Detection Algorithm of Semantic Inconsistency for Fuzzy Ontology

Merging

Yonghong Luo1,a*, Weiming Liu1,b

1School of Civil Engineering and Transportation, South China University of Technology, Guangzhou

510641, China

acourse_c@163.com, blwm_64@sina.cn

Key words: Fuzzy Ontology, Ontology Merging, Membership Grade, Semantic Inconsistency

Mapping

Abstract. Fuzzy ontology can be built to effectively deal with uncertainty and ambiguity for

domain knowledge modeling. Merging multiple fuzzy local ontologies may implement semantic

integration of multiple data sources and semantic interoperability between heterogeneous systems in

distributed environment. In order to solve the problem of semantic inconsistency mappings for

fuzzy ontology merging system, we proposed a detection algorithm of semantic inconsistency

mapping which includes sub detection methods of circular semantic inconsistency, subclass-of

axiom redundancy semantic inconsistency, attribute membership semantic inconsistency and

disjoint axioms redundancy semantic inconsistency. With the detection algorithm of semantic

inconsistency, we establish fuzzy ontology merging system in experiment.

Introduction

It is clear that precise ontology model can’t fully describe uncertain knowledge and vague

information in many application domains. However, the fuzzy ontology [1] with fuzzy concepts is

an extension of the domain ontology with crisp concepts. It is more suitable to describe the domain

knowledge than domain ontology for solving the uncertainty reasoning problems. It is necessary to

introduce fuzzy ontology for representing fuzzy information. Nowadays, applied ranges of fuzzy

ontology has been gradually extended, but constructing fuzzy ontology with different languages,

methods and descriptive ways might result in heterogeneity of multiple ontologies in the same

domain. So, semantic fusion of multiple data sources and semantic interoperability between

heterogeneous systems in distributed environment such as grid computing [2] and cloud computing

[3] usually can be implemented through integrating multiple fuzzy local ontologies. However,

ontology merging is one of the valid ways for ontology integration.

Currently there are few works on fuzzy ontology merging. Authors in literature [4, 5] proposed

an integration method of fuzzy ontology based on consensus in which integrated algorithm have not

be verified by experiments, so the accuracy and effectiveness of fuzzy ontology integration can not

be ensure. Literature [6] used concept lattice gluing to merge fuzzy ontologies, and proposed a

method of fuzzy ontology merging based on fuzzy concept gluing. We all know that precise

ontology merging methods are semi-automatic, and are not suitable for merging fuzzy ontologies

due to lack of means to handle uncertain, vague information. So, there are still no effective

solutions to solve the problem of semantic inconsistencies for concept mapping in fuzzy ontologies

merging system at present. For the various inconsistencies, fuzzy ontology merging systems need

more human intervention to repair inconsistency relations between concepts. In order to effectively

solve the problem of semantic inconsistency in fuzzy ontologies merging system, and strive to

Applied Mechanics and Materials Vols. 530-531 (2014) pp 407-412Online available since 2014/Feb/27 at www.scientific.net© (2014) Trans Tech Publications, Switzerlanddoi:10.4028/www.scientific.net/AMM.530-531.407

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realize full automation of fuzzy ontology merging, we present a detection algorithm which can find

out all of semantic inconsistencies.

Description of Fuzzy Ontology

Definition 1. A fuzzy ontology can be defined as a 7-tuple form: ( )FO C, A, V, U, R, Z, I=

where C is set of concepts, a concept is fuzzy concept if it contains at least one attribute with

membership grade or it is set of fuzzy set; A is set of attributes belonging to the concepts in C. An

attribute is a fuzzy attribute if its value is a fuzzy set; V is the domain of A. V also can be expressed

as a set of attribute values, and aV Va A∈= ∪ where Va is the domain of the attribute a; U is set of

membership functions, and the range of each membership function is a concrete fuzzy concept; R

is set of fuzzy relations between concepts: 1 2 nR {R , R , ,R }= … where ( ]iR C C 0,1⊂ × × for i=1,

2,…, n. A relation is then a set of pairs of concepts with a weight representing the degree to which

the relationship should belong; Z is set of axioms, which can be interpreted as integrity constraints

or relationships between instances and concepts; I is set of instances. An instance is fuzzy instance

if its corresponding concept is fuzzy concept.

Definition 2.A fuzzy concept FC can be defined as a triple form: FC FC FCFC (A ,V ,f )= where

FCA A⊂ is set of attributes describing the concept, including fuzzy attributes and precise attributes;

FCV V⊂ is the union of each attribute domain; fFC

is set of fuzzy functions which represent the

degrees that the attributes of fuzzy concepts describe their concept instances, [ ]FC FCf :A 0,1→ .

Definition 3. A concrete fuzzy concept cfc C⊆ can be defined as a 4-tuple from:

cfc cfc cfc cfccfc (V ,V ', L ,f )= where cfc is the unique identifier of concrete fuzzy concept; cfcV V⊆ is

the domain of concrete fuzzy concept cfc; cfcV ' (0,1]⊆ presents fuzzy values of the concrete set

Vcfc; cfcL V⊆ models linguistic qualifiers, which is determined by the strength of the attribute

value in Vcfc; cfcf U∈ is a membership function on Vcfc, cfc cfc cfcf :V V '→ , ( )cfc cfc cfcv V ,f v V '∀ ∈ ∈ .

Fuzzy Ontology Mapping

According to the definitions of concrete fuzzy concept (property), the similarity degree between

fuzzy concepts or fuzzy properties depends on their fuzzy sets. We assume that { }1 AFC A,V,f= and

{ }2 A'FC A',V',f= are the concrete fuzzy concepts. V and V’ are the concrete sets. A and A’ are the

corresponding fuzzy sets of V and V’, respectively. We specify U V V'= ∪ , the operations between

fuzzy subsets can be defined as follows:

' ', ( ) min{ ( ), ( )}A A A Ax U f x f x f x∩

∀ ∈ = (1)

408 Advances in Measurements and Information Technologies

' ', ( ) max{ ( ), ( )}A A A Ax U f x f x f x∪

∀ ∈ = (2)

' ', ( ) min{ ( ),1 ( )}A A A Ax U f x f x f x−

∀ ∈ = − (3)

' ', ( ) min{1 ( ), ( )}A A A Ax U f x f x f x−

∀ ∈ = − (4)

A-A’ is the fuzzy subset of elements that belong to A and not to A’, A’-A is the fuzzy subset of

elements that belong to A’ and not to A. So, the similarity degree between attribute (concept) a1 and

attribute (concept) a2 can be calculated with the following expressions:

1 2 ' 1 2 '( , ) sup ( ) ( , ) sup{min{ ( ), ( )}}x U A A A ASim a a f x Sim a a f x f x∈ ∩

= → = (5)

1 2 ' 1 2 '( , ) sup ( ) ( , ) sup{max{ ( ), ( )}}x U A A A ASim a a f x Sim a a f x f x∈ ∪

= → = (6)

1 2 ' 1 2 '( , ) sup ( ) ( , ) sup{min{ ( ),1 ( )}}x U A A A ASim a a f x Sim a a f x f x∈ −

= → = − (7)

1 2 ' 1 2 '( , ) sup ( ) ( , ) sup{min{1 ( ), ( )}}x U A A A ASim a a f x Sim a a f x f x∈ −

= → = − (8)

Therefore, the similarity degree of attributes between fuzzy concepts of FC1 and FC2 can be

calculated as follows:

1 2 )

11 2

1 2

( , )( , )

max(| |,| |)

n FC FC

i iiSim a a

AS FC FCFC FC

==∑

(9)

where 1aFC and 2aFC are attributes belonging to FC1 and FC2, respectively; |FC1| and |FC2| denote

the number of attributes for FC1 and FC2, respectively; n is the maximum value of |FC1| and |FC2|.

The calculation of similarity degree between fuzzy concepts usually needs to take into multiple

important factors of concepts account. Here we determine the final similarity degree between fuzzy

concepts through aggregating name similarity, semantic similarity and attribute similarity.

In this paper, name similarity and semantic similarity [7] between FC1 and FC2 are denoted as

NS(FC1, FC2) and SS(FC1, FC2),respectively.

Finally, the integrated similarity degree between fuzzy concepts can be calculated as follows:

1 2 1 2 1 21 2

( , ) ( , ) ( , )( , )

3

a n sw AS FC FC w NS FC FC w SS FC FCS FC FC

× + × + ×= (10)

Where wa, wn and ws denote the weight of attribute similarity, name similarity and semantic

similarity respectively, and a n sw w w 1+ + = .

With the formula of similarity degree between fuzzy concepts, we can easily implement the

initial mapping algorithm between fuzzy ontologies.

Detection algorithm of semantic inconsistency

Definition 4 Circular Semantic Inconsistency: Given that C D⊆ in FO1 and ' 'D C⊆ in FO2,

Mapping(C,C’) and Mapping(D,D’) are semantically inconsistent as they violate the subsumption

rule with respect to GCIs(Generalized Concept Inclusions) or PCS(Property Subsumption Criteria)

of local ontologies FO1 and FO2. We call the semantic inconsistency of Mapping(C,C’) and

Mapping(D,D’) as Circular Semantic inconsistency.

Applied Mechanics and Materials Vols. 530-531 409

Definition 5 Semantic Inconsistency of Subclass-of Axiom Redundancy: Given that concept C

and concept D belong to direct parent-child relation (D subclassof C) in FO1, concept C’ and

concept D’ belong to indirect parent-child relation (D’ subclassof K, K subclassof C’) in FO2.

Mapping(C,C’) and Mapping(D,D’) violate the sunsumption rule of GCIs. We call the semantic

inconsistency of Mapping(C,C’) and Mapping(D,D’) as Semantic Inconsistency of Subclass-of

Axiom Redundancy.

Definition 6 Semantic Inconsistency of Attribute Membership: Given that the attribute set of

concept C is A, the attribute set of concept C’ is A’. Mapping(C, C’) violates consistent attribute

membership rule if ( ) ( )a au C u C'≠ for a (a A A')∀ ∈ ∩ . We call the semantic inconsistency of

Mapping(C, C’) as semantic inconsistency of attribute membership.

Definition 7 Semantic Inconsistency of Redundant Disjoint Axioms: Given that concept C

disjoints with concept D in FO1, but C’ overlap D’ in FO2. Mapping(C,C’) and Mapping(D,D’)

violate the satisfiability rule of knowledge. We call the semantic inconsistency of Mapping(C,C’)

and Mapping(D,D’) as Semantic Inconsistency of Redundant Disjoint Axioms.

The detection algorithm of semantic inconsistency is as follows:

Input: mapping_list, FO1, FO2

Output: different type sets of inconsistency mapping

Begin

step1: n=mapping_list.length;

step2: for (i=0; i<n; i++)

step3: Mapping mp=mapping_list (i);

step4: if (mp.depth1! = mp.depth2) then

step5: selected_mapping.add (mp);

step6: end if

step7: end for

step8: k=selected_mapping.length;

step9: for (i=0; i<k-1; i++) do

step10: Mapping mp1=selected mapping (i);

step11: for (j=i+1; j<k; j++) do

step12: Mapping mp2=selected_mapping (j);

step13: if (mp1.depth1<mp2.depth1 and mp1.depth2>mp2.depth2) then

step14: if (FO1.checkParentChildRelation (mp1.name1, mp2.name1) or

FO2.checkParentChildRelation (mp2.name2, mp1.name2) then

step15: adding (mp1, mp2) to circular_incon [];

step16: end if

step17: end if

step18: end for

step19: end for

step20: for (i=0;i<n-1;i++) do

step21: Mapping mp1=mapping_list(i);

step22: for (j=i+1; j<n; j++) do

step23: Mapping mp2=mapping_list (j);

step24: if (FO1.checkDirectPraentChildRelation (mp1.name1, mp2.name1) and

FO2.checkIndirectPraentChildRelation (mp1.name2, mp2.name2) then

step25: adding (mp1, mp2) to subclassof_redundant_incon [];

410 Advances in Measurements and Information Technologies

step26: end if

step27: end for

step28: end for

step29: for (i=0; i<n; i++) do

step30: Mapping mp=mapping_list (i);

step31: for ∀a in (mp.Concept1.A ∩ mp.Concept2.B) do

step32: if ua (mp.Concept1)! = ua (mp.Concept2) then

step33: adding mp into membership_inconsistency [];

step34: end if

step35: end for

step36: end for

step37: for (i=0; i<n; i++) do

step38: Mapping mp1=mapping_list (i);

step39: if hasDisjointConcepts (mp1.name1) then

step40: Discon [] = GetDisjointSiblings (mp1.name1);

step41: end if

step42: for (j=0; j<Discon.length; j++)

step43: Mapping mp2=searchMapping (Discon[j]);

step44: if ((mp1.name2 and mp2.name2) are NotDisjointWith or Equivalence or

ParentChildRelation or hasCommonClass) then

step45: adding (mp1, mp2) to redundant_disjoint_incon [];

step46: end if

step47: end for

step48: end for

End.

Conclusions

Due to the differences on tool and language for different R&D teams to construct fuzzy ontologies,

especially on the understanding and description of concept structure, so that it can lead to the

heterogeneity of various fuzzy ontologies in the same domain. Therefore, we have presented a

detection algorithm for the semantic inconsistency of concepts mapping in this paper, and

implemented a fuzzy ontology merging system (FOMS). FOMS only need to analyze the results of

initial mapping solution, then can find out all of the semantic inconsistency mappings. In the future,

we will highlight the following works: (1) Modifying the detection algorithm of semantic

inconsistency to improve the merging accuracy for FOMS. (2) Studying on the detection algorithm

of semantic inconsistency for fuzzy ontologies described with different languages.

References

[1] Changshing Lee, Zhiwei Jian, Linkai Huang, A fuzzy ontology and its application to news

summarization, J. Man, and Cybernetics, Part B: Cybernetics.35(5) 859-880.

[2] P. Brezany, A. Woehrer, A. M. Tjoa, The Grid: vision, technology development and applications,

J. e & i Elektrotechnik und Informationstechnik. 123(6)251-258.

[3] V. V. Arutyunov, Cloud computing: Its history of development, modern state, and future

considerations, J. Scientific and Technical Information Processing. 39(3)173-178.

Applied Mechanics and Materials Vols. 530-531 411

[4] Trong Hai Duong, Ngoc Thanh Nguyen, Fuzzy Ontology Integration Using Consensus to Solve

Conflicts on Concept Level, in: Proceedings of New Challenges for Intelligent Information and

Database Systems, Daegu, 2011, pp. 33-42.

[5] Hai Bang Truong, Ngoc Thanh Nguyen, A Multi-attribute and Multi-valued Model for Fuzzy

Ontology Integration on Instance Level, in: Proceedings of the 4th Asian conference on Intelligent

Information and Database Systems.Kaohsiung, 2012, pp.187-197.

[6] Li Zhaohai, Li Guanyu, Concept lattice gluing based fuzzy ontology merging method, J.

Application Research of Computers. 29(7) 2517-2519.(In Chinese).

[7] Ming Mao, Yefei Peng, Michael Spring, An adaptive ontology mapping approach with neural

network based constraint satisfaction, J. Web Semantics. 8(1) 14-25.

412 Advances in Measurements and Information Technologies

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