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Knowledge Base Revision in Description Logics

Guilin Qi, Weiru Liu, and David A. Bell

School of Electronics, Electrical Engineering and Computer Science

Queen’s University Belfast, UK

email: {G.Qi, W.Liu, DA.Bell}@qub.ac.uk

Abstract. Ontology evolution is an important problem in the Seman-

tic Web research. Recently, Alchourr´on, G¨ardenfors and Markinson’s

(AGM) theory on belief change has been applied to deal with this prob-

lem. However, most of current work only focuses on the feasibility of

the application of AGM postulates on contraction to description logics

(DLs), a family of ontology languages. So the explicit construction of a

revision operator is ignored. In this paper, we ﬁrst generalize the AGM

postulates on revision to DLs. We then deﬁne two revision operators in

DLs. One is the weakening-based revision operator which is deﬁned by

weakening of statements in a DL knowledge base and the other is its re-

ﬁnement. We show that both operators capture some notions of minimal

change and satisfy the generalized AGM postulates for revision.

1 Introduction

Ontologies play a crucial role for the success of the Semantic Web [6]. One of

the challenging problems for the development of ontology is ontology evolution,

which is deﬁned as the timely adaptation of an ontology to the arisen changes

and the consistent management of these changes [10]. Ontology evolution is a

very complex process, i.e. it consists of six phases [27]. In this paper, we consider

an important phase called semantics of change phase, which prevents inconsis-

tencies by computing additional changes that guarantee the transition of the

ontology into a consistent state [27]. A center problem in this phase is incon-

sistency handling. There are various forms of inconsistencies, such as structural

inconsistency, logical inconsistency and user-deﬁned inconsistency. Among them,

logical inconsistency in ontology evolution has attached lots of attention in recent

years, where ontologies are represented by logical theories, such as description

logics [21, 1, 8, 11, 10, 14, 19, 25].

AGM’s theory of belief change [9] has been widely used to deal with logical

inconsistency resulting from revising a knowledge base by newly received infor-

mation. There are three types of belief change, i.e. expansion,contraction and

revision. Expansion is simply to add a sentence to a knowledge base; contraction

requires to consistently remove a sentence from a knowledge base and revision is

the problem of accommodating a new sentence to a knowledge base consistently.

Alchourr´on, Gardenfors and Markinson proposed a set of postulates to character-

ize each belief change operator. The application of AGM’ theory to description

2 Guilin Qi, Weiru Liu, and David A. Bell

logics is not trivial because it is based on the assumptions that generally fail for

DLs [7]. For example, a DL is not necessarily closed under the usual operators

such as ¬and ∧[8]. In [7, 8], the basic AGM postulates for contraction were

generalized to DLs and the feasibility of applying the generalized AGM theory

of contraction to DLs and OWL was studied. However, no explicit belief change

operators were proposed in their papers. Furthermore, they did not consider the

application of AGM postulates for revision in DLs.

In this paper, we ﬁrst generalize the AGM postulates for revision to DLs.

Instead of discussing the feasibility of applying the postulates, we propose two

revision operators in DLs. One is the weakening-based revision operator which is

deﬁned by weakening of statements in a DL knowledge base. Since the weakening-

based revision operator may result in counterintuitive results in some cases, we

propose an operator to reﬁne it. We show that both operators capture some no-

tions of minimal change and satisfy the generalized AGM postulates on revision.

This paper is organized as follows. Section 2 gives a brief review of description

logics. In Section 3, we generalize the G¨ardenfors postulates on revision to DLs.

We then propose two revision operators and discuss their logical properties in

Section 4. In Section 5, we have a brief discussion on related work. Finally, we

conclude the paper in Section 6 and give some further work.

2 Description logics

In this section, we will introduce some basic notions of Description Logics (DLs),

a family of well-known knowledge representation formalisms [3]. To make our

approach applicable to a family of interesting DLs, we consider the well-known

DL ALC [26], which is a simple yet relatively expressive DL. Let NCand NR

be pairwise disjoint and countably inﬁnite sets of concept names and role names

respectively. We use the letters Aand Bfor concept names, the letter Rfor

role names, and the letters Cand Dfor concepts. >and ⊥denote the universal

concept and the bottom concept respectively. The set of ALC concepts is the

smallest set such that: (1) every concept name is a concept; (2) if Cand D

are concepts, Ris a role name, then the following expressions are also concepts:

¬C(full negation), CuD(concept conjunction), CtD(concept disjunction),

∀R.C (value restriction on role names) and ∃R.C (existential restriction on role

names).

An interpretation I= (∆I,·I) consists of a set ∆I, called the domain of I,

and a function ·Iwhich maps every concept Cto a subset CIof ∆Iand every

role Rto a subset RIof ∆I×∆Isuch that, for all concepts C,D, role R, the

following properties are satisﬁed:

(1) >I=∆Iand ⊥I=∅, (¬C)I=∆I\CI,

(2) (CuD)I=CI∩DI, (CtD)I=CI∪DI,

(3) (∃R.C)I={x|∃ y s.t.(x, y)∈RIand y∈CI},

(4) (∀R.C)I={x|∀y(x, y)∈RIimplies y∈CI}.

A DL knowledge base consists of two components, the terminological box

(TBox) and the assertional box (ABox). A TBox is a ﬁnite set of terminological

Lecture Notes in Computer Science 3

axioms of the form CvD(general concept inclusion or GCI for short) or C≡D

(equalities), where Cand Dare two (possibly complex) ALC concepts. An inter-

pretation Isatisﬁes a GCI CvDiﬀ CI⊆DI, and it satisﬁes an equality C≡D

iﬀ CI=DI. It is clear that C≡Dcan be seen as an abbreviation for the two

GCIs CvDand DvC. Therefore, we take a TBox to contain only GCIs. We

can also formulate statements about individuals. We denote individual names as

a, b, c. A concept (role) assertion axiom has the form C(a) (R(a, b)), where Cis

a concept description, Ris a role name, and a, b are individual names. To give

a semantics to ABoxs, we need to extend interpretations to individual names.

For each individual name a,·Imaps it to an element aI∈∆I. The mapping

·Ishould satisfy the unique name assumption (UNA), that is, if aand bare

distinct names, then aI6=bI. An interpretation Isatisﬁes a concept axiom C(a)

iﬀ aI∈CI, it satisﬁes a role axiom R(a, b) iﬀ (aI, bI)∈RI. An ABox contains a

ﬁnite set of concept and role axioms. A DL knowledge base Kconsists of a TBox

and an ABox, i.e. it is a set of GCIs and assertion axioms. An interpretation

Iis a model of a DL (TBox or ABox) axiom iﬀ it satisﬁes this axiom, and it

is a model of a DL knowledge base Kif it satisﬁes every axiom in K. In the

following, we use M(φ) (or M(K)) to denote the set of models of an axiom φ(or

DL knowledge base K). Kis consistent iﬀ M(K)6=∅. Two DL knowledge bases

K1and K2are said to be element-equivalent iﬀ there is a bijectin ffrom K1to

K2such that for every φin K1,M(f(φ)) = M(φ). Let Kbe an inconsistent DL

knowledge base. A set K0⊆Kis a conﬂict of Kif K0is inconsistent, and any

sub-knowledge base K00⊂K0is consistent. Given a DL knowledge base Kand a

DL axiom φ, we say K entails φ, denoted as K|=φ, iﬀ M(K)⊆M(φ). We use

KB to denote the set of all possible DL knowledge bases.

3 Generalizing the AGM Postulates for Revision to DLs

Let Lbe a propositional language constructed from a ﬁnite alphabet Pof propo-

sitional symbols using the usual operators ¬(not), ∨(or) and ∧(and). An in-

terpretation is a mapping from Pto {true, f alse}. A model of a formula φis an

interpretation that makes φtrue in the usual sense. M(φ) denotes the set of all

the models of φ. A formula φis satisﬁable if M(φ)6=∅. We denote the classical

consequence relation by `. Two formulas φand ψare equivalent, denoted as

φ≡ψiﬀ M(φ) = M(ψ). In [17], AGM postulates for revision are rephrased as

follows, where ◦is a revision operator which is a function from a pair of formulas

ψand µto a new formula denoted by ψ◦µ.

(R1) ψ◦µ`µ

(R2) If ψ∧µis satisﬁable then ψ◦µ≡ψ∧µ

(R3) If µis satisﬁable then ψ◦µis also satisﬁable

(R4) If ψ1≡ψ2and µ1≡µ2then ψ1◦µ1≡ψ2◦µ2

(R5) (ψ◦µ)∧φimplies ψ◦(µ∧φ)

(R6) If (ψ◦µ)∧φis satisﬁable then ψ◦(µ∧φ) implies (ψ◦µ)∧φ

We ﬁrst deﬁne a revision operator in DLs. Before that, we need to introduce

the notion of a disjunctive DL knowledge base (or DKB) in [19], which is deﬁned

4 Guilin Qi, Weiru Liu, and David A. Bell

as a set of DL knowledge bases. In the following, a DL knowledge base is viewed

as a disjunctive DL knowledge base which contains a single DL knowledge base.

In propositional logic, disjunction ∨is a very important connective used to de-

ﬁne revision operators. For example, the result of Dalal’s revision operator is

(syntactically) in disjunction form [5]. However, DL languages do not allow dis-

junctions of TBox statements with ABox statements. The semantics of DKBs is

deﬁned as follows [19]:

Deﬁnition 1. A DKB Kis satisﬁed by an interpretation I(or Iis a model of

K) iﬀ ∃K∈K such that I |=K.Kentails φ, denoted K |=φ, iﬀ every model of

Kis a model of φ.

Let DKB denote a set of (disjunctive) DL knowledge bases. A revision oper-

ator in DLs can be deﬁned as follows.

Deﬁnition 2. A knowledge base revision operator (or revision operator for short)

in DLs is a function ◦:DKB×KB → DKB which satisﬁes the following condi-

tion: K◦K0|=φ, for all φ∈K0.

That is, both the original knowledge base and the resulting knowledge base can

be a DKB, Whist the newly received knowledge base must be an ordinary DL

knowledge base (i.e. it is not a DKB).

We next generalize postulates (R1)-(R6) to DLs. The generalization is not

as trivial as we have thought. The problem is that both the original knowledge

base the result of revision may be a disjunctive DL knowledge base. To generalize

(R1)-(R6), we need to deﬁne the conjunction of a disjunctive DL knowledge base

and an ordinary DL knowledge base. A more simple way to generalize AGM

postulates is to deﬁne them in a model-theoretic way as follows.

It is clear that (R1)-(R3) can be generalized in the following way. Let Kbe

a (disjunctive) DL knowledge base and K0be a DL knowledge base, we have

(G1) K◦K0|=φfor all φ∈K0

(G2) If M(K)∩M(K0)6=∅, then M(K◦K0) = M(K)∩M(K0)

(G3) If K0is consistent, then M(K◦K0)6=∅

(G1) guarantees that the new information is inferred from the revised knowl-

edge base. (G2) requires that when there is no conﬂict between Kand K0, the

result of revision be equivalent to the “union” of Kand K0, i.e. the set of its

models are M(K)∩M(K0). (G3) is a condition preventing a revision from intro-

ducing unwarranted inconsistency.

The postulate (R4) is the principle of irrelevance of syntax. Its generalization

has the following form:

(G4) If M(K) = M(K1) and M(K0) = M(K2), then M(K◦K0) = M(K1◦K2).

(G4) requires that the revised knowledge base be independent of the syntax

of both original knowledge bases and new information. The rule (R4) (and its

generalization (G4)) is (are) very strong condition(s) because many syntax-based

revision operators in propositional logic do not satisfy it. It is interesting to

consider a weakened version of (G4) as follows.

Lecture Notes in Computer Science 5

(G4)0If K1and K2are element-equivalent and M(K0

1) = M(K0

2), then M(K1◦K0

1)

=M(K2◦K0

2).

Finally, (R5) and (R6) are generalized as follows.

(G5) M(K◦K0)∩M(K00)⊆M(K◦(K0∪K00 ))

(G6) If M(K◦K0)∩M(K00) is not empty, then M(K◦(K0∪K00))

⊆M(K◦K0)∩M(K00)

We have the following deﬁnition.

Deﬁnition 3. A revision operator ◦is said to be AGM compliant if it satisﬁes

(G1-G6). It is quasi-AGM compliant if it satisﬁes (G1)-(G3), (G4)0, (G5-G6).

4 Revision Operators for DLs

4.1 Deﬁnition

In this subsection, we propose a revision operator for DLs and provide a semantic

explanation of it.

In this paper, we only consider inconsistencies arising due to objects being

explicitly introduced in the ABox. That is, suppose Kand K0are the original

knowledge base and the newly received knowledge base respectively, then for

each conﬂict Kcof K∪K0,Kcmust contain an ABox statement. For example,

we exclude the following case: > v ∃R.C ∈Kand > v ∀R.¬C∈K0. The

handling of conﬂicting axioms in the TBox has been discussed recently in [25,

22]. In this paper, we discuss the resolution of conﬂicting information which

contains assertional axioms in the context of knowledge revision.

In order to deﬁne our approach, we need to extend ALC with nominals O

(also called individual names [24]). A nominal has the form {a}, where ais

an individual name. It can be viewed as a powerful generalization of DL ABox

individuals. The semantics of {a}is deﬁned by {a}I={aI}for an interpretation

I. Nominals are very important expressions and they are included in many

important DLs, such as SHOQ [13].

We give a method to weaken a GCI ﬁrst.

Deﬁnition 4. Let CvDbe a GCI. A weakened GCI (CvD)weak of CvDhas

the form (Cu¬{a1}u...u¬{an})vD, where nis the number of individuals to be

removed from C. We use d((CvD)weak) = nto denote the degree of (CvD)weak.

It is clear that when d((CvD)weak ) = 0, (CvD)weak =CvD. The idea of

weakening a GCI is similar to weaken an uncertain rule in [4]. That is, when a

GCI is involved in conﬂict, instead of dropping it completely, we remove those

individuals which cause the conﬂict.

The weakening of an assertion is simpler than that of a GCI. The weakened

assertion φweak of an ABox assertion φ=C(a) is of the form φweak =>(a) or

φweak =φ. When φweak =>(a), we have I |=φweak for all I. Therefore, when

φweak =>(a), we simply delete φ. Indeed, we denote φweak by >(a) when φis to

be deleted for convenience of theoretical analysis. The degree of φweak, denoted

as d(φweak ), is deﬁned as d(φweak) = 1 if φweak =>(a) and 0 otherwise.

6 Guilin Qi, Weiru Liu, and David A. Bell

Deﬁnition 5. Let Kand K0be two DL knowledge bases. Suppose K0is consis-

tent and K∪K0is inconsistent. A DL knowledge base Kweak,K0is a weakened

knowledge base of K w.r.t K0if it satisﬁes:

–Kweak,K 0∪K0is consistent, and

–There is a bijection ffrom Kto Kweak,K0such that for each φ∈K,f(φ)is

a weakening of φ.

The set of all weakened base of K w.r.t K0is denoted by W eakK0(K).

Example 1. Let K={bird(tweety), birdvf lies}and K0={¬f lies(tweety)},

where bird and f lies are two concepts and tweety is an individual name. It is

easy to check that K∪K0is inconsistent. Let K1={>(tweety), bir dvflies},

K2={bird(tweety), birdu¬{tweety}vf lies}, then both K1and K2are weak-

ened bases of K w.r.t K0.

The degree of a weakened base is deﬁned as follows.

Deﬁnition 6. Let Kweak,K 0be a weakened base of a DL knowledge base K w.r.t

K0. The degree of Kweak,K0is deﬁned as

d(Kweak,K 0) = Σφ∈Kweak,K0d(φ)

In Example 1, we have d(K1) = d(K2) = 1.

We now deﬁne a revision operator.

Deﬁnition 7. Let Kbe a (disjunctive) DL knowledge base, and K0be a newly

received DL knowledge base. The result of weakening-based revision of Kw.r.t

K0, denoted as K◦wK0, is deﬁned as follows: If K0is inconsistent, then K◦wK0=

{K∪K0:K∈K}; Otherwise,

K◦wK0=[

K∈K

{K0∪Kweak,K 0:Kweak,K0∈W eakK0(K), and

6 ∃Ki∈W eakK0(K), d(Ki)< d(Kweak,K 0)}.

If K0is inconsistent, the result of revision is an inconsistent disjunctive DL

knowledge base. When K0is consistent, the result of revision of Kby K0is

a disjunctive DL knowledge base consisting of DL knowledge bases which are

unions of K0and a weakened base of a DL knowledge base Kin Kwith the

minimal degree. In the following, we assume that the original knowledge bases are

ordinary DL knowledge base. This assumption is used to simply our discussions.

We next consider the semantic aspect of our revision operator.

Deﬁnition 8. Let Wbe a non-empty set of interpretations and I ∈ W,φa

DL axiom, and Ka DL knowledge base. If φis an assertion, the number of

φ-exceptions eφ(I)is 0 if Isatisﬁes φand 1 otherwise. If φis a GCI of the

form CvD, the number of φ-exceptions for Iis:

eφ(I) = ½|CI∩(¬DI)|if CI∩(¬DI)is ﬁnite

∞otherwise. (1)

The number of K-exceptions for Iis eK(I) = Σφ∈Keφ(I). The ordering ¹Kon

Wis: I ¹KI0iﬀ eK(I)≤eK(I0), for I0∈ W.

Lecture Notes in Computer Science 7

The deﬁnition of φ-exception originates from Deﬁnition 6 in [19]. However, in

[19], it is used to deﬁne an ordering ¹π

Kon a set of interpretations with the same

pre-interpretation π= (∆π, dπ), where ∆πis a domain and dπis a denotation

function which maps every individual name ato a diﬀerent element in ∆π.

We give a proposition to give a semantic explanation of our weakening-based

revision operator.

Proposition 1. Let Kbe a consistent DL knowledge base. K0is a newly received

DL knowledge base. ◦wis the weakening-based revision operator. We then have

M(K◦wK0) = min(M(K0),¹K).

Proposition 1 says that the models of the resulting knowledge base of our revision

operator are models of K0which are minimal w.r.t the ordering ¹Kinduced by

K. So it captures some kind of minimal change. All proofs of this paper can be

found in [23].

Example 2. Let K={∀hasC hild.RichHuman(Bob), hasC hild(Bob, M ary),

RichHuman(Mary), hasChild(Bob, T om)}. Suppose we now receive new infor-

mation K0={hasChild (Bob, J ohn),¬RichHuman(John)}. It is clear that

K∪K0is inconsistent. Since ∀hasChild. RichH uman(Bob) is the only assertion

axiom involved in conﬂict with K0, we only need to delete it to restore consis-

tency, that is, K◦wK0={>(Bob), hasC hild(Bob, M ary), RichH uman(Mary),

hasChild(Bob, T om), hasC hild(Bob, J ohn),¬RichHuman (J ohn)}.

We have the following proposition.

Proposition 2. Given two DL knowledge bases Kand K0. The weakening-based

revision operator is not AGM-compliant but it is quasi-AGM compliant, that is,

it satisﬁes postulates (G1), (G2), (G3), (G40), (G5) and (G6).

4.2 Reﬁned weakening-based revision

In the weakening-based revision, to weaken a conﬂicting assertion axiom, we

simply delete it. The problem for this method of weakening is that it does not

take the constructors of description languages, such as conjunction (u) and value

restriction (∀R.C), into account. This may result in counterintuitive conclusions.

In Example 2, after revising Kby K0using the weakening-based operator, we

cannot infer that RichHuman(T om) because ∀hasChild.RichHuman(Bob) is

discarded, which is counterintuitive. From hasChild(Bob, T om) and ∀hasChild.

RichHuman(Bob) we should have known that RichHuman(T om) and this as-

sertion is not in any conﬂict of K∪K0. The solution for this problem is to treat

John as an exception and that all children of Bob other than John are rich hu-

mans.

For an ABox assertion of the form ∀R.C(a), it is weakened by dropping

some individuals which are related to the individual aby the relation R, i.e.

its weakening has the form ∀R.(Ct {b1, ..., bn})(a), where bi(i= 1, n) are

individuals.

We give another example to illustrate the problem of the weakening method.

8 Guilin Qi, Weiru Liu, and David A. Bell

Example 3. Let K={birduflies(tweety), bird(chirpy)}and K0={¬flies(twee

ty)}. Clearly, birduflies(tweety) is in conﬂict with ¬f lies(tweety) in K0. Let

φ=birduf lies(tweety). The weakening of φis φweak =>(tweety).

In Example 3, to weaken φ, we simply delete it. However, bird(tweety), which

can be inferred from K, is not responsible for any conﬂict of K∪K0. Therefore, it

is counterintuitive to delete it. This intuition is based on the assumption of the

independence of concept names. That is, we take concept names as the “basic

unit of change”.

Before deﬁning the new weakening method, we need to deﬁne an atomic

concept.

Deﬁnition 9. A concept is an atomic concept iﬀ it is either a concept name or

is of one of the forms {a},∀R.C or ∃R.C, where ais an individual name and

Cis a (complex) concept.

We assume that each concept Coccurring in the original DL knowledge

base Kis in conjunctive normal form, i.e., C=C1u...uCnsuch that Ci=

Ci1t...tCim, where Cij is either an atomic concept or the negation of a con-

cept name. Conjunctive normal forms can be generated by the following steps.

First, we transform the concept Cinto its negation normal form by the follow-

ing equalities: ¬¬Ci≡Ci,¬(CiuDi)≡ ¬Cit¬Di,¬(CitDi)≡ ¬Ciu ¬Di,

¬(∃R.Ci)≡ ∀R.¬Ci,¬(∀R.Ci)≡ ∃R.¬Ci. Second, we move disjunction in-

ward and conjunction outward according to De Morgan’s law: C1t(C2uC3)≡(C1

tC2)u(C1tC3). Suppose C(a)∈K, where Cis a concept in conjunctive normal

form, we assume that each concept assertion C(a) is decomposed into φ1, ...,

φnsuch that φi= (Ci1t...tCim)(a). Note that acannot be moved inside the

disjunction constructor because disjunction of ABox assertions is not allowed in

DLs.

We now deﬁne a new weakening method. The idea is that we weaken a con-

cept assertion by weakening its atomic concepts. That is, we have the following

deﬁnition.

Deﬁnition 10. Let φ=R(a, b)be a role assertion. A weakened relation as-

sertion φweak of φis deﬁned as φweak =>R(a, b)or φweak =φ, where >Ris

interpreted as >I

R=∆I×∆Ifor each interpretation I= (∆I,·I). Let φ=C(a)

be a concept assertion. A weakened concept assertion φweak of φis deﬁned re-

cursively as follows:

1) if C=Aor ¬Afor a concept name A, then φweak =>(a)or φweak =φ,

2) if C=∃R.D, then φweak =>(a)or φweak =φ,

3) if C=∀R.D, then φweak =∀R.(Dt {b1, ..., bn})(a)or >(a),

4) if C={b}, where bis an individual name, then φweak =>(a)or φweak =φ,

5) if C=Ci1t...tCim, where Cij is either an atomic concept or the negation of

an atomic concept, then φweak = ((Ci1)weak t...t(Cim)weak )(a)1if (Cij)weak 6≡>

for all jand φweak =>(a)otherwise,

1According to 1), 2), 3), and 4), we have (Cij)weak =>or Cij if Cij is either a

concept name or the negation of a concept name or of the form ∃R.D or {b}, and

(Cij )weak =∀R.(Dt {b1, ..., bn}) if Cij is of the form ∀R.D.

Lecture Notes in Computer Science 9

Let us explain the part 5) of Deﬁnition 10. Since the concept of φis in disjunctive

form, if there exists a Cij such that (Cij)w eak ≡ >, then C≡ >. That is,

the weakening of a disjunct concept of φmay inﬂuence the weakening of other

disjuncts. When weakening a role assertion, we introduce the top role. However,

in implementation, the top role does not exist in the resulting knowledge base

because the role assertion is simply deleted if the role name is weakened into

the top role. In this paper, we only consider the reﬁnement of the weakening of

ABox assertions. Similarly, we can also reﬁne the weakening of TBox axioms.

We next deﬁne the degree of a weakened assertion.

Deﬁnition 11. Let φ=R(a, b), then d(φweak ) = 1 if φweak =>R(a, b)and 0

otherwise. Let φ=C(a), then d(φ)is deﬁned recursively as follows:

1) if C=Aor ¬Afor a concept name A, then d(φweak) = 1 if φweak =>(a)

and 0 otherwise,

2) if C=∃R.C, then d(φweak) = 1 if φweak =>(a)and 0 otherwise,

3) if C=∀R.D, then d(φweak ) = nif φweak =∀R.(Dt {b1, ..., bn})(a)and +∞

otherwise,

4) if C={b}, where bis an individual name, then d(φweak) = 1 if φw eak =>(a)

and 0 otherwise,

5) if C=Ci1t...tCim, where Cij is either an atomic concept or the negation of

an atomic concept, then d(φweak) = max{d(((Cij )w eak)(a)) : j= 1, ..., m},

In part 5) of Deﬁnition 11, we use max (instead of sum) to determine the degree

of an assertion in “disjunction” form. This deﬁnition agrees with the semantic

interpretations of disjunction in many logics such as fuzzy logic and possibilistic

logic.

We call the weakened base obtained by applying weakening of GCIs in Deﬁ-

nition 4 and weakening of assertions in Deﬁnition 10 as a reﬁned weakened base.

We then replace the weakened base by the reﬁned weakened base in Deﬁnition 7

and get a new revision operator, which we call a reﬁned weakening-based revision

operator which is denote by ◦rw. Let us go back to Example 2 again. Accord-

ing to our discussion before, ∀hasChild.Rich Human(Bob) is the only assertion

axiom involved in the conﬂict in Kand John is the only exception which makes

∀hasChild.RichHuman(Bob) in conﬂict with K0, so K◦rwK0={∀hasC hild.

(RichHumant{John})(Bob), hasChild(B ob, Mary ), RichHuman(M ary),

hasChild(Bob, T om), hasC hild(Bob, J ohn),¬RichHuman(J ohn)}. We can then

infer that RichHuman(Tom) from K◦rwK0.

We consider another example. Let K={((∀R.C)tD)(a), R(a, b)}and K0=

{¬D(a),¬C(b)}, where Cand Dare concept names. Clearly, K∪K0is inconsis-

tent. We can either weaken ((∀R.C)tD)(a) or R(a, b) to restore consistency. To

weaken R(a, b), we can simply delete it, i.e. its weakening has the form >R(a, b).

We have d(>R(a, b)) = 1. For φ= ((∀R.C )tD)(a), we should weaken ∀R.C in-

stead of D. This is because if we weaken Dto >then (∀R.C)tDalso needs to be

weakened to >. In this case, we have d(φweak) = +∞. In contrast, if we weaken

(∀R.C)tDto (∀R.(Ct{b}))tD, then Ddoes not need to be weakened. In this

case, we have d((∀R.(Ct{b}))tD)(a)) = 1 and d(φweak ) = 1. Therefore, there

10 Guilin Qi, Weiru Liu, and David A. Bell

are two weakened bases of K w.r.t K 0, i.e. K1={((∀R.(Ct{b}))tD)(a), R(a, b)}

and K2={((∀R.C)tD)(a)}.

To give a semantic explanation of the reﬁned weakening-based revision op-

erator, we need to deﬁne a new ordering between interpretations.

Deﬁnition 12. Let Wbe a non-empty set of interpretations and I ∈ W,φa

DL axiom, and Ka DL knowledge base. If φis a concept assertion, then the

number of φ-exceptions for Iis deﬁned recursively as follows:

1) if φ=A(a)or ¬A(a)for a concept name A, then eφ

r(I) = 0 if I |=φand 1

otherwise,

2) if φ=∃R.C(a), then eφ

r(I) = 0 if I |=φand 1 otherwise,

3) If φis an assertion of the form ∀R.C(a), the number of φ-exceptions for I

is:

eφ

r(I) = ½|RI(aI)∩(¬CI)|if RI(aI)∩(¬CI)is ﬁnite

∞otherwise, (2)

where RI(aI) = {b∈∆I: (aI, b)∈RI}.

4) If φ={b}(a), where bis an individual name, then eφ

r(I) = 0 if I |=φand 1

otherwise,

5) φ= (Ci1t...tCim)(a), where Cij is either an atomic concept or the negation

of an atomic concept, then eφ

r(I) = max{eCij (a)

r(I) : j= 1, ..., m}.

If φis a role assertion, then eφ

r(I) = 0 if I |=φand 1 otherwise.

If φis a GCI of the form CvD, the number of φ-exceptions for Iis:

eφ

r(I) = ½|CI∩(¬DI)|if CI∩(¬DI)is ﬁnite

∞otherwise. (3)

The number of K-exceptions for Iis eK

r(I) = Σφ∈Keφ

r(I). The reﬁned ordering

¹r,K on Wis: I ¹r,K I0iﬀ eK

r(I)≤eK

r(I0), for I0∈ W.

The following proposition gives the semantic interpretation of the reﬁned

weakening-based revision operator.

Proposition 3. Let Kbe a consistent DL knowledge base. K0is a newly received

DL knowledge base. ◦rw is the reﬁned weakening-based revision operator. We then

have

M(K◦rw K0) = min(M(K0),¹r,K).

Proposition 3 says that the reﬁned weakening-based operator can be accom-

plished with minimal change. The proof is similar to that of Proposition 1.

Proposition 4. Let Kbe a consistent DL knowledge base. K0is a newly received

DL knowledge base. We then have

M(K◦rw K0)⊆M(K◦wK0).

By Example 3, K◦rw K0and K◦wK0are not equivalent. Thus, we have shown

that the resulting knowledge base of the reﬁned weakening-based revision con-

tains more information than that of the weakening-based revision. However, the

Lecture Notes in Computer Science 11

reﬁned weakening-based revision need to convert every ABox assertion to its

conjunctive normal form. In some cases this conversion can lead to an expo-

nential explosion of the size of the ABox assertion. So the sizes of the revised

DL knowledge bases of the reﬁned weakening-based operator are exponentially

larger than those of the weakening-based operator in the worst case.

The reﬁned weakening-based revision operator is still not AGM compliant.

Proposition 5. Given two DL knowledge bases Kand K0. The reﬁned weakening-

based revision operator is not AGM-compliant but it is quasi-AGM compliant.

5 Related Work

The importance of applying AGM theory on belief change to terminological sys-

tems has not been fully recognized until recent years. In his book [20], Nebel

considered the revision problem in terminological logics in 1990. He proposed

some revision operators based on several existing approaches on modiﬁcation of

a terminological knowledge base. When deﬁning his revision operator, he pre-

sumed that the terminological knowledge is more relevant than the assertional

knowledge. Recently, some work has been done to analyze the feasibility of ap-

plying AGM theory on belief change to DLs [16, 7, 8]. However, none of them

considers the explicit construction of a revision operator. Furthermore, they

did not consider the application of AGM postulates for revision in DLs where

knowledge bases instead of knowledge sets are considered. The work in [16, 8] is

based on the coherence model, i.e. both the original and the revised knowledge

bases should be knowledge sets which are knowledge bases closed under logi-

cal consequence. In [7], Fuhrmann’s postulates for knowledge base contraction is

generalized to DLs. One may wonder if we can establish the relationship between

revisions and contractions via the Levi and Harper identities. However, the prob-

lem is that Levi and Harper identities are not applicable in DLs [8]. In [19], some

revision operators were proposed for revising a stratiﬁed DL knowledge base. The

semantic aspects of these revision operators are also considered. To deﬁne their

operators, an extra expression in DLs, called cardinality restrictions on concepts,

is necessary. In contrast, our operators are based on nominals. Since cardinality

restrictions can be encoded as nominals, our revision operators can be seen as

a reﬁnement of the revision operators in [19]. In [14], a general framework for

reasoning with inconsistent ontologies was given based on concept relevance. A

problem with their framework is that they do not consider the structure of DL

language. For example, when a GCI is in conﬂict in a DL knowledge base, it is

deleted to restore consistency. Our work is also related to the work in [1], where

Reiter’s default logic is embedded into terminological representation formalisms.

In their paper, conﬂicting information is also treated as exceptions. To deal with

conﬂicting default rules, they instantiated each rule using individuals appearing

in the ABox and applied two existing default reasoning methods to compute all

extensions. This instantiation step is not necessary for our revision operators.

Furthermore, in [1], the resolution of conﬂicting ABox assertions was not consid-

ered. This work is also related to the work on updating DL ABoxes in [15]. They

12 Guilin Qi, Weiru Liu, and David A. Bell

showed that in any standard DL in which nominals and the ”@” constructor

are not expressible, updated ABoxes cannot be expressed. They only consider

a simple form of ABox update where the update information contains possibly

negated ABox assertions that involve only atomic concepts and roles.

6 Conclusions and Further Work

In this paper, we have discussed the problem of applying AGM theory of be-

lief revision to DLs. We ﬁrst generalized the reformulated AGM postulates for

revision to DLs. Then two revision operators were proposed by weakening as-

sertion axioms and GCIs. We showed that both revision operators satisfy the

generalized postulates and capture some notions of minimal change.

Several problems are left as further work. First, none of our revision opera-

tors is AGM compliant, that is, they do not satisfy (G4). We are looking for a

revision operator satisfying all the AGM postulates. Second, to implement our

revision operators, an important problem is to detect GCIs and and assertions

which are responsible for the conﬂict. Some existing techniques on debugging

of unsatisﬁable classes (such as [25, 22]) can be adopted or generalized to deal

with this problem. We will develop tableaux-based algorithms for implement-

ing our revision operators. Based on the results in [25], it is expected that the

computational complexity of our operators may not increase the complexity of

consistency checking in the DL under consideration.

7 Acknowledge

We would like to thank the anonymous reviewers for their useful comments which

have helped us to improve the quality of this paper.

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