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Int. J. of Metadata, Semantics and Ontologies,, Vol. x, No. x, 2015 1
Multi-level Networked Knowledge:Representation and DL-Reasoning
Sihem KlaiLabGed, Department of Computer Science,University of Badji Mokhtar of Annaba, Po-Box 12, 2300, Algeriaqquad E-mail: [email protected]
Antoine ZimmermannEcole Nationale Superieure des Mines, FAYOL-ENSMSE, LSTI, F-42023Saint-Etienne, France
E-mail:[email protected]
Med Tarek KhadirLabGed, Department of Computer Science,University of Badji Mokhtar of Annaba, Po-Box 12, 2300, Algeria
E-mail: [email protected]
Abstract: Integrating pre-existing, heterogeneous, and complementary ontologies, andexploiting them jointly in reasoning remains a major challenge. Ontology alignmentsmake explicit the correspondences between terms from different ontologies and mustbe taken into account in reasoning. Two forms of correspondences can be introduced:mappings represent predefined relations such as subsumption, equivalence, or disjointness,that have a fixed semantics in all interpretations; links can relate complementaryontologies by introducing terms defined by experts, and their semantics varies accordingto interpretations. Different experts can introduce different terms according to their pointsof view, which brings semantically heterogeneous links. Thus, integrating pre-existingnetworks of aligned ontologies requires aligning terminologies from different alignments, soas to form higher level alignments. This generates networked knowledge that can in turnbe aligned with other networked knowledge. As a resultn we talk of multi-level networkedknowledge, a concept that we formalise here and for which we propose a possible formalsemantic for automating reasoning tasks. This semantic consists in reducing reasoningon networked knowledge to reasoning over DL formalisms for which we have reasoningprocedures. The proposed approach is implemented and tested in order to compare results,for different networks.
Keywords: networked knowledge; ontologies; ontology alignments; DL-reasoning.
1 Introduction
In information systems, and more recently in WebSemantic, a number of heterogeneous, independentlydeveloped ontologies may be exploited in a singleapplication that needs to share some knowledge. Theseontologies are developed in different contexts and maywell cover complementary domains.
In order to overcome the heterogeneity problem,complementary knowledge may be introduced in orderto describe correspondences between ontologies to beexploited. These correspondences, represent relationsbetween entities (terms or formulas) belonging todifferent ontologies. This set of correspondences istermed ontology alignment.
In order to exploit, during reasoning, a number ofheterogeneous ontologies as well as correspondences, asimple solution consists in viewing the ontology system
as a unique global ontology. Therefore, each localontology as well as each alignment, is then considered asa knowledge complement over a larger domain. Takinginto account all this knowledge, i.e., ontologies andalignments, may be performed using a fusion processobtaining a centralised system, or using distributedreasoning algorithms based on classical logic, (as shownin: SomeWhere [1], SomeRDFS [3] and SomeOWL [2]).Such approaches consider ontologies and alignmentsdescribing a unique global theory, however, presentinginconvenient if the ontologies to be combined are highlyheterogeneous. They will, therefore, describe differentcontexts and points of view, potentially incompatible.The other possible solution consists in managing the setof ontologies as well as the corresponding alignmentsas a complex semantic network, where each node isrepresented by an ontology formalizing a given domain,
Copyright c© 2008 Inderscience Enterprises Ltd.
Copyright c© 2009 Inderscience Enterprises Ltd.
2 S. Klai et al.
with a different context than the ones given by therest of the ontologies within the network. Such anapproach, needs strong formalism modeling the alreadyaligned ontology network, offering specific algorithmsand techniques for contextual reasoning.
In that sense, a number of formalisms have beenproposed to model an aligned ontology network forcontextual reasoning. These formalisms may be dividedinto two categories based on definition and applicationpurposes.
Alignment is a major concern, as it constitutesan important element of the complete system,distinguishing two major types of correspondences inorder to define them. The first type is for instance givenby Distributed Description Logic [6] and IDDL [27])that define relations, termed proposed mapping, in orderto reduce semantic heterogeneity problems betweenterms and entities belonging to different ontologies.These correspondences are associated with a predefinedset of relations such as subsumption, equivalence,disjunction, etc. where the given semantic is fixed for all
interpretations (e.g., mt:belong⊥←→ eq:belong).
The second type of alignment is used to linkontologies covering complementary domains, it is thecase of E-connection [21]. It is represented by inter-ontological links between entities, termed simply links
(e.g., pr:T1compose←→ eq:FD1). This type of relations is
defined by experts in the context of domain ontologycombination, as well as semantic representation ofcontext links.
In this paper, we focus on proposing a formalismwhich supports both alignment types in order to permitheterogeneous ontology combination associated withdifferent contexts covering complementary domains.Regarding the second point of difference, whichis the application or treatment of alignments, themajority of proposed formalisms, such as DistributedDescription Logic [6], E-connection [21] and Package-based Description Logic [4]), consist in integratingalignment as external knowledge for the correspondingtarget ontology. The alignment is then, defined andexploited following the target ontology point of view.In order to ensure reasoning over the ontology network,each ontology must be enriched with a reasoningmechanism which support external knowledge.
The other way of looking at the problem is toconsider alignments to be exploited at a higher levelindependent from local ontologies and termed globallevel [27]. In this approach, the alignment language maybe more expressive than the languages defining localontologies, allowing better alignment reuse. However,only mappings are considered in this work, and noproposition concerning the integration of links at theglobal level is made.
Links are, supposedly, introduced by experts.However, this may be unfeasible if experts covering allontologies domain cannot be found. Therefore, if distinctpairs of ontologies are aligned by different experts with
different terms and points of view, then it is likely thatthe heterogeneity problem will needs to be consideredthis time between links.
The proposed contribution consists in a newformalism termed:”Multi Level Networked Knowledge(MLNK)”.
The organisation of the rest of the article is asfollows: Section 2 describes a scenario representingan ontology network aligned using heterogeneous pairsontology alignments. Section 3 presents the state ofthe art for Multi level formalisms with similar worksdiscussed. Section 4, describes the syntax formalism ofthe MLNK components. In Section 5, a DescriptionLogic interpretation of a given node have been studiedin the goal of applying reasoning strategies. Finaly,section 6 depicts the implementation and evaluation ofthe proposed approach, followed by a general conclusion.
2 Motivation example
In this section, a real life application example of gasturbine ontological representation is presented. Due totheir wide usage in electricity production, gas turbineare often found in the center of large power systemsthat need to be managed in terms of knowledge andmaintenance. Four ontologies describing gas turbine havebeen developed for the purpose of this example, namely:
• an ontology for equipment (eq), modeling theturbine technical and hierarchical knowledge.These information are provided by the constructorand contains 5033 concepts, where each conceptdescribes an equipment or turbine component,such as the concept flame-detector given byinstance FD1;
• an ontology termed (Pr), modeling spare parts,such as the concept trim given by the instance T1;
• an ontology for modeling the position of theequipment in the turbine hierarchy (zn);
• An ontology created from an existing databasemt, using a semi automatic approach, covering,maintenance operations (both preventive and afterbreakdown). The mt ontology exploits the firstontologies (eq),Pr and zn) in order to providedetails on equipments and spare parts concernedby maintenance operations.
Exploiting these ontologies requires their alignment andthe integration of the latter in all reasoning or searchstrategies. For this purpose a number of alignement toolshave been applied in order to provide mappings such as:
mt:belong⊥←→ eq:belong between (mt, eq) ontologies
pair and pr:trimv←→ eq:instrumentation between
(pr, eq). These sets of correspondences (or mappings)are enriched in a semi-automatic manner using links aswell as consulting domain experts, an expert for a pair
Multi-level Networked Knowledge: Representation and DL-Reasoning 3
Ontologies Axiomseq: flame-detector(FD1)
flame-detector v ∃belong.instrumentation
pr: trim(T1)
zn: zone(ANNA1TG01)
mt: intervention(I1)team(TE1)intervene(TE1, I1)member v ∃belong.team
Alignments
Aeq-zn: eq:FD1part-of←→ zn:ANNA1TG01
Apr-eq: pr:trimv←→ eq:instrumentation
pr:T1compose←→ eq:FD1
Amt-eq: mt:I1concern←→ eq:FD1
eq:belong⊥←→ mt:belong
AApr-eq-Aeq-zn : pr-eq:compose≡←→ eq-zn:part-of
Table 1 an excerpt of ontologies and associatedalignments
of ontologies. This operation revealed the existence ofsemantic’s heterogeneity problems between alignmentsinter ontologies, and more precisely between links. Asan example of heterogeneity, the appearance of differentnames within the alignment pair Apr-eq and Aeq-zn,it is Apr-eq:compose and Aeq-zn:part-of , these linkshave similar semantic. it is clear that reasoning onthe set of ontologies and their existing alignments,semantic heterogeneity problem between links need tobe solved. For the previous case, this is done insertingan equivalence relation between links Apr-eq:compose etAeq-zn:part-of becomes necessary.
Example 1: an excerpt of ontologies and associatedalignments are presented in Table 1.
In order to solve semantic heterogeneity problembetween links, it is essential to represent and understandthe semantics of the MLNK. Further than semanticconnections of a MLNK representing local knowledge,we may find semantic connections between alignmentsthemselves, Figure 1 represents the turbine exampleshowing alignment levels.
3 State of Art and Related Work
In our best knowledge, no Multi Level Networkedknowledge formalism has ever been proposed anddeveloped. The problem of aligning alignment
II
T
flame-detectorI(FD1)flame-detectorIIIIIIIbelong.instrumentation E
zone(ANNA1TG01) intervention(I1),Iteam(TE1)iintervene(TE1,I1)memberIIIIIIIIbelong.teamE
pr eq zn mt
pr:T1IconsistIeq:FD1 eq:FD1IIpart-ofIIzn:ANNA1TG01 mt:I1IIconcernIIeq:DF1IIIImt:belongIIIIIIIeq:belong
trimI(T1)
pr-eq eq-zn mt-eq
pr-eq--eq-znpr-eq:composeIIIIIIIeq-zn:part-of=
Figure 1 Knowledge representation levels
vocabularies is a novelty within our contribution.However, in this section only works related to theproposed approach are presented.
Research performed in order to ensure a contextualreasoning in a distributed environment, may be classifiedinto three main research categories: Aligned knowledgenetworks, contextual modelling and modular ontologies.
3.1 Aligned Knowledge Networks
Aligned Knowledge Networks (AKN), are formed withontologies linked to each other by an alignmentprocess. Many formalisms specifies their syntax andsemantic, generally using correspondences following theaforementioned definition. If we consider mappings only(leaving out links) then we can site, on the one hand, thefollowing formalisms: Distributed Description Logic [6]and Integrated Distributed Description Logics [27],however with different semantics. On the other hand,Package-based Description Logics [4], even if the laterdoes not use a mapping type correspondences it may beexpressed using syntax in AKN terms.
Links are clearly shown in the formalism E-connection [21] but can be used in formulas implyingterms of more than two ontologies. Combining links andmapping type relations in a single formalism is addressedin [26].
However, none of the aforementioned formalismsadresses the issue of links terms alignements between twodistinct knowledge networks. Each of them, is associatedwith reasoning procedures, sometimes implemented sucha in: Drago [24] implements a pair to pair reasoning forDDL; DRAOn [11] implements a distributed reasoningfor IDDL. Resoaning in E-connection, was implementedin a former version of the famous DL Pellet reasoner [25].Note that a multi-level network interpretation in DL, donot implies that DL suffice to interpret it.
All the aforementioned formalisms are based on DL.Other local logics are used to formalise AKN. In [17],the RDF formalism [8] is considered and a knowledgenetwork is represented using quadruplet (c, s, p, o)where, c is a context identifier of the triplet (s, p, o)that is, an identifier of ontology in which it belongs.Bridges rules are defined between quadruplet of differentcontexts in order to allow knowledge reasoning throughcontexts. This formalism is also associated to a reasoning
4 S. Klai et al.
procedure using forward chaining, working only ifbridges rules between contexts are acyclic. At a differentlevel, Distributed First Order Logic [15], uses firstorder logic as a local logic completed with bridge rulescompleting DDL ones. Once again, these formalisms takeinto account only one level of heterogeneous knowledgeintegration.
3.2 Contextual knowledge modeling
A number of works on contextual reasoning, at theopposite of our proposal, do not use correspondencesas defined here. Indeed, knowledge from multiplecontexts are jointly exploited via a meta descriptionof the contexts themselves. Established relationsbetween contexts play a similar role, ensured byalignments between ontologies, however not expressingcorrespondences between entities from differentontologies. Klarman [19] proposes such a formalismstructured on two levels. The first one concerns themeta ontology, expressed using DL, where contexts areconsidered as instances. The second level focus on localknowledge detailing contexts. This formalism is given bythe CLC
LOlanguage, which is a composition of the object
language LO and the contextual language LC . Theimplementation of this work, gave the WORKFLOWsystem which allows reasoning on both levels andthe answer to SPARQL requests. Alternatively, thementioned formalism permits also the integration oftime as well as contextual knowledge origins. These lateraspects are out of the scope of this paper and will notbe addressed.
Contextualized Knowledge Repository (CKR [16])introduces the notion of context class and generaliseknowledge propagation by introducing the eval operatorin order to represent concept’s extension or expressinga role given in a different context. CKR is alsostructured on two levels; with the top one describingmeta knowledge on contexts over a number of dimensions(spatial, temporal, etc.) and a lower level defining localcontexts containing valid instances within the context.In order to encourage context reuse, each contextknowledge are organised in a set of modules. CKR isbased on three languages: The meta language LΓ, theobject language LΣ and Le
Σ which is the extension of LΣ
including the eval expressions in LΣ. All three languagesare DL based.
Joseph and Serafini [18], proposed also a two levelformalism (meta and object knowledge) similar to CKR,however founded on RDF instead of DL. The strengthis put on RDF knowledge depository constituted of atriplet set. The Contextualised Knowledge Base (CKB)is a pair 〈M,G〉 where M constitute meta knowledgewhich includes dimension graph (time, location, etc.),contexts definitions as well as their relations.G is either,a RDF graph set or a context graph representing thetriplet ensemble valid within the context. This graph set,represent the object level of CKB. Reasoning is thenperformed over both levels, applying already defined
inference rules, in order to ensure knowledge propagationthrough both levels as well as through contexts.
Once again, these formalisms do not resolveheterogeneity problem at a higher level. Indeed, contextknowledge introduce specific terms to the meta languagewhich may be different from one knowledge network toanother.
3.3 Modularity
The topology of MLNKs permits the representation ofa set of knowledge in a modular manner. Each alignedontology constitute a module, where alignements permitto link a number of modules in order to form a largerone, when itself can be used.
It seems therefore, interesting to compare ourapproach to existing works in the domain of modularontologies. Indeed, previous described formalisms (DDL,E-connection, P-DL, IDDL) are described as modularontology languages and compared as such [7].
A part of works on modularization undertakedetection and decomposition of large monolithicontologies into modules [9], along with proprietiespermitting modular reasoning [20]. This approach willnot be treated here, however, construction of modularontologies starting with heterogeneous modules willbe investigated. In that field, Distributed OntologyLanguage (Distributed Ontology Language [22]) allowsthe combination of a number of ontologies with thedescription of the existing mappings between them,along with providing metadata on the ontology set. Thislanguage has been submitted for standardization withinthe Object Management Group1.
It is worth noticing that the formal semanticof the DOL is not yet defined in a unique form.This is also the case for MLNK. Indeed, authorsproposed three possible semantics [23], following thedistributed semantic analysis provided by Zimmermannand Euzenat [28].Again, the language does not addressthe heterogeneity highest level.
Modular construction may also be treated as asoftware engineering problem [14], where authors definea module using: (1) Its internal ontology, (2) interfacespermitting to expose certain terms and axioms in orderto be reused by other modules, (3) Import modulesand (4) alignment between them. These, will permit ahierarchical construction similar to the notion of multilevel knowledge, as a composed module may be in turnimported with other alignments. However, this work doesnot consider alignment terms (i.e., links). Authors do notspecify a unique semantic, but focus on a set of possiblesemantics that may comply with description logics suchas DDL or P-DL. This approach has been implementedin a tool named ModOnto [5].
In the same manner, [12] propose a modular ontologyformalism based on interfaces (Interface-Based modularOntology Formalism ou IBF). In this case relations ofmapping and link types are expressed in terms of axioms
Multi-level Networked Knowledge: Representation and DL-Reasoning 5
between the entities of different modular ontologies,using interfaces in order to encapsulate ontology axioms.
4 Multi-level Networked Knowledge syntax
Representation and formalization of MLNK implies onthe one hand, represent each component of this networkand formalize the semantics and on the other hand,that the relationships between these components can berepresented. This section, is dedicated to the syntacticrepresentation of a multi-level networked knowledgecomponents, like ontologies, alignments and knowledgenodes.
4.1 Knowledge representation languages andontologies
A knowledge representation language L, is definedby a syntax (how formulas are expressed) and asemantic (the meaning and sense of formulas). Syntax ischaracterized by a number of symbols and constructionrules, permitting after combination to express wellformed formulas. Description Logic, constitute a set ofrepresentation languages that may be used to representa knowledge domain in a structured and formal manner.
We then speak of signatures or vocabulary in orderto design structured terms which are subsets of a givenlanguage symbols. Each signature permits the definitionof a set of formulas defined by the used language, anda set of formulas constructed from a common signatureform an ontology.
Vocabulary: A vocabulary is a structured set ofterms.
Example 2: In Description Logics, a vocabulary isa triple 〈NI , NC , NR〉 where NI is a set of individualnames, NC is a set of concept names and NR is a set ofrole names.
In any ontology language, the signature of anontology is a vocabulary.
Ontology: Let L a knowledge representation language.An ontology O of L is a pair 〈Σ(O), A(O)〉 where Σ(O)is a signature of L and A(O) is a sub set of formulaswhich can be constructed with Σ(O). Formulas in A(O)are called proper axioms of O.
Local ontologies or knowledge sources in a multi-levelnetworked knowledge are linked using alignments
4.2 Alignment language
An alignment LA language permits the description ofcorrespondences between two vocabularies. It is alsocaracrterised by a syntax (how correspondences areexpressed) and a semantic (how correspondences areinterpreted). The syntax of LA is defined by:
• a set of terms , called links, specific to thealignment language noted V (LA);
• a function E(LA), which associate to eachsignature of a representation language L, a set ofentities that can be aligned;
• a set of relation’s symbols R(LA).
Thus, the syntax of an alignment language LA is definedby the triplet 〈V (LA), E(LA), R(LA)〉, note 〈V,E,R〉when no ambiguity exists. Two types of correspondencesmight be defined as mapping and link correspondences.
mapping correspondence: Let V1 and V2 twoaligned vocabularies and LA 〈V,E,R〉 an alignmentlanguage. A mapping correspondence is a triplet〈e1, e2, r〉 noted e1
r←→ e2 where:
• e1 ∈ E(V1) and e2 ∈ E(V2) are matchable entities;
• r ∈ R denotes an existent relation between e1 ande2.
Others define it as a 4− tuple 〈e1, e2, r, n〉 where n ∈[0, 1] is a confidence value. We drop this component sinceit does not play a role in our formalisation.
Refering to example 1, eq:belong⊥←→ mt:belong is
a mapping correspondence. belong term can be foundin both ontologies vocabulary, for instance eq and mt,formalized in description logic, with different meanings.mappings are constructed using the set of operators R ={v,≡⊥,∈=}
link correspondence: Let V1 and V2 two alignedvocabularies and LA 〈V,E,R〉 an alignment language. A
link correspondence is a formula in the form e1l←→ e2
where:
• e1 ∈ E(V1) and e2 ∈ E(V2) are matchable entities;
• l ∈ V denotes an existent relation between e1 ande2.
Always refering to example 1, eq:FD1part-of←→
zn:ANNA1TG01 and mt:I1concern←→ eq:FD1 are link
correspondences. Terms part-of and concern donot appear in the ontologies vocabularies, they wereintroduced at the alignment level in order to linkdifferent vocabularies entities. The alignment, now,possesses its own vocabulary and therefore may bealigned with other vocabularies in order to avoidheterogeneity problems.
Alignment: Let V1 and V2 be two vocabularies. Analignment of V1 and V2 is a tuple Λ = 〈V, κ, λ〉 where:
• V is an alignment vocabulary;
• κ is a set of mapping correspondences, e1r←→ e2
where e1 ∈ E(V1), e2 ∈ E(V2) et r ∈ R;
6 S. Klai et al.
• λ is a set of link correspondences, e1l←→ e2 where
e1 ∈ E(V1), e2 ∈ E(V2) and l ∈ V ;
Example 3:
• In DDL or in IDDL, alignments are between theontologies signatures and the sets V , λ are allempty.
• In E-connections, cross-ontology knowledge caninvolve terms from more than two ontologies.However, if one restricts to E-connection axioms ofthe form 〈Ei〉j(ai, bj), where Ei is a link relation,ai is an individual in ontology Oi and bj isan individual in ontology Oj , then this can berepresented as a correspondence in λ, with l beinga term in the alignment vocabulary (the set κ isempty).
4.3 Knowledge node
Once the basic component of a MLNK are introduced, inthe section 4, it is now possible to introduce the notion ofknowledge node, which generalize the notion of ontology.Informally, an ontology is a level 0 knowledge node,while all knowledge node of level m > 0, is constructedfrom a number of nodes with inferior level, linked usingalignment (figure 2). Formally the node is defined as:
Knowledge node: A knowledge node is a pair K =〈VK , AK〉 where VK is a vocabulary, also written Voc(K)and both VK and AK are defined recursively:
• an ontology O is a knowledge node with vocabularyVoc(O) = Sig(O) and AK is the set of axioms;
• for n ≥ 1, if K1, . . . ,Kn are knowledge nodes withvocabularies Voc(K1), . . . ,Voc(Kn), and for alli, j ∈ [1, n], Λij is an alignment of Voc(Ki) andVoc(Kj), then K = 〈VK , AK〉 is a knowledge nodewith the vocabulary:
VK =⋃
i,j∈[1,n]
{ij : l | l ∈ Voc(Λij)} ∪⋃
i∈[1,n]
{i : e | e ∈ Voc(Ki)}
and AK = 〈(Ki)i∈[1,n], (Λij)i,j∈[1,n]〉.
If a knowledge node includes only ontologies andontology alignments, we call it a network of alignedontologies. If a knowledge node is neither a singleontology, nor a network of aligned ontologies, we call ita multi-level networked knowledge base (see Figure 2).
5 Multi-level networked knowledgesemantics
The representation of MLNK was defined independentlyof any language and can support multiple semantics. Inthis paper, we focus on the description logic semanticswhich is a way of MLNK interpreting, leading to a formalsemantic representation allowing centralized reasoning.
pr eq znlevel 0
K2=eq K3=zn
K4 K5
K1=pr
AK1-K2 AK2-K3
level 1
K6
K2=eq K3=zn
K4 K5
K1=pr
AK4-K5
level 2AK2-K3AK1-K2
Figure 2 Recursive representation of nodes
5.1 DL Syntax and Semantics
A DL ontology is composed of concepts, roles andindividuals, as well as axioms built out of these elements.A concept is either a primitive concept A, or, givenconcepts C,D, roleR, individuals a1, . . . , ak, and naturalnumber n, ⊥, >, C tD, C uD, ∃R.C, ∀R.C, ≤ nR.C,≥ nR.C, ¬C or {a1, . . . , ak}. A role is either a primitiverole P , or, given roles R and S, R t S, R u S, ¬R, R−,R ◦ S and R+.
Interpretations are pairs 〈∆I , ·I〉, where ∆I is anon-empty set (the domain of interpretation) and·I is the function of interpretation such that for allprimitive concepts A, AI ⊆ ∆I , for all primitive rolesP , P I ⊆ ∆I ×∆I , and for all individuals a, aI ∈∆I . Interpretations of complex concepts and roles isinductively defined by ⊥I = ∅, >I = ∆I , (C tD)I =CI ∪DI , (C uD)I = CI ∩DI , (∃R.C)I = {x|∃y.y∈CI ∧ 〈x, y〉∈RI}, (∀R.C)I = {x|∀y.〈x, y〉∈RI ⇒ y∈CI}, (≤ nR.C)I = {x|]{y∈CI |〈x, y〉∈RI} ≤ n}, (≥nR.C)I = {x|]{y∈CI |〈x, y〉∈RI} ≥ n}, (¬C)I = ∆I \CI , {a1, . . . , ak} = {aI1, . . . , aIk}, (R t S)I = RI ∪ SI ,(R u S)I = RI ∩ SI , (¬R)I = (∆I ×∆I) \RI , (R−)I ={〈x, y〉|〈y, x〉∈RI}, (R ◦ S)I = {〈x, y〉|∃z.〈x, z〉∈RI ∧〈z, y〉∈SI} and (R+)I is the reflexive-transitive closureof RI .
Axioms are either subsumption C v D, sub-roleaxioms R v S, instance assertions C(a), role assertionsR(a, b) and individual identities a = b, where C andD are concepts, R and S are roles, and a and b areindividuals. An interpretation I satisfies axiom C v Dif and only if CI ⊆ DI ; it satisfies R v S if and only ifRI ⊆ SI ; it satisfies C(a) if and only if aI ∈CI ; it satisfiesR(a, b) if and only if 〈aI , bI〉∈RI ; and it satisfies a = bif and only if aI = bI . When I satisfies an axiom α, it isdenoted by I |= α.
An ontology O is composed of a set of terms(primitive concepts/roles and individuals) called thesignature of O and denoted by Σ(O), and a set of axiomsdenoted by Ax(O). An interpretation I is a model of anontology O if and only if for all α∈Ax(O), I |= α. In thiscase, we write I |= O. The set of all models of an ontologyO is denoted by Mod(O). A semantic consequence of an
Multi-level Networked Knowledge: Representation and DL-Reasoning 7
ontology O is a formula α such that for all I∈Mod(O),I |= α.
5.2 DL-Multi-level networked knowledge
A MLNK is presented as a unique ontology in DL.This latter is constructed from one node or more, linkedwith each other’s using alignments. The singularityof such ontology, is that its elements are prefixedusing the index of the corresponding knowledge sources,in order to distinguish between entities coming fromdifferent knowledge source. The representation of suchontology implies the introduction of newer notions suchas combined signatures and combined set of formulas.
In this section, it will be demonstrated that reasoningon knowledge networks comes to apply a reasoningalgorithm in DL. Before defining the new notions ofcombined signature and combined set of formulas whichare the component of the DL ontology generated froma multi level knowledge node, a complementary functionpermitting the indexing of the ontology elements is firstlydefined.
index the element of ontology: Let i an indice, wedefine the function prefix on the terms, axioms andontologies as follows:
• prefix(X, i) = {i:X} where X is an atomic concept,atomic role or an individual;
• prefix(Σ, i) = {prefix(X, i) | X ∈ Σ} where Σ is asignature of ontology and X is an atomic concept,atomic role or an individual;
• prefix(2, i) = 2 where 2 ∈ {⊥,>};
• prefix(X2Y, i) = prefix(X, i)2prefix(Y, i) where2 ∈ {t,u} and X,Y can be two concepts or tworoles;
• prefix(2R.C, i) = 2prefix(R, i).prefix(C, i) where2 ∈ {∃,∀} and R is a role and C is an concept;
• prefix(2n.R, i) = 2nprefix(R, i) where 2 ∈ {≤,≥}and R is a role;
• prefix(¬X, i) = ¬prefix(X, i) where X is an conceptor a role;
• prefix(R2, i) = prefix(R, i)2 where 2 ∈ {+,−} andR is a role;
• prefix(X v Y, i) = prefix(X, i) v prefix(Y, i) whereX and Y are two concepts or two roles;
• prefix(C(a), i) = prefix(C, i)(i:a) where C is aconcept and a is an individual;
• prefix(R(a, b), i) = prefix(R, i)(i:a, i:b) where R is arole, a, b are an individuals;
• prefix(a = b, i) = {i:a = i:b} where a, b are anindividuals;
• prefix(F, i) = {prefix(f, i) | f ∈ F} where F is a setof description logic axioms;
• prefix(O, i) = 〈prefix(Σ(O), i), prefix(F (O), i)〉where O is a description logic ontology.
A combined signature is an ontology signature resultingfrom the transformation of a MLNK. It consists in theunion of local node signatures as well as alignmentvocabularies used to link them.
Combined signature: Let N a multi-level knowledgenode, the combined signature Σcomb is definedrecursively:
• if N is an ontology then Σcomb(N) = Σ(N) =〈C,R,U〉;
• if N is a multi-level knowledge node compound ofsubnodes and alignments between them, for n ≥ 1,we have Σcomb(N1), . . . ,Σcomb(Nn) combinedsignatures of the local nodes and for all i, j ∈[1, n], Vij is the alignment vocabulary betweenΣcomb(Ni) and Σcomb(Nj) and prefix given inthe definition 5.1 . We define role a functionwhich assigned for all link of Vij a specificrole in description logic. Then Σcomb(N) =⋃
i∈[1,n]{prefix(Σcomb(Ni), Ni)} ∪ {role(l) | l ∈Vij}.
The combined set of formulas is the set of formulas ofthe resulting ontology from the network transformation.It is the union of local node formulas as well asgenerated formulas from correspondences. Firstly, thefunction associating each correspondence in to an axiomis defined.
Correspondence transformation in to axiom:Let Aij for i, j ∈ [1, n] an alignment between a node iand a node j. We define trans a function which assignfor each correspondence of Aij a DL axiom:
• trans({i:A v←→ j:B}) = {prefix(A, i) vprefix(B, j)};
• trans({i:A ≡←→ j:B}) = {prefix(A, i) ≡prefix(B, j)};
• trans({i:A ⊥←→ j:B}) = {prefix(A, i) v¬prefix(B, j)};
• trans({i:u ∈←→ j:A}) = {prefix(A, j)(i:u)};
• trans({i:u =←→ j:u′}) = {i:u = j:u′};
• trans({i:u l←→ j:u′}) = {role(l)(i:u, j:u′)};
• trans({i:A l←→ j:B}) = {prefix(A, i) v∃role(l).prefix(B, j)}
where A, B, u et u′ are the matchable entities and l is alink.
8 S. Klai et al.
Combined set of formulas : Assume N a multi-level knowledge node. Combined formulas Fcomb forcombined signature Σcomb is defined recursively:
• if N is an ontology then Fcomb(N) = A(N) the setof N axioms;
• if N is a multi-level knowledge node compoundof sub nodes N1, . . . , Nn and alignmentsbetween a pair of nodes Aij, for all i, j ∈ [1, n]Fcomb(N1), . . . , Fcomb(Nn) are a sets of local nodesaxioms, trans(Aij) is a function which assign foreach correspendence of Aij a description logicaxiom (see definition 5.3) and prefix the functionwhich assigns indices to elements of th sub nodes(see definition 5.1). Then, combined formulasFcomb(N) =
⋃i∈[1,n]{prefix(Fcomb(Ni), Ni)} ∪⋃
i,j∈[1,n]{trans(c) | c ∈ Aij}.
From these definition, we can currently introduce aDL-multi-level networked knowledge.
DL Multi-level networked knowledge: AssumeN a multi-level knowledge node, we define FDL afunction which transform recursively the node N in toDL ontology:
• if N is an ontology then FDL(N) = N ;
• if N is a multi-level knowledge node compound ofsub nodes Ni for i ∈ [1, n] and alignments betweena pair of nodes Aij for i, j ∈ [1, n] then FDL(N) =FDL〈{Ni}, {Aij}〉 = 〈Σcomb(N), Fcomb(N)〉 whereΣcomb(N) is a combined signature and Fcomb(N)is a combined formulas.
In Tables 2, 3 definitions 5.1 to 5.5 will be applied inorder to generate a DL ontology from the MLNK givenin example 1.
Remark : Node K6 is composed of nodes K4 andK5, each including node K2. Terms in K2 are thereforeduplicated as they can be represented in different pointof views in K4 and K5.
An interpretation I in DL is assigned to a DLontology generated from the transformation of a givenmulti level knowledge node. The satisfaction relation ata node level, depends on the satisfactory relation in DL.
knowledge node interpretation: A DL-interprtation of multi-level knowledge node X is adescription logic interpretaion for combined signature ofX.
knowledge node satisfaction relation: Let X amulti-level knowledge node, I a DL-interpretation of Xand FDL(X) a function given in definition 5.5. Then IDL-satisies X (denoted I |=N−DL X) if and only if Isatisfies FDL(X) in description logic. In this case, we cansay I is a DL-model of X.
Nodes Combined signature
K1=pr Σcomb(K1) = {trim, T1}K2=eq Σcomb(K2) = {flame-detector,
instrumentation,FD1, belong}K3=zn Σcomb(K3) = {zone, ANNA1TG01}K4= Σcomb(K4) = {k1:trim, k1:T1,
{K1,K2, k2:flame-detector,
AK1-K2} k2:instrumentation,
k2:FD1, k2:belong,
compose}
K5= Σcomb(K5) = {k2:flame-detector,
{K2, K3, k2:instrumentation, k2:FD1,
AK2-K3} k2:belong, k3:zone,
k3:ANNA1TG01,
part-of
K6= Σcomb(K6) = {k4:k1:trim,
{k4,K5, k4:k1:T1,k4:k2:flame-detector,
AK1-K2} k4:k2:belong, k4:k2:FD1,
k4:k2:instrumentation ,
k5:k3:zone, k5:k3:ANNA1TG01 ,
k5:k2:belong, k5:k2:instrumentation ,
k5:k2:flame-detector, k5:k2:FD1k4:compose, k5:part-of
Table 2 Combined signatures
Example 4: Let a node X = 〈{O1, O2}, {A12}〉 with
O1 = {C v D}, O2 = {C ≡ B} and A12 = {1:C⊥←→
2:C} O1 model is an interpretation of signature {C,D}and O2 model is an interprtation of signature {C,B}. Xtransformation into DL-ontology:
• prefix(O1, 1) = {1:C v 1:D};
• prefix(O2, 2) = {2:C ≡ 2:B};
• trans({1:C⊥←→ 2:C}) = {prefix(C, 1) v
¬prefix(C, 2)};
• FDL(X) = {1:C v 1:D, 2:C ≡ 2:B, 1 : C v ¬2 : C};
X model is an interpretation I of combined signature{1:C, 1:D, 2:C, 2:B} then we have I |=N−DL X if Isatisfies FDL(X) and I satisfies FDL(X) if I |= 1:C v 1:D,I |= 2:C ≡ 2:D and I |= 1 : C v ¬(2 : C) with:
• I |= 1:C v 1:D if and only if (1:C)I ⊆ (1:D)I
• I |= 2:C ≡ 2:D if and only if (2:C)I ≡ (2:B)I
• I |= 1 : C v ¬(2 : C) if and only if (1 : C)I ⊆¬(2 : C)I
Automatic reasoning on a given node X, is limitedto reasoning on the generated ontology FDL(X), wherethe coherence of X is deducted from the coherenceof FDL(X). The knowledge level structure permits touse the implication relation, between nodes of differentlevels.
Multi-level Networked Knowledge: Representation and DL-Reasoning 9
Nodes Combined axioms
K1=pr Fcomb(K1) = {trim(T1)}K2=eq Fcomb(K2) = {flame-detector(FD1),
flame-detector v∃belong.instrumentation}
K3=zn Fcomb(K3) = {zone(ANNA1TG01)}K4= Fcomb(K4) = {k1:trim(k1:T1),
{K1,K2, k2:flame-detector(k2:FD1),
AK1-K2} k2:flame-detector v∃(k2:belong).(k2:instrumentation),
compose(k1:T1, k2:FD1) }
K5= Fcomb(K5) = {k2:flame-detector(k2:FD1),
{K2, K3, k2:flame-detector v ∃(k2:belong)
AK2-K3} (k2:instrumentation),k3:zone(k3:ANNA1TG01),
part-of(k2:FD1, k3:ANNA1TG01)}
K6= Fcomb(K6) = {k4:k1:trim(k4:k1:T1)
{k4,K5, k4:k2:flame-detector v ∃(k4:k2:belong).
AK1-K2} (k4:k2:instrumentation),k4:k2:flame-detector(k4:k2:FD1),
k5:k3:zone(k5:k3:ANNA1TG01),
k4:compose(k1:T1, k2:FD1),
k5:part-of(k2:FD1, k3:ANNA1TG01),
k5:k2:flame-detector(k5:k2:FD1),
k5:k2:flame-detector v ∃(k5:k2:belong).
(k5:k2:instrumentation),k4:compose ≡ k5:part-of
Table 3 Combined axioms
Implication relation: Assume X, Y be two multi-level knowledge nodes, it is said that X DL-implies Y ifand only if all DL-models of X are DL-models of Y .
Property 1: Let X a global knowledge node,including local nodes N1, . . . , Nk, prefix the functionaffecting a node prefix to all its terms, then X impliesall local prefixed nodes noted (X |=N−DL prefix(Ni) forall i ∈ [1, k]) however the contrary is not implied. It issaid that a node X implies directly its local nodes in theparticular case where, local node terms are disjoint, asthis case do not necessitate the addition of prefixes toterms in order to be distinguished.
Proof Let X = 〈{Ni}, {Aij}〉 and an interpretationI is X DL-model noted I |=N−DL X and I |=N−DL
X if I |= FDL(X) ⇒ I |= FDL({Ni}, {Aij})⇒ I |=⋃i=1..n prefix(Ni, Ni) ⇒ I |= prefix(Ni, Ni) for all
i=1..n
Example 5: Let a node X = 〈{O1, O2}, {A12}〉with O1 = {C v D} et O2 = {C ≡ B} O1 modelis an interpretation of signature {C,D}, O2 modelis an interprtation of signature {C,B} and Xmodel is an interpretation of combined signature{1:C, 1:D, 2:C, 2:B} prefix(O1, 1) = {1:C v 1:D}
prefix(O2, 2) = {2:C ≡ 2:B} Applying property, X nodeimplies prefix(O1, 1) and implies prefix(O2, 2), becauseX |=N−DL 1:C v 1:D but X does not satisfy C v D andX |=N−DL 2:C ≡ 2:B but X does not satisfy C ≡ B.
And prefix(O1, 1) does not implies X becauseprefix(O1, 1) does not satisfy 2:C ≡ 2:B
A multi Level Knowledge node transformationpermits to link it to DLs which includes many reasoningoperational tools such as: Fact++2, Pellet3, HermiT4,exploited when reasoning on a node.
6 Implementation and Experimentation
In this section, the implementation of a JAVA interfacefor the purpose of reasoning on a MLNK using DLsemantics, presented in section5.2.This GUI, namedDescription Logic for Multi-Level Networked KnowledgeReasoner (DLMLNKR), permits reasoning either toverify consistency, testing axiom’s satisfaction or forinferring new knowledge using Pellet or hermit reasoners.
6.1 Architecture
DL MLNKR accepts as inputs ontologies on OWL/RDFformat, as well as alignments over a number of levelssaved in RDF files under the format API Alignment [13],expanded to stock links. The steps of the DL MLNKRarchitecture (See Figure 3), are cited and describedbelow:
• Loading source ontologies specifing theirlocalisations or IRIs;
• Load alignments, this operation consists in loadingalignment saved in RDF files ”results”, resultingfrom alignment discovery tools available on theWorld Wide Web. Alignment may be enriched in asemi-automatic manner using links;
• Fusioning loaded ontologies obtaining a global one;
• Alignement parsing, this consists in extractingmappings and links. Links are transformed inspecific roles and mappings in axioms, these arethen inserted in the global ontology;
• Testing the consistency of the global ontology aswell as testing axioms satisfaction;
• Inferring new knowledge permitting, thus,alignment composition transformed in axioms;
• Construction of a knowledge node, and saving itfor further alignments with other ontologies and/ornetworks.
10 S. Klai et al.
pp
Loading
ontologies
Loading
alignementsp
Sourcepontologies
MRDF/OWLpfilesN
Alignments
MRDFpfilesN
mergingpof
ontologies
-pAlignmentpParsing;
-pTransformationpofpcorrespondencesp
pppinptopaxioms;
-pIntegrationpofpaxiomspinpapglobal
pppontology.
Global
ontologyp
Reasoning
-Pellet
-pHermitinferredp
axiomsp
Results
DLMLNKpR
Figure 3 Architecture reasoning DLMLNKR
6.2 Experimentation and evaluation
Tests were performed on complex ontologieswith their respective alignments obtained fromthe Anatomy, Conference and benchmark:”http://oaei.ontologymatching.org/2014/”. Alignmentwere enriched using a semi-automatic manner withlinks and correspondences between alignments. Table 4,describes the size of the used ontologies and alignmentsconstituting the MLNK. Through these tests, theimpact of ontologies size will be shown, along with theontologies number impact and level impact during thetransformation of the MLNK in a DL ontology andconsistency check.
The evaluation put the focus on the execution timeof the two main operations, for instance:
• The transformation of the MLNK in a soleglobal ontology, focusing on the source ontologiesfusion time (fus) and the parsing time (pars)divided into recovery of correspondences time fromalignments, transfer time into axioms as well asinsertion time of these axioms into the globalontology;
• The consistency test time for the resulting ontology(Consist).
6.2.1 Impact size
The tests were performed on Anatomy ontologies withcorresponding alignments (see Table 4). In order tomeasure the mpact of size, we conducted first testson Approximately 25% then Approximately 50% andfinally 100% of ontologies human, mouse, and alignmentAhuman−mouse. The test results are shown in the table 5.
6.2.2 Impact number
The tests were performed on Conference ontologieswith corresponding alignments (see Table 4). In orderto measure the impact of ontologies and alignmentsnumber, we started with a network consisting of twoontologies and alignment for a first case and for eachcase, we have integrated into the network a new ontologyand corresponding alignments where integrated to thenetwork. The test results are shown in table 6.3.
Example Ontologies/alignments Size
Anatomy Human 3.27 MBmouse 1.33 MBAhuman−mouse 371 kB
Conference cmt 27.8kBconference 38.4kBconfOf 36.5kBedas 100kBekaw 35.1kBiasted 70kBsigkdd 15.1 kBOntologies alignment 65.76kB
Benchmarks 101 71.5kB103 80.4kB104 46.1kBA101−103 44.4kBA101−104 49.3kBA101−103−101−104 4.45kB
Table 4 Ontologies/alignments size
6.2.3 Impact level
The tests were performed on benchmark ontologieswith corresponding alignments for three possibletypologies.The first typology (T1) consist in loading theset of ontologies and alignments in a sole operation,and then reason on the complete set. The secondtypology, (T2), consists in creating a node for a pair ofontologies with their respective alignment, then reasonon the network constituted by the pair. The thirdtypology (T3) consists in reasoning on the knowledgenode constructed and aligned using the third ontology.The three typologies are depicted in figure4, and Thetest results are shown in the table 7.
6.3 Discussion
The test results (table 5) showed that the size of theontologies and alignments has no impact on the transferof MLNK as well as the consistency test in DL ontologyobtained. It is clear that the increase in the size ofontologies and alignment advances the transfer time andtest consistency time. We notice the same for testing theimpact on the evolution of the number of ontologies andalignments on the network. The results in (table 6.3)show that the number has no significant impact on thetransfer and on the consistency test. As against thecomplexity of ontology influences largely on the testconsistency, it’s the case5 and case6 where integrationof ontology ’iasted’ despite its size is smaller than ’edas’ontology size, but the test consistency time is muchhigher. This is confirmed by testing cases7 where thenetwork consists of cmt, iasted, Acmt−iasted and case8with cmt, edas, Acmt−edas. The (table 7) shows that thetest consistency time is approximately equal for differentMLNK levels (level 0, Level 1 and Level 2) but thetransfer time in to DL ontology is progressing with theevolution of the level, so there is no impact on the level.
Multi-level Networked Knowledge: Representation and DL-Reasoning 11
Ontologies/alignments fus pars Total Consist
(ms) (ms) (ms) (ms)
' 25% Human 152 248 400 332' 25% mouse' 25% Ahuman−mouse
' 50% Human 293 490 783 540' 50% mouse' 50% Ahuman−mouse
100% Human 570 821 1391 1290100% mouse100% Ahuman−mouse
Table 5 One-level networked knowledge example forimpact size
T1
O1 O2 O3 A12 A23 AN1-N2
N1
O1 O2 O3 A12 A23 AN-23
T2
N2 N
T3
O1 O2 O3 A12 A23 A12-23
Figure 4 DLMLNKR Typologies
The typologies change is the only one to have an impacton the transfer of MLNK in to a DL ontology. The resultsin the (table 7) show that typology2 (T2) is the bestfor the transfer of the network concerning Level 1 andLevel 2 (levels that contain alignments). But, for theconsistency test, we notice that the typology3 (T3) givesslightly better results compared with other typologiesconsistency test.
7 Conclusions
In this paper a formalism capable of reasoning ona network of heterogeneous, complementary alignedontologies, is presented. The alignment posses propervocabulary and necessitate, sometimes, to be alignedat different levels which represent the novelty ofthe presented formalism called MLNK. The semanticinterpretation of the formalism is based on the existingparadigm, disposing of complete reasoning procedures,along with operational tools such as description logic DL,used in this case. The proposed approach is implementedand tested in order to compare results, for differentnetworks. As future work, and on one hand, a morecomplete implementation is planned, using a differentsemantic in order to make distributed reasoning, forinstance DDL. However it will still be interesting toview the MLNK semantic, in a manner where a formalsemantic is directly constructed on the structure ofthe MLNK, and propose then a correct and completereasoning algorithm better adapted to the structure.On the other hand, an extension of the XMAP++
alignment tool proposed in [10] in order to permitthe automatic discovery of correspondences between
case
Con
fere
nce
onto
logi
es/a
lign
men
tfus
(ms)
pars
(ms)
Tota
lConsist
(ms)
case
1cm
t,co
nfe
ren
ce,A
cm
t−conference
30
50
80
320
case
2cm
t,co
nfe
ren
ce,
con
fOf
40
130
170
330
case
3cm
t,co
nfe
ren
ce,
con
fOf,
edas
,60
260
320
440
case
4cm
t,co
nfe
ren
ce,
con
fOf,
edas
,eka
w,
70
882
892
450
case
5cm
t,co
nfe
ren
ce,
con
fOf,
edas
,eka
w,i
aste
d,
80
917
997
3090
case
6cm
t,co
nfe
ren
ce,
con
fOf,
edas
,ek
aw,
iast
ed,
sigk
dd
,130
1101
1231
3110
case
7cm
t,ia
sted
,Acm
t−ia
sted
60
80
140
2280
case
8cm
t,ed
as,A
cm
t−ekaw
40
60
100
360
Table
6O
ne-
level
net
work
edknow
ledge
exam
ple
for
impact
num
ber
12 S. Klai et al.
(a) (b)
Figure 5 (a) Impact size results. (b) Impact number results.
(a) (b)
Figure 6 (a) Impact typologies for Level0. (b) Impact typologies for level1.
(a) (b)
Figure 7 (a) Impact typologies Level2. (b) Impact level for Typology1.
(a) (b)
Figure 8 (a) Impact level for Typology2. (b) Impact level for Typology3.
Multi-level Networked Knowledge: Representation and DL-Reasoning 13
fus(
ms)
pars
(ms)
pars
(ms)
pars
(ms)
Tota
l(m
s)Consist
(ms)
(101
-103
)(1
01-1
04)
(101
-103
-101
-104
)
leve
l0
T1
(101,1
03,1
04)
=70
70
460
T2
(101,1
03)
=50
170
460
(101,1
04)
=40
(N1,N
2)
=80
T3
(101,1
03)
=50
120
440
(N1,1
04)
=70
leve
l1
T1
(101,1
03,1
04)
=70
950
1030
2050
470
T2
(101,1
03)
=50
430
1070
450
(101,1
04)
=40
470
(N1,N
2)
=80
T3
(101,1
03)
=50
430
1180
450
(N1,1
04)
=60
640
leve
l2
T1
(101,1
03,1
04)
=10
060
010
5010
002650
450
T2
(101,1
03)
=50
430
1140
460
(101,1
04)
=40
470
(N1,N
2)
=80
70T
3(1
01,1
03)
=50
430
1790
400
(N1,1
04)
=60
660
590
Table
7M
ult
i-le
vel
net
work
edknow
ledge
exam
ple
for
impact
level
and
typ
olo
gie
s
alignments over different MLNK levels as well as semi-automatic insertion of links , is planned. Consider thepossibility of exploiting alignments at a certain levelmay help discovering alignments on a superior one, andimprove alignment techniques.
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