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

**Towards** **Formal** **Comparison** **of** **Ontology****Linking**, **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, Slovakiahomola@fmph.uniba.sk, 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 modulars**of**tware 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 h**and**, 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** dem**and**, that would deal with duplicity, temporal unavailability**and** 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, roles**and** individuals are possibly related. **Mapping** enables for knowledge reuse **and**also 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 **of**this 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 **of**them 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 st**and**ard one.

Industry**Ontology**IndustryFood&Drink**Ontology**Alcoholic-BeverageBeverageS**of**tDrinkFood&Drink-ProducerproducesWineBeerSodaproducesMaltBasedS**of**tDrinkBreweryWineryproduces(a)Business**Ontology**Law**Ontology**LegalEntityCompanycontractedByemployedByConsultantEmployeeLegalPersonNaturalPerson(b)CompanyModuleCorporationPartonomyModulepartOfdepartmentOfemployeeOfExecutive-BoardmemberOfdepartmentOfmemberOfemployeeOfExecutive(c)**Mapping**LinksImportsSubsumptionRolesFig. 1. **Ontology** combination paradigms: a) ontology linking; b) ontology mapping; c)ontology importing.

On the other h**and**, 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 **of**RBoxes 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 syntacticshorth**and** for the pair **of** bridge rules i : X −→ ⊑ j : Y **and** i : X −→ ⊒ j : Y . Thesymbol B st**and**s 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 formulae**and** (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 j**and** 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 h**and** 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, **and**only 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 **of**Horrocks [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 alphabet**of** 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 **of** → **and** 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 I**of** 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 k**and** N k I k. An ij-property axiom is**of** the form P 1 ⊑ P 2 , where P 1 , P 2 ∈ ɛ ij . An ij-property box R ij is a finite set**of** 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, **and**P, 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, each**of** 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, **and**later 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 h**and**le nominals, hence P-DL ontologieswith nominals cannot be reduced. On the other h**and**, 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 = ∅, **and**hence 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 h**and**,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 **and**DDL ɛ (F,I,RC,RP), we now have a formal pro**of** 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 **and**alphabet 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 **and**links 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 **of**fers 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 **of**fer no such features.Second, to our best knowledge it is not possible to reduce the richer flavours**of** 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 **of**the 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 h**and**, 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 **of**links (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 h**and**, 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, **of**fering a versatile apparatus **of**ontology 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** **of**fers means to resolve possible heterogeneityin the modeling approaches used by different ontologies.Finally, P-DL **of**fers an ontology importing paradigm, which is very familiarto importing in s**of**tware engineering. Thus P-DL is intuitive **and** easily understoodalso by users without deep underst**and**ing **of** ontology integration issues.Its strong semantics deals with many modeling issues, such as transitive propagation**of** 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.

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