Revision of DL-Lite Knowledge Bases

Revision of DL-Lite Knowledge Bases

Zhe Wang, Kewen Wang, and Rodney Topor Griffith University, Australia

Abstract. We address the revision problem for knowledge bases (KBs) in De- scription Logics (DLs). This problem has received much attention in the ontology management and DL communities, but the existing proposals are restricted in sev- eral ways. In this paper we develop a formal framework for revision of DL-Lite KBs, using techniques that are analogous to those for model-based revision in propositional logic. However, unlike propositional logic, a DL-Lite KB can have infinitely many models, which makes it hard to define and compute revision in terms of models. For this reason, we first develop an alternative semantic charac- terization for DL-Lite by introducing the concept of a feature and then define a specific revision operator for DL-Lite KBs based on features (instead of models).

In contrast to previous approaches, we tackle the problem of revision between KBs, and the result of revision is always an unique DL-Lite KB. We also present an algorithm for computing KB revisions in DL-Lite.

1 Introduction

An ontology is a formal description of a common vocabulary for a domain of interest and the relationships between terms built from the vocabulary. Ontologies have been ap- plied in a wide range of practical domains such as medical informatics, bio-informatics and, more recently, the Semantic Web. Description Logics (DLs) have been used as underlying formalisms of most ontology languages recently. A DL-based ontology is usually expressed as a DL-knowledge base (KB), which consists of a TBox (termino- logical box) and an ABox (set of assertions).

DL-Lite [1, 3] is specifically designed as a family of lightweight DLs with rela- tively restricted expressive power but efficient ontological reasoning algorithms. In par- ticular, logics in the DL-Lite family have polynomial time computational complexity with respect to standard reasoning tasks, and LogSpace data complexity with respect to complex query answering. With reasonable expressive power and efficient reason- ing support, especially on answering complex queries over large amounts of data, the DL-Lite family has proved to be an appealing formalism for ontology development.

A challenge in ontology maintenance is that ontologies are not static but may evolve over time. Because of limitations in the initial design, changes in users’ requirements, need to reflect changes in the real world, and so on, there is a need to revise ontologies in accordance with the new knowledge. Recently, intensive interest has been shown on revision/update in DLs. Earlier approaches concentrated on adapting revision postu- lates for propositional logic to DLs but no specific revision operators were provided, e.g. [4, 5]. Recently, there have been several attempts to define specific revision/update operators for DLs including [7, 9, 11–16]. In particular, an update operator for DL-Lite

ABoxes is defined in [7]. In fact, most specific revision/update operators for DLs can deal only with ABoxes, like [7, 9, 11], or only with TBoxes, like [13, 14]. As shown in [8, 14, 16], thekernel revisionoperator for propositional logic can be adapted to most DLs in a straightforward way, but the result of revision is not an unique KB in most cases and thus arbitrary choices by the user are required.

It is worth mentioning that update is traditionally distinguished from revision in that update addresses changes of the actual state of the world (e.g., that resulting from an action) whereas revision addresses the incorporation of new knowledge about the world. In this paper, we focus on the revision problem.

In propositional logic, the area ofbelief revision[6] is well developed, with nu- merous proposals and techniques for knowledge base revision. Among them, Satoh’s revision operatorϕ∗sµ[17] is well known. In his approach, the set of models of a re- vised knowledge baseK⁰are those models ofK⁰that are ‘closest’ to those of the initial knowledge baseK, where the distance between two models is based on their symmetric difference.

However, such model-based approaches cannot be applied directly to KB revision in DLs. The difficulties in defining and computing model-based revision of KBs in DLs are that: (1) unlike a propositional theory, a KB in a DL may have infinitely many models; (2) a model of a KB can be infinite; and (3) given a collectionM of models, there may not exist an unique KBKsuch thatM is exactly the set of all models ofK.

In this paper, we aim to define a rational revision operator for DL-Liteboolontologies in a way analogous to the model-based approaches in propositional logic. In order to achieve this goal, we first define the concept offeaturesfor DL-Litebool, which precisely capture most important semantic properties of DL-Litebool KBs, and are always finite.

We adapt the techniques of model-based revision in propositional logic to the revision of DL-Litebool KBs, and define a specific revision operator based on the distance between features. We also show that the revision operator possesses desirable properties, and can be computed through syntactic methods.

Although our approach is based on DL-Litebool [1], we remark that the belief revi- sion method we propose also applies to other logics in DL-Lite. In contrast to previous approaches, we address the problem of defining and computing revision of a DL-Lite KB by another KB. Also, the result of our revision is always an unique KB in DL-Lite.

Due to space limitation, proofs are omitted in this paper but a longer version of this paper with complete proofs can be found athttp://www.cit.gu.edu.au/

˜kewen/revision_long.pdf.

2 DL-Lite Family

Asignatureis a finite setS = S_C ∪ S_R∪ S_O∪ S_N whereS_C is the set of atomic concepts,S_Ris the set of atomic roles,S_O is the set of individual names (or, objects) andSN is the set of natural numbers inS. We assume1is always inSN.>and⊥will not be considered as atomic concepts or atomic roles. Formally, given a signatureS, a

DL-Litebool languageLhas the following syntax:

R←−P |P⁻

B←− > | ⊥ |A| ≥n R C←−B | ¬C|C1uC2

wheren ≥ 1,A ∈ SC andP ∈ SR.B is called abasic concept andC is called ageneral concept. We write∃R as a shorthand for≥ 1 R, and let R⁺ = P, where P ∈ SR, wheneverR=PorR=P⁻.

Aliteralis a basic concept or its negation. The set of all literals onSis denoted as LitS, and the set of all basic concepts is denoted asLit⁺_S.

A DL-Litebool TBoxT is a finite set of concept inclusions of the formC1 v C2, whereC₁andC₂are general concepts. A DL-Litebool ABoxAis a finite set of mem- bership assertions of the formC(a), R(a, b), wherea, bare individual names. We call C(a)a concept assertion andR(a, b)a role assertion. A DL-Litebool knowledge base (KB) is a pairK=hT,Ai.

The semantics of a DL-Lite KB is given by interpretations. An interpretationIis a pair(∆^I,·^I), where∆^Iis a non-empty set called thedomainand·^Iis an interpretation function which associates each atomic conceptAwith a subsetA^Iof∆^I, each atomic roleP with a binary relation P^I ⊆ ∆^I ×∆^I, and each individual nameawith an element a^I of∆^I such thata^I 6= b^I for each pair of individual namesa, b(unique name assumption).

The interpretation function·^Ican be extended to general concept descriptions:

(P⁻)^I={(a^I, b^I)|(b^I, a^I)∈P^I} (≥n R)^I={a^I | |{b^I|(a^I, b^I)∈R^I}| ≥n}

(¬C)^I=∆^I\C^I (C1uC2)^I=C₁^I∩C₂^I

An interpretationI is a model of inclusionC1 vC2ifC₁^I ⊆C₂^I;Iis a model of an assertionC(a)ifa^I ∈C^I;I is a model of an assertionR(a, b)) if(a^I, b^I)∈R^I).

I is a model of a TBoxT (or ABoxA) ifI is a model of each inclusion inT (resp., each assertion inA). Two inclusions (or resp., assertions, TBoxes, ABoxes) are said to beequivalentif they have exactly the same models.

Iis a model of a KBhT,Ai, ifIis a model of bothT andA. A KBKis consistent if it has at least one model. Two KBsK1,K2that have the same models are said to be equivalent, denotedK1 ≡ K2. A KBKlogically implies an inclusion or assertionα, denotedK |=α, if all models ofKare also models ofα. We will useMod(K)to denote the set of models of a KBK.Sig(K)is the signature ofK.

As in propositional logic, a DL-Litebool conceptCcan always be transformed into an equivalent disjunction of conjunctions of literal concepts (DNF).

3 Features in DL-Lite

In this section, we introduce the concept of features in DL-Lite_bool, which provides an alternative semantic characterization for DL-Litebool. An advantage of semantic fea- tures over models is that the number of all features for a DL-Litebool knowledge base

is finite and each feature is finite as we will see later. These finiteness properties make it possible to recast key approaches to revision for classical propositional logic into DL-Lite_bool.

Features for DL-Litebool are based on the notion oftypesdefined in [10].

In the following sections, we assumeS is a fixed signature. Recall that aS-literal (or, literal) is a basic concept onSor its negation.LitS is the set of allS-literals.

AS-type is a subsetτofLitScontaining>and satisfying the following conditions:

– for everyS-literalL,L∈τiff¬L6∈τ;

– for anym, n∈ SN withm < n,≥n R∈τimplies≥m R∈τ;

– for anym, n∈ SN withm < n,¬(≥m R)∈τimplies¬(≥n R)∈τ.

A typeτcan be equivalently represented by the conjunction of all literals inτ, which we also call atypewhen the context is clear and no confusion is caused. An important observation is that every general conceptCoverS corresponds to a setTs(C)of S- types, in light of the fact thatCcan be transformed into an equivalent disjunction of S-types. For example, conceptAu(Bt ∃P)can be transformed into(AuBu ∃P)t (Au ¬Bu ∃P)t(AuBu ¬∃P).

Informally speaking, types correspond to propositional models whereasTs(C)cor- respond to the set of models of formula C. For simplicity, we assume that a type τ consists of only its basic concepts, following similar common practices for proposi- tional models. For example, letS_C={A, B},S_R={P}, andS_N ={1,3}. Thenτ= {A,∃P,≥3P,∃P⁻}is a type instead ofτ = { >, A, ¬B,∃P,≥ 3P,∃P⁻,¬(≥

3P⁻)}.

We say a typeτsatisfies a conceptCifτ ∈Ts(C), andτsatisfies concept inclusion C₁ vC₂ifτ ∈Ts(¬C₁tC₂). In this way, we define thatτsatisfies a TBoxT if it satisfies each inclusion inT.

However, in order to capture the membership relations in a KB, we need to extend the notion of types and thus defineHerbrand setsin DL-Lite.

Definition 1. A S-Herbrand set H is a finite set of assertions of the form B(a) or P(a, b), wherea, b∈ SO,B ∈Lit⁺_S,P∈ SR, satisfying the following conditions

1. for eacha∈ SO, ifB1(a), . . . , Bk(a)are all the concept assertions aboutainH, then the set{B1, . . . , Bk}forms anS-type.

2. for eachP ∈ SR, ifP(a, bi)(1≤i≤n) are all the assertions inHof such form, then for anym∈ SN withm≤n,(≥m P)(a)is inH;

3. for eachP ∈ SR, ifP(bi, a)(1≤i≤n) are all the assertions inHof such form, then for anym∈ SN withm≤n,(≥m P⁻)(a)is inH.

Informally,B(a)6∈ HwithB∈Lit⁺_S means that¬B(a)holds inH, andP(a, b)6∈

HwithP ∈ SR means thata, bare not related byP inH. Thus, Herbrand sets are different from ABoxes by adopting a close world assumption. It is not hard to see that, conditions 2 and 3 in Definition 1 preserve consistency of a Herbrand set. And condi- tion 2 in Definition 1 requires that≥n P ∈ Himplies≥m P ∈ Hfor allm∈ S_N withm < n. These conditions are introduced mainly for simplifying the definition of our revision operator in the following section.

For simplicity, we denote the group of all the concept assertionsB₁(a), . . . , B_k(a) aboutainHasτ(a), whereτ ={B₁, . . . , B_k}is a type. We sayτ(a)is inHifB_i(a) is inHfor all1≤i≤k.

Herbrand sets can be used to capture membership relations in a KB. We say a Her- brand setHsatisfies a membership assertionC(a)ifτ(a)is inHandτ ∈Ts(C); and Hsatisfies assertionP(a, b)(resp.,P⁻(b, a)) ifP(a, b)(resp.,P⁻(b, a)) is in H.H satisfies an ABoxAifHsatisfies all the assertions inA.

To provide an alternative characterization for reasoning in a KB, we could use pairs hτ,Hi, whereτis a type andHis a Herbrand set, to replace standard interpretations, such thathτ,Hisatisfies KBhT,AiifτsatisfiesT andHsatisfiesA. To this end, we want it to guarantee thatKis consistent iff there exists some pairhτ,HisatisfyingK.

However, the following example shows that this approach does not work.

Example 1. LetK=h{ ∃P⁻v ⊥ },{ ∃P(a)}ibe a DL-Lite_boolKB. Obviously,Kis inconsistent and thus has no model. However, letS ={P, a,1}andτ ={∃P}, then hτ,{τ(a)}isatisfiesK.

The problem with using pairs hτ,Hi as alternative semantic characterization for DL-Lite is that a single typeτis not sufficient to capture the semantic connection be- tween the TBox and the ABox of a KB. Our investigation shows that it is plausible to use a pair of a set of types and a Herbrand set, as an alternative semantic characteriza- tion for a DL-LiteboolKB. Using sets of types as semantic characterizations of DL-Lite TBoxes is also suggested in [10].

Thus, we introduce the definition offeaturesas follows.

Definition 2. Given a signatureS, afeature onS, or simplyS-feature, is defined as a pairF =hΞ,Hi, whereΞ is a non-empty set ofS-types andHaS-Herbrand set, such that the following conditions are satisfied:

1. for eachP ∈ SR,∃P ∈S

Ξ iff ∃P⁻∈S Ξ;

2. for eacha∈ SOandτ(a)inH, s. t.τis anS-type,τ ∈Ξ.

Informally, the first condition in Definition 2 says that∃Pis a satisfiable concept in F if and only if∃P⁻ is also a satisfiable concept. And the second condition requires that the interpretation of the ABox must be in accordance to the TBox.

We use a featureF=hΞ,Hias an alternative for an interpretation in DL-Litebool: – Fsatisfies an inclusionC1vC2overS, ifΞ⊆Ts(¬C1tC2).

– Fsatisfies an assertionC(a)overS, ifτ(a)is inHandτ∈Ts(C).

– Fsatisfies assertionP(a, b)(resp.,P⁻(b, a)) overS, ifP(a, b)is inH.

We sayFis amodel featureof DL-LiteboolKBKifFsatisfies each inclusion and each assertion inK.ΩKdenotes the set of all model features ofK.

From the definition of features, a model feature is finite and the number of model features of a KB is also finite.

Example 2. Consider the KBK =hT,Ai, whereT ={Av ∃P, B v ∃P,∃P⁻ v B, AuBv ⊥,≥2P⁻ v ⊥ }andA={A(a), P(a, b)}. It is shown in [2] thatKis a KB having no finite model.

An infinite modelI of K is defined as follows: ∆^I = {d_a, d_b, d₁, d₂, d₃. . .}, a^I = d_a andb^I = d_b; the concept A is interpreted as a singleton {d_a} and B as

{d_b, d₁, d₂, d₃. . .}; and rolePis interpreted as{(d_a, d_b),(d_b, d₁),(d₁, d₂), . . . ,(d_i, d_i+1), . . .}.

TakeS =Sig(K) ={A, B, P,1,2, a, b}. The (finite) model feature ofKthat cor- responds toI isF = hΞ,Hiwhere the Herbrand setH ={τ₁(a), τ₂(b), P(a, b)} andΞ ={τ1, τ₂}withτ₁={A,∃P}andτ₂={B,∃P,∃P⁻}.

Given an inclusion or assertionαoverS =Sig(K), we defineK |=_f αif for each F ∈ Ω_K,F satisfiesα. Given two KBsK₁ andK₂, letS = Sig(K₁∪ K₂), define K₁|=_f K₂ifΩ_K1 ⊆Ω_K2; andK₁≡_f K₂ifΩ_K1 =Ω_K2.

The following two results show that model features do capture the semantic proper- ties of DL-Litebool KBs.

Proposition 1. LetKbe a DL-Litebool KB andS=Sig(K). Then 1. Kis consistent iffKhas a model feature;

2. for any concept inclusionC1vC2overS,K |= (C1vC2)iffK |=f (C1vC2);

3. for any membership assertionC(a)(resp.,R(a, b)) overS,K |=C(a)iffK |=f

C(a)(resp.,K |=R(a, b)iffK |=f R(a, b));

Proposition 2. LetK1,K2be two DL-Litebool KBs andS=Sig(K1∪ K2). Then 1. K1|=K2iffK1|=_fK2;

2. K1≡ K2iffK1≡f K2.

In summary, the advantages of model features over standard models can be de- scribed as follows. First, each model feature of a KB is finite in structure. Second, there are a finite number of model features for each KB. Third, each non-empty set of model features determines a unique KB up to KB equivalence. In particular, given a non-empty setΩofS-features, we can always construct an unique DL-Litebool KBKoverSsuch thatΩ⊆Ω_KandΩ_Kis minimal among all supersetsΩ_K⁰ ofΩ. In Section 5, we will introduce a method for constructing such a DL-Lite_boolKB from a set of features.

4 Feature-based Revision

In this section, we introduce a specific revision operator for DL-Lite KBs based on model features rather than models.

Before introducing the definition of our feature-based revision operator, we first ex- tend the definition of symmetric difference4so that it can be used for model features.

Recall thatS₁4S2= (S₁−S2)∪(S2−S1)for any two setsS₁andS₂. Given twoS- featuresF1=hΞ1,H1iandF2=hΞ2,H2i, thedistancebetweenF1andF2, denoted F₁4F₂, is a pairhΞ₁4Ξ₂,H₁4H₂i. Note thatH₁4H₂may not be a Herbrand set anymore. For example, let S = {P, a, b₁, b₂, b₃,1,2,3},H₁ = {∃P(a), P(a, b₁)}

andH₂ = {∃P(a), (≥ 2 P)(a), P(a, b₂), P(a, b₃)}. Then we getH₁4H₂ = {(≥

2 P)(a), P(a, b1), P(a, b2), P(a, b3)}, which is not a Herbrand set because it does not contain∃P(a)and(≥3P)(a).

To compare two distances, we could defineF₁4F₂ ⊆_d F₃4F₄ ifΞ₁4Ξ₂ ⊆_d Ξ₃4Ξ₄ and H₁4H₂ ⊆_d H₃4H₄, where F_i = hΞ_i,H_ii for i = 1,2,3,4; and F₁4F₂ ⊂_d F₃4F₄ifF₁4F₂ ⊆_d F₃4F₄ andF₃4F₄ 6⊆_d F₁4F₂. However, we will show in an example later that such a measure is too weak for KB revision and can- not preserve enough information. Instead, we set a preference on Herbrand sets over type sets:F14F2⊂dF34F4iff

1. H14H2⊂ H34H4, or

2. H₁4H₂=H₃4H₄andΞ₁4Ξ₂⊂Ξ₃4Ξ₄.

Note that here we set a preference on Herbrand set over type set, such that asser- tions in the original ABox are prior to be preserved than inclusions in the TBox. The rationale behind this preference is that assertions represent more specific information than inclusions and thus from the view point of knowledge representation, should have higher priority.

According to our new semantic characterization for DL-LiteboolKBs, the semantics of each KBKis determined by the setΩ_Kof all model features ofK.

Given two KBsKandK⁰, to define our revision operatorK ◦ K⁰, we need to specify the subset ofΩ_K⁰ that is closest toΩ_K.

LetΩ1andΩ2be two sets of features. Then we define σ(Ω1, Ω2) ={ F1∈Ω1|there existsF2∈Ω2s.t.

(F₁⁰4F₂⁰)6⊂d(F14F2)for allF₁⁰ ∈Ω1andF₂⁰ ∈Ω2}

Now we are ready to define our revision operator in terms of model features.

Definition 3. LetK,K⁰ be two DL-Litebool KBs andS =Sig(K ∪ K⁰). The(feature- based) revisionofKbyK⁰is defined as the DL-Litebool KBK ◦ K⁰such that ifΩK=∅, thenΩK◦K⁰=ΩK⁰; otherwise

1. σ(Ω_K0, Ω_K)⊆Ω_K◦K0, and

2. Ω_K◦K⁰ is minimal in the sense that, for any DL-LiteboolKBK⁰⁰satisfying the above condition 1, Ω_K◦K⁰ ⊆Ω_K⁰⁰.

Note that the definition is slightly different from Satoh’s asσ(Ω_K⁰, Ω_K)may not correspond exactly to a DL-Lite_boolKB. By requiringΩ_K◦K⁰to be the minimal superset ofσ(Ω_K⁰, Ω_K), we defineK ◦ K⁰in a conservative manner. Another option is to define Ω_K◦K⁰ to be a maximal subset ofσ(Ω_K⁰, Ω_K). However, asΩ_K◦K⁰ can be very small if defined this way, there is a risk that unexpected logical consequences are introduced.

Thus, it is more plausible to use supersets instead of subsets.

For any two DL-Litebool KBsK1,K2that are both revisions ofKbyK⁰, we have Ω_K1 =Ω_K2. By Proposition 2, this definition uniquely defines the result ofK ◦ K⁰up to equivalence. We will show in Section 5 thatK ◦ K⁰always exists.

Example 3. Consider a DL-Litebool KB

K=h {PhD vStudent,Student v ¬∃teaches},{PhD(Tom)} i.

The TBox ofKspecifies that PhD students are students, and students are not allowed to teach any courses, while the ABox states thatTom is a PhD student. Suppose PhD students are actually allowed to teach, and we want to reviseKwithK⁰ =h {PhD v

∃teaches},∅ i.

ThenF⁰=h {τ1},{τ1(Tom)} i, whereτ1={Student}is a model feature ofK⁰. TakeF=h {τ1, τ2}, {τ2(Tom)} iwhereτ2={PhD,Student}. It can be verified that Fis a model feature ofKand it is closest toF⁰.

The distance betweenF and F⁰ is a minimal one among those between model features ofKandK⁰. Thus,F⁰is a model feature ofK ◦ K⁰.

In fact, we can show thatK◦K⁰ish {PhDvStudent,PhDv ∃teaches,Studentu

∃teaches vPhD},{Student(Tom)} i.

In Example 3, the new inclusion added causes conceptPhD to be unsatisfiable, i. e.,PhD must be interpreted as an empty set, and thus contradictsPhD(Tom). The contradiction is resolved through revision by replacingStudent v ¬∃teaches with Student u ∃teaches vPhD. Also,PhD(Tom)is weaken to beStudent(Tom), be- cause¬∃teaches(Tom)is also (implicitly) expressed inK. However, if we treat type sets and Herbrand sets equally in measuring distances, as we can show,Student(Tom) will be lost from the result of revision.

The standard AGM postulates (R1) – (R6) for propositional belief revision have been adapted to DLs in [15]. However, the authors present the postulates in terms of models of KBs. In the following, we reformulate them using KB combinations and entailments, in a manner analogous to classical AGM postulates.

(R1) K ◦ K⁰|=K⁰;

(R2) ifK ∪ K⁰is consistent, thenK ◦ K⁰ ≡ K ∪ K⁰; (R3) ifK⁰is consistent, thenK ◦ K⁰is consistent;

(R4) ifK1≡ K2andK⁰₁≡ K⁰₂, thenK1◦ K₁⁰ ≡ K2◦ K⁰₂;

(R5) ifΩ_K◦K⁰ =σ(Ω_K⁰, Ω_K), then(K ◦ K⁰)∪ K⁰⁰|=K ◦(K⁰∪ K⁰⁰);

(R6) if(K ◦ K⁰)∪ K⁰⁰is consistent, thenK ◦(K⁰∪ K⁰⁰)|= (K ◦ K⁰)∪ K⁰⁰.

The following theorem states the first five postulates are satisfied by our revision operator.

Theorem 1. The revision operator defined in Definition 3 satisfies postulates (R1) – (R5).

Note that (R5) is conditional, and in this sense varies from its corresponding classi- cal AGM postulate. We remark that the condition in (R5) cannot be removed. That is, (K ◦ K⁰)∪ K⁰⁰|=K ◦(K⁰∪ K⁰⁰)is not always satisfied. Also, (R6) is not always satisfied by our revision operator, as with Satoh’s revision operator in propositional logic. This can be seen from the following example.

Example 4. LetK = h {A v B, B v C}, {A(a)} i, K⁰ = h {B v A, A v

¬C}, {(AtC)(a)} i, andK⁰⁰=h ∅, { ¬B(a)} i. ThenK ◦ K⁰ =h {AvB, Bv A, Av ¬C}, {(AtC)(a)} i, and after combined withK⁰⁰,(AtC)(a)in the ABox is replaced withC(a). However,K◦(K⁰∪K⁰⁰) =h {BvC, BvA, Av ¬C}, {(At C)(a)} i. That is,(K ◦ K⁰)∪ K⁰⁰6|=K ◦(K⁰∪ K⁰⁰)andK ◦(K⁰∪ K⁰⁰)6|= (K ◦ K⁰)∪ K⁰⁰.

5 Algorithm for Computing Revision

In this section, we investigate the problem of computing revision and as a result, present an algorithm that can compute the result of revision syntactically.

The revision operator is based on model features of KBs. In what follows, we in- troduce a method for computing all the model features of a DL-Litebool KBK, by first computing the assembling of all the model features, which can be obtained directly fromKthrough syntactic transformations.

Given two featuresF₁ =hΞ₁,H₁iandF₂ =hΞ₂,H₂i, we define theassembling ofF₁andF₂asF₁⊕ F₂=hΞ₁∪Ξ₂,H₁⊕ H₂i, whereH₁⊕ H₂is called amultiple Herbrand set, consisting of:

– {τ₁, τ₂}(a), for eacha∈ S_O,τ₁(a)∈ H₁andτ₂(a)∈ H₂; and – P(a, b), fora, b∈ S_O,P(a, b)∈ H₁andP(a, b)∈ H₂.

In a multiple Herbrand set,{τ1, τ₂}can be viewed as a disjunction of typesτ₁and τ₂. Since role disjunction is not allowed in DL-Litebool, we only need to consider those role assertions appearing in bothH₁andH₂.

Given a DL-Litebool KBK, we useLΩ_K =hΞ_K,M_Kito denote the assembling of all the model features ofK, whereΞ_Kis a type set andM_Ka multiple Herbrand set.

For simplicity, given{τ1, . . . , τn}(a)in a multiple Herbrand setM, we also represent it asΞ(a)withΞ ={τ1, . . . , τn}.

Example 5. Recall the KBKin Example 2.Ξ_Kconsists of nine types:{ ∅, τ1, τ2,{A,∃P,≥ 2P},{B,∃P}, {B,∃P,≥2P,(∃P⁻)},{∃P,(≥2P)} }, where a type containing a basic concept in brackets, e.g.,(∃P⁻), represents two types, one containing∃P⁻and the other not. The multiple Herbrand setMKconsists of{τ1,{A,∃P,≥ 2 P} }(a), {τ2,{B,∃P,≥2P,∃P⁻} }(b), andP(a, b).

To compute the model features of a KBK, a naive method is to check for eachS- featureFwhetherFsatisfiesK. However, we can show thatLΩ_Kcan be computed directly fromKthrough syntactic transformations, andΩ_Kcan be easily obtained from LΩ_K.

Now we show how to computeLΩ_K for a DL-Litebool KBKthrough syntactic method (ref. Figure 1).

Note that 2 and 3 of Step 3 ensure thatMis a multiple Herbrand set, and thus, together with Step 2, ensurehΞ,Miis an assembling of features.

Theorem 2. Given a consistent DL-Litebool KBK, Algorithm 1 always returnsL ΩK. The following result shows that all the model features ofKcan be easily obtained fromLΩ_K.

Proposition 3. Given a DL-LiteboolKBK, ifL

Ω_K=hΞ_K,M_Ki, then for any feature F =hΞ,Hi,Fis a model feature ofKiff the following conditions are satisfied:

1. Ξ⊆Ξ_K;

2. τa∈Ξa, for eacha∈ SO,τa(a)∈ HandΞa(a)∈ M_K; and

Algorithm 1 (Compute the assembling of all the model features of a DL-LiteboolKB) Input:A DL-Litebool KBK=hT,Aiand a signatureS.

Output:L ΩK. Method:

Step 1.ComputeS-type setΞ =T

(C₁vC₂)∈T Ts(¬C1tC2).

Step 2.Eliminate fromΞall the typesτsuch that∃R∈τfor someRbut∃R⁻6∈S Ξ.

Step 3.Initially, takeM = {P(a, b) | P(a, b)orP⁻(b, a) ∈ A }. Then, for eacha ∈ SO, perform the following procedure:

1. add>(a)intoA, and computeΞa=T

C(a)∈ATs(C)∩Ξ;

2. for eachP ∈ SR, ifnis the number ofP-successors ofainMandmthe largest number inSNwithm≤n, then eliminate fromΞaall the types not containing≥ m P;

3. for eachP ∈ SR, ifnis the number ofP-predecessors ofainMandmthe largest number inSNwithm≤n, then eliminate fromΞaall the types not containing≥ m P⁻; 4. addΞa(a)intoM.

Step 4.ReturnhΞ,MiasL ΩK.

Fig. 1.Compute assembling of model features.

3. P(a, b)∈ H, for eachP(a, b)∈ M_K.

From Proposition 3,Ω_Kis the set consisting of all the features satisfying 1 – 3 of Proposition 3.

GivenΩ_KandΩ_K⁰, we can select those features inΩ_K⁰that are closet toΩ_K, i. e., σ(Ω_K⁰, Ω_K), through the definition. To provide a complete algorithm for our revision operator, we only need a method to constructK ◦ K⁰fromσ(Ω_K⁰, Ω_K).

Each KB uniquely determines an assembling of features, that is, the assembling of all its model features. The converse is also true, as shown in the following proposition.

Proposition 4. Given a setΩof features, there always exists a DL-Litebool KBKsuch that

1. L

Ω_K=L Ω;

2. Ω⊆Ω_K; and for any DL-Litebool KBK⁰withΩ⊆Ω_K⁰,Ω_K⊆Ω_K⁰.

The above proposition suggests that,K ◦ K⁰ can be constructed fromLΩ_K◦K⁰. Also, to computeL

Ω_K◦K⁰, we can avoid computing all the features inΩ_K◦K⁰, but obtain it directly fromσ(Ω_K⁰, Ω_K). Indeed, we haveL

Ω_K◦K⁰ =L

σ(Ω_K⁰, Ω_K).

Finally, we present an algorithm for DL-Litebool KB revision in Figure 2, which follows from the definition.

Theorem 3. Given two consistent DL-Litebool KBsKandK⁰, Algorithm 2 always re- turnsK ◦ K⁰.

The problem of computing the revision is decidable although it may be exponential in the worst case.

Algorithm 2 (Compute the result of revising a DL-LiteboolKB) Input:Two DL-LiteboolKBsKandK⁰,S=Sig(K ∪ K⁰).

Output:K ◦ K⁰. Method:

Step 1.Use Algorithm 1 to computeL

ΩKandL ΩK⁰. Step 2.ComputeΩKandΩK⁰through Proposition 3.

Step 3.Obtainσ(ΩK⁰, ΩK)and computeL

ΩK◦K⁰ =L

σ(ΩK⁰, ΩK). SupposeL

ΩK◦K⁰ = hΞ,Mi.

Step 4.Construct fromhΞ,Mithe DL-Litebool KBhT,Aiwith T ={ l

B∈τ

Bu l

B6∈τ

¬Bv ⊥ | B∈Lit⁺_S;τis anS-type s.t.τ6∈Ξ},

and

A={( G

τ∈Ξ_a(a)

(l

B∈τ

Bu l

B6∈τ

¬B))(a) | a∈ SO, Ξa(a)∈ M } ∪ {P(a, b)∈ M }.

Step 5.ReturnhT,AiasK ◦ K⁰.

Fig. 2.Compute revision.

6 Conclusion

Approaches on adapting classical model-based revision to DLs have proposed in [13, 15]. In [15], the author discuss a special case in KB revision, that is, when inconsistency is caused by ABox assertions. The distances between two models are defined through the number of individuals whose membership differ in the two models. A TBox revi- sion operator that can handleincoherenceis introduced in [13], in which the distance between two models are defined through the number of atomic concepts they disagree.

In both of these two approaches, the results of revision are not single KBs, butdis- junctive KBs. Also, when applying to Example 3 (with only the TBox revised in the second approach), neither of these two approaches hasStudent(Tom)as a logical con- sequence of the revision. Same problem also occurs in those kernel revision operators with disjunctive KBs as the results of revision.

We have developed a formal framework for revising general KBs (with no specific restriction) in DL-Litebool, based on the notion of features. We have defined a specific revision operator, which is a natural adaptation of revision approaches from proposi- tional logic. We have also developed algorithms for computing KB revisions in DL- Lite_bool, and the result of our revision is always an unique DL-Lite_bool KB. We note that other propositional revision operators, e. g., Dalal’s revision, belief contraction and update can also be easily adopted in our framework.

There are still several problems remaining for future work. One is to extend our ap- proach to KB revision in other DLs. The major difficulty here is that a general concept in most expressive DLs cannot be transformed into an equivalent disjunction of con- junctions of literal concepts. Another problem is to look at applications of our revision operator in nonmonotonic reasoning problems in DLs.

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