complexity of deciding KB satisfiability in languagesDL-Lite^HN_α under the restrictions(inter1)and(inter2), respectively.

We adapt the proof presented in [3], where a DL-Lite^HN_bool KB K = (T,A) is encoded into a sentence K^‡e in the one-variable first-order logic QL¹. We use a slightly longer but simpler encoding. Everya_i∈ob(A) is associated to the individual constanta_iofQL¹, and every concept nameA_ito the unary predicate A_i(x). For each concept≥q RinKwe introduce a fresh unary predicateE_qR(x).

For each role name P_k ∈role^±(K), two individual constants dp_k and dp⁻_k are introduced, as representatives of the objects in the domain and range of P_k, respectively. The encodingC^∗ of a conceptC is defined inductively:

⊥^∗=⊥, (A_i)^∗=A_i(x), (≥q R)^∗=E_qR(x),

>^∗=>, (¬C)^∗=¬C^∗(x), (C₁uC₂)^∗=C₁^∗(x)∧C₂^∗(x).

TheQL¹ sentence encoding the knowledge baseK is defined as follows:

K^‡e = ∀x

T^∗(x) ∧ T^R(x) ∧ ^

R∈role^±(K)

R(x) ∧ δR(x)

∧ A^‡e.

FormulasT^∗(x), theδ_R(x), forR∈role^±(K), andT^R(x) encode the TBoxT: T^∗(x) = ^

C₁vC₂∈T

C₁^∗(x)→C₂^∗(x)

, δR(x) = ^

q,q⁰∈Q^R_T, q⁰>q

Eq⁰R(x)→EqR(x) ,

T^R(x) = ^

Rv^∗_TR⁰

^

q∈Q^R_T

EqR(x)→EqR⁰(x) ,

whereQ^R_T contains 1, allqsuch that≥q Roccurs inT and allQ^R_T⁰, forR⁰v^∗_T R.

SentenceA^‡e encodes the ABoxA:

A^‡e= ^

Ak(ai)∈A

Ak(ai) ∧ ^

¬Ak(ai)∈A

¬Ak(ai) ∧ ^

a,a⁰∈ob(A) R⁰v^∗_TR, R⁰(a,a⁰)∈A

Eq^e_R,aR(a) ∧ ^

¬Pk(ai,aj)∈A R(a_i,a_j)∈A, Rv^∗_TP_k

⊥,

whereq^e_R,ais the maximum number inQ^R_T such that there areq^e_R,amany distinct a_i withR_i(a, a_i)∈ A and R_i v^∗_T R. For each R ∈role^±(K), we also need the following formula expressing the fact that the range ofRis not empty whenever its domain is non-empty:

R(x) =E1R(x)→inv(E1R(dr)),

whereinv(E₁R(dr)) isE₁P_k⁻(dp⁻_k) ifR=P_k andE₁P_k(dp_k) ifR=P_k⁻. Lemma 1. A DL-Lite^HN_bool knowledge base under restriction(inter2) is satisfi- able iff the QL¹-sentenceK^‡e is satisfiable.

Proof. (Sketch) The only challenging direction is (⇐). To prove it, we adapt the proofs of Theorem 5.2 and Lemma 5.14 in [3]. The idea of the proof is to construct aDL-Lite^HN_bool interpretationI, fromM, the minimal Herbrand model ofK^‡e. We denote the interpretations of unary predicates P and constantsa ofQL¹ in M byP^M anda^M, respectively. LetD=ob(A)∪ {dp_k, dp⁻_k |P_k∈role(K)}be the domain ofM. ThenI = (∆^I,·^I) is defined inductively:∆^I =S∞

m=0Wm, such that W0 is the setD0 =ob(A), and for every ai ∈ ob(A), a^I_i = a^M_i . Each set Wm+1,m≥0, is constructed by adding toWmfresh copies of certain elements fromD\ob(A). The extensionsA^I_k of concept namesAk are defined by taking A^I_k ={w∈∆^I |M|=A^∗_k[cp(w)]}, (1) wherecp(w) is the elementd∈D of whichwis a copy.

The interpretation for each rolePk, is defined inductively asP_k^I =S∞ m=0P_k^m, whereP_k^m⊆Wm×Wm, along with the construction of∆^I. The initial interpre- tation for each role name Pk is defined as follows:

P_k⁰={(a^M_i , a^M_j )∈W0×W0|R(ai, aj)∈ AandRv^∗_T Pk}. (2) For everyR∈role^±(K), therequiredR-rankr(R, d) ofd∈Dis defined by taking r(R, d) = max {0} ∪ {q∈Q^R_T |M|=EqR[d]}

. Theactual R-rank rm(R, w) of a point w∈∆^I at stepmis

rm(R, w) =

(]{w⁰∈Wm+1|(w, w⁰)∈P_k^m+1}, ifR=Pk, ]{w⁰∈Wm+1|(w⁰, w)∈P_k^m+1}, ifR=P_k⁻.

Assume thatW_mandP_k^m,m≥0, have been already defined. LetW_m+1=∅and P_k^m+1=∅, for each role nameP_k. If we hadr_m(R, w) =r(R, cp(w)), for each role Randw∈W_m, then the interpretation we need would be constructed. However, the actual rank of some points could still be smaller than the required rank. We cure these defects by addingR-successors for them. Note that the ‘curing’ process for a givenw andR, not only increases the actualR-rank ofw, but also all its R⁰-ranks, for all R v^∗_T R⁰. At this point we adapt the construction in [3] to obtain the interpretationI we are intending. For eachPk ∈role(K), we consider two sets of defects inP_k^m: Λ^m_k ={w∈Wm\W_m−1|rm(Pk, w)< r(Pk, cp(w))}

andΛ^m−_k ={w∈Wm\Wm−1|rm(P_k⁻, w)< r(P_k⁻, cp(w))}.

In each equivalence class [R] = {S | S ≡^∗_T R} we select a single role, a representative. Let G = (Rep^∗_T, E) be a directed graph such that Rep^∗_T is the set of representatives and (R, R⁰)∈EiffRis a proper sub-role ofR⁰. Clearly,G is a directed acyclic graph and so, by a topological sort, one can assign to each representative a unique number smaller than the number of all its descendants in G. We use the ascending total order induced onGwhen choosing an elementP_k inRep^∗_T, and extend in that wayW_mandP_k^mtoW_m+1andP_k^m+1, respectively.

(Λ^m_k ) Letw∈Λ^m_k, q=r(P_k, cp(w))−r_m(P_k, w),d=cp(w). There isq⁰ ≥q >0 with M |= E_q⁰P_k[d]. Then, M |= E₁P_k[d] and M |= E₁P_k⁻[dp⁻_k]. In this case we takeq fresh copies w₁⁰, . . . , w⁰_q of dp⁻_k, add them to W_m+1 and for each 1≤i≤q, setcp(w⁰_i) =dp⁻_k, add the pairs (w, w⁰_i) to eachP_j^m+1 with P_k v^∗_T P_j and the pairs (w⁰_i, w) to eachP_j^m+1 withP_k⁻v^∗_T P_j (note that by adding pairs toP_j^m+1 we change its the actual rank);

(Λ^m−_k ) This rule is the mirror image of (Λ^m_k):Pk anddp⁻_k are replaced withP_k⁻ anddpk, respectively.

We need to show that, for allw∈∆^I and all≥q RinT,

(a1) if≥q Roccurs positively inT thenM|=EqR[cp(w)] impliesw∈(≥q R)^I; (a2) if≥q Roccurs negatively inT thenw∈(≥q R)^I impliesM|=EqR[cp(w)].

Consider firstw∈W0. It should be clear that actualR-rank ofw r0(R, w) ≤ rank(R,A) +P

R⁰∈dsubT(R)rank(R⁰,T)

and so, by (inter2), the total number of R-successors before we cure the de- fects does not exceed ubound(R,T). If ubound(R,T) = ∞ then there are no negative occurrences of≥q R with q≥2 and, although may have r_m(R, w)≥ r(R, cp(w)) after curing the defects of R, both (a₁)and (a₂)hold. Otherwise, we have ubound(R,T) + 1∈Q^R_T and so, by (inter2), maxQ^R_T >rank(R,T) + rank(R,A), whence r₀(R, w)<maxQ^R_T. So, asr(R, cp(w))≤lbound(R,T) and lbound(R,T)<ubound(R,T) <maxQ^R_T, after curing the defect, we will have rm(R, w) = r(R, cp(w)), for all m >0, and both (a1) and(a2)hold. The case withw∈Wm₀\Wm₀−1, form0>0 is similar, only now

rm₀(R, w) ≤ 1 +P

R⁰∈dsubT(R)rank(R⁰,T).

Finally, we show thatI |=ϕfor each ϕ∈ K. Forϕ=A_k(a_i),ϕ=¬A_k(a_i) the claim is by the definition ofA^I_k. Forϕ=¬P_k(a_i, a_j), we have (a_i, a_j)∈P_k^I iff (a_i, a_j)∈P_k⁰ iff R(a_i, a_j)∈ A and Rv^∗_T P_k. By induction on the structure of concepts and (a1) and (a2), one can show that I |= C1 v C2 whenever M |= ∀x(C₁^∗(x) → C₂^∗(x)), for each ϕ = C1 v C2. Finally, I |= ϕ holds by definition in caseϕ=R1vR2∈ T.

Theorem 1. Under restriction (inter2), checking KB satisfiability is NP- complete in DL-Lite^HN_bool, PTime-complete in DL-Lite^HN_horn and NLogSpace- complete in both DL-Lite^HN_krom andDL-Lite^HN_core.

We now consider the case where the restriction(inter1) is imposed on the interaction between sub-roles and number restrictions. In presence of an ABox, (inter2)restricts the number ofR-successors in the ABox, which appears to be a strong constraint on the instances of the ABox. On the other hand, the less restrictive condition(inter1), which does not impose any bound onR-successors in the ABox, does not come for free, as shown by the following theorem:

Theorem 2. Under restriction(inter1), checking KB satisfiability isNP-hard even in DL-Lite^HN_core.

Proof. We show that graph 3-colorability can be reduced to KB satisfiability.

Let G= (V, E) be a graph with verticesV and edges E and {r, g, b} be three colors. Consider the following KB K = (T,A) with role names vi and w and object nameso, r, g, band thexi, for each vertexvi∈V:

T ={≥(|V|+ 4)wv ⊥} ∪ {vivw, B1v ∃vi, B2u ∃v_i⁻v ⊥ |vi∈V} ∪ {∃v_i⁻u ∃v_j⁻v ⊥ |(v_i, v_j)∈E},

A={B1(o), w(o, r), w(o, g), w(o, b)} ∪ {w(o, xi), B2(xi)|vi∈V}.

It can be shown thatKis satisfiable iffGis 3-colorable.

4 Reasoning with Attributes

In this section we study the effect of extendingDL-Litewith attributes. In par- ticular, we show that for the Bool, Horn and core cases the addition of attributes does not change the complexity of KB satisfiability.

Theorem 3. KB satisfiability is NP-complete in DL-Lite^{N A}_bool,PTime-complete inDL-Lite^{N A}_horn and NLogSpace-complete inDL-Lite^{N A}_core.

Under restriction (inter2), checking KB satisfiability is NP-complete in DL-Lite^{HN A}_bool , PTime-complete in DL-Lite^{HN A}_horn and NLogSpace-complete in DL-Lite^{HN A}_core .

Proof. (Sketch) We encode a DL-Lite^{HN A}_α KB K = (T,A) in a QL¹ sentence K^‡a in a way similar to the translation used in Lemma 1. Denote byval(A) the set of all value names that occur inA. Similarly to roles, we define the setsQ^U_T

of natural numbers for all occurrences of ≥q U (including sub-attributes). We need a unary predicateE_qU(x), for each attribute nameU andq∈Q^U_T, denoting the set of objects with at least q values of attributeU. We also need, for each attribute nameU and each datatypeT, a unary predicatesU T(x), denoting all objects that may have attribute U values only of datatype T. Following this intuition, we extend·^∗ by the following two statements:

(≥q U)^∗=E_qU(x) and (∀U.T)^∗=U T(x).

TheQL¹ sentence encoding the KBK is defined as follows:

K^‡a = K^‡e ∧ ∀x

T^U(x)∧ ^

U∈att(K)

δU(x)∧α¹_U(x)∧α²_U(x)

∧ A^‡a ∧ A^‡2a,

where K^‡e is as before,T^U(x),δU(x) and A^‡a are similar toT^R(x),δR(x) and A^‡e, but rephrased for attributes and their inclusions. The new types of ABox assertions require the following formula:

A^‡2a= ^

U_k(a_i,v_j)∈A

^

datatypeT

U T(a_i)→T v_j

∧ ^

T(v_j)∈A

T v_j,

whereT vj is a propositional variable for each datatypeT and eachvj∈val(A).

The two additional formulas, α¹_U(x) and α²_U(x), capturing datatype inclusions and disjointness constraints are:

α¹_U(x) = ^

TvT⁰∈T

U T(x)→U T⁰(x) ,

α²_U(x) = ^

TuT⁰v⊥∈T

U T(x)∧U T⁰(x)∧E1U(x)→ ⊥

∧ ^

v∈val(A)

T v∧T⁰v→ ⊥ .

We would like to note here that the formulaα²_U(x) for disjoint datatypes demon- strates a subtle interaction between attribute range constraints∀U.T and mini- mal cardinality constraints∃U.

We show thatKis satisfiable iff theQL¹-sentenceK^‡ais satisfiable. For (⇐), letM|=K^‡a. We construct a modelI= (∆^I_O∪∆^I_V,·^I) ofKsimilarly to the way we proved Lemma 1 but this time datatypes will have to be taken into account:

let∆^I_O be defined inductively as before and∆^I_O =val(A)∪V. The set V will be constructed starting from val(A) in order to ‘cure’ the attribute successors as follows. For each datatypeT and each attributeU, let

T⁰={v∈val(A)|M|=T v} and U⁰={(a, v)|U(a, v)∈ A}.

For every attribute U ∈ att(K), we can define the required U-rank r(U, d) of d∈D and the actualU-rankr₀(R, w) ofw∈∆^I_Oas before, treatingU as a role name (the only difference is that there will be only one step, and so, the actual rank is needed only for step 0). We can also consider the equivalence relation induced by the sub-attribute relation in T, then we can choose representatives

and a linear order on them respecting the sub-attribute relation of T. We can start from the smaller attributes and ‘cure’ their defects. Let w ∈ ∆^I_O and q = r(U,cp(w))−r₀(U, w) > 0. Take q fresh elements v₁, . . . , v_q, add those fresh values to V, add pairs (w, v1), . . . ,(w, vq) toU⁰ and addv1, . . . , vq toT⁰ for each datatype T with M |= U T[cp(w)]. Let U^I and T^I be the resulting relations. Now, it can be shown that if M|=K^‡a thenI |=ϕ for everyϕ∈ K.

We only note here that fresh valuesvjcannot be added to two disjoint datatypes T andT⁰ because of formulaα²_U(x).

Now, given a KB with a Bool or Horn TBox,K^‡a is a universal one-variable formula or a universal one-variable Horn formula, respectively, which immedi- ately gives theNPandPTimeupper complexity bounds for the Bool and Horn fragments. The NLogSpace upper bound for KBs with core TBoxes is not so straightforward becauseα²_U(x) is not a binary clause. In this case we note that K^‡ais still a universal one-variable Horn formula and therefore,K^‡ais satisfiable iff it is true in the ‘minimal’ model. The minimal model can be constructed in the bottom-to-top fashion by using only positive clauses of K^‡a (i.e., clauses of the form∀x(B₁(x)∧ · · · ∧B_k(x)→H(x))) and then checking whether the negative clauses ofK^‡a (i.e., clauses of he form∀x(B₁(x)∧ · · · ∧B_k(x)→ ⊥)) hold in the constructed model. By inspection of the structure of K^‡a, one can see that all its positive clauses are in fact binary, and therefore, whether an atom is true in its minimal model or not can be checked inNLogSpace.

It is of interest to note that the complexity of KB satisfiability increases in the case of Krom TBoxes:

Theorem 4. KB satisfiability is NP-complete in DL-Lite^{N A}_krom, and so, in DL-Lite^{HN A}_krom even under(inter)and(inter2).

Proof. (Sketch) The proof exploits the ternary disjointness formula α²_U(x) in K^‡a. In fact, ifT uT⁰ v ⊥ ∈ T then the following concept inclusion, although not in the syntax ofDL-Lite^{N A}_krom, is a logical consequence ofT (cf.α²_U(x)):

∀U.Tu ∀U.T⁰u ∃U v ⊥.

Using such ternary intersections one can encode 3SAT. Let ϕ =Vm

i=1C_i be a 3CNF, where theC_i are ternary clauses over variablesp₁, . . . , p_n. Now, suppose p_j1

i ∨ ¬p_j2 i ∨p_j3

i is theith clause ofϕ. It is equivalent to¬p_j1 i ∧p_j2

i ∧ ¬p_j3

i → ⊥

and so, can be encoded as follows:

T_i¹uT_i²v ⊥, ¬A_j1

i v ∀U_i.T_i¹, A_j2

i v ∀U_i.T_i², ¬A_j3 i v ∃U_i, where theA₁, . . . , A_n are concept names for the variables p₁, . . . , p_n, andU_i is an attribute and T_i¹ and T_i² are datatypes for the ith clause (note that Krom concept inclusions of the form ¬B vB⁰ are required, which is not allowed in the core TBoxes). LetT consist of all such inclusions for clauses inϕ. It can be seen thatϕis satisfiable iffT is satisfiable.

5 Conclusions

We studied two different extensions of theDL-Litelogics. First,local attributes allow to use the same attribute associated to different concepts with different datatype range restrictions. We showed that the extension with attributes is harmless with the only notable exception of the Krom fragment, where the com- plexity rises fromNLogSpacetoNP.

Second, we consider weak syntactic restrictions on interaction between cardi- nality constraints and role inclusions and study their impact on the complexity of satisfiability. For example, under(inter)[3], roles with sub-roles cannot have maximum cardinality constraints. We present two alternative restrictions, which coincide without ABoxes, and show that the complexity of TBox satisfiability under them coincides with the complexity of TBox satisfiability without role in- clusions. However, if we want to preserve complexity of KB reasoning, condition (inter2) imposes a bound on the number R-successors in the ABox. Indeed, under the weaker condition (inter1) complexity of KB satisfiability rises to at leastNP(even for the core fragment).

As a future work, we intend to fill the gaps in Table 1 and, in particular, to see whether theNP-hardness results have a matching upper bound. We are also working on query answering in the languages with attributes.

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