Knowledge Base Revision in Description Logics

Abstract

Ontology evolution is an important problem in the Semantic Web research. Recently, Alchourrón, Gärdenfors and Markinson’s
(AGM) theory on belief change has been applied to deal with this problem. 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 refinement. We show that both operators capture some notions of minimal change and satisfy the generalized AGM postulates
for revision.

Full text

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.
References
1. F. Baader and B. Hollunder. Embedding defaults into terminological knowledge
representation formalisms, Journal of Automated Reasoning, 14(1):149-180, 1995.
2. F. Baader, M. Buchheit, and B. Hollander. Cardinality restrictions on concepts.
Artificial Intelligence, 88:195-213, 1996.
3. F. Baader, D.L. McGuiness, D. Nardi, and Peter Patel-Schneider. The Descrip-
tion Logic Handbook: Theory, implementation and application, Cambridge Univer-
sity Press, 2003.
4. S. Benferhat, and R.E. Baida. A stratified first order logic approach for access
control. International Journal of Intelligent Systems, 19:817-836, 2004.
5. M. Dalal. Investigations into a theory of knowledge base revision: Preliminary re-
port, Proc. of AAAI’88, 3-7, 1988.
6. T. Berners-Lee, J. Hendler, and O. Lassila. The semantic web, Scientific American,
284(5):3443, 2001.
7. G. Flouris, D. Plexousakis and G. Antoniou. Generalizing the AGM postulates:
preliminary results and applications. In Proc. of NMR’04, 171-179, 2004.

Lecture Notes in Computer Science 13
8. G. Flouris, D. Plexousakis and G. Antoniou. On applying the AGM theory to DLs
and OWL, In Proc. of ISWC’05, 216-231, 2005.
9. P. G¨ardenfors, Knowledge in Flux-Modeling the Dynamic of Epistemic States, The
MIT Press, Cambridge, Mass, 1988.
10. P. Haase and L. Stojanovic. Consistent evolution of OWL ontologies. In Proc. of
2nd European Semantic Web Conference (ESWC’05), 182-197, 2005.
11. P. Haase, F. van Harmelen, Z. Huang, H. Stuckenschmidt, and Y. Sure. A frame-
work for handling inconsistency in changing ontologies, In Proc. of ISWC’05,
LNCA3729, 353-367, 2005.
12. S.O. Hansson. In denfence of base contraction. Synthese, 91: 239-245, 1992.
13. I. Horrocks, and U. Sattler. Ontology reasoning in the SHOQ(D) description logic,
In Proc. of IJCAI’01, 199-204, 2001.
14. Z. Huang, F. van Harmelen, and A. ten Teije. Reasoning with inconsistent ontolo-
gies, In Proc. of IJCAI’05, 254-259, 2005.
15. H. Liu, C. Lutz, M. Miliˇci´c, and F. Wolter. Updating description logic ABoxes. In
Proc. of KR’06, 2006.
16. S.H. Kang and S.K. Lau. Ontology revision using the concept of belief revision.
In Proc. of 8th International Conference on Knowledge-base Intelligent Information
and Engineering Systems (KES’04), 261-267, 2004.
17. H. Katsuno and A.O. Mendelzon. Propositional Knowledge Base Revision and
Minimal Change, Artificial Intelligence, 52(3): 263-294, 1992.
18. C. Lutz, C. Areces, I. Horrocks, and U. Sattler. Keys, nominals, and concrete
domains, Journal of Artificial Intelligence Research, 23:667-726, 2005.
19. T. Meyer, K. Lee, and R. Booth. Knowledge integration for description logics, In
Proc. of AAAI’05, 645-650, 2005.
20. B. Nebel. Reasoning and Revision in Hybrid Representation Systems, LNAI 422,
Springer Verlag, Berlin, Heidelberg, New York, 1990.
21. B. Nebel. What is Hybrid in Hybrid Representation and Reasoning Systems?, In
F. Gardin and G. Mauri and M. G. Filippini, editors, Computational Intelligence
II: Proc. of the International Symposium Computational Intelligence 1989, North-
Holland, Amsterdam, 217-228, 1990.
22. B. Parsia, E. Sirin and A. Kalyanpur. Debugging OWL ontologies, In Proc. of
WWW’05, 633-640, 2005.
23. G. Qi, W. Liu and D.A. Bell. Knowledge base revision in description logics. Avail-
able at http://www.cs.qub.ac.uk/ G.Qi/qlb06e.pdf
24. A. Schaerf. Reasoning with individuals in concept languages. Data and Knowledge
Engineering, 13(2):141-176, 1994.
25. S. Schlobach, and R. Cornet. Non-standard reasoning services for the debugging
of description logic terminologies, In Proc. of IJCAI’2003, 355-360, 2003.
26. M. Schmidt-Schauß, and G. Smolka. Attributive Concept descriptions with com-
plements, Artificial Intelligence, 48:1-26, 1991.
27. L. Sto janovic. Methods and Tools for Ontology Evolution, PhD thesis, University
of Karlsruhe, 2004.