On Prototypes for Winslett's Semantics of DL-Lite ABox Evolution

On Prototypes for Winslett’s Semantics of DL-Lite ABox Evolution

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

last_name@inf.unibz.it

Abstract. Evolution of Knowledge Bases expressed in Description Logics (DLs) proved its importance. Most studies on evolution in DLs have focused on model- based approaches to evolution semantics and in particular on Winslett’s semantics (WS). It was understood that evolution under WS even in tractable DLs, such asDL-Lite, suffers from inexpressibility, i.e., the result of evolution cannot be expressed in the same logics. In this work we show which combination ofDL- Litelogical constructs is responsible for the inexpressibility and explain reasons for such a behaviour. We present novel techniques, based on what we called prototypes, to capture Winslett’s evolution inFO[2]forDL-LiteR. We also discuss which fragments ofDL-LiteRare closed under evolution.

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, 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 a new knowledgeN into an existing KBK so 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, e.g., by deleting parts ofKconflicting withN. Different choices for changes are possible, corresponding to different approaches tosemanticsfor KB evolution [3,4,5].

One approach to evolution semantics that proved its importance isWinslett’s seman- tics(WS) [6], which is anupdatesemantics in terms of Katsumo and Mendelzon [4], and was originally proposed for propositional theories. Under this semantics the result of evolutionK N is aset of modelsofN that are minimally distanced from models ofK, where the distance is based on symmetric difference between models (see Section 3 for

details). Since the result of evolutionK Nis 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 N is the main challenge in a study of evolution under WS. WhenKandN are propositional theories, representing K N is well understood [5], while it becomes dramatically more complicated as soon asKandN are first-order, e.g., DL KBs [7].

In this work we study how WS can be applied to evolution of KBs under the following two assumptions. First, we assume that bothKandN are written in a language of the DL-Litefamily [8]. The focus onDL-Lite is not surprising sinceDL-Lite is tightly connected with conceptual data models and it is the basis of OWL 2 QL, a tractable OWL 2 profile.Second, we assume thatN is a new ABox and the TBox ofKshould remain the same after the evolution. That is, we study a so-calledABox evolution. ABox evolution is important for areas, e.g., bioinformatics, where the structural knowledge TBox is well crafted and stable, while ABox facts about specific individuals may get changed, or/and new facts can be inserted in the ABox. These ABox changes should be reflected in KBs in a way that the TBox is not affected.

There are several works on WS for bothDL-Liteand more expressive DLs. Liu, Lutz, Milicic, and Wolter studied Winslett’s evolution in expressive DLs [7], for KBs with empty TBoxes. Most of DLs they considered are not closed under WS and in order to close these logics they used “@” operator. Poggi, Lembo, De Giacomo, Lenzerini, and Rosati applied WS toDL-Lite[9] and proposed an algorithm to compute the result of evolution. It turned out that their algorithm is wrong, i.e. it is neither sound, nor complete [10]. Actually, such an algorithm cannot exist since Calvanese, Kharlamov, Nutt, and Zheleznyakov showed that, e.g.,DL-LiteFRis not closed under WS of evolu- tion [11], that is, there areKandN such thatK N is not axiomatizable in this family.

Recently [12] we introducedprototypes, which are in a way generalization of the notion of canonical model, and proposed a way to capture some fragments ofDL-LiteinFO[2], a fragment of first-order logic that uses two variables only.

Current work extends the preliminary results of [12]. Our goals here are (i) to clarify our prototype-based techniques which was only sketched in [12], (ii) to extend the techniques to widerDL-Litefragments,

(iii) to gain a better understanding on which fragments ofDL-Liteare closed under WS and how to approximate evolution results inDL-Lite.

We would also like to promote prototypes since we believe they are an useful tool to study evolution of ontologies and might be not only ofDL-Liteones.

In Sections 2 and 3 we defineDL-LiteRand ABox evolution under WS. In Section 4 we give an intuition of our approach to capture WS of evolution forDL-LiteRKBs using prototypes andFO[2]theories. In Sections 5 and 6 we formalize the approach. Finally, we discuss properties and approximation of these theories.

2 DL-Lite

R

We introduce some basic notions of DLs (see [1] for more details). We consider a logic DL-LiteRofDL-Litefamily of DLs [8,13].DL-LiteRhas 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. A DLknowledge 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 [14,15].

The semantics of DL-Lite KBs is given in the standard way: using first orderinter- pretationsI, all over the same countable domain∆. We assume that∆contains the constants andc^I=c, i.e., we adoptstandard names. Alternatively, we view interpreta- tions as sets of atoms:A(a)∈ Iiffa∈A^IandP(a, b)∈ Iiff(a, b)∈P^I.

Definitions ofIbeing amodelof an ABox or a TBox assertionF, denotedI |=F, and a KBK, denotedI |=K, are standard, as well as the notion ofsatisfiability. We use Mod(K)to denote the set of all models ofK. We use entailment on KBsK |=K⁰in the standard sense. An ABoxA T-entailsan ABoxA⁰, denotedA |=T A⁰, ifT ∪ A |=A⁰, andAisT-equivalenttoA⁰, denotedA ≡^T A⁰, ifA |=T A⁰andA⁰|=T A.

The deductiveclosure of a TBoxT, denotedcl(T), is the set of all TBox assertionsF such thatT |=F. For satisfiable KBsK= (T,A), afull closure ofA(wrtT),fcl_T(A), is the set of all membership assertionsf (both positive and negative) overadom(K) such thatA |=T f. Clearly, inDL-LiteRbothcl(T)andclT(A)are computable in time quadratic in, respectively,|T |, i.e., the number of assertions ofT, and|T ∪ A|. For the ease of exhibition and wlg we assume that all TBoxes and ABoxes are closed.

Ahomomorphismhfrom a modelIto a modelJ is a structure-preserving mapping from∆ to∆ satisfying:(i) h(a) = a for every constant a;(ii) if α ∈ A^I (resp., (α, β)∈P^I), thenh(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 modelI ofK, denoted asIK^canor justI^canwhenKis clear from the context, is a model ofKwhich can be homomorphically embedded in every model ofK[8].

3 Winslett’s Semantics for Evolution of Knowledge Bases

We start with ABox evolution of single models under Winslett’s semantics. LetK= (T,A)be aDL-LiteR KB,I a model ofK, andN a new ABox satisfiable with T.

Evolution of a modelI ofK is based on the symmetric difference :S₁ S₂ = (S1\S₂)∪(S2\S₁), and defined as follows. The (result of)evolution ofI withN

under Winslett’s semantics (WS) [9], denotedI N, is the set of modelsJ such that:

(i) J ∈Mod(T ∪ N), and

(ii) there is no modelJ⁰ ∈Mod(T ∪ N)satisfyingI J⁰(I J

Note that in Case(i)we haveModof bothT andN, which means that the evolution preserves both the old TBox and the new knowledge. Case(ii)guarantees the principle of minimal change [5]. We extend the definition to KBs:

The result ofevolution ofKwithN under WS, denotedK N, is the following set of models:

K N =∪^I∈Mod(K)I N.

In terms of [10], WS corresponds toL^a⊆semantics, i.e., local model-based semantics based on atoms and set inclusion.

The input for evolution is two finite syntactic objects: a KBK= (T,A)and a new informationN, while the outputK N is a set of models, which is an infinite object for DL-LiteR. Indeed,K N is in general infinite. One can easily come up with examples whereK N has an infinite number of infinite models. These observations imply that storingK N is infeasible and in practice one would like to represent the evolution as a KBK⁰. Moreover, one would like to stay within the same formalism and expressK⁰ inDL-LiteR. Formally, we say that a logicLisclosedunder Winslett’s evolution if for everyKandN inL, the result of evolutionK N is expressible inL, that is, there is a KBK⁰ = (T,A⁰)inLsuch thatMod(K⁰) =K N.

Example 1. Consider the followingDL-LiteKBK¹= (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. The following models belong toI 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)}.

Indeed, all the 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 betweenIandJ(I)is minimal.

What are these modifications? Since in everyJ(I)the new assertionC(b)holds and (Cv ¬∃R⁻)∈ T¹, there should be noR-atoms withb-fillers at the second coordinate inJ(I). Hence, the necessary modifications ofI are either to drop (some of) theR- atomsR(a, b)andR(x, b), or to modify (some of) them, by substituting theb-fillers with another ones, while keeping the elementsaandxon the first position. The model J⁰corresponds to the case when bothR-atoms are dropped, while inJ¹andJ²only R(a, b)is dropped andR(x, b)is modified toR(x, d)andR(x, e), respectively. Note that the modification inR(x, b)leads to a further change in the interpretation ofCin bothJ¹andJ², namely,C(d)andC(e)should be dropped, respectively.

4 Prototypes for Winslett’s Semantics

We first present a general discussion on issues with capturing WS inDL-Lite, then give an intuition of our approach for capturing it inFO[2], and finally give an example of how the approach works. In the next section we formalize the approach.

ABox Evolution of aDL-LiteKBKwith an ABoxN is the set of modelsK N that may not have a canonical one [12]. This immediately yields thatK N cannot be described (aka axiomatized) in any language of theDL-Litefamily.

Example 2. We now illustrate the lack of canonical models inK¹ N¹from Example 1.

One can verify that any modelJ^canthat can be homomorphically embedded intoJ⁰,J¹, andJ²is such thatA^Jcan=R^Jcan=∅, ande, d /∈C^Jcan. It is easy to check that such a model does not belong toK¹ N¹. Hence, there is no canonical model inK N and it is inexpressible inDL-Lite.

K � N S0 S1

S³ S2

J0 J1 J²

J3

Mod(K(J⁰)) Mod(K(J1)) Mod(K(J2))

Mod(K(J³))

Fig. 1. Graphical representation of our approach to capture the result of evolution under WS.

A closer look at setsK N for differentKandN gave a surprising outcome: all of them satisfy the following property.

Theorem 3. K N can be divided (but in general not partitioned) into finitely many subsetsS⁰, . . . ,Sⁿof models, where eachSⁱhas a canonical modelJⁱ. Each of these canonical models is a minimal element inK N wrt homomorphisms.

We called theseJⁱsprototypes[12]. Thus, capturingK N in some logics boils down to(i)capturing eachSⁱwith some theoryK^Siand(ii)taking the disjunction across allK^Si. This will give the desired theoryK⁰ =K^S1∨ · · · ∨ K^Snthat capturesK N. Unfortunately, some ofK^Siare notDL-Litetheories (while they areFO[2]theories, see Section 5 for details).

We constructK⁰in two steps. First, we constructDL-LiteRKBsK(Jⁱ)for eachJⁱ such thatK(Jⁱ)is a sound approximations ofSⁱs, that is,Sⁱ⊆Mod(K(Jⁱ)). Second, based onKandN, we construct anFO[2]formulaΨ, which cancels out all the models inMod(K(Jⁱ))\ Sⁱ, that is,K^S0∨ · · · ∨ K^Sn=Ψ∧(K(J⁰)∨ · · · ∨ K(Jⁿ)).

To get a better intuition on our approach, consider Figure 1, where the result of evolutionK Nis depicted as the figure with solid-line borders (each point within the figure is a model inKN). Assume thatKN can be divided in four subsetsS⁰, . . . ,S³. To emphasize this fact,K N looks similar to a hand with four fingers, where each finger represents anSⁱ. Consider the left part of Figure 1. Each ofSⁱs has a canonical model depicted as a star. UsingDL-Lite_R, we can provide KBsK(J⁰), . . . ,K(J³)that are sound approximation of correspondingSⁱs. We depict the modelsMod(K(Jⁱ))as ovals with dashed-line boarders. Consider the right part of Figure 1. In this figure we depict in grey the modelsMod(K(Jⁱ))\ Sⁱthat are cut off byΨ.

Before proceeding to the next section where we formalize our approach, we introduce prototypes formally.

Definition 4. LetKbe a DL-Lite_RKB andN be an ABox. Aprototypal set forK N is a minimal subset

J

={J⁰, . . . ,Jⁿ}ofK N satisfying the property:

for everyJ ∈ K N there isJⁱ∈

J

^{such that}Jⁱ,→ J.

We call everyJⁱ ∈

J

^aprototype forK N. Note that prototypes generalize canonical models in the sense that every set of models with a canonical one, sayMod(K)for a DL-LiteRKBK, has a prototype, which is exactly the canonical model.

5 Computing Winslett’s Semantics When No Roles Interact

We first discuss some of the reasons of WS inexpressibility in our examples andDL-Lite_R.

BZP(K,N)

1. J0:=Align(I^can,N)∪ N, whereI^canis the canonical model ofK.

2. For eachR(a, b)∈AA(K,N), doJ0:=J0\ {R(a, b)},

if there is noR(a, β)∈ A \AA(K,N)doJ0:=J0\root^atT(∃R(a)).

3. ReturnJ0.

Fig. 2.The procedure of building zero-prototype

Dual-Affection of Roles. As we discussed in the previous section and illustrated in Example 1, sets of modelsK N that result from Winslett’s evolution do not have canonical models. We now give an intuitionwhyinK N canonical models are missing.

Observe that in Example 1 the roleRis affected by the old TBoxT¹as follows:

(i) T¹places(i.e., enforces theexistenceof)R-atoms in the evolution result, and on oneof coordinates of theseR-atoms, there are constants from specific sets, e.g., Av ∃RofT¹enforcesR-atoms with constants fromAon the first coordinate, and (ii) T¹forbidsR-atoms inK¹ N¹with specific constants on theothercoordinate,

e.g.,∃R⁻ v ¬CforbidsR-atoms withC-constants on the second coordinate.

Due to thisdual-affection(both positive and negative) of the roleR inT¹, we were able to provide an ABoxA¹ andN¹, which together triggered the case analyses of modifications on the modelI, that is,A¹ andN¹ weretriggersforR. Existence of such an affectedRand triggersA¹andN¹madeK¹ N¹inexpressible inDL-LiteR. Therefore, we now learn how to detect dually-affected roles in TBoxes and how to understand whether these roles are triggered by an ABox and a new (ABox) information.

Formally, letT be a TBox, a roleRisdually-affectedinT if for some conceptsA andBit holds thatT |=Av ∃RandT |=∃R⁻v ¬B. LetN be an ABox satisfiable withT, then a dually-affected roleRistriggeredbyN if there is a conceptBsuch that T |=∃R⁻v ¬BandN |=T B(b)for some constantb. The setTR(T,N)(or simply TR) is the set of all roles (dually-affected inT) that are triggered byN.

Description Logics DL-Lite_R^I. We now show a restriction ofDL-Lite_Rfor which we later present an algorithm to capture WS using prototypal set.DL-Lite_R^I (whereIstands for (mutual)independenceof roles) is a restriction ofDL-LiteR in 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.

Some interaction is still possible: role projections may contain the same concept. This restriction allows us to analyze evolution affecting roles independently for every role.

Components for Computation. We now introduce several notions and notations that we further use in the description of our algorithm. Analignmentof a modelIwithN, denotedAlign(I,N), is the interpretation:

Align(I,N) ={f |f ∈ Iandf is satisfiable withN }.

An auxiliary set of atomsAA(Auxiliary Atoms) that, due to evolution, should be deleted from the original KB and have some extra condition on the first coordinate is:

AA(T,A,N) ={R(a, b)∈fcl_T(A)| T |=Av ∃R,A |=T A(a),N |=T ¬∃R⁻(b)}. For the setTRwe define the set offorbidden atomsFA[T,A,N](Ri)of the original ABox as:

{D(c)∈fcl_T(A)| ∃R_i⁻(c)∧D(c)|=T ⊥,N 6|=T D(c), andN 6|=T ¬D(c)}.

BP(K,N,J0) 1. J^:={J0}.

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

for eachR= (Ri₁, . . . , Ri_k)such thatDj(cj)∈FA(Ri_j)forj= 1, . . . , kdo for eachB= (Ai1, . . . , Ai_k)such thatAj∈SC(Rj)do

J[D,R,B] :=h J0\Sk

i=1rootT(Di(ci))i

∪Sk i=1

h

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

J:=J∪ {J[D,R,B]}.

3. ReturnJ.

Fig. 3.The procedure of building prototypes inDL-LiteR^I based on the zero prototypeJ0

Consequently, the set of forbidden atoms for the entire KB(T,A)andN is FA(T,A,N) =∪^Ri∈TRFA(T,A,N)(Ri).

In the following we omit the arguments(T,A,N)whenever they are clear from the context. For a roleR, the setSC(R), whereSCstands forsub-concepts, is a set of those concepts which areimmediatelyunder∃Rin the concept hierarchy generated byT: SC(R) ={A| T |=Av ∃Rand there is noA⁰s.t.T |=AvA⁰andT |=A⁰ v ∃R}. Iff is an ABox assertion, thenroot^at_T(f)is a set of all the atoms thatT-entailf. For example,A(x)∈root^at_T(∃R(x))ifT |=Av ∃R.

We are ready to proceed to construction of prototypes.

Constructing Zero-Prototype. The procedureBZP(K,N)(Build Zero Prototype) in Figure 2 constructs the main prototypeJ⁰forKandN fromDL-Lite_R^I, which we call zero-prototype. Based onJ⁰we will construct all the other prototypes. To buildJ⁰one has to align the canonical model ofKwithN, and then delete from the resulting set of atoms all the auxiliary atomsR(a, b)(fromAA(K,N)). In the case whennoR(a, β)for some constantβsuch thatR(a, β)∈AA(K,N)is in the canonical model, we also delete atomsroot^at_T(∃R(a)), since their presence in the model and the absence ofR-atoms with aat the first coordinate would contradict the TBox.

Constructing Other Prototypes. The procedureBP(K,N,J⁰)(Build Prototypes) of constructing

J

for the case ofDL-Lite_R^I, takes J⁰ and manipulates with it by first dropping atoms fromFAand then adding atoms in order to compensate the dropped ones so that the result is an evolvedmodelunder WS. It can be found in Figure 3

We conclude the discussion on the algorithms with a theorem:

Theorem 5. LetK= (T,A)be a DL-Lite_R^I KB, andN a DL-LiteRABox consistent withT. Then the setBP(K,N, BZP(K,N))is a prototypal set forK N.

Continuing with Example 1, it is easy to check that the prototypal set forK¹andN¹ is{J⁰,J¹,J²,J³}, whereJ⁰,J¹, andJ²are described in the example and

J³: A^I ={x, y}, C^I ={b}, R^I ={(x, d),(y, e)}.

We proceed to correctness of BP in capturing evolution inDL-Lite_R^I, where we use the following setFC[T,A,N](Ri) ={c|D(c)∈FA[T,A,N](Ri)}, that collects all the constants that participate in the forbidden atoms.

Theorem 6. LetK= (T,A)be a DL-Lite_R^I KB,N a DL-LiteRABox consistent withT, andBP(K,N, BZP(K,N)) ={J⁰, . . . ,Jⁿ}is a prototypal set forK N. Then

K N =Mod(T)∩Mod(A⁰∨. . .∨ Aⁿ)∩Mod(Φ∧Ψ), whereA_iis a DL-LiteRABox such thatJⁱis a canonical model for(T,Aⁱ), and

Φ= ^

R_i∈TR

^

c_j∈FC[R_i]

∀x.ˆ`

Ri(x, cj)→(root^atT(∃Ri(x))6=∅)´

∧

∀y.(Ri(x, cj)∧Ri(x, y)→y=cj)],

Ψ= ^

R(a,b)∈S_at

∃R(a)→root^atT(∃R(a))∩fcl_T(A).

TheAⁱ mentioned in Theorem 6, can be constructed in the similar way that the corresponding prototypesJⁱ, taking the original ABoxAinstead ofI^can. Note that an ABox may include a negative literals, like¬B(c). Those should be treated in the same way that the positive literal (atoms) are. We will denote such an ABox asA[Jⁱ].

Theorem 7. A prototypeJⁱis a canonical model of the KB(T,A[Jⁱ]).

Continuing with Example 1, the ABoxesA[J⁰]andA[J¹]are as follows:

A[J⁰] ={C(d), C(e), C(b)}; A[J¹] ={A(x), C(e), C(b), R(x, d)}. A[J²]andA[J³]can be built in the similar way. Note that onlyA[J⁰]is inDL-Lite_R, while writingA[J¹], . . . ,A[J³]requires variables in ABoxes. Variables, also known assoft constants, are not allowed inDL-LiteR ABoxes, while present inDL-LiteRS

ABoxes. Soft constantsxare constants not constrained by the Unique Name Assumption:

it is not necessary thatx^I =x. SinceDL-LiteRSis tractable andFOrewritable [13], expressingA[J¹]inDL-LiteRSinstead ofDL-LiteRdoes not affect tractability.

6 Computing Winslett’s Semantics with Roles Interaction

The algorithm BP for constructing prototypal set works only when roles do not interact.

The following example illustrates that it does not work in a general case.

Example 8. Consider a KBK²= (T²,A²)and a new ABoxN²={C(b)}: TBoxT²: ∃R⁻v ¬∃P⁻, ∃R⁻v ¬C, Av ∃R, B v ∃P;

ABoxA²: R(a, b), A(a), R(f, g), A(f), P(c, d), B(c), C(e).

One can check that the following modelJ⁰is inK² N²:

A^J0 ={y}, B^J0 ={z}, C^J0 ={b, e}, R^J0 ={(y, d)}, P^J0 ={(z, g)}. At the same time, BP overK²andN²returns the following four prototypes only:

A^Ji B^Ji C^Ji R^Ji P^Ji

i= 0 {f} {c} {b, e} {(f, g)} {(c, d)}

i= 1 {f, x} {c} {b} {(f, g),(x, e)} {(c, d)}

i= 2 {f, y} ∅ {b, e} {(f, g),(y, d)} ∅ i= 3 {f, x, y} ∅ {b} {(f, g),(x, e),(y, d)} ∅

wherexandyare fresh constants. It is easy to see that none ofJⁱs is homomorphically embeddable inJ⁰. Thus, BP does not captureJ⁰and it is incomplete.

BPrec(K,N) 1. ComputeJ^:=BP(K,N, BZP(K,N)).

2. Repeat

J^0:=J;

for eachJ ∈J⁰doJ:=J∪BP(K,N ∪S,J),

whereS ={f∈ J |a fresh constant from∆\adom(K)appears inf};

untilJ⁼J⁰. 3. ReturnJ.

Fig. 4.The procedure of building prototypes inDL-LiteR

6.1 Recursive BP Algorithm.

For generalDL-LiteR KBs, BP algorithm does return prototypes but not all of them.

The reason is: when, while constructing prototypes with BP, we delete a forbidden atom (an atom fromFA), it may trigger another dually-affected role and such triggering may require further modifications, which are not accounted by BP. In order to compute all prototypes we should run BP recursively: considering the prototypes obtained at the previous step as zero ones. We present a recursive algorithmBP_rec for building prototypes for generalDL-Lite_R KBs in Figure 4. The following theorem shows the correctness of the algorithm.

Theorem 9. LetK = (T,A)be a DL-LiteRKB andN a DL-LiteR ABox consistent withT. Then the algorithmBP_rec(K,N)terminates and returns the finite set which is a prototypal set forK N.

We illustrateBP_recon the following example.

Example 10. Consider KBK²= (T²,A²)and a new ABoxN²from Example 8. Let us computeBPrec(K²,N²). First we runBP(K,N,J⁰)and it returns four prototypes:J⁰, J¹,J², andJ³(see Example 8). Now we apply theBPprocedure toJ¹,J², andJ³. It is easy to see thatBP(K,N ∪ {A(x), R(x, e)},J¹) =∅, since no role atom except forR(a, b)was affected. ConsiderBP(K,N ∪ {A(y), R(y, d)},J²): it consists of the only prototypeJ⁴:

A^J4 ={y}, B^J4 ={z}, C^J4 ={b, e}, R^J4 ={(y, d)}, P^J4 ={(z, g)}. The uniqueness of the prototype follows from the fact that the role atom that was affected inJ²isP(c, d)andFA[T,A,N ∪ {A(y), R(y, d)}](P) ={∃R⁻(g)}. Finally, runningBP(T,N ∪ {A(y), R(y, d), B(z), P(z, g)},J⁴)we obtain a prototypeJ⁵:

A^J5 ={y, v}, B^J5 ={z}, C^J5 ={b}, R^J5 ={(y, d),(v, e)}, P^J5 ={(z, g)}. Note thatBP(T,N ∪{A(y), R(y, d), B(z), P(z, g), A(v), R(v, e)},J⁵) =∅. Anal- ogously,J⁶can be obtained by runningBP(K,N ∪{A(x), A(y), R(x, e), R(y, d)},J³):

A^J6={x, y}, B^J6 ={z}, C^J6 ={b}, R^J6 ={(x, e),(y, d)}, P^J6={(z, g)}. Thus, the prototypal set

J

^forK N is{Jⁱ}⁶i=0.

We conclude with the theorem thatBP_recgives a sound approximation for WS.

Theorem 11. LetK= (T,A)be a DL-LiteRKB,N a DL-LiteRABox consistent with T, andBP_rec(K,N) ={J⁰, . . . ,Jⁿ}is a prototypal set forK N. Then

K N ⊆Mod(T)∩Mod(A⁰∨. . .∨ Aⁿ)∩Mod(Φ∧Ψ),

whereAiis a DL-LiteRABox such thatJⁱis a canonical model for(T,Aⁱ)andΦand Ψ are as they defined in Theorem 6.

6.2 Closure Under Evolution and Approximation

Next theorem allows us to approximate results of evolution under WS, sinceFO[2]is decidable.

Theorem 12. K N under WS for KBs in DL-LiteRcan be captured inFO[2].

As a future work we are going to study ways to approximate the resultedFO[2]

theories inDL-Lite.

Finally, we discuss cases when the result of Winslett’s evolution is expressible inDL-LiteR. The following formulas appearing in Theorem 6 are not expressible in DL-Lite_R:(i)the disjunction of the ABoxesA⁰∨. . .∨ Aⁿ and(ii)formula Φ∧Ψ. The disjunction of ABoxes becomes expressible when it is of the length one, i.e., there is the only prototype:J⁰. The last statement yields thatFC = ∅and therefore Φis always true. The formulaΨ becomes trivially true whenAA=∅, i.e., for every atom R(a, b)∈fcl_T(A)eitherN 6|=T ¬∃R⁻(b)orroot^at_T(∃Ri(ai))∩fcl_T(A) =∅. As one can see, the condition of expressibility of the result inDL-LiteR(emptiness ofFAand AA), depends on a TBox, an ABox, and a new information. Hence, if we do a chain of evolution, at some step the result may be not expressible inDL-Lite_R. Since TBox stays unchangeable, to guarantee the expressibility we need to find TBoxesT such that (T,A) N is expressible inDL-LiteRfor everyAandN. A condition that guarantees the emptiness ofFAandAAis: for every roleR∈Σ(K∪N)at least one of the following items holds:(1)there is no conceptCsuch thatT |=∃R⁻ v ¬C, or(2)there is no conceptAsuch thatT |=A v ∃R. The former conditions gives thatTR =∅ since N 6|=T ¬∃R⁻(b), which leads toFA=AA=∅. The latter one yields thatSC(R) =∅, thereforeTRagain is empty.

As a practical summary of this section, given a KBKand a new ABoxN, one can check (in polynomial time) whether any dually-affected role is “triggered” byN. If it isnotthe case, one can compute (in polynomial time) an evolved KBK⁰ that exactly capturesK N. Otherwise, it is the case thatK N is inexpressible inDL-LiteR. Thus, one can compute anFO[2]theory that capturesK N and then approximate it in DL-Lite_R, by, for example, dropping all the notDL-Lite_Rformulas. We will not focus on approximation in this paper.

7 Conclusion

We studied how to capture ABox evolution for DL-Lite_R under WS. In general the result of evolution requires constructs that are not present inDL-LiteR, and even not inDL-Lite, such as disjunction. Moreover, in general the result of evolution, which is a set of models, does not even have a canonical model, which should always exist for

anyDL-Litetheory. It turned out that the inexpressibility is caused by a condition on the TBox level, which we called dual-interaction: by pairs of assertions of the formAv ∃R and∃R⁻ v ¬B. In order to capture evolution results in the presence of dual-interactions, we introduced prototypes. Our approach is based on the observation that evolution results can be divided into a finite number of subsets and each of them has a canonical model, i.e., a prototype. These subsets can be captured by theories guided by prototypes and the disjunction of these theories, compensated with two formulas, captures evolution results and is inFO[2]. We proved that this technique works forDL-LiteR. We are currently working on efficient approximation of the obtainedFO[2]theory inDL-Lite and on extending results to capture evolution for otherDL-Litelanguages.

Acknowledgements We are thankful to Diego Calvanese and Werner Nutt for insightful discussions. The authors are supported by EU projects ACSI (FP7-ICT-257593), Ontorule (FP7-ICT-231875). The first author was supported by the ERC FP7 grant Webdam (agreement n. 226513). Part of this work was carried out while the first author was visiting Department of Computer Science of Oxford University.

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