Understanding Inexpressibility of Model-Based ABox Evolution in DL-Lite

of Model-Based ABox Evolution in DL-Lite

Evgeny Kharlamov, and Dmitriy Zheleznyakov KRDB Research Centre, Free University of Bozen-Bolzano, Italy

last_name@inf.unibz.it

Abstract. Evolution of Knowledge Bases (KBs) expressed in Description Logics (DLs) proved its importance. Recent studies of evolution in DLs mostly focussed on model-based approaches. They showed that evolution of KBs in tractable DLs, such asDL-Lite, suffers from inexpressibility, i.e., the result of evolution cannot be captured inDL-Lite. What is missing in these studies isunderstanding: in which DL-Litefragments evolution can be captured, what causes the inexpressibility, which logics is sufficient to express evolution, and whether one can approximate it inDL-Lite. This paper provides some understanding of these issues for both update and revision. We found whatDL-Liteformulas make evolution inexpressible and how to capture evolution in their absence. We introduce the notion of prototypes that gives an understanding of how to capture evolution for a richDL-Litefragment inFO[2]. Decidability ofFO[2]gives possibility for approximations.

1 Introduction

Description Logics (DLs) provide excellent mechanisms for representing structured knowledge by means of Knowledge Bases (KBs)Kthat are composed of two compo- nents: TBox (describes intensional or general knowledge about an application domain) and ABox (describes facts about individual objects). DLs constitute the foundations for various dialects of OWL, the Semantic Web ontology language.

Traditionally DLs have been used for modelingstatic and structural aspects of application domains [1]. Recently, however, the scope of KBs has broadened, and they are now used also for providing support in the maintenance andevolutionphase of information systems. This makes it necessary to studyevolution of Knowledge Bases[2], where the goal is to incorporate new knowledgeN into an existing KBKso as to take into account changes that occur in the underlying application domain. In general,N is represented by a set of formulas denoting those properties that should be true after K has evolved, and the result of evolution, denotedK N, is also intended to be a set of formulas. In the case whereN interacts withKin an undesirable way, e.g., by causing the KB or relevant parts of it to become unsatisfiable,N cannot simply be added to the KB. Instead, suitable changes need to be made inKso as to avoid this undesirable interaction. Different choices for changes are possible, corresponding to different approaches to semantics for KB evolution [3,4,5].

An important group of approaches to evolution semantics, that we focus in this paper, is calledmodel-based(MBAs). Under MBAs the result of evolutionK N is aset of

modelsofN that are minimally distanciated from models ofK. Depending on what the distance between models is and how to measure it, eight different MBAs were introduced (see Section 3 for details). SinceK N is a set of models, whileKandN are logical theories, it is desirable to representK N as a logical theory using the same language as forKandN. Thus, looking for representations ofK Nis the main challenge in studies of evolution under MBAs. WhenKandN are propositional theories, representingK N is well understood [5], while it becomes dramatically more complicated as soon asK andN are first-order, e.g., DL KBs [6].

Model based evolution of KBs whereK andN are written in a language of the DL-Litefamily [7] has been recently extensively studied [6,8,9]. The focus onDL-Liteis not surprising sinceDL-Liteis the basis of OWL 2 QL, a tractable OWL 2 profile. It has been shown that for every of the eight MBAs one can findDL-LiteKandN such that K N cannot be expressed inDL-Lite[10,11], i.e., DL-Liteisnot closedunder MBA evolution. This phenomenon was also noted in [6,10] for some of the eight semantics.

What is missing in all these studies of evolution forDL-Liteisunderstandingof:

(1) What fragments ofDL-Liteare closed under model-based evolutions? WhatDL-Lite formulas are responsible for inexpressibility of model-based evolutions?

(2) What are sufficient extensions ofDL-Liteto capture model-based evolutions of DL-LiteKBs? How to capture the evolutions ofDL-LiteKBs in these extensions?

(3) ForDL-LiteKandN, is it possible and how to do “good” approximations ofK N? In this paper we study the problems(1)-(3)for so-calledABox evolution, i.e.,N is a new ABox and the TBox ofKshould remain the same after the evolution. ABox evolution is important for areas, e.g., bioinformatics, where the structural knowledge TBox is well crafted and stable, while ABox facts about individuals may get changed.

These ABox changes should be reflected in KBs in a way that the TBox is not affected.

Our study covers both the case of ABox updates and ABox revision [4].

In Sections 2 and 3 we defineDL-Lite_Rand ABox evolution. In Section 4 we study relationships between different MBAs. In Section 5 we introduceDL-Lite⁺_R, a restriction on DL-Lite_R where disjointness of concepts with role projections is forbidden. We show thatDL-Lite⁺_R is closed under most of MBA evolutions and provide tractable algorithms computingK N. In Section 6 we study two important MBAs where distance between models is based on atoms and prove thatDL-Lite⁺_Ris a sufficient borderline of expressibility for these semantics: the class of KBs that are inDL-Lite_Rbut not in DL-Lite⁺_Ris not closed under these semantics. We also introduce and studyDL-Lite_R^I that restrictsDL-Lite_R, extendsDL-Lite⁺_R and not closed under MBAs. In order to capture evolution ofDL-Lite_R^I KBs inFO, we introduce prototypes, based on which we show that the evolution can be expressed inFO[2](restriction of first-order logics that only makes use of two variables). Finally, we argue that decidability ofFO[2]gives possibilities for approximations ofK N. Proofs can be found in [12].

2 DL-Lite

R

We introduce some basic notions of DLs, (see [13] for more details). We consider a logic DL-Lite_RofDL-Litefamily of DLs [7,14].DL-Lite_Rhas the following constructs for (complex)conceptsandroles:(i)B::=A| ∃R,(ii)C::=B| ¬B,(iii)R::=P |P⁻,

where A andP stand for anatomic concept and role, respectively, which are just names. Aknowledge base(KB)K = (T,A)is compound of two sets ofassertions:

TBoxT, andABoxA.DL-Lite_RTBox assertions areconcept inclusion assertionsof the formB vCandrole inclusion assertionsR1 vR2, while ABox assertions are membership assertionsof the formA(a),¬A(a), andR(a, b). Theactive domainofK, denotedadom(K), is the set of all constants occurring inK. TheDL-Litefamily has nice computational properties, for example, KB satisfiability has polynomial-time complexity in the size of the TBox and logarithmic-space in the size of the ABox [15,16].

The semantics of DL-Lite KBs is standard and based on first order interpretations, all over the same countable domain∆. AninterpretationIis a function·^Ithat assigns to each C a subsetC^I of∆, and toR a binary relationR^I over∆ in such a way that(¬B)^I = ∆\B^I,(∃R)^I = {a | ∃a⁰.(a, a⁰)∈ R^I}, and(P⁻)^I ={(a2, a1) | (a1, a2)∈P^I}. We assume that∆contains the constants and thatc^I=c, i.e., we adopt standard names. Alternatively, we view an interpretation as a set of atoms and say that A(a)∈ Iiffa∈A^IandP(a, b)∈ Iiff(a, b)∈P^I. An interpretationI is amodel of a membership assertionA(a)(resp.,¬A(a)) ifa∈A^I(resp.,a /∈A^I), ofP(a, b)if (a, b)∈P^I, and of an inclusion assertionD1vD2ifD₁^I⊆D₂^I.

As usual, we useI |=Fto denote thatIis a model of an assertionF, andI |=K to denote thatI |=Ffor each assertionFinK. We useMod(K)to denote the set of all models ofK. A KB issatisfiableif it has at least one model. We use entailment on KBs K |=K⁰in the standard sense. An ABoxA T-entailsan ABoxA⁰, denotedA |=_T A⁰, ifT ∪ A |= A⁰, andAisT-equivalenttoA⁰, denotedA ≡T A⁰ ifA |=_T A⁰ and A⁰|=_T A.

The deductiveclosure of a TBoxT (of an ABoxA), denotedcl(T)(resp.,cl_T(A)), is the set of all TBox (resp., positive ABox) assertionsF such that T |= F (resp., A |=_T F). Clearly inDL-Lite_R cl(T)(andcl_T(A)) is computable in time quadratic in the number of assertions ofT, i.e.,|T |, (resp.,|T ∪ A|). In our work we assume that all TBoxes and ABoxes are closed, while results are extendable to arbitrarily KBs.

For satisfiable KBsK = (T,A)afull closure ofA, denotedfcl_T(A), is the set of all membership assertionsf(both positive and negative) overadom(K)such thatA |=_T f.

Ahomomorphismhfrom a model I to a modelJ is a mapping from∆ to∆ satisfying:(i)h(a) =afor every constanta;(ii)ifα∈A^I(resp.,(α, β)∈P^I), then h(α)∈A^J (resp.,(h(α), h(β))∈P^J) for everyA(resp.,P). We writeI ,→ J if there is a homomorphism fromI toJ. A canonical modelIofK, denoted asI_K^canor just I^canwhenKis clear from the context, is a model ofKwhich can be homomorphically embedded in every model ofK[7].

3 Evolution of Knowledge Bases

This section is based on [10]. LetK= (T,A)be aDL-Lite_RKB andN a “new” ABox.

We study how to incorporateN’s assertions intoK, that is, howKevolvesunderN [2].

More practically, we studyevolution operatorsthat takeKandN as input and return, possibly inpolynomial time, aDL-Lite_RK⁰= (T,A⁰)(with the same TBox asK) that captures the evolution, and which we callthe (ABox) evolution ofKunderN. Based on the evolution principles of [10], we requireKandK⁰to be satisfiable and coherent.

Mod(K) :I0 I1

J0 J1J2 J3

Mod(T ∪ N) : L ⊆ #

s a global:G

local: set: number:

atoms:

symbols:

Approach

Type of distance Distance is

built upon

Fig. 1. Left: measuring distances between models and finding local minimums.

Right: three-dimensional space of approaches to model-based evolution semantics.

Model-Based Semantics of Evolution. In model-based approaches (MBAs), the result of evolution of a KBKwrt new knowledgeN is a setK N of models. The idea of MBAs is to choose asK Nsome models of(T,N)depending on their distance toK’s models. Katsuno and Mendelzon [4] considered two ways, so calledlocalandglobal, of choosing these models ofN, where the first choice corresponds toknowledge update and the second one toknowledge revision.

The idea of the local approaches is to consider all modelsIofKand for eachIto take those modelsJ of(T,N)that are minimally distant fromI. Formally,

K N = [

I∈Mod(K)

I N, whereI N = arg min

J ∈Mod(T ∪N)

dist(I,J).

wheredist(·,·)is a function whose range is a partially ordered domain andarg minstands for theargument of the minimum,that is, in our case, the set of modelsJ for which the value ofdist(I,J)reaches its minimum value, givenI. The distance functiondistvaries from approach to approach and commonly takes as values either numbers or subsets of some fixed set. To get a better intuition of the local semantics, consider Figure 1, left, where we present two modelI⁰andI¹of a KBKand four modelsJ⁰, . . . ,J³of(T,N).

We represent the distance between a model ofKand a model ofT ∪ Nby the length of a line connecting them. Solid lines correspond to minimal distances, dashed lines to distances that are not minimal. In this figure{J⁰}= arg min_{J ∈{J0,...,J3}}dist(I⁰,J) and{J²,J³}= arg min_{J ∈{J0,...,J3}}dist(I¹,J).

In the global approach one choses models ofN that are minimally distant fromK: K N = arg min

J ∈Mod(N)

dist(Mod(K),J), (1)

wheredist(Mod(K),J) = min_I∈Mod(K)dist(I,J). Consider again Figure 1, left, and assume that the distance betweenI⁰ andJ⁰ is the global minimum, hence,{J⁰} = arg min_{J ∈{J0,...,J3}}dist({I⁰,I¹},J).

Measuring Distance Between Interpretations. The classical MBAs were developed for propositional theories [5], where interpretations were sets of propositional atoms, two distance functions were introduced, respectively based on symmetric difference “ ” and on the cardinality of symmetric difference:

dist_⊆(I,J) =I J and dist#(I,J) =|I J |, (2)

whereI J = (I \ J)∪(J \ I). Distances underdist_⊆are sets and are compared by set inclusion, that is,dist_⊆(I¹,J¹)≤dist_⊆(I²,J²)iffdist_⊆(I¹,J¹)⊆dist_⊆(I²,J²).

Finite distances underdist#are natural numbers and are compared in the standard way.

One can extend these distances to DL interpretations in two different ways. One way is to consider interpretationsI,J as sets ofatoms. ThenI J is again a set of atoms and we can define distances as in Equation (2). We denote these distances asdist^a_⊆(I,J) anddist^a_#(I,J). Another way is to define distances at the level of the concept and role symbolsin the signatureΣunderlying the interpretations:

dist^s_⊆(I,J) ={S ∈Σ|S^I6=S^J}, and dist^s_#(I,J) =|{S∈Σ|S^I 6=S^J}|. Summing up across the different possibilities, we have three dimensions, which give eight semantics of evolution according to MBAs by choosing: (1) thelocalor theglobal approach, (2)atomsorsymbolsfor defining distances, and (3)set inclusionorcardinality to compare symmetric differences. In Figure 1, right, we depict these three dimensions.

We denote each of these eight possibilities by a combination of three symbols, indicating the choice in each dimension, e.g.L^a#denotes the local semantics where the distances are expressed in terms of cardinality of sets of atoms.

Closure Under Evolution. LetDbe a DL andM one of the eight MBAs introduced above. We sayDisclosed under evolution wrtM (or evolution wrtM isexpressible inD) if for any KBsKandN written inD, there is a KBK⁰written inDsuch that Mod(K⁰) =K N, whereK N is the evolution result under semanticsM.

We showed in [10,11] thatDL-Lite is not closed under any of the eight model based semantics. The observation underlying these results is that on the one hand, the minimality of change principle intrinsically introduces implicit disjunction in the evolved KB. On the other hand, sinceDL-Liteis a slight extension of Horn logic [17], it does not allow one to express genuine disjunction (see Lemma 1 in [10] for details).

LetM be a set of models that resulted from the evolution of(T,A)withN. A KB (T,A⁰)is asound approximationofM ifM ⊆Mod(T,A⁰). A sound approximation (T,A⁰)isminimalif for every sound approximation(T,A⁰⁰)inequivalent to(T,A⁰), it

holdsMod(T,A⁰⁰)6⊂Mod(T,A⁰), i.e.(T,A⁰)is minimal wrt “⊆”.

4 Relationships Between Model-Based Semantics

LetS¹andS²be two evolution semantics andDa logic language. ThenS¹is subsumed byS²wrtD, denoted(S¹ 4sem S²)(D), or justS¹ 4sem S²whenDis clear from the context, ifK S1N ⊆ K S2N for all satisfiable KBsKandN written inD, where K SiN denotes evolution underSⁱ. Two semanticsS¹andS²areequivalent(wrtD), denoted(S¹ ≡^sem S²)(D), if(S¹ 4^sem S²)(D)and(S² 4^sem S¹)(D). Further in this section we will considerKandN written inDL-Lite_R. The following theorem shows the subsumption relation between different semantics, which we also depict with solid arrows in Figure 2. The figure is complete in the following sense: there is a solid path between any two semanticsS¹andS²iff there is a subsumptionS¹4^semS².

Mod(K) :

Mod(K � N) :

I^min I1

J^min J1

G#^s L^s#

L^s_⊆ G^s_⊆

G#^a

G_⊆^a L^a_⊆

L^a#

� →

� →

Fig. 2. Left: Subsumptions for evolution semantics.

“ ”: inDL-Lite_R(Theorem 1). “ ”: inDL-Lite⁺_R(Theorems 6, 7).

Dashed frame surrounds semantics under whichDL-Lite⁺_Ris closed.

Right: commutation diagram forL^a_⊆semantics, “I ⇒ J” denotes thatJ ∈ I N. Theorem 1. Letβ ∈ {a, s}andα∈ {⊆,#}. Then

Gα^β4sem L^βα, G#^β 4semG_⊆^β, and L^s#4sem L^s⊆.

Proof. Letdist be one of the four considered distances, EG = K N wrtGα^β and EL =K N wrtL^βαbe corresponding global and local semantics based ondist. For given Kand N, letJ⁰ ∈ EG. Then, there isI⁰ |= K such that for everyI⁰⁰ |= K andJ⁰⁰ |=T ∪ N itdoes notholddist(I⁰⁰,J⁰⁰) dist(I⁰,J⁰).In particular, when I⁰⁰ = I⁰, there is no J⁰⁰ |= T ∪ N such that dist(I⁰,J⁰⁰) dist(I⁰,J⁰), which yields thatJ⁰ ∈ arg minJ ∈Mod(T ∪N)dist(I⁰,J), and J⁰ ∈ EL. We conclude that:

G#^a 4^sem L^a#, G_⊆^a 4^semL^a_⊆, G#^s 4^semL^s#, G_⊆^s 4^sem L^s_⊆.

Now considerE^#=KNwrtL^β#, which is based ondist_#, andE⊆=KNwrtL^β⊆, which is based ondist_⊆. AssumeJ⁰ ∈ E^#andJ⁰ 6∈ E⊆. Then, from the former assump- tion we conclude existence ofI⁰ |=Ksuch thatJ⁰ ∈arg minJ ∈Mod(T ∪N)dist#(I⁰,J).

From the latter assumption, J⁰ ∈ E/ ⊆, we conclude existence of a modelJ⁰⁰ such thatdist_⊆(I⁰,J⁰⁰) ( dist_⊆(I⁰,J⁰). This yields thatdist#(I⁰,J⁰⁰) dist#(I⁰,J⁰), which contradicts the fact thatJ⁰ ∈ E^#, assuming thatdist_⊆(I⁰,J⁰)is finite. Thus, E^#4sem E⊆as soon asdist_⊆(I,J)is finite. This finiteness condition always holds for whenβ =ssince the signature ofK ∪ Nis finite. It is easy to check thatdist_⊆(I,J) may not be finite whenβ=a, hence,L^a#64^sem L^a_⊆.

Similarly one can show thatG#^s 4semG_⊆^s. It also holds thatG#^a 4semG_⊆^a due to the

finite model property inDL-Lite_R. ut

5 Evolution in DL-Lite

⁺_R

Here we consider a restriction ofDL-Lite_R, which we callDL-Lite⁺_R. A KBK= (T,A) is in DL-Lite⁺_R if it is in DL-Lite_R and T 6|= ∃R v ¬B for any role R and any conceptB. Intuitively, “+” emphasizes absence of disjointness for roles (only positive inclusions involving roles are permitted).DL-Lite⁺_Ris defined semantically, while one can syntactically check (in quadratic time), given aDL-Lite_RKBK, whetherKis in DL-Lite⁺_R: one computes a closure ofK, checks that no assertions of the form∃Rv ¬B are in the closure and if it is the case, thenKis inDL-Lite⁺_R.DL-Lite⁺_Ris an extension of RDFS ontology language (of itsFOfragment).DL-Lite⁺_Radds to RDFS the ability of expressing disjointness of concepts (A1v ¬A²) and mandatory participation (Av ∃R).

SinceDL-Lite⁺_RrestrictsDL-Lite_R, the semantics relations from Theorem 1) are also correct forDL-Lite⁺_R. We now study what further4sem-relations hold inDL-Lite⁺_R.

INPUT : consistent KBs(T,A)andN

OUTPUT: a setA⁰⊆fcl_T(A)of ABox assertions A⁰:=∅;S:=fcl_T(A);

1

repeat

2

choose someφ∈ S; S:=S \ {φ};

3

if{φ} ∪fclT(N)is consistentthenA⁰:=A⁰∪ {φ}

4

untilS=∅;

5

Algorithm 1: AlgorithmAlignAlg((T,A),N)forA⁰deterministic computation

5.1 Capturing Atom-Based Evolution

We first study evolution under L^a_⊆. LetI be an interpretation and N a knowledge base. AnalignmentofIwithN, denotedAlign(I,N), is the interpretationJ ={f | f ∈ Iandf is satisfiable withN }. There is a tight connection (Lemma 2) between an alignment of a model and its evolution underL^a⊆semantics. Moreover, alignment preserves homomorphic relationship on interpretations (Lemma 3). LetNbe a set of membership assertions andIan interpretation. A unionI ∪Ndenotes the modelI ∪JN, whereJN ={f |f ∈ N andfis positive}.

Lemma 2. LetK = (T,A)and(T,N)be satisfiable DL-Lite⁺_R KBs, andI |= K. ThenI N ={Align(I,N)∪ N }, underL^a⊆semantics.

Lemma 3. LetKandN be satisfiable DL-Lite⁺_R KBs andI be a model ofK. Then Align(I_K^can,N),→Align(I,N).

Lemmas 3 and 2 imply that(I^can N),→(I N)for everyI ∈Mod(K)under L^a_⊆, where we apply,→to the sets(I^can N)and(I N), instead of their single models. We depict this phenomenon in Figure 2, right: evolution preserves homomorphic embeddability ofI^canintoI. This fundamental property implies thatI^can N consists of the universal model ofK NwrtL^a_⊆.

Consider an algorithmAlignAlg(see Algorithm 1) that inputsK,N, and returns the alignmentAlign(I^can,N): it drops all the assertions offcl_T(A)contradictingN and keeps the rest. UsingAlignAlgwe can computeK N:

Theorem 4. LetK= (T,A)and(T,N)be satisfiable DL-Lite⁺_RKBs. Then:

K N =Mod(T,AlignAlg(K,N)∪ N)(wrtL^a_⊆).

Note that since computation ofAlignAlg(T,A,N)is polynomial in A andN, computation ofK N is also polynomial.

Example 5. ConsiderT = {B0 v B, B v ¬C},A = {C(a)}, and N = B(a).

Then,fcl_T(A) ={C(a),¬B0(a),¬B(a)}. The alignment of(T,A)withN is the set AlignAlg((T,A),N) ={B(a),¬B(a)}. Hence, the result of evolution under every the model-based semantics with atom-based distance is(T,{B(a),¬B0(a)}).

Relationships Between Atom-Based Semantics. Next theorem shows that evolutions wrt all four model-based semantics on atoms coincide wrtDL-Lite⁺_R. We depict these relations between semantics in Figure 2 using a dashed arrow, e.g., betweenL^a_⊆ 4semG#^a. Note that there is a path with solid or dashed lines between any two semantics if and only if there is a subsumption wrtDL-Lite⁺_R.

Theorem 6. L^a#≡^semL^a_⊆≡^sem G#^a ≡^semG_⊆^a.

Theorems 6 and 4 imply that inDL-Lite⁺_Rone can useAlignAlgto compute evolution under all MBAs on atoms.

5.2 Capturing Symbol-Based Evolution

Observe that symbol-based semantics behave differently from atom-based ones: two local semantics (on set inclusion and cardinality) coincide, as well as two global ones, while there is no subsumption between local and global ones, as depicted in Figure 2.

Theorem 7. The following relations hold:

(i) L^a⊆4semG#^s, whileG#^s 64^semL^a⊆;

(ii) L^s⊆≡^semL^s#, andG⊆^s ≡^semG#^s, whileL^s⊆64^sem G^s#.

As a corollary of Theorem 7, the algorithm of computingK N of Theorem 4 in general does not work for computing evolution under any of the symbol-based semantics.

At the same time what it outputs is a complete approximation of all symbol-based semantics, while it approximates global semantics better than the local ones.

Consider the algorithmSymAlgin Figure 2 that will be used for evolutions on symbols. It works as follows: for every atomφinN it checks whetherφsatisfies a conditionΠ (Line 4). If it does, the algorithm deletes all those literals that share their concept name withφ. Both local and global semantics have their ownΠ:Π_G andΠ_L. Capturing Global Semantics. Π_G(φ)checks whetherφofN T-contradictsA:Π_G(φ)is true iff¬φ∈fcl_T(A)\AlignAlg((T,A),N). Intuitively,SymAlgfor global semantics works as follows: having contradiction betweenN andAonφ=B(c), the change of B’s interpretation is inevitable. Since the semantics traces changes on symbols only, and Bis already changed, one can drop fromAall the assertions of the formB(d). Clearly, SymAlg(K,N, Π_G)can be computed in time polynomial in|A ∪ K|. The following theorem shows correctness of this algorithm.

Theorem 8. LetK= (T,A)and(T,N)be satisfiable DL-Lite⁺_RKBs. Then DL-Lite⁺_R is closed underG_⊆^s andG#^s, and for bothK N =Mod(T,SymAlg(K,N, Π_G)), Capturing Local Semantics. Observe thatL^s⊆andL^s#are not expressible inDL-Lite⁺_R. Theorem 9. DL-Lite⁺_Ris not closed underL^s⊆andL^s#semantics.

To compute the minimal sound approximations under local semantics on symbols, we use algorithmSymAlgin Figure 2 with the followingΠ_L:Π_L(φ)is true iffφ /∈ S¹. That is,Π_Lchecks whether the ABoxT-entailsA(c)∈fcl_T(N), and if it does, the

INPUT : consistentDL-Lite⁺_RKB(T,A)and ABoxN, a formula propertyΠ OUTPUT: a setA⁰⊆fcl_T(A)∪fcl_T(N)of ABox assertions

A⁰:=∅;S1:=AlignAlg((T,A),N);S2:=fcl_T(N);

1

repeat

2

choose someφ∈ S2; S2 :=S2\ {φ};

3

ifΠ(φ) =TRUEthenS1:=S1\ {φ⁰|φandφ⁰have the same concept name}

4

untilS2=∅;

5

A⁰:=S1∪fclT(N)

6

Algorithm 2: AlgorithmSymAlg((T,A),N, Π)for deterministic computation of K N underG⊆^s andG#^s semantics and minimal sound approximation underL^s⊆

andL^s#semantics

algorithm deletes all the assertions fromfcl_T(A)that share the concept name withA(c).

This property is weaker than one for global semantics, since it is easier to get changes in interpretation ofAby choosing a model ofKwhich does not includeA(c). Clearly, SymAlg(K,N, Π_L)can be computed in time polynomial in|A ∪ K|. The following theorem shows correctness of the algorithm.

Theorem 10. LetK= (T,A)and(T,N)be satisfiable DL-Lite⁺_RKBs. Then the KB (T,SymAlg(K,N, Π_L))is a minimal sound approximation ofK N underL^s_⊆andL^s#.

6 Evolution in DL-Lite

_R^I

In the previous section we considered evolution in DL-Lite⁺_R. Here we show that DL-Lite⁺_R is essentially a maximal fragment ofDL-Lite_R closed under atom-based evolution, and present a wide fragment of DL-Lite_R, namely DL-Lite_R^I, that is not closed under atom-based evolution while the evolution can be captured inFO[2].

The following theorem shows that the restriction not to have roles involved into negation relation is essential, and even a minimal violation leads to inexpressibility of evolution. The minimal way to violate conditions ofDL-Lite⁺_Ris to have two assertions:

Av ∃Rand∃R⁻v ¬Centailed from a TBox.

Theorem 11. Let assertions{A v ∃R, ∃R⁻ v ¬C} be entailed from a DL-Lite_R TBoxT, for someRandC, Then there exist ABoxesAandN such that(T,A) N is inexpressible in DL-Lite_RunderL^a⊆and underL^a#.

The following example provides reasons whyDL-Lite⁺_Rrestrictions are important for expressibility of atom-based MBAs. We will further use this example to show how to captureL^a_⊆evolution inFOfor KBs violatingDL-Lite⁺_Rrestrictions.

Example 12. Consider the followingDL-LiteKBsK¹= (T¹,A¹)andN¹={C(b)}: T¹={Av ∃R,∃R⁻v ¬C}; A¹={A(a), C(e), C(d), R(a, b)}. Consider the following modelIofK¹:

I: A^I ={a, x}, C^I ={d, e}, R^I ={(a, b),(x, b)},

wherex∈∆\adom. We now show that the following models belong toI N¹, this will be used further to captureK¹ N¹.

J⁰: A^I =∅, C^I ={d, e, b}, R^I =∅, J¹: A^I ={x}, C^I ={e, b}, R^I ={(x, d)}, J²: A^I ={x}, C^I ={d, b}, R^I ={(x, e)}, J³: A^I ={a, x}, C^I ={b}, R^I ={(a, d),(x, e)}, J⁴: A^I ={x}, C^I ={d, e, b}, R^I ={(x, β1)},

whereβ1∈∆\adom(K¹). Clearly all five models satisfyN¹andT¹. To see that they are inI N¹observe that every modelJ(I)∈(I N¹)can be obtained fromIby making modifications that guarantee thatJ(I)|= (N¹∪ T¹)and that the distance betweenI andJ(I)is minimal. What are these modifications? Since in every suchJ(I)the new knowledgeC(b)holds and(Cv ¬∃R⁻)∈ T¹, there should be noR-arc incoming into binJ(I), hence the necessary modification ofI is to forbid theR-arcs going from aand fromxto point intob. Where inJ(I)should this R-arcs point in? There are only three mutually exclusive cases for each ofaandx:(i)there is noR-arc inJ(I) that points froma(resp.,x), we simply drop it fromI,(ii)it points to some element β ∈∆\ {d, e, b}, that is,J(I)|=R(a, β).(iii)it points to an elementγof{d, e}, that is,J(I)|=R(a, γ). Case (i) for bothaandxcorresponds toJ⁰, hence,J⁰∈ I N¹. Case (ii) for bothaandxcorresponds toJ⁴, hence,J⁴∈ I N¹. Case (i) and Case (iii) foracorresponds toJ¹andJ², hence, bothJ¹andJ²are inI N¹. Case (iii) for both aandxcorresponds toJ³, hence,J³∈ I N¹. There are clearly more modelsI N¹ than these five.

To show thatK¹N¹is inexpressible inDL-Lite_Rconsider a modelJ⁵∈Mod(T¹,N¹) which is not inK¹ N¹, while it gives a property of models fromK¹ N¹that is respon- sible for inexpressibility ofK¹ N¹.

J⁵: A^I ={x}, C^I ={e, b}, R^I ={(x, d),(x, β1)}.

Observe thatJ⁴ is closer toI thatJ⁵, hence J⁵ 6∈ (I N¹). Indeed, since for ev- ery modelI⁰ ∈ Mod(K¹)it holds thatI⁰ |=C(d)andI⁰ 6|= R(x, d), we have that {C(d), R(x, d)} ⊆ I⁰ J⁵. At the same timeJ⁴|=C(d)andJ⁴6|=R(x, d), whileJ⁴ andJ⁵agree on all the atoms butC(d)andR(x, d). Hence,(I⁰ J⁴)((I⁰ J⁵)and J⁵6∈(I N¹). Moreover, sinceI⁰is an arbitrary model ofK¹,J⁵∈/(K¹ N¹).

Let us make a closer look atJ⁵: it extendsJ¹with the atomR(x, β1), and this is the reason why it is not in the result of evolutionK¹ N¹. This observation gives a restriction on all the modelsJ onK¹ N¹: if inJ there is oneR-arc from somexinto d, thenJ has no other arcs fromx. In a way, we have a functionality restriction on the roleR, when it connects two specific elements of the domain:xandd. The same kind of functionality holds forxande, and both of them are not expressible inDL-Lite. This functionality forxandd, and forxandecan be formally written as the followingφand ψ, respectively:

φ=∀x∀y.[R(x, d)∧R(x, y)→y=d], ψ=∀x∀y.[R(x, e)∧R(x, y)→y=e]. Since every model ofK¹ N¹should satisfyφandψ, andDL-Liteis a slight extension of Horn logics [17],K¹ N¹cannot be captured inDL-Lite.

This example shows that DL-Lite_R is not closed under model-based evolution.

Besides the fact that bothφ andψ are inexpressible inDL-Lite_R, while should be entailed by the result of evolution, there is another observation also responsible for inexpressibility ofK¹ N¹: it has no canonical model. At the same time, one can see that the setK¹ N¹can be divided in four subsets, where each has a canonical model. We now show that these four canonical models, which we callprototypesforK¹ N¹, can be used to captureK¹ N¹inFO[2], in a similar way as we used the unique canonical model of the evolution result inDL-Lite⁺_Rto capture the evolution inDL-Lite⁺_R. Definition 13. LetKbe a DL-Lite_RKB andN be an ABox. Aprototypal set forK N is a minimal subset

J

= {J¹, . . . ,Jⁿ} ofK N satisfying the property: for every J ∈ K N there isJⁱ∈

J

^{such that}_Ji,→ J. EveryJⁱ∈

J

^{is a}prototype forK N.

Continuing with Example 12, one can check that the prototypal set

J

^for_K1 N¹ consists of four models:{J⁰,J¹,J²,J⁶}, where

J⁶: A^I ={x1, x2}, C^I ={b}, R^I ={(x1, d),(x2, e)}. Note thatJ⁶,→ J³andJ⁰,→ J⁴.

We now introduce a restriction ofDL-Lite_Rfor which the result of evolution under L^a_⊆ can be captured inFOusing prototypes.DL-Lite_R^I (whereI stands for (mutual) independenceof roles) is a restriction ofDL-Lite_Rin which TBoxesT satisfy: for any two rolesRandR⁰,T 6|=∃Rv ∃R⁰andT 6|=∃Rv ¬∃R⁰. That is, we forbid direct role interaction (subsumption and disjointness) between role projections. The interaction is still possible but in a simple manner only, e.g., projections may contain the same concept. This restriction allows us to analyze evolution that affects roles independently for every role. For the ease of exhibition of the following procedure constructing

J

^{, we}

further assume that inDL-Lite_R^I for each roleRthere is exactly one conceptARsuch thatK |=ARv ∃R. As forDL-Lite⁺_R, one can syntactically check (in quadratic time), given aDL-Lite_RKBK, whetherKis inDL-Lite_R^I using the closure ofK.

Iff is an ABox assertion, thenroot_T(f)is a set of all the ABox assertions, that T-entailf. The procedureBP(K,N)(whereBP stands for build prototypes) of con- structing

J

for the case ofDL-Lite_R^I, generalizes what was done in Example 12 in the following way: In Items 1 and 2 it takes into account all the roles and concepts that may be interpreted differently in different resulted models (R,AandCin our example) and all the constants that could not be present at the second coordinate of those roles in the models of the original KB (eanddin our example). In Items 3, 4, and 5 it builds the prototypeJ⁰. Finally in Item 6 it builds the rest of prototypes basing on J⁰. A pseudocode of the procedureBP(K,N)is the following:

1. S^R={R1, . . . , Rn}is a set of all roles such that (i) N |=_T ¬R⁻_i (bi)for somebiand

(ii) for everyRi(ai, bi)infcl_T(A), an atomAR(ai)∈fcl_T(A)andR(ai, b⁰_i)∈/ Align(fcl_T(A),N), whereb⁰_i∈ {/ bj}ⁿj=1.

S^at=Sn

j=1{Rj(a, bj)|Rj(a, bj)∈fcl_T(A)}. 2. FA(Ri)is equal to the following set

{D(c)∈fcl_T(A)| ∃R⁻_i (c)∧D(c)|=_T ⊥andN 6|=_T D(c), andN 6|=_T ¬D(c)}. Then FA=S

Ri∈SRFA(Ri), FC={c|D(c)∈FA},

where the acronymsFAandFCstand for forbidden atoms and constants, respectively.

3. I⁰:=Align(I^can,N)∪ N, whereI^canis the canonical model ofK. 4. For eachR(a, b)∈ S^at, doI⁰ :=I⁰\ {AR(a)}.

5. J⁰:=I⁰,

J

:={J⁰}.

6. For each subsetD={D1(c1), . . . , Dk(ck)} ⊆FAdo

for eachR= (Ri1, . . . , Rik)such thatDj(cj)∈FA(Rij)forj= 1, . . . , kdo J[D,R] :=h

J⁰\Sk

i=1root_T(Di(ci))i

∪Sk i=1

cl_T(R⁰_i(xi, ci))∪ {AR⁰_i(xi)} , where allxi’s are different constants from∆\adom(K), fresh forI^can.

J

:=

J

∪ {J[D,R]}.

Theorem 14. LetK= (T,A)be a DL-Lite_R^I KB, andN a DL-Lite_RABox consistent withT. Then the setBP(K,N)is a prototypal set forK N.

We proceed to correctness of BP in capturing evolution inDL-Lite_R^I.

Theorem 15. LetK= (T,A)be a DL-Lite_R^I KB,N a DL-Lite_RABox consistent with T, andBP(K,N) ={J⁰, . . . ,Jⁿ}is a prototypal set forK N. Then

K N =Mod(T)∩Mod(A¹∨. . .∨ Aⁿ)∩Mod(Φ∧Ψ), whereAiis a DL-Lite_RABox such thatJⁱis a canonical model for(T,Aⁱ), and

Φ= ^

cj∈FC

∀x.[(R(x, cj)→AR(x))∧ ∀y.(R(x, cj)∧R(x, y)→y=cj)],

Ψ = ^

R(a,b)∈Sat

∃R(a)→AR(a).

Note that Theorem 4 about computation ofK N forDL-Lite⁺_Ris a particular case of Theorem 15 whenFC=∅and there is just one prototype. Next theorem allows us to approximate result of evolution by aDL-Lite_RKB, sinceFO[2]is decidable.

Theorem 16. K N underL^a_⊆for KBs in DL-Lite_R^I is inFO[2].

7 Conclusion

We studied expressibility of ABox evolution (for both update and revision) over two subfamilies ofDL-Lite_R:DL-Lite⁺_RandDL-Lite_R^I, that both extend RDFS. The first one is closed under most of the evolution semantics, while the second one is not closed even under MBAs were distance is based on atoms. We isolated conditions on TBox assertions that lead to inexpressibility: pairs of assertions of the formA v ∃Rand∃R⁻ v ¬C bring inexpressibility. Note that this condition is similar to the notion of unexpected facts introduces in [10], for formula-based semantics of evolutions. ForDL-Lite⁺_Rwe provided algorithms how to compute semantics that are expressible, and how to approximate those that are not. ForDL-Lite_R^I we captured local model-based semantics, where the distance between models defined on atoms, inFO[2]. For this purpose we introduced prototypes and showed how they can be used to capture results of evolution.

It is the first attempt to provide an understanding of inexpressibility of MBAs for DL-Liteevolution. Without this understanding it is unclear how to proceed with the

study of evolution in more expressive DLs and what to expect from MBAs in such logics.

Moreover, isolating the smallest fragment ofFOthat is sufficient to capturedDL-Lite evolution is important in understanding whether it is possible and how to do reasonable approximations of evolution results. We also believe that our techniques of capturing semantics based on prototypes give a better understanding of how MBAs behave on FOtheories. We are currently working on extending the results to capture evolution for generalDL-Lite_RKBs.

Acknowledgements We are thankful to Diego Calvanese, Balder ten Cate, and Werner Nutt for discussions. The authors are supported by EU projects ACSI (FP7-ICT-257593), Ontorule (FP7-ICT-231875), and ERC FP7 grant Webdam (agreement n. 226513).

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