Towards Formal Comparison of Ontology Linking, Mapping and ...

Towards Formal Comparison of Ontology Linking, Mapping and ...

Towards Formal Comparison of OntologyLinking, Mapping and ImportingMartin Homola 1,2 and Luciano Serafini 11 Fondazione Bruno Kessler,Via Sommarive 18, 38123 Trento, Italy2 Comenius University,Faculty of Mathematics, Physics and Informatics,Mlynská dolina, 84248 Bratislava,, serafini@fbk.euAbstract. Multiple distributed and modular ontology representationframeworks have recently appeared. They typically extend DescriptionLogics (DL), with new constructs to represent relations between entitiesacross several ontologies. Three kinds of constructs appear in the literature:link properties, found in E-connections, semantic mapping, foundin Distributed Description Logics (DDL), and semantic imports, used inPackage-based Description Logics (P-DL). In this work, we aim towardsformal comparison of the expressive power of these frameworks, and thusalso the ontology combination paradigms that they instantiate. Reductionfrom DDL to E-connections is already known. We present two newreductions, from P-DL to DDL and vice versa. These results show thatthere are similarities between these frameworks. However, due to the factthat none of the reductions is unconditional, it cannot be claimed thatany of the three approaches is strictly more expressive than another.1 IntroductionThere are multiple reasons behind the research in modular and distributed ontologies.Introduction of modularity into ontology engineering, inspired by modularsoftware engineering, calls for organizing ontologies into modules which couldthen be reused and combined in novel ways and thus the whole ontology engineeringprocess will be facilitated and simplified. Another motivation comes fromthe Semantic Web vision, where ontologies are seen as a central ingredient, buton the other hand, decentralization and duplicity of knowledge sources is seen asvery important too [1]. The centralized, monolithic treatment of ontologies, predominantin the mainstream theoretical research on ontology representation [2],is not sufficient; a novel decentralized and distributed ontology representationapproach is of demand, that would deal with duplicity, temporal unavailabilityand also occasional inconsistence of the various ontological data sources.The effort to achieve such an ontological representation breaks down intotwo problems: what are the requirements for an encapsulated and reusable ontologymodule, and how to combine such modules and enable reasoning with

the combined ontology. There are two main research directions. The first is tocombine the modules simply by means of union and to find out the requirementsunder which such combination is feasible [3,4,5]. The second direction, on whichwe focus in this paper, imposes none or only very basic requirements on themodules and provides new constructs in order to combine the modules. Representationframeworks of this kind typically work with multiple DL [2] ontologies,and they provide novel constructs to interlink these ontologies. Each frameworkemploys slightly different constructs; three ontology combination paradigms aredistinguished: ontology linking, ontology mapping and ontology importing.Ontology linking allows individuals from distinct ontologies to be coupledwith links. A strict requirement is that the domains of the ontologies that arebeing combined are disjoint. See Fig. 1 a), where an ontology of companies isinter-linked with another ontology of products using the link produces. Linksallow for complex concepts to be constructed, and they basically act as crossontologyroles. The linking paradigm is employed by E-connections [6].Ontology mapping, in contrast, allows to relate ontologies on the same domainor on partially overlapping domains. Special mapping constructs indicatehow elements from different ontologies are semantically related. Concepts, rolesand individuals are possibly related. Mapping enables for knowledge reuse andalso for resolution of semantic heterogeneity between ontologies that model thesame domain but each from a different perspective. See Fig. 1 b), where mappingis expressed between two ontologies which possibly cover same entities butone models business relations and the other one models legal relations of theseentities. In this paper we take a look at DDL [7], another notable instance ofthis paradigm are Integrated Distributed Description Logics [8].Ontology importing allows a subset of concepts, relations and individualsdefined in one ontology to be imported into another ontology where they arethen reused. The importing takes care of propagating also the semantic relationsthat exist between these entities in their home ontology into the ontology thatimports them. In Fig. 1 c), several roles are imported from a dedicated ontologymodule that deals with partonomy. Importing has been studied by Pan et al.[9]. The P-DL framework [10] falls under this paradigm.This work aims towards comparison of the expressive power of these approaches.Some comparisons have already appeared in the literature. Some ofthem stay on qualitative or intuitive level [11,12]. Formal comparisons of theexpressive power are less frequent. To our best knowledge, only reduction that isknown is between DDL and E-connections [13,6]. In this paper, we extend theseefforts by producing two new reductions, from P-DL to DDL (with a specificallyadjusted semantics), and vice versa. These results show, that there are certainsimilarities between P-DL and DDL, however, the semantics of P-DL is stronger,and it cannot be claimed that one of the frameworks is more expressive than theother. Similarly, one has to be careful when interpreting the reduction betweenDDL and E-connections [13,6], because also this result assumes a DDL semanticswhich is not currently considered the standard one.

IndustryOntologyIndustryFood&DrinkOntologyAlcoholic-BeverageBeverageSoftDrinkFood&Drink-ProducerproducesWineBeerSodaproducesMaltBasedSoftDrinkBreweryWineryproduces(a)BusinessOntologyLawOntologyLegalEntityCompanycontractedByemployedByConsultantEmployeeLegalPersonNaturalPerson(b)CompanyModuleCorporationPartonomyModulepartOfdepartmentOfemployeeOfExecutive-BoardmemberOfdepartmentOfmemberOfemployeeOfExecutive(c)MappingLinksImportsSubsumptionRolesFig. 1. Ontology combination paradigms: a) ontology linking; b) ontology mapping; c)ontology importing.

On the other hand, these results provide us with valuable insight on the relationbetween the paradigms of ontology linking, ontology mapping and ontologyimport, and can possibly guide the user that is to pick an appropriate formalismfor a particular application.2 Distributed Description LogicsDDL [7,14,15] follow the ontology mapping paradigm. They allow to connectmultiple DL KB with bridge rules, a new kind of axioms that represents themapping. An important point is that bridge rules are directed. Typically, DDLare built over SHIQ. For the lack of space we cannot introduce SHIQ fully,please see the paper by Horrocks et al. [16].Assume an index set I. Sets N C = {N Ci } i∈I , N R = {N Ri } i∈I and N I = {N Ii }will be used for concept, role and individual names respectively. A distributedTBox is a family of TBoxes T = {T i } i∈I , a distributed RBox is a family ofRBoxes R = {R i } i∈I , a distributed ABox is a family of ABoxes A = {A i } i∈I ,such that K = 〈T i , R i , A i 〉 is a SHIQ KB built over symbols from N Ci , N Ri ,and N Ii . By i : φ we denote that φ is a formula from K i . A bridge rule from i toj is an expression of one of the forms:i : X−→ ⊑j : Y , i : X −→ ⊒j : Y , i : a ↦−→ j : b ,where X and Y are either both concepts or both roles, and a, b are two individuals,in the respective language. The expression i : X −→ ≡ j : Y is a syntacticshorthand for the pair of bridge rules i : X −→ ⊑ j : Y and i : X −→ ⊒ j : Y . Thesymbol B stands for a set of bridge rules, such that B = ⋃ i,j∈I,i≠j B ij whereB ij contains only bridge rules from i to j. A DDL KB over I is K = 〈T, R, A, B〉with all four components ranging over I.A distributed interpretation I = 〈{I i } i∈I , {r ij } i,j∈I,i≠j 〉 consists of local interpretations{I i } i∈I and domain mappings {r ij } i,j∈I,i≠j (notation: r ij (d) ={d ′ | 〈d, d ′ 〉 ∈ r ij } and r ij (D) = ⋃ d∈D r ij(d)). For each i ∈ I the local interpretationis of the form I i = 〈 〉∆ Ii , ·Ii . In DDL, ∆I imay also be empty. In such acase we call I i a hole and denote it by I i = I ɛ . A distributed interpretation Isatisfies elements of K (denoted by I |= ɛ ·) according to the following clauses:1. I |= ɛ i : φ if I i |= φ;2. I |= ɛ T i if I |= ɛ i : φ for each φ ∈ T i ;3. I |= ɛ T if I |= ɛ T i for each i ∈ I;4. I |= ɛ R i if I |= ɛ i : φ for each φ ∈ R i ;5. I |= ɛ R if I |= ɛ R i for each i ∈ I;6. I |= ɛ A i if I |= ɛ φ for each φ ∈ A i ;7. I |= ɛ A if I |= ɛ A i for each( i ∈) I;8. I |= ɛ i : X −→ ⊑ j : Y if r ij XI i ⊆ YI j( );9. I |= ɛ i : X −→ ⊒ j : Y if r ij XI i ⊇ YI j( );10. I |= ɛ i : a ↦−→ j : b if b Ij ∈ r ij aI i ;11. I |= ɛ B if I |= ɛ φ for all axioms φ ∈ B.

A distributed interpretation I is a distributed model, also ɛ-model, of K, ifI |= ɛ T, I |= ɛ R, I |= ɛ A and I |= ɛ B (denoted I |= ɛ K). I is a d-model of K, ifI |= ɛ K and it contains no hole (denoted I |= d K). A DDL KB K is i-consistent,if it has an ɛ-model with I i ≠ I ɛ ; it is globally consistent if it has a d-model.A concept C is ɛ-satisfiable (d-satisfiable) with respect to K i of K, if thereis an ɛ-model (d-model) I of K such that C Ii ≠ ∅. A formula i : φ is ɛ-entailed(d-entailed) with respect to K, if in every ɛ-model (d-model) of K we have Isatisfies i : φ. This is denoted by K |= ɛ i : φ (K |= d i : φ).The semantics corresponding to ɛ-satisfiability and ɛ-entailment is the actualstate of the art in DDL. It enjoys many reasonable properties [7,14]. We willcall this semantics DDL ɛ . The notions of d-satisfiability and d-entailment (thesemantics DDL d ) are rather auxiliary. As usual, entailment of subsumption formulaeand (un)satisfiability are interreducible and both reducible into decidingi-consistence of a KB (DDL ɛ ) or global consistence of a KB (DDL d ).DDL ɛ and DDL d permit any domain relations [7], but also alternate semanticswith restricted domain relations have been investigated [17,18]. DDL ɛ whichrequires domain relations to be partial functions is denoted by DDL ɛ (F). DDL ɛwith injective domain relations is denoted by DDL ɛ (I). DDL ɛ with compositionallyconsistent domain relations (r ij ◦ r jk = r ik for any distinct i, j, k ∈ I) isdenoted by DDL ɛ (CC). DDL ɛ with restricted compositionality (r ij ◦ r jk = r ikfor any distinct i, j, k ∈ I, if there is a directed path of bridge rules from i to jand from j to k) is denoted by DDL ɛ (RC). DDL ɛ with role-preserving domainrelation (if (x, y) ∈ R Ii than r ij (x) ≠ ∅ =⇒ r ij (y) ≠ ∅), for each R thatappears on the left hand side of any bridge rule from i to j, will be denotedby DDL ɛ (RP). Variants with DDL d and combinations are possible (e.g., lateron we will discuss DDL ɛ (F,I,RC,RP), the semantics with partially functional,injective, restricted-compositional and role-preserving domain relations).In addition, we will assume that local alphabets are mutually disjoint, andonly atomic concepts are used in bridge rules. This is a normal form of DDLwhich is equivalent to the full version without these restrictions.3 Package-based Description LogicsP-DL instantiate the ontology import paradigm. They allow a subset of terms tobe imported from one DL KB to another (in P-DL these ontology modules arecalled packages). The review is based on the recent publication of Bao et al. [19]which builds on top of SHOIQ resulting into P-DL language SHOIQP. Thespace does not permit us to introduce SHOIQ formally, please see the work ofHorrocks [20].A package based ontology is any SHOIQ ontology P which partitions into afinite set of packages {P i } i∈I , using an index set I. Each P i uses its own alphabetof terms N Ci ⊎N Ri ⊎N Ii (concept, role and individual names, respectively). Thealphabets are not mutually disjoint, but for any term t there is a unique homepackage of t, denoted by home (t). The set of home terms of a package P i ∈ Pis denoted by ∆ Si . A term t occurring in P i is a local term in P i if home (t) = i,

otherwise it is a foreign term in P i . If t is a foreign term in P j , and home (t) = i,ttthen we write P i → P j . If P i → P j for any term t, then also the package P jimports the package P i (denoted P i → P j ). By → ∗ we denote the transitiveclosure ofand by Pj∗ the set P j ∪ {P i |i → ∗ j}.When P i → P j , the symbol ⊤ i occurring within P j represents the importeddomain of P i . Also in this case a novel contextualized negation constructor ¬ i isapplicable within P j ¬ i (it is however a mere syntactic sugar and can be ruledout as we always have ¬ i C ≡ ⊤ i ⊓ ¬ j C).〉A distributed interpretation of P is a pair I =〈{I i } i∈I , {r ij } ∗ i→j , such thateach I i = 〈 〉∆ Ii , ·Ii is an interpretation of the local package Pi and each r ij ⊆∆ Ii ×∆ Ij is a domain relation between ∆ Ii and ∆ Ij . A distributed interpretationI is a model of {P i } i∈I , if the following conditions hold:1. there is at least one i ∈ I such that ∆ Ii ≠ ∅;2. I i |= P i ;3. r ij is an injective partial function, and r ii is the identity function;4. if i ∗ → j and j ∗ → k, then r ik = r ij ◦ r jk (compositional consistency);5. if i t → j, then r ij (t Ii ) = t Ij ;6. if i R → j and (x, y) ∈ R Ii than r ij (x) ≠ ∅ =⇒ r ij (y) ≠ ∅ (role preserving).The three main reasoning tasks for P-DL are consistency of KB, conceptsatisfiability and concept subsumption entailment with respect to a KB. Theseare always defined with respect to a so called witness package P w ∈ P. A packagebasedontology P is consistent as witnessed by a package P w of P, if there exists amodel I of Pw ∗ such that ∆ Iw ≠ ∅. In this case we also say that P is w-consistent.A concept C is satisfiable as witnessed by a package P w of P, if there exists amodel I of Pw ∗ such that C Iw ≠ ∅. A subsumption formula C ⊑ D is valid aswitnessed by a package P w of P (denoted P |= C ⊑ w D), if for every model Iof Pw ∗ we have C Iw ⊆ D Iw .4 E-connectionsThe E-connections framework represents the ontology linking paradigm. Althoughmany flavours of E-connections are known, for sake of simplicity weintroduce the language C E (SHIQ) [6,21,22]. This language allows to connectmultiple ontologies expressed in SHIQ [16] with links.Assume a finite index set I. For i ∈ I let N Ci and N Ii be pairwise disjointsets of concepts names and individual names respectively. For i, j ∈ I, i and jnot necessarily distinct, let ɛ ij be sets of properties, not necessarily mutuallymdisjoint, but disjoint with respect to N kmC kand N k I k. An ij-property axiom isof the form P 1 ⊑ P 2 , where P 1 , P 2 ∈ ɛ ij . An ij-property box R ij is a finite setof ij-property axioms. The combined property box R contains all the propertyboxes for each i, j ∈ I.

Given some i ∈ I, A ∈ N Ci , two i-concepts C and D, and a j-concept Z, andP, S ∈ ɛ ij , S is simple (see [16,22]), the following are also i-concepts:⊥ i |⊤ i |A|¬C|C ⊓ D|C ⊔ D|∃P.Z|∀P.Z| n S.Z| n S.Z .A combined TBox is a tuple K = {K i } i∈I where each K i is a finite set of i-local GCI axioms of the form C ⊑ D, where C, D are i-concepts. A combinedABox A = {A i } i∈I ∪ A E is a set of local ABoxes A i , each comprising of a finitenumber of i-local concept assertions C(a) and role assertions R(a, b), where Cis an i-concept, R ∈ ɛ ii , a, b ∈ N Ii . A E is a finite set of object assertions, eachof the form a · E · b, where E ∈ ɛ ij , a ∈ N Ii , b ∈ N Ii . A combined KB is a tripleΣ = 〈K, R, A〉, where each component ranges over the same index set I.Now we focus on the semantics. Given some KB Σ = 〈K, R, A〉 with indexset I, a combined interpretation is a triple I = 〈{∆ Ii } i∈I , {·Ii } i∈I , {·Iij } i,j∈I 〉,where ∆ Ii ≠ ∅, for i ∈ I, and ∆ Ii ∩∆ Ij = ∅, for i, j ∈ I, i ≠ j. The interpretationfunctions provide denotation for i-concepts (·Ii ) and for ij-properties (·Iij ).Each ij-property P ∈ ɛ ij is interpreted by P Iij ⊆ ∆ Ii × ∆ Ij . I satisfies anIij-property axiom P 1 ⊑ P 2 , if P ij I1 ⊆ P ij 2 . Each i-concept C is interpreted byC Ii ⊂ ∆ Ii ; ⊤ i = ∆ Ii and ⊥ i = ∅, and denotation of complex i-concepts mustsatisfy the constraints as given in Table 1. I satisfies an i-local GCI C ⊑ D(denoted by I |= C ⊑ D), if C Ii ⊆ D Ii . I satisfies an i-local concept assertionC(a) (denoted I |= C(a)), if a Ii ∈ C Ii ; it satisfies an i-local role assertion R(a, b)(denoted I |= R(a, b)), if 〈 〉a Ii , b Ii ∈ RI ii; I satisfies an object assertion a · E · b(denoted I |= a · E · b), a ∈ N Ii , b ∈ N Ij , E ∈ ρ ij , if 〈 〉a Ii , b Ij ∈ EI ij.X X I i¬C ∆ I i\ C I iC ⊓ D C I i∩ D I iC ⊔ D C I i∪ D I i∀P.Z {x ∈ ∆ I i| (∀y) (x, y) ∈ P I ij=⇒ y ∈ Z I j}∃P.Z {x ∈ ∆ I i| (∃y) (x, y) ∈ P I ij∧ y ∈ Z I j} n S.Z I i = {x ∈ ∆ I i| ♯{y | (x, y) ∈ S I ij} ≥ n} n S.Z {x ∈ ∆ I i| ♯{y | (x, y) ∈ S I ij} ≤ n}Table 1. Semantic constraints on complex i-concepts in E-connections.Finally, a combined interpretation I is a model of Σ = 〈K, R, A〉 (denotedby I |= Σ), if I satisfies every axiom in K, R and A. An i-concept is satisfiablewith respect to Σ, if Σ has a combined model I, such that C Ii ≠ ∅. We haveΣ |= C ⊑ D, for two i-concepts C and D, if in each combined model I of Σ,C Ii ⊆ D Ii . Both reasoning tasks are inter-reducible, as usual.

5 Relating DDL to P-DLIn this section, we present two reductions. First is from P-DL into DDL, andlater on also vice versa from DDL into P-DL. As we have learned, P-DL usea rather strongly restricted semantics, hence we will use DDL ɛ (F,I,RC,RP) inorder to make the reduction possible.Theorem 1. Given a SHIQP ontology P = {P i } i∈I , with the importing relationP i → P j . Let us construct a DDL KB K = 〈T, R, A, B〉 overtI:– T i := {φ | φ ∈ P i is a GCI axiom} ∪ {⊤ i ≡ ⊤}, for each i ∈ I;– R i := {φ | φ ∈ P i is a RIA axiom}, for each i ∈ I;– A i := {φ | φ ∈ P i is an ABox assertion}, for each i ∈ I;– B ij := {i : C −→ ≡Cj : C | P i → Pj } ∪ {i : R −→ ≡Rj : R | P i → Pj } ∪ {i : a ↦−→aj : a | P i → Pj }, for each i, j ∈ I.Given any i ∈ I, P is i-consistent if and only if K is i-consistent underDDL ɛ (F,I,RC,RP).It trivially follows that also the decision problems of satisfiability and subsumptionentailment are reducible, since they are reducible into i-consistence.Corollary 1. Deciding satisfiability and subsumption entailment in P-DL reducesinto deciding i-consistence in DDL under the semantics DDL ɛ (F,I,RC,RP).We will now show, that under this strong semantics of DDL, also a reductionin the opposite direction is possible.Theorem 2. Let K = 〈T, R, A, B〉 be a DDL KB over some index set I. Weconstruct a package-based ontology P = {P i } i∈I . Each package P j contains aunion of the following components:1. T j ∪ {C ⊑ G | i : C −→ ⊑ j : G ∈ B} ∪ {G ⊑ C | i : C −→ ⊒ j : G ∈ B};2. R j ∪ {R ⊑ S | i : R −→ ⊑ j : S ∈ B} ∪ {S ⊑ R | i : R −→ ⊒ j : S ∈ B};3. A j ∪ {a = b | i : a ↦−→ j : b ∈ B}.In addition, P uses the following imports:C– P i → Pj if either i : C −→ ⊑ j : G ∈ B or i : C −→ ⊒ j : G ∈ B, for anyi, j ∈ I, for any i-concept C and for any j-concept G;R– P i → Pj if either i : R −→ ⊑ j : S ∈ B or i : R −→ ⊒ j : S ∈ B, for any i, j ∈ I,for any i-role R and for any j-role S;a– P i → Pj if i : a ↦−→ j : b ∈ B for any i, j ∈ I, for any i-individual a and forany j-individual b.Given any i ∈ I, P is i-consistent if and only if K is i-consistent underDDL ɛ (F,I,RC,RP).Also the remaining decision problems are reducible, because they are reducibleinto i-consistence in DDL.

Corollary 2. Deciding satisfiability of concepts and subsumption entailmentwith respect in DDL ɛ (F,I,RC,RP) reduces into deciding i-consistence in P-DL.As we see, P-DL are closely related to DDL ɛ (F,I,RC,RP). More precisely,these results show, that it is possible to implement importing with bridge rules,and vice versa, it is possible to simulate bridge rules with imports and additionallocal axioms, if the requirements placed on domain relations are the same.These results do not imply, however, that the two frameworks have the sameexpressivity, for two reasons. First, in terms of expressive power, none of thereductions is complete. DDL does not handle nominals, hence P-DL ontologieswith nominals cannot be reduced. On the other hand, we deal with simplifiedversion of DDL in this paper. We did not provide any reduction for DDL ontologieswith heterogeneous bridge rules [15], and as these bridge rules essentiallyrequire domain relations that are not functional, it is hard to imagine that anyreduction is possible.The second reason is the fact that the DDL semantics used by the reductionsis considerably stronger than what is commonly understood as appropriate DDLsemantics. DDL ɛ (RC), a weaker version of DDL ɛ (F,I,RC,RP) has been studied[18], and it has been showed that it does not satisfy all the desiderata commonlyplaced on DDL [7,14]. More specifically, this semantics may behave unexpectedly,in the presence of an accidental inconsistency in one of the ontologiesthat are combined [18]. The problem also occurs with the stronger semanticsDDL ɛ (F,I,RC,RP), and we will show that it also occurs in P-DL.Example 1. Consider a package-based ontology P consisting of three packagesP 1 , P 2 and P 3 with the following imports:P 1C→ P2 , P 2D→ P3 , P 1E→ P3 .This is a very basic P-DL setting, each of the three packages imports one concept.Furthermore, let us assume that all tree packages are empty (i.e., P 1 = P 2 =P 3 = ∅). This means, that the three concepts C, D and E are unrelated. It iseasy to show, that if P 2 becomes inconsistent, for some reason, then the importedconcept D becomes unsatisfiable in P 3 . This is quite intuitive, since D is importedfrom P 2 . However, as we will show, if P 2 becomes inconsistent, also E becomesunsatisfiable in P 3 which we consider rather unintuitive, since this concept isimported from P 1 and is unrelated to D.Let P 2 be inconsistent. For simplicity, let us assign P 2 :={⊤ ⊑ ⊥}. Due to∗ ∗the first two imports we get that P 1 → P2 and P 2 → P3 , and so the semanticsimplies that in any model of P, it must be the case that r 13 = r 12 ◦r 23 . However,since P 2 is inconsistent, ∆ I2 = ∅ in every model, and hence r 12 ◦ r 23 = ∅, andhence also r 13 = ∅. And so, we also get E I3 = r 13 (E I1 ) = ∅.The DDL ɛ semantics does not exhibit this problem [14]. On the other hand,DDL ɛ has problems with transitive propagation of the effects of bridge rules,which is not a problem for DDL ɛ (RC), nor DDL ɛ (F,I,RC,RP) [18]. In the P-DL setting this reformulates as the problem of transitive propagation of the

imported semantic relations, and due to correspondence between P-DL andDDL ɛ (F,I,RC,RP), we now have a formal proof that it never appears in P-DL.This clearly marks the difference between the two approaches.6 On Relation of DDL and E-connectionsThe relation of DDL and E-connections has been studied in the literature. It isknown, that under certain assumptions, it is possible to reduce a DDL KB intoE-connections and reason equivalently in the latter formalism. More specifically,for a DDL KB with bridge rules between concepts and between individuals, thereasoning problems associated with d-entailment are reducible [6,13].Theorem 3 ([6,13]). Assume a DDL KB K = 〈T, R, A, B〉 with index set I andalphabet N C = {N Ci } i∈I , N R = {N Ri } i∈I and N I = {N Ii } i∈I which contains nobridge rules between roles. Let N C ′ = N C , N I ′ = N I , ɛ ′ ii = N Ri, for each i ∈ I.Let ɛ ′ ij = {E ij}, for each i, j ∈ I, i ≠ j, where E ij is a new symbol.Let us construct Σ ′ = 〈K ′ , R ′ , A ′ 〉, a C E (SHIQ) KB built over the index setI and the vocabulary N C ′ , N I ′ and {ɛ ′ ij } i,j∈I, as follows:– K i ′ := T i ∪ {∃E ij .C ⊑ G | j : C −→ ⊑ i : G ∈ B} ∪ {H ⊑ ∃E ij .D | j : Di : H ∈ B}, for each i ∈ I;– R ′ i := R i, for each i ∈ I;– A ′ i := A i, for each i ∈ I;– A ′ E := {a · E ij · b | i : a ↦−→ j : b ∈ B}.It follows, that K is globally consistent if and only if Σ ′ is consistent.⊒−→As a straightforward consequence of the theorem, also the remaining reasoningtasks related to global consistency are reducible.Corollary 3. Deciding d-satisfiability and d-entailment of subsumption with respectto a DDL KB reduces into deciding the consistence of KB in E-connections.This shows, that there is some similarity between bridge rules in DDL andlinks in E-connections. The reduction especially shows, what is the exact relationbetween bridge rules on one side, and links on the other one. However it cannotbe claimed that E-connections are more expressive than DDL based on thisresult. This is for two reasons.First, the reduction is for the DDL d semantics. In DDL, however, “the semantics”is currently DDL ɛ , which allows holes and satisfies a number of desiredproperties [7,14]. In particular, DDL ɛ offers effective means to deal with accidentalinconsistency, and hence it is possible to reason with a DDL KB, even ifsome local ontologies are inconsistent. E-connections offer no such features.Second, to our best knowledge it is not possible to reduce the richer flavoursof DDL that include either bridge rules between roles or heterogeneous bridgerules between concepts and roles [15].These findings are well in line with the different purpose for which each ofthe formalisms was designed. E-connections work with carefully crafted local

modules with non-overlapping modeling domains, while DDL allow to connectoverlapping ontologies in which part of the terminology is modeled on both sidesalthough possibly differently.7 ConclusionThe novel results of this paper are the reductions between P-DL and DDL (witha specifically adjusted semantics) and vice versa. This result suggests that thesetwo frameworks have many similarities, and under certain assumptions, importsused in P-DL can be expressed by bridge rules used in DDL, and the otherway around. On the other hand, these results also point out, that the actualdifference between the two formalisms is in the fact that P-DL uses much strongersemantics than DDL. These semantics significantly differ, and hence it cannotbe claimed that one of these frameworks is more expressive than the other.Similarly, it cannot be claimed that E-connections are more expressive thanDDL, based on the reduction given by Kutz et al. [6,13]. This reduction is notgiven for the DDL semantics with holes, which is currently considered the mostappropriate one, but for a simplified semantics instead. In addition, for somemore complex DDL constructs no reduction is known in the literature. Whatthe reduction does show, is how the concept of bridge rules and the concept oflinks (used in E-connections) are related.We conclude, that these results are well in line with the particular purpose forwhich each of the formalisms has been designed and is suitable for. E-connectionsare particular well suited for combining multiple ontologies with separated localdomains. This separation of domains, which has to be maintained, may also serveas a reasonable guide during modular ontology development.On the other hand, sometimes we wish to combine and integrate also ontologieswith partially overlapping domains. Especially in heterogeneous distributedknowledge environments, such as the Semantic Web, this is unavoidable. In suchapplications, DDL seems to be the most suitable, offering a versatile apparatus ofontology mapping, which allows concepts, roles and individuals to be associatedfreely according to the need of the application. DDL is robust enough to dealwith accidental inconsistency and offers means to resolve possible heterogeneityin the modeling approaches used by different ontologies.Finally, P-DL offers an ontology importing paradigm, which is very familiarto importing in software engineering. Thus P-DL is intuitive and easily understoodalso by users without deep understanding of ontology integration issues.Its strong semantics deals with many modeling issues, such as transitive propagationof imported relations, however, as the reduction suggests, it may possiblybehave unexpectedly in the presence of accidental inconsistency in the system.Acknowledgement. Martin Homola is supported from the project VEGAno. 1/0668/10 of Slovak Ministry of Education and Slovak Academy of Sciences.Extended version of this paper will shortly appear as a technical report.

References1. Berners-Lee, T., Hendler, J., Lassila, O.: The semantic web. Scientific American284(5) (2001) 34–432. Baader, F., Calvanese, D., McGuinness, D., Nardi, D., Patel-Schneider, P., eds.:The Description Logic Handbook. Cambridge University Press (2003)3. Cuenca Grau, B., Horrocks, I., Kazakov, Y., Sattler, U.: A logical framework formodularity of ontologies. In: IJCAI 2007. (2007)4. Ghilardi, S., Lutz, C., Wolter, F.: Did I damage my ontology? A case for conservativeextensions in description logics. In: KR 2006, AAAI Press (2006)5. Lutz, C., Walther, D., Wolter, F.: Conservative extensions in expressive descriptionlogics. In: IJCAI 2007. (2007)6. Kutz, O., Lutz, C., Wolter, F., Zakharyaschev, M.: E-connections of abstractdescription systems. Artificial Intelligence 156(1) (2004) 1–737. Borgida, A., Serafini, L.: Distributed description logics: Assimilating informationfrom peer sources. Journal of Data Semantics 1 (2003) 153–1848. Zimmermann, A.: Integrated distributed description logics. In: DL-2007. Volume250 of CEUR WS. (2007)9. Pan, J.Z., Serafini, L., Zhao, Y.: Semantic import: An approach for partial ontologyreuse. In: WoMo-06. Volume 232 of CEUR WS. (2006)10. Bao, J., Honavar, V.: Ontology language extensions to support collaborative ontologybuilding. In: ISWC2004. Volume 3298 of LNCS., Springer (2004)11. Wang, Y., Haase, P., Bao, J.: A survey of formalisms for modular ontologies. In:IJCAI’07, SWeCKA Workshop. (2007)12. Bao, J., Caragea, D., Honavar, V.: On the semantics of linking and importing inmodular ontologies. In: ISWC 2006. Volume 4273 of LNCS., Springer (2006)13. Cuenca Grau, B., Kutz, O.: Modular ontology languages revisited. In: IJCAI’07,SWeCKA Workshop. (2007)14. Serafini, L., Borgida, A., Tamilin, A.: Aspects of distributed and modular ontologyreasoning. In: IJCAI-05. (2005)15. Ghidini, C., Serafini, L.: Mapping properties of heterogeneous ontologies. In:WoMo-06. Volume 232 of CEUR WS. (2006)16. Horrocks, I., Sattler, U., Tobies, S.: Practical reasoning for expressive descriptionlogics. In: LPAR’99. Number 1705 in LNAI, Springer (1999) 161–18017. Homola, M., Serafini, L.: Towards distributed tableaux reasoning procedure forDDL with increased subsumption propagation between remote ontologies. In: KR2008 Ontology Workshop, Australian Computer Society (2008)18. Homola, M., Serafini, L.: Augmenting subsumption propagation in distributeddescription logics. Applied Artificial Intelligence 24 (2010) 137–17419. Bao, J., Voutsadakis, G., Slutzki, G., Honavar, V.: Package-based descriptionlogics. In: Modular Ontologies: Concepts, Theories and Techniques for KnowledgeModularization. Volume 5445 of LNCS. Springer (2009) 349–37120. Horrocks, I., Sattler, U., Tobies, S.: Practical reasoning for very expressive descriptionlogics. J. Interest Group in Pure and Applied Logic 8(3) (2000) 239–26421. Cuenca Grau, B., Parsia, B., Sirin, E.: Working with multiple ontologies on thesemantic web. In: ISWC2004. Volume 3298 of LNCS., Springer (2004)22. Parsia, B., Cuenca Grau, B.: Generalized link properties for expressive E-connections of description logics. In: AAAI-2005, AAAI Press (2005)

More magazines by this user
Similar magazines