- In the extensive usage of ontologies envisaged by the Semantic Web there is a compelling need for expressing mappings between the components of heterogeneous ontologies. These mappings are of many different forms and involve the different components of ontologies. State of the art languages for ontology mapping enable to express semantic relations between homogeneous components of different ontologies, namely they allow to map concepts into concepts, individuals into individuals, and properties into properties. Many real cases, however, highlight the necessity to establish semantic relations between heterogeneous components. For example to map a concept into a relation or vice versa. To support the interoperability of ontologies we need therefore to enrich mapping languages with constructs for the representation of heterogeneous mappings. In this paper, we propose an extension of Distributed Description Logics (DDL) to allow for the representation of mapping between concepts and relations. We provide a semantics of the proposed language and show its main logical properties.1
In the extensive usage of ontologies envisaged by the Semantic Web there is a compelling need for expressing mappings between different and heterogeneous ontologies. These mappings are of many different forms and involve the different components of ontologies.
Most of the formalisms for distributed ontology integration, which are based on the p2p architecture [12], provide a language to express semantic relations between concepts belonging to different ontologies. These classes of languages are usually called mapping languages [11], [9]. These formalisms can express that a concept, say MarriedMan, in Ontology 1 is equivalent to the concept Husband in Ontology 2, or that the concept Benedict in Ontology 3 is more specific that the concept Relative in Ontology 4. Few mapping languages allow also to express semantic relations between properties in different ontologies (see [7], [5]). However, to the best of our knowledge, none of the existing approaches support mappings between properties and concepts. Such mappings are necessary to express the semantic relations between two ontologies, when the information represented as a concept in the former is represented as a relation in the latter, or vice versa. As a practical example consider two ontologies. The first one is the ontology http://www.daml.org/2001/01/gedcom/gedcom which contains the concept Family and the property spouseIn. Family represents the set of families, and spouseIn relates a Human with the family in which he/she is one of the spouses. The second one is the ontology http://ontologyportal.org/translations/SUMO.owl, which contains the relation spouse, which represents the relationship of marriage between two Humans. In integrating these two ontologies one would like to state, for instance, that every family in the first ontology can be mapped into a married couple in the second ontology, or in other words, that that the concept Family can be mapped into the relation spouse.
The goal of this paper is to extend a language for ontology mapping, an to introduce mechanisms for the representation of heterogeneous mappings between ontologies. We focus on the mappings between concepts and relations as they provide a challenging example of ontology mismatch, we describe their formal semantics and we study their main properties. We adopt the formal framework of Distributed Description Logics (DDL) [10] because it is a formalism which is explicitly constructed to state expressive relations between heterogeneous languages and domains of interpretation. Summarizing, the claimed contributions of this paper are: (i) an expressive mapping language for heterogeneous ontologies; (ii) a clear semantics for the proposed mapping language, and (iii) an investigation of its basic logical properties.
The paper is structured as follows: in Section II we motivate our work with a detailed example. In Section III we provide an extension of DDL to represent heterogeneous mappings. In Section IV we study the main properties of the proposed logic. We end with related work (Section V) and some final remarks (Section VI).
Let us consider two ontologies from the web. The first is an extensive ontology describing the domain of Geography developed by a corporation2, the second is the DAML+OIL representation of the 2001 CIA World Fact Book3. We call the first Ontology 1, and the second Ontology 2. Looking at the ontologies in detail we have noticed that both need to 2See http://reliant.teknowledge.com/DAML/Geography.daml 3See http://www.daml.org/2001/12/factbook/factbook-ont represent the notion of geographic coordinates. Figures 1 and 2 report the definition of geographic coordinates from the two ontologies.
While the two ontologies are interested in describing the geographic coordinate system of Earth, they have specific views on how to describe this domain of knowledge. Rather than giving a detailed formalization of the example, we focus on the key elements that are affected by the representation of geographical coordinates in the two ontologies. Ontology 1 does not contain an explicit notion of position on hearth (or geographical coordinate), and expresses positions in terms of Latitude and Longitude (see Figure 1). Ontology 2 takes a different perspective and expresses geographical coordinates as a specific concept LatLon, which has two properties represented by the roles latitude and longitude which are numbers of type Double (see Figure 2). A graphical representation of two different representations of geographical coordinates inspired by the definitions in Figures 1 and 2 is given in Figure 3. Despite the different choices made by the ontology designers, there is clear relation between Ontology 1 and Ontology 2, and in particular between the concepts Latitude and Longitude and the properties (roles) latitude and longitude, respectively.
State of the art formalisms for the representation of distributed ontologies and reasoning, would do very little with this example, except perhaps identifying that Latitude and Longitude in Ontology 1 are related with Double in Ontology 2 (assuming that Latitude and Longitude are represented as doubles also in Ontology 1). But this is an hardly informative mapping, as it does not capture the essential fact that both ontologies are describing the geographic coordinate system of Earth.
Description Logic (DL) has been advocated as the suitable formal tool to represent and reason about ontologies.
Distributed Description Logic (DDL) [3], [10] is a natural generalization of the DL framework designed to formalize multiple ontologies interconnected by semantic mappings. As defined in [3], [10], Distributed Description Logic provides a syntactical and semantical framework for formalization of multiple ontologies pairwise linked by semantic mappings. In DDL, ontologies correspond to description logic theories (Tboxes), while semantic mappings correspond to collections of bridge rules (B).
In the following we recall the basic definitions of DDL as defined in [10], and we extend the set of bridge rules, introducing new semantic mappings between distributed ontologies.
A. Distributed Description Logics: the syntax
Given a non empty set I of indexes, used to identify ontologies, let {DLi}i∈I be a collection of description logics4.
For each i ∈ I let us denote a T-box of DLi as Ti. In this paper, we assume that each DLi is description logic weaker or
4We assume familiarity with Description Logic and related reasoning systems, described in [1]. at most equivalent to SHIQ. Thus a T-box will contain all the information necessary to define the terminology of a domain, including not just concept and role definitions, but also general axioms relating descriptions, as well as declarations such as the transitivity of certain roles.
We call T = {Ti}i∈I a family of T-Boxes indexed by I.
Intuitively, Ti is the description logic formalization of the i-th ontology. To make every description distinct, we will prefix it with the index of ontology it belongs to. For instance, the concept C that occurs in the i-th ontology is denoted as i : C.
Similarly, i : C ⊑ D denotes the fact that the axiom C ⊑ D is being considered in the i-th ontology.
Semantic mappings between different ontologies are expressed via collections of bridge rules. In the following we use A and B as placeholders for concepts and R and S as placeholders for relations.
Definition 1: A bridge rule from i to j is an expression defined as follows:
⊒ i : A −→ j : B
⊑ i : A −→ j : B
⊒ i : R −→ j : S
⊑ i : R −→ j : S
⊒ i : A −→ j : R
⊑ i : A −→ j : R
⊒ i : R −→ j : A
⊑
i : R −→ j : A
(concept-onto-concept bridge rule)
(concept-into-concept bridge rule)
(role-onto-role bridge rule)
(role-into-role bridge rule)
(concept-onto-role bridge rule)
(concept-into-role bridge rule)
(role-onto-concept bridge rule)
(role-into-concept bridge rule)
(
Bridge rules do not represent semantic relations stated from an external objective point of view. Indeed, there is no such global view in the web. Instead, bridge rules from i to j express relations between i and j viewed from the subjective point of view of the j-th ontology. Let us discuss the different mapping categories.
a) Homogeneous bridge rules: Bridge rules (
⊒ concept bridge rule i : A −→ j : B expresses the fact that, according to j, A in i is more general than B in j. Therefore, bridge rules from i to j provide the possibility of translating into j’s ontology (under some approximation) the concepts of a foreign i’s ontology. Note, that since bridge rules reflect a subjective point of view, bridge rules from j to i are not necessarily the inverse of the rules from i to j, and in fact <rdfs:Class rdf:ID= "Latitude"> <rdfs:subClassOf
rdf:resource = "http://reliant.teknowledge.com/DAML/SUMO.owl#Region"/> <rdfs:label>latitude</rdfs:label> <rdfs:label>parallel</rdfs:label> <rdfs:comment>Latitude is the class of Regions, associated with areas on the Earth’s surface, which are parallels measured in PlaneAngleDegrees from the Equator.</rdfs:comment> </rdfs:Class> <rdfs:Class rdf:ID= "Longitude"> <rdfs:subClassOf
rdf:resource ="http://reliant.teknowledge.com/DAML/SUMO.owl#Region"/> <rdfs:label>longitude</rdfs:label> <rdfs:label>meridian</rdfs:label> <rdfs:comment>Longitude is the class of Regions, associated with areas on the Earth’s surface, which are meridians measured in PlaneAngleDegrees from the PrimeMeridian through GreenwichEnglandUK.</rdfs:comment> </rdfs:Class> ⊒ i : Article −→ j : ConferencePaper expresses the fact that, according to ontology j, the concept Article in ontology i is more general than its local concept ConferencePapers, while the bridge rules
⊑ i : Article −→ j : Article
⊒
i : Article −→ j : Article
say that, according to ontology j, the concept Article in
ontology j is equivalent to its local concept Article. Bridge
rules (
For example, the bridge rule:
⊑ i : marriedTo −→ j : partnerOf says that according to ontology j, the relation marriedTo in ontology i is less general than its own relation partnerOf.
b) Heterogeneous bridge rules: Bridge rules (
⊒ 1 : Latitude −→ 2 : latitude says that according to ontology 2, concept Latitude in ontology 1 is more general than its own relation latitude. That is all latitudes in its own ontology have a corresponding Latitude in ontology 1.
Bridge rules (
Bridge rule (
⊑ 1 : spouse −→ 2 : Family states that every married couple in ontology 1, can be mapped into a family in ontology 2. Similarly the bridge rule. Similarly the bridge rule
⊒ 1 : WorksFor −→ 2 : WorkingContract states that every working contract in ontology 2 corresponds to some working relation in ontology 1.
Bridge rules (
T = h{Ti}i∈I , Bi consists of a collection {Ti}i∈I of T-boxes, and a collection B = {Bij }i6=j∈I of bridge rules between them.
B. Distributed Description Logics: the semantics
The semantic of DDL, which is a customization of Local Models Semantics [6], [7], assigns to each ontology Ti a local interpretation domain. The first component of an interpretation of a DTB is a family of interpretations {Ii}i∈I , one for each T-box Ti. Each Ii is called a local interpretation and consists of a possibly empty domain ΔIi and a valuation function ·Ii , which maps every concept to a subset of ΔIi , and every role to a subset of ΔIi × ΔIi . The interpretation on the empty domain is denoted with the apex ǫ.
Notice that, in DL, interpretations are defined always on a non empty domain. Therefore Iǫ is not an interpretation in DL. In DDL however we need to provide a semantics for partially inconsistent distributed T-boxes, i.e. DTBs in which some of the local T-boxes are inconsistent. Iǫ provides an “impossible interpretation” which can be associated to inconsistent Tboxes. Indeed, Iǫ satisfies every axiom X ⊑ Y (also ⊤ ⊑ ⊥) since X Iǫ = ∅ for every concept and role X .
The second component of the DDL semantic are families of domain relations. Domain relations define how the different T-box interact and are necessary to define the satisfiability of bridge rules.
Definition 3: A domain relation rij from ΔIi to ΔIj is a subset of ΔIi × ΔIj . We use rij (d) to denote {d′ ∈ ΔIj | hd, d′i ∈ rij }; for any subset D of ΔIi , we use rij (D) to denote Sd∈D rij (d); for any R ⊆ ΔIi × ΔIi we use rij (R) to denote Shd,d′i∈R rij (d) × rij (d′).
A domain relation rij represents a possible way of mapping the elements of ΔIi into its domain ΔIj , seen from j’s perspective. For instance, if ΔI1 and ΔI2 are the representation of time as Rationals and as Naturals, rij could be the round off function, or some other approximation relation. This function has to be conservative w.r.t., the order relations defined on Rationals and Naturals. Domain relation is used to interpret homogeneous bridge rules according with the following definition.
Definition 4 (Satisfiability of homogeneous bridge rules): The domain relation rij satisfies a homogeneous bridge rule w.r.t., Ii and Ij , in symbols hIi, rij , Ij i |= br, according with the following definition: 1) hIi, rij , Ij i i : A −⊑→ j : B, if rij (AIi ) ⊆ BIj 2) hIi, rij , Ij i i : A −⊒→ j : B, if rij (AIi ) ⊇ BIj where A and B are either two concept expressions or two role expressions.
Domain relations do not provide sufficient information to evaluate the satisfiability of heterogeneous mappings. Intuitively, an heterogeneous bridge rule between a relation R and a concept A connects a pair of objects related by R with an object which is in A. This suggests that, to evaluate heterogeneous bridge rules from roles in i to concepts in j one needs a relation that maps pair of objects in ΔIi into objects of ΔIj , and to evaluate a heterogeneous bridge rule from concepts in i to roles in j one needs a relation that maps objects in ΔIi into pairs of objects in ΔIj .
Definition 5: A concept-role domain relation crij from ΔIi to ΔIj is a subset of ΔIi ×ΔIi ×ΔIj . A role-concept domain relation rcij from ΔIi to ΔIj is a subset of ΔIi ×ΔIj ×ΔIj . We use crij (d) to denote {hd1, d2i ∈ ΔIj ×ΔIj | hd, d1, d2i ∈ crij }; for any subset D of ΔIi , we use crij (D) to denote Sd∈D crij (d). We use rcij (hd1, d2i) to denote {d ∈ ΔIj | hd1, d2, di ∈ rcij }; for any subset R of ΔIi × ΔIi , we use rcij (R) to denote Shd1,d2i∈R rcij (hd1, d2i).
Domain relation crij represents a possible way of mapping elements of ΔIi into pairs of elements in ΔIj , seen from j’s perspective. For instance, if ΔI1 and ΔI2 are the representation of geographical coordinates as in Figure 4, cr12 could be the function mapping latitude values into the corresponding latitudes. For instance, by setting
cr12(TropicOfCancerI1 ) = {hx, 23.27i ∈ latitudeI2 } we can represent the fact that the tropic of cancer is associated with pairs of objects hx, yi such that y is the latitude of x and y is equal to 23.27 (the latitude of the tropic of cancer). Vice-versa a domain relation rcij represents a possible way of mapping a pair of ΔIi into the corresponding element in ΔIj . For instance, if the pair DJohnI1 , MaryI1 E ∈ spouseI1 , then the fact that
rc12(JohnI1 , MaryI1 ) = family23I2 represents the fact that family23 is the family containing the married couple of John and Mary.
Definition 6 (Satisfiability of heterogeneous bridge rules): The concept-role domain relation crij satisfies a concept to role bridge rule w.r.t., Ii and Ij , in symbols hIi, crij , Ij i |= br, according with the following definition: 1) hIi, crij , Ij i i : A −⊑→ j : R, if crij (AIi ) ⊆ RIj 2) hIi, crij , Ij i i : A −⊒→ j : R, if crij (AIi ) ⊇ RIj where A is a concept expression of i and R a role expression of j.
The role-concept domain relation rcij satisfies a role to concept bridge rule w.r.t., Ii and Ij , in symbols hIi, rcij , Ij i |= br, according with the following definition: 1) hIi, rcij , Ij i i : R −⊑→ j : A, if rcij (RIi ) ⊆ AIj 2) hIi, rcij , Ij i i : R −⊒→ j : A, if rcij (RIi ) ⊇ AIj where A is a concept expression of j and R a role expression of i.
I = h{Ii}i∈I , {rij }i6=j∈I , {crij }i6=j∈I , {rcij }i6=j∈I i of a DTB T consists of local interpretations Ii for each Ti on local domains ΔIi , and families of domain relations rij , crij and rcij between these local domains.
Definition 8 (Satisfiability of a Distributed T-box): A distributed interpretation I satisfies the elements of a DTB T according to the following clauses: for every i, j ∈ I 1) I i : A ⊑ B, if Ii A ⊑ B 2) I Ti, if I i : A ⊑ B for all A ⊑ B in Ti 3) I Bij , if 4) I • hIi, rij , Ij i satisfies all the homogeneous bridge rules in Bij , • hIi, crij , Ij i satisfies all the concept-to-role bridge rules in Bij , • hIi, rcij , Ij i satisfies all the role-to-concept bridge rules in Bij
T, if for every i, j ∈ I, I Ti and I Bij
Definition 9 (Distributed Entailment and Satisfiability): T i : C ⊑ D (read as “T entails i : C ⊑ D”) if for every I, I T implies I d i : C ⊑ D. T is satisfiable if there exists a I such that I T. Concept i : C is satisfiable with respect to T if there is a I such that I T and CIi 6= ∅.
In this section we enunciate the most important properties of the extended version of DDL and for each result we provide an example that explains why this property is desirable. We assume familiarity with Description Logics, and in particular with SHIQ. Symbols, ⊔, ⊓, and − denote the usual union, intersection, and inverse operators of Description Logics. Similarly, ∃R.C is used to denote the existential restriction. ⊒ i : A −→ j : G, and I
⊑ i : B −→ j : H, then
I
Ontology 1
Theorem 2 (Concept into/onto concept): If I i : A −⊒→ j : G, and I i : Bk −⊑→ j : Hk for 1 ≤ k ≤ n (with n ≥ 0), then:
Theorem 3 (Concept into/onto role): If I i : A −⊒→ j : R, and I i : Bk −⊑→ j : Sk for 1 ≤ k ≤ n (with n ≥ 0), then: where A and Bk (1 ≤ k ≤ n) are concepts, R and Sk (1 ≤ k ≤ n) are roles, and X is any arbitrary concept.
Example 3: Let us assume that a distributed interpretation I satisfies the following bridge rules:
⊒ • i : Parent −→ j : hasChild,
⊑ • i : Mother −→ j : motherOf, and
⊑ • i : Father −→ j : fatherOf, where Parent, Mother and Father are concepts, while hasChild, motherOf and fatherOf are relations. Theorem 3 guarantees that if I i : Parent ⊑ Mother ⊔ Father, then
LatLon la t i t u d e l o n g i t u d e where holdsSeasTicketOf, Person and JuventusFan are concepts, while SeasonalTicketHolder, supporterOf and JuventusFan are relations. Corollary 1 allows to infer that if I i : holdsSeasTicketOf ⊑ supporterOf then I i : SeasonalTicketHolder ⊑ Person ⊔ JuventusFan and I i : SeasonalTicketHolder ⊑ Person ⊓ JuventusFan. ⊒ i : A −→ j : into marriages. Moreover, assume we also have a bridge rule mappings wives in ontology 1 into women in ontology 2 as follows:
R, and I i : B −⊑→ j : S, then:
Example 6: Let us assume that a distributed interpretation I satisfies the following bridge rules:
⊒ • i : Mother −→ j : motherOf
⊑ • i : Parent −→ j : hasChild, where Parent, and Mother are concepts, while hasChild, and motherOf are relations. Corollary 2 ensures that if I i : Mother ⊑ Parent, then I j : motherOf− ⊑ hasChild−.
and I i : S −⊑→ j : Q, then: ⊒ i : R −→ j : P ,
Example 7: Let us assume that a distributed interpretation I satisfies the following bridge rules between roles: ⊒ • i : marriedTo −→ j : partnerOf, and
⊒ • i : livingWith −→ j : friendOf.
If I i : marriedTo ⊑ leavingWith−, then I j : partnerOf ⊑ FriendOf−.
Theorem 5 (Role union): If the DL includes role union R⊔ S, then Theorem 2 can be generalised to all bridge rules, with the only constraint of A and Bk and G and Hk families of homogeneous elements, that is either families of concepts or families of roles.
All the mapping languages described in [11] do not support full heterogeneous mappings. In general, however, mapping languages support a limited version of heterogeneous mappings. For instance, in [5], it is possible to express the mapping ∀x.(∃y.R1(x, y) → C2(x)) or, similarly, in the original version of DDL one can state the mapping ⊑ 1 : ∃R.⊤ −→ 2 : C
However, the encoding of heterogeneous mappings shown above is not very expressive and its usage can also lead to undesirable consequences. For instance, assume a relation IsMarried exists in ontology 1, and a concept Marriage exists in ontology 2. Assume we want to impose that the relation IsMarried in ontology 1 is equivalent to the concept Marriage in ontology 2, and we only have mappings as in Equation (22). Then, we can only state mappings of the form: (21) (22) ⊑ 1 : ∃IsMarried.⊤ −→ 2 : Marriage
⊒ 1 : ∃IsMarried.⊤ −→ 2 : Marriage But these mappings express something rather different from our initial goal as they map single elements of a couple ⊑ 1 : Wife −→ 2 : Woman
Wife ⊑ ∃IsMarried.⊤ together with the axiom in ontology 1 stating that a wife is a married entity. From all this we can infer in ontology 2 that a wife is a marriage, i.e.,
Wife ⊑ Marriage
This undesirable conclusion reflects the fact that in mapping the two ontologies, we have identified the participants of a relation, (the married person) with the relation itself (the marriage). To avoid this bad behavior, Omelayenko claims in [8] that mappings between classes and properties are not relevant from an application point of view. We believe that the examples shown in the paper provide a convincing evidence that this is not the case, and that an appropriate formalization of heterogeneous mappings can avoid some of the problems mentioned above.
An effort towards the formalization of heterogeneous mappings between concepts and relations in the area of federated databases is described in [2]. In this work the authors define five types of correspondences between concepts and properties. If A is a concept and R is a relation, they consider the following correspondences: • A is equivalent to R; • A is more general to R; • A is less general to R; • A and R do overlap; • A and R do not overlap.
The semantics of the correspondences above can be expressed by the following mappings: • ∀x.(A(x) ↔ ∃y.R(y, x)); • ∀x.(∃y.R(y, x) → A(x)); • ∀x.(A(x) → ∃y.R(y, x)); • ∃x.(A(x) ∧ ∃y.R(y, x)); • ∀x.(A(x) → ¬∃y.R(y, x)).
This semantics is similar to the encoding described in Equation (21). The only difference is that it considers the range of the relation R in place of the domain. Therefore it suffers of problems similar to the ones shown above for Equation (21).
Different forms of mappings (bridge rules) have been studied in other formalisms strictly related to DDL, such as COWL [4] and DFOL [7]. Both formalisms do not address the problem of heterogeneous mappings and should therefore be extended in this direction.
The language and the semantics presented in this paper constitute a genuine contribution in the direction of the integration of heterogeneous ontologies. The language proposed in this paper makes it possible to directly bind a concept with a relation in a different ontology, and vice-versa. At the semantic level we have introduced a domain relation that maps pairs of object into objects and vice-versa. This also constitute a novelty in the semantics of knowledge integration. We have showed the main formal properties of the mapping language, and we have left the complete characterization of the logic for future work.