inconsistent ontology revision based on ontology constructs
TRANSCRIPT
Expert Systems with Applications 37 (2010) 7269–7275
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Expert Systems with Applications
journal homepage: www.elsevier .com/locate /eswa
Inconsistent ontology revision based on ontology constructs
Yinglong Ma a,b,*, Shaohua Liu b, Beihong Jin b, Gang Xu b
a Department of Computer Science, North China Electric Power University, DeShengMen Wai ZhuXinZhuang, Beijing 102206, Chinab Technology Center of Software Engineering, Institute of Software, Chinese Academy of Sciences, P.O. Box 8718, Beijing 100190, China
a r t i c l e i n f o a b s t r a c t
Keywords:OntologyOntology reuseSemantic metricsOntology evaluationOntology inconsistency revision
0957-4174/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.eswa.2010.04.001
* Corresponding author at: Department of ComElectric Power University, DeShengMen Wai ZhuXinZhTel./fax: +86 10 51963385.
E-mail addresses: [email protected], [email protected] (S. Liu), [email protected] (G. Xu).
1 www.geneontology.org/.2 swoogle.umbc.edu/.
Assessing the quality of ontologies is important to reuse/integrate ontologies. In this paper, we propose anapproach to revise ontology inconsistency by measuring the relevant ontology constructs such as seman-tic partitions, axiom fanouts per concept, minimal incoherence-preserving subsets, incoherence impactvalue per axiom and preferable revision weight per concept in the ontology being measured. We also pro-pose several algorithms to compute the measurement values of these ontology constructs, which can beused to pinpoint and rank which axioms and concepts are most likely to cause the ontology inconsistency.Our approach can help determine which concepts should be preferentially modified and therefore enablesautomatic revision of ontology inconsistency in the context of a dynamic and evolving environment.
� 2010 Elsevier Ltd. All rights reserved.
1. Introduction der to keep in accord with application requirements (Oberle, Staab,
Ontologies play a key role to provide shared knowledge modelsto semantic driven application systems. They not only serve as thefoundation of the Semantic Web (Berners-Lee, Hendler, & Lassila,2001) but also are being applied to content management, andinformation integration and knowledge management applications.The use of ontologies provides semantic driven applications withextensive reasoning capabilities, and thus improves query answer-ing and facilitates integration and cooperation between systems. Agood example of domain ontology is the gene ontology (GO) fromthe Open Biology Ontologies.1 However, construction of ontologiesis time-consuming and arduous. Fortunately, an increasing numberof ontologies representing different domain knowledge have beenmade available from Web by using the ontology search engine suchas Swoogle.2 The high cost of constructing and maintaining ontol-ogies encourages their sharing and reuse.
In order to reuse/integrate ontology knowledge, manyresearches treated ontologies as static knowledge. Once the ontol-ogy was constructed and deployed, the knowledge captured by theontology will be changed no longer. However, ontologies in real-world applications will inevitably be changed and updated becauseapplication requirements of enterprises adopting ontologies astheir conceptual backbone are always continually changed andmodified. There is a need for ontologies to continually evolve in or-
ll rights reserved.
puter Science, North Chinauang, Beijing 102206, China.
[email protected] (Y. Ma),c.cn (B. Jin), xugang@otcaix.
Studer, & Volz, 2005; Stojanovic et al., 2002). In the situation, it israther challenging for ontology engineers to reuse/integrate theontologies with evolving characters. At least two aspects shouldbe considered for reuse of evolving ontologies. On one hand, differ-ent evolving versions of an ontology will come into being alongwith changes of application requirements. It is very important forontology users to select an ontology most suitable to satisfy theirapplications. On the other hand, as changes to an ontology maypossibly bring about inconsistency of the ontology, it is crucialfor an approach supporting ontology evolutions to guarantee thatontologies evolve from one consistent state into another consistentone (Plessers & Troyer, 2006). Ontology engineers urgently need amethod to detect and resolve inconsistencies of ontologies.
In recent years, measuring and evaluating ontologies havereceived more attention because it is necessary to evaluate ontol-ogies for ontology integration/reuse. Many ontology measureswere proposed to assess the quality of ontologies and thereforetracked their subsequent evolution (Vrandecic & Sure, 2007). Byusing different types of ontology metrics to assess ontology qual-ity, ontology engineers can rank and further select the ontologiesmost suitable to satisfy their application requirements. They caneven modify and improve the design of ontologies by means ofmeasuring (inconsistent) ontologies.
Most research in the field of ontology reuse concerning measure-ment of evolving ontologies focused on ranking and selection of con-sistent ontologies. In this paper, we propose an approach to reviseontology inconsistency by measuring the relevant ontology con-structs such as semantic partitions, axiom fanouts per concept, min-imal incoherence-preserving subsets, incoherence impact value peraxiom and preferable revision weight per concept in the ontologybeing measured. Several algorithms are proposed to compute themeasurement values of these ontology constructs, through which
7270 Y. Ma et al. / Expert Systems with Applications 37 (2010) 7269–7275
we can pinpoint and rank which axioms and concepts are mostlikely to cause the ontology inconsistency. Our approach can helpusers determine which concepts should be preferentially modifiedand enables automatic revision of ontology inconsistency in thecontext of a dynamic and evolving environment.
This paper is structured as follows. Section 2 briefly introducesthe description logics and inconsistency checking. Section 3 is theoverview of our approach. In Section 4, we discuss how to measurethe relevant ontology constructs and the relevant algorithms. InSection 5, we present a semantic measurement driven approachto pinpoint and rank ontological axioms and concepts most likelyto cause ontology inconsistencies. Section 6 discusses how to re-vise ontology inconsistencies by using relative semantic metricsin addition to measurement information. Section 7 discusses re-lated work, and Section 8 is the conclusion.
Fig. 1. Overview of the proposed approach.
2. Description logics and inconsistency checking
Description Logics (DL) (Baader, Calvanese, McGuinness, Nardi,& Patel-Schneider, 2003) can be regarded as a well-founded ontol-ogy representation language. This paper will not give the detailsabout DL. We refer interested readers to the literature (Baaderet al., 2003). DL represents domain knowledge by defining the rel-evant concepts of the domain. A knowledge base K can be definedas K ¼ ðT ;AÞ, where T and A are TBox and Abox, respectively.TBox and ABox represent the sets of terminological axioms andindividual assertions, respectively. In TBox, the basic descriptionsare atomic concepts and atomic roles. Based on atomic concepts,complex concept descriptions can be constructed by iterativelyapplying constructors such as intersection (u), union (t), negationð:Þ, value restriction ("R�C) and existential quantification ($R�C).Axioms express how concepts and roles are related to each other.Generally, an axiom is of the form C v D or C � D, where C and Dare concept descriptions. C v D if concept C is subsumed by con-cept D. C � D if C v D and D v C. An ABox is a set of assertions ofthe form C(a) or R(a,b), where R is a role, and a,b are individuals.For an interpretation I ¼ ðDI ; �I Þ; �I maps every atomic concept Ato a subset AI # DI , and every atomic role R to a binary relationRI # DI � DI , where DI is the domain, and �I is the interpretationfunction. I satisfies C v D if CI # DI and satisfies C � D if CI ¼ DI .I satisfies C(a) if aI 2 CI and satisfies R(a,b) if ðaI ; bI Þ 2 RI . I is amodel of K if I is the model of T and A. I is a model of T ðAÞ ifit satisfies all axioms (assertions) in T ðAÞ. A concept C in a TBoxT is satisfiable if there exists a model of T such that CT – ;. A TBoxT is incoherent if there exists a concept in T which is unsatisfiable.A knowledge base K is consistent if there is a model of K.
For a knowledge base K ¼ ðT ;AÞ, the tableau algorithm can beused to check the satisfiability of a concept in T and consistency ofA w.r.t T . Its basic principle of checking the satisfiability of aconcept C is to gradually build a model I of C such that CI is notempty. As a result, a tree-like model of concept can be built bydecomposing concept C by using tableau expansion rules. The tab-leau algorithm terminates when either no more rules can be appli-cable, or when a clash occurs. A clash is the form fCðaÞ;:CðaÞg or{6nS,PnS}. Based on the tableau algorithm, a satisfiable conceptrefers the one that no more rules can be applied and no clashesoccurred. We refer interested readers to the literature (Schlobach,Huang, Cornet, & van Harmelen, 2007) for some tableau expansionrules of satisfiability of ALC concepts.
3. Overview of the approach proposed
In this paper, we propose an approach to revise an inconsistentontology by measuring relevant ontology constructs. In the follow-ing, we will overview our approach, which is shown in Fig. 1.
For the ontology being analyzed, we define some ontology con-structs beforehand. They include semantic partition, fanouts perconcept, minimally incoherent-preserving sub-TBox (MIPS), inco-herence impact value per axiom and preferable revision concept.
When the ontology being analyzed is loaded for inconsistencychecking, if it is incoherent, then we can compute metric valuesof relevant ontology constructs. According to the metric values ob-tained, we can revise the ontology and make it coherent. Specifi-cally speaking, we first compute semantic partitions of theontology. Next, a corresponding SMIS (i.e. the set of MIPSs) canbe obtained with respect to each incoherent semantic partitionby using Algorithm 2 (see Section 4). We can further compute inco-herence impact value per axiom in each SMIS. For each incoherentsemantic partition, we can construct a corresponding concept sub-sumption weighted tree (CSWT), and further use shortest pathalgorithm to pinpoint and rank the concepts most likely to causeontology incoherence. At last, we will obtain the preferable revi-sion weight value of each concept.
For the revision of TBox incoherence, the modification of theincoherent ontology can be made by weakening or strengtheningthose concepts with preferable revision weight value in eachsemantic partition. We also propose a set of rules that guide us toappropriately revise ontology. For the revision of ABox inconsis-tency, we find out the axioms from which C(a) and D(a) can be de-rived by keeping track of the tableau algorithm for satisfiabilitychecking of instance a for each clash including {C(a),D(a)}. Thenwe will decide which concepts should be revised to make the ABoxconsistent by measuring fanouts of relevant concepts.
4. Measuring ontology constructs
Definition 1 (Unfoldable axioms). For any axiom a of the formA v B or A � B, a is unfoldable iff A is a named concept that isatomic and unique, and B does not contain direct or indirectreference to A.
The definition of unfoldable axioms here is originally fromunfoldable TBox (Schlobach et al., 2007). In fact, all axioms in aknowledge base (ontology) can be transformed into unfoldableones. For examples, in the axiom of the form A v B, if A is not aconcept name, i.e. A is not a defined concept, then we can definea unique concept name C such that C � A and C v B. For the axiom
Y. Ma et al. / Expert Systems with App
of the form A � B, there will be three situations: (1) if both A and Bare not concept names, then we can define a concept name C suchthat C � A and C � B; (2) if A is not a concept name and B is a con-cept name, then we transform A � B to B � A; and (3) if both A andB are concept names, then we replace all occurrences of B with Aand further delete B from the knowledge base. If there are conceptnames A,A1, . . .,An such that A v A1,A1 v A2, . . .,An�1 v An andAn v A, then we replace all occurrence of each Ai (1 6 i 6 n) withA and further delete all Ai (1 6 i 6 n) from the knowledge base inorder to avoid cycle subsumption.
Note that in the following content, we assume that all axioms inthe knowledge base being measured have been transformed intounfoldable ones.
Definition 2 (Set of defined concepts). For a TBox T , the set ofdefined concepts of T is denoted CT ¼ fC1; . . . ;Cng, where each Ci
(1 6 i 6 n) is a named concept that is atomic and unique.Note that in an original knowledge base consisting of unfoldable
axioms, it is possible that not all complex concepts have been de-fined and assigned a unique concept name. For example, if a knowl-edge base contains the unfoldable axioms A v "R�(C u D), complexconcepts "R�(C u D) and C u D have not been assigned a unique con-cept name. In the paper, we require to normalize and name all com-plex concepts. We propose an algorithm to name all possiblecomplex concepts, which is shown in algorithm as follows.
1. For a knowledge base K ¼ ðT ;AÞ, all atomic concepts and theconcept in the left side of each axioms in T are tagged andmoved into CT .
2. For each axiom in T , if the axiom is of the form A � CC or A v CC,where CC is not a named complex concept, then name(CC). Notethat for the axiom of the form A � CC, the name of CC just istagged as A that is atomic and unique. The algorithm name(CC)also can further name complex concepts contained in CC. It canbe performed in an iterative way, which is shown in Algorithm1. In each iteration, the named concepts are moved into CT .
3. At last, CT consists of all possible defined (named) concepts.
Algorithm 1. Algorithm name(C)
Input: C is an unnamed complex concept inK ¼ ðT ;AÞ; CT is set of defined concepts of KOutput: CT1 if C is of the form :B then name (B);2 else tagged (C) and move C into CT3 switch c do4 case is of the form A u B5 if not tagged (A) then name (A);6 if not tagged (B) then name (B);7 end8 case is of the form A t B9 if not tagged (A) then name (A);10 if not tagged (B)then name (B);11 end12 case is of the form "R�B13 if not tagged (R)then tagged (R);14 if not tagged (B) then name (B);15 end16 case is of the form $R�B17 if not tagged (R)then tagged (R);18 if not tagged (B)then name (B);19 end20 end
Definition 3 (Directly-subsumed-by). Let T be a TBox. 8C;D 2
CT ; C is directly subsumed by D, denoted directly-subsumed-by(C,D), if and only if 8C;D 2 CT ðC v D ^ :9C0 2 CT ðC0 v D^ C v C0ÞÞ.Definition 4 (Axiom fanouts). 8C 2 CT , the axiom fanouts of C aredenoted a set AFC = {D1, . . . ,Dm}, where for each Di ð1 6 i 6m 6 jCT jÞ, directly-subsumed-by (Di,C) holds, and jCT j representsthe number of defined concepts in CT .
Example 1. Let T ¼ fA v B; A � C t D; E � C u Dg, then CT ¼fA; B; C; D; Eg. According to Definition 4, we will find that conceptsC, D and E are directly subsumed by A. So, AFA = {C,D}. E is notdirectly subsumed by A because there exists the concept C or Dsuch that E v C ^ C v B or E v D ^ D v B.
It is possible for an ontology to contain two or more differentsources causing inconsistencies. We focus on incoherence in theTBox T of an ontology, and give the definition of incoherence valueof its a sub-TBox.
Definition 5. The function iv : 2T ! f0; 1g assigns a value to eachT 0# T , where
lications 37 (2010) 7269–7275 7271
ivðT Þ0 ¼0 if T 0 is coherent or T 0 is empty
1 otherwise
(
Example 2. Look at a simple ontology example in DL syntax. Tcontains axioms as follows: 1: A v B, 2: A v :B, 3: C v E u F; 4:E v "s�G u D; 5: F v 9s � :G. For sake of simplicity, we refer the axi-oms by their numbers. There are 32 ð2jT jÞ possible subsets of T ,where T 0 ¼ f1; 2g and T 00 ¼ f3; 4; 5g such that ivðT 0Þ ¼ 1 andivðT 00Þ ¼ 1. In fact, for any T 000 # T such that T 0 # T 000 orT 00 # T 000; ivðT 000Þ ¼ 1.
By using Definition 5, we continue to introduce the MIPS defini-tion from Schlobach et al. (2007).
Definition 6. (Minimally incoherence-preserving sub-TBox, MIPS). Fora sub-TBox T 0# T ; T 0 is a minimal incoherence-preserving sub-TBox of T iff the following conditions hold:
1. ivðT 0Þ ¼ 12. ivðT 00Þ ¼ 0 for every T 00 � T 0
Schlobach et al. illustrate two algorithms to calculate MIPS(Schlobach et al., 2007). If the TBox T has a small number of axi-oms, we give an alternative Algorithm 2 to obtain the set of allMIPSs in T . The set is denoted SMIS. In the algorithm, once an inco-herent sub-TBox with the smallest cardinality has been obtained,its all supersets will be ignored and need not perform incoherencechecking. This will reduce complexity of incoherence checkingbased on logical reasoning.
Algorithm 2. Obtaining MIPSs of T
Input: The TBox TOutput: SMIS1 SMIS = ;;2 for all T 0# T , sorted by cardinality of T 0 do3 if 9T 00 2 SMIS such that T 00 � T 0 then go the next
iteration;4 if ivðT 0Þ ¼ 1 then move T 0 to SMIS;5 end6 return SMIS;
7272 Y. Ma et al. / Expert Systems with Applications 37 (2010) 7269–7275
Within an ontology with a large number of MIPSs, some onto-logical modules cannot be effectively congregated together so asto achieve a close and unambiguous sharing. The more MIPSs an
ontology has, the more difficult its sharing is. It is necessary anddesirable to reduce ontology incoherence.Example 3. We revise the T in Example 2 by adding some axiomssuch that T contains such axioms as follows: 1: A v B, 2: A v :B, 3:C v E u F; 4: E v "s�G u D; 5: F v 9s � :G; 6: D v A. We will find thatT 1 ¼ f1; 2g and T 2 ¼ f3; 4; 5g are the MIPSs. But the sub-TBoxT 3 ¼ f1; 2; 6g is not a MIPS.
Definition 7 (Axiom signature). For an axiom a of the form A v B, ifB 2 CT is an atomic concept, then the signature of axiom a is denotedSignature(a) = {A,B}. If B is a tag of a complex concept expression andcontains named concept B1, . . .,Bn, where Bi 2 CT for each 1 6 i 6 n,then the signature of a is Signature(a) = {A,B,B1, . . .,Bn}.
Definition 8 (Semantic relevance of axioms). For a TBox T , two axi-oms aandb are semantically relevant, denoted SemRelevant(a,b), iff9A; B 2 CT ðA 2 SignatureðaÞ ^ B 2 SignatureðbÞÞ such that thereexists a subsumption relation between A and B by using certainreasoning paths of concept subsumption.
Definition 9 (Semantic partition). A sub-TBox T 0 is a semanticpartition of TBox T if and only if the following conditions hold:
1. T 0 # T2. 8a 2 T 0 and 8b 2 T n T 0, a and b are semantically irrelevant.3. there is no T 0 � T 00 such that T 00 is a semantic partition of T .
Example 4. Look back to Example 2. The semantic partitions of Tare as follows: T 0 ¼ f1;2g and T 00 ¼ f3; 4; 5g.
According to Definition 9, we present Algorithm 3 to obtain allontology partitions of the TBox T .
Algorithm 3. Obtaining semantic partitions of T
Input: The TBox TOutput: PartS1 PartS[] = ;;2 nosp = 0;3 for all a 2 T and not labeled(a) do4 label a;5 move a to PartS[nosp];6 for all b 2 T and not tagged(b) do7 if SemRelevant(a,b) then8 label b;9 move b to PartS[nosp];10 end11 end12 nosp++;13 end14 return PartS;
We know that semantically relevant axioms cannot becontained in two different semantic partitions. Meanwhile, all axi-
oms in same MIPS are semantically relevant. So, it is not difficult toobtain the following proposition.Proposition 1. The same MIPS cannot be contained in differentsemantic partitions.
This can help separate a knowledge base into multiple semanticpartitions so as to reduce complexity of computing relevant ontol-ogy constructs.
Definition 10 (Incoherence impact value per axiom). Let SMIS ={MIS1,MIS2, . . .,MISn} be the set of MIPSs of T . For 1 6 j 6m and1 6m, ij,k 6 n, the incoherence impact of each axiom a in T can bedefined by a function mapping imp : T ! N such that
impðaÞ ¼m if for all i1; . . . ; ij; . . . ; im such that a 2 MISij
0 if for each MISk 2 SMIS such that a R MISk
�We give Algorithm 4 to compute incoherence impact of each
axiom.
Algorithm 4. Computing the imp value of each axiom in T
Input: A non-empty TBox T , and SMIS—the set ofMIPSsOutput: IMP[]1 j = 0;2 for all a 2 T do3 i = 04 for all mis 2 SMIS do5 if a 2mis then i++;6 end7 IMP[j++] = (a, i)8 end9 return IMP[];
Example 5. Look back to the example in Example 3. We findimp(1) = imp(2) = imp(3) = imp(4) = imp(5) = 1 and imp(6) = 0. Eachaxiom in T at most belongs to one MIPS of T .
5. Pinpointing axioms and concepts causing incoherence
In the section, we will use the example from the literature (Lam,Pan, Sleeman, & Vasconcelos, 2006), which is shown in Example 6.But note that the example ontology should be transformed accord-ing to a series of steps of ontology normalization. For example,complex concepts should be named in an iterative way.
Example 6
1: PGCourse v :UGCourse,2: UGCourse v Course,3: PGCourse v Course,4: PGStudent v Student,5: UGStudent v Student,6: MphilStudent v PGStudent,7: PhDStudent v PGStudent,8: MScStudent v PGStudent,9: MphilStudent v "take�Course,
10: MScStudent v "take� (PGCourse u Exam),11: PhDStudent v "take�PGCourse,12: Student v $register�Dept u $take�UGCourse,13: PGStudent v 9take � ð:ExamÞ
5.1. Pinpointing axioms causing incoherence
The axioms causing incoherence can be pinpointed by theMIPSs in a TBox, which was introduced by Schlobach (2005).Mildly different from the method, our method first computessemantic partitions of the ontology being analyzed. Then, we com-pute MIPSs for each semantic partition obtained. This probably re-duces computational complexity of computing MIPSs.
Y. Ma et al. / Expert Systems with Applications 37 (2010) 7269–7275 7273
In Example 6, there are two semantic partitions in the ontology.They are as follows: T 0 ¼ f1;2;3g and T 00 ¼ f4;5;6;7;8;9;10;11;12;13g. According to Algorithm 3, we find that the sub-ontology T 0 is coherent and has no MIPSs. There are three MIPSsin T 00, which are as follows. mips1 = {4,7,11,12}, mips2 = {8,10,13}and mips3 = {4,8,10,12}.
Although the axioms causing semantic incoherence can bedetermined by using the method of Schlobach (2005) or Algorithm3, it is not enough for ontology engineers to revise the related axi-oms. It is desirable for ontology engineers to determine which con-cepts are very likely to cause ontology incoherence so as tofacilitate automatic revisions of evolving ontologies.
5.2. Pinpointing and ranking concepts causing incoherence
5.2.1. Concept subsumption tree
Definition 11 (Concept subsumption weighted tree). Let SMIS ={mis1,mis2, . . .,misn} be the set of all MIPSs in a semantic partition.Let N ¼ [n
1misi. A concept subsumption tree is denotedCSWT = (CN ;E;W), where CN is the set of defined concepts in theSMIS, E # CN � CN is the set of axioms in N and W is the set ofweight values, and "a 2 E, imp(a) 2W.
It is not difficult to find that the defined concepts and axioms ina TBox are nodes and edges in a CSWT, respectively. The weight va-lue of each edge just is the incoherence impact value of the axiomcorresponding to the edge. Intuitionally speaking, a CSWT can re-flect unsatisfiability propagates, i.e. one unsatisfiable conceptmay cause many other concepts to become unsatisfiable as well.It is desirable for ontology engineers to determine which conceptsare most likely to cause incoherence in satisfiability propagates soas that they can directly pinpoint and revise them. Here, we con-sider the CSWT as an undirected graph because we concentrateon semantic interconnections between concepts rather than thedirection of semantics. In the following, we first give the distancebetween two concepts in an undirected CSWT.
Definition 12 (Distance between two concepts). Let C,D be any twoconcepts in CSWT ¼ ðCN ;E;WÞ, i.e. C;D 2 CN . Let Path(C,D) be theset of paths between C and D. "p 2 Path(C,D), Ep # E represents
Fig. 2. An example of CSWT.
Table 1Normalized distances of concepts in Fig. 2.
Concept(C) X Y M N
normdw(C) 1.25 1.89 1.786 1.31w(C) 0.94 1.44 1.308 0.987
the set of edges in path p. d(p) = jEpj represents the number ofedges in path p. dwðpÞ ¼
Pe2Ep
impðeÞ represents the sum of theweights of edges in path p. "p 2 Path(C,D), dw(C,D,p) represents thedistance between concepts C and D, where
dwðC;D;pÞ ¼P
e2Ep
impðeÞ !
=dðpÞ ifdðpÞ– 0
0 ifdðpÞ ¼ 0
8><>:
Note that in a CSWT, the axiom of the form A � B describing
equivalent classes is not explicitly shown because an axiomdescribing equivalent classes can be regarded as a node in theCSWT.Next, we use the shortest path algorithm similar to the litera-tures (Dijkstra, 1959; Johnson, 1977) to compute the shortest pathbetween any two concepts in the CSWT. We then normalize thedistance by the number of concepts reachable by the given conceptin the undirected CSWT.
Definition 13. Normalized distance of a given conceptFor anundirected CSWT, let dw(C,D) = minp2Path(C,D){dw(C,D,p)} representsthe shortest distance between concepts C and D. Let RC be the setof concepts reachable by the given concept C in the CSWT. Thenormalized distance of the given concept C is denoted normdwðCÞ ¼P
D2RCdwðC;DÞ
� �=jRC j.
The values of normalized distance of concepts can be used todetermine which concepts are most likely to cause incoherencein satisfiability propagates in a CSWT. Intuitionally speaking, the
larger the normalized distance of a concept is, the more likelythe concept is to cause incoherence in satisfiability propagates.Taking Example 6, we build the CSWT for all MIPSs in semanticpartition T Prime (see Section 5.1), which is shown in Fig. 2, whereX � "take�PGCourse, Y � 9register � Dept u 9take � UGCourse;M �9take � :exam, N � "take�(PGCourse u exam). We can compute thenormalized distances of these concepts, which is shown in Table 1.From Table 1, we find that concept Y has the largest normalizeddistance. In the case, we say that the concept should be preferablyaddressed to resolve semantic incoherence. Furthermore, we arguethat fanouts of each unsatisfiable concept also should be consideredbecause the concepts with more fanouts can be thought of as hot-spots in the ontology. This is similar to Kleinberg’s algorithm(Kleinberg, 1999) and the PageRank algorithm (Brin & Page,1998). We will pinpoint and rank concepts according to the follow-ing equation. For a TBox T and the set of defined concepts CT , thepreferable revision weight of each concept C 2 CT is denoted w(C),where
wðCÞ ¼ k � normdwðCÞ þ ð1� kÞ � jAFC j=maxD2CfjAFDjg ð1Þwhere 0 < k < 1 is a real number and allows fine tuning of the con-cept ranking criteria. Table 1 shows the w values of these conceptswhen k ¼ 2
3.
6. Inconsistent ontology revision
6.1. Incoherent TBox revision
In the following, we give the following algorithm to revise theincoherence in a given TBox T .
Student PGStudent PhDStudent MScStudent
1.786 1.643 1.443 1.691.420 1.429 1.073 1.124
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1. Compute all semantic partitions of T . For each semantic parti-tion, if it is incoherent, then compute its SMIS.
2. For each SMIS, build its CSWT. And then further pinpoint andrank the unsatisfiable concepts that will be preferably revisedaccording to Eq. (1).
3. Select a concept C to be preferably revised from the CSWT,and further select the subsumption axiom a = maxC2Signature(b)
{imp(b)}.4. If C is in the left (right) side of a, then appropriately strengthen
(weaken) C such that the number of MIPSs in SMIS willdecrease.
5. If the semantic partition being revised still is incoherent,then go step 3 to continue to revise other ranked concepts.
6. If all semantic partitions are coherent, then the algorithm ter-minates. Otherwise, go to step 2 for next SMIS.
In the algorithm, concepts possibly can be strengthenedor weakened. We use the method similar to the literatures(Plessers & Troyer, 2006). A concept A is weakened either byremoving A or by replacing it with an appropriate superclass ofA. A concept A is strengthened by replacing A with an appropriatesubclass of A. We propose a set of rules for strengthening andweakening concepts so as to resolve ontology incoherence. Here,if concept A is in the left side of a, it is denoted left(A,a). Similarly,if concept A is in the right side of a, then it is denoted right(A,a).The rules for strengthening and weakening concepts are asfollows.
left(A,a)
— A defined concept A can be strengthened by replacing Awith some subclass of A.— If there are no any appropriate subclass of A to strengthenA, we find out the defined concept B in the right side of a, i.e.right(B,a). Then the defined concept B can be weakened byreplacing B with some superclass of B.— If there still are no any appropriate superclass of B toweaken B, then delete axiom a.right(A,a)
— A defined concept A can be weakened by replacing A withsome superclass of A.— If there are no any appropriate superclass of A to weakenA, we find out the defined concept B in the left side of a, i.e.left(B,a). Then the defined concept B can be strengthened byreplacing B with some subclass of B.— If there still are no any appropriate subclass of B tostrengthen B, then delete axiom a.Note that the superclasses and subclasses of the replacingconcepts also must be the defined classes. If there are no anyappropriate superclass (resp. subclass) of A to weaken (resp.strengthen) A, this means that A has no defined superclasses (resp.subclasses) or such concept replacements for concept A will not re-duce the number of MIPSs in the revised ontology.
Take the example in Fig. 2 to illustrate our concept replace-ment. First, we compute w(C) value for each defined conceptin the CSWT. According to Table 1, We obtain that the conceptY has the largest preferable revision weight, and thus it shouldbe preferably revised so as to reduce incoherent degree of therevised ontology. Because Y is in the right side of axiom 12, itshould be weakened by its a superclass of Y. According to Defi-nition 4, we compute fanouts AFY = {Y1,Y2}, where Y1 � $regis-ter�Dept and Y2 � $take�UGCourse. It is appropriate for theweakening of Y to replace Y with Y1 because such a replacementcan reduce the number of MIPSs. If Y2 is used, the number ofMIPSs is not reduced.
6.2. Inconsistent ABox revision
In the subsection, we assume that the TBox is coherent. We re-vise ABox inconsistency through the following two steps. First, wepinpoint the axioms causing inconsistency. Specifically speaking,for each clash including {C(a),D(a)}, by keeping track of the tableaualgorithm for satisfiability checking of instance A, we find out theaxioms from which C(a) and D(a) can be derived. The set of theseaxioms is denoted S. Then, we compute fanouts of each conceptin set S, and rank these concepts according to their fanout values.Second, we select the concept with the highest fanout value, andthe subsumption axiom including the concept. Then we can revisethe concept by strengthening or weakening it in order to resolveinconsistency.
Taking Example 6, we revise the ontology by replacing conceptY with Y1 and removing concept M (i.e. removing axiom 13). As aresult, the TBox of the revised ontology will be coherent. We as-sume that the ABox of the coherent ontology includes the follow-ing assertions: {MScStudent(m), take(m,c), UGCourse(c)}. We findthat the ABox is inconsistent with respect to the TBox becausethe clash {PGCourse(c),UGCourse(c)} occurs. In the first step, wedetermine the set S of axioms causing the inconsistency by keepingtrack of tableau algorithm for satisfiability checking of instance c,i.e. S ¼ fN � 8take � ðPGCourse u examÞ; MScStudent v N; PGCoursev :UGCourseg. It is not difficult to find that jAFNj = 1, jAFPGCoursej = 0and jAFMScStudentj = 0. Because concept N has the highest fanout va-lue and is in the right side of subsumption axiom MScStudent v N,we weaken N by replacing N with its an appropriate superclass N1,where N1 � "take�(Course u exam). In the situation, the consistencyin the ABox will be resolved.
7. Related work
In the context of changing and dynamic Web, ontologies willinevitably bring about internal inconsistencies with the evolutionof ontologies. To our knowledge, our work is most relevant to thework by Deng, Haarslev, and Shiri (2007); Qi and Hunter (2007).They aim to detect and resolve incoherence/inconsistency in anontology by measuring inconsistent/incoherent degree of theontology. Some measures also were proposed. In order to measureincoherent ontologies, Qi and Hunter (2007) proposed two classesof metrics to measuring so-called ontology incoherence in ontolo-gies. The first class of metrics measures unsatisfiable conceptnames for comparing them. The second class of metrics aims tomeasure terminology axioms for comparing ontologies. They alsoreported some preliminary but interesting experimental results.However, authors did not further tell us how we can modify inco-herence and inconsistency in an ontology by using these metrics tomeasure ontology incoherence. Deng et al. presented an approachto measure inconsistencies in an ontology by using the so-calledShapley value (Deng et al., 2007). By measuring inconsistency inan ontology, authors can determine which axioms or which partsof these axioms should be removed or revised to make the ontol-ogy consistent. We compare our work with these work. First, wepropose some ontology constructs for resolving ontology inconsis-tency/incoherence. These constructs are defined from differentperspectives, and cover semantic partitions, fanouts of concepts,and inconsistency degree of axioms. In contrast, the measures pro-posed in Deng et al. (2007); Qi and Hunter (2007) only measureunsatisfiable concept names and terminology axioms. Second, intheir papers, the details how to apply their metrics to revise incon-sistent/incoherent an ontology are sketchy. Compared with thesework, our work not only defines some ontology metrics, but alsoproposes some relevant algorithms to apply our ontology metricsto revise inconsistent ontologies.
Y. Ma et al. / Expert Systems with Applications 37 (2010) 7269–7275 7275
There also were some methods that were proposed to resolveinconsistency/incoherence in ontologies. In SWOOP (Kalyanpur,Parsia, Sirin, & Grau, 2006), some strategies were used to rank erro-neous axioms according to their importance and further rewritethese axioms. Lam et al. proposed some methods to handle ontol-ogy inconsistency by ranking and rewriting axioms (Lam et al.,2006). Authors first found Maximally Concept-Satisfiable Subon-tologies (MCSSs), then proposed knowledge heuristics based onontology structure and patterns of usage of the ontology for select-ing the MCSS which is most likely to be correct. Plessers and Troyer(2006) proposed an algorithm to detect axioms causing a logicalinconsistency by extending the tableau algorithm. Then a set ofrules was proposed and used to resolve the detected inconsistency.Haase et al. proposed an approach to localize an inconsistency byusing the so-called a Minimal inconsistent sub-ontology (Haase &Stojanovic, 2005). An unsatisfiable concept can be resolved byremoving one axiom from the minimal inconsistent sub-ontology.However, this approach cannot determine the true cause of theinconsistency. Furthermore, it possibly bring about the loss ofsome useful ontology information. In contrast, Schlobach et al. pre-sented the concepts of MUPS and MIPS, through which we can pin-point the axioms causing incoherence in an ontology (Schlobachet al., 2007). Our work has used the concept of MIPS fromSchlobach et al. (2007). In belief revision community, some workalso was made to deal with inconsistency such as Hansson andWassermann (2002), Meyer et al. (2005). Different from theseworks, our work focuses on measuring some ontology constructsto resolve inconsistency/incoherence in an ontology during thecourse of ontology revision.
8. Conclusion
In this paper, we propose an approach to revise inconsistency inan ontology by measuring the relevant ontology constructs in theontology being measured. Our approach can help determine whichconcepts should be preferentially modified, and enables automaticrevision of ontology inconsistency in the context of a dynamic andevolving environment.
Acknowledgements
This work is partially supported by the National ‘‘863” High-Tech Program under Grant (No. 2007AA04Z148), the National Nat-ural Science Foundation of China under Grant (No. 60703036) andthe Fundamental Research Funds for the Central Universities.
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