Ontology matching and alignment is a key mechanism for linking the diverse datasets and ontologies arising in the Semantic Web. We show that category theory provides the powerful abstractions needed for a uniform treatment at various levels: semantics, language design, reasoning and tools. The Distributed Ontology Language DOL is extended in a natural way with constructs for networks of ontologies. We in particular show how the three semantics of Zimmermann and Euzenat can be uniformly and faithfully represented using these DOL language constructs. Finally, we summarise how the DOL alignment features are currently being implemented in the Ontohub/Hets ecosystem, including support for the OWL and Alignment APIs.
Ontology matching and alignment is a key mechanism for linking the diverse datasets
and ontologies arising in the Semantic Web. Matching based on statistical methods
is a relatively developed eld, with yearly competitions since 2004 comparing the
various strengths and weaknesses of existing algorithms [
Ontology alignments express semantic correspondences between the entities of dierent ontologies. The correspondences of an alignment can be various relations, like equivalence, subsumption, disjointness or instance between entities of the ontologies, which can be named entities, like classes, roles, individuals, function symbols etc. or even complex concepts or terms.
The problem of giving an interpretation to alignments in terms of the semantics
of the ontologies is complicated by the fact that the domains of interpretation of
the two ontologies may be incompatible. Dierent ways of dealing with this problem
exist in the literature. The rst solution, called simple semantics in [
A major problem with these approaches is their diversity. There exist some
attempts for unication, which however remain unsatisfactory: there is no common
syntax, no common semantic framework, and no common tool support. In this work,
we show how category theory can provide such a unifying framework at various levels,
improving previous related work [
The general representation and reasoning framework that we propose includes: 1) a declarative language to specify networks of ontologies and alignments, with independent control over specifying local ontologies and complex alignment relations, 2) the possibility to align heterogeneous ontologies, and 3) in principle, the possibility to combine dierent alignment paradigms (simple/integrated/contextualised) within one network.
Through category theory, we obtain a unifying framework at various levels: semantic level We give a uniform semantics for distributed networks of aligned ontologies, using the powerful notion of colimit, while reecting properly the semantic variation points indicated above. (meta) language level We provide a uniform notation (based on the distributed ontology language DOL) for distributed networks of aligned ontologies, spanning the dierent possible semantic choices. reasoning level Using the notion of colimit, we can provide reasoning methods for distributed networks of aligned ontologies, again across all semantic choices. 1 tool level The tool ontohub.org provides an implementation of analysis and reasoning for distributed networks of aligned ontologies, again using the powerful abstractions provided by category theory. logic level Our semantics is given for the ontology language OWL, but due to the abstraction power of the framework, it easily carries over to other logics used in ontology engineering, like RDFS, rst-order logic or F-logic.
This shows that category theory is not only a powerful abstraction at the semantic level, but can properly guide language design and tool implementations and thus provide useful abstraction barriers from a software engineering point of view.
The distributed ontology language DOL is a metalanguage in the sense that it
enables the reuse of existing ontologies as building blocks for new ontologies using a
variety of structuring techniques, as well as the specication of relationships between
ontologies. One important feature of DOL is the ability to combine ontologies that
are written in dierent languages without changing their semantics. A formal
specication of the language can be found in [
The general picture is then as follows: existing ontologies can be integrated as-is into the DOL framework. With our new extended DOL syntax, we can specify dierent kinds of alingments. From such an alignment, we construct a graph of ontologies and morphisms between themin a way depending on the chosen alignment framework. Sometimes, this step also involves transformations on the ontologies, such as relativisation of the (global) domain using predicates. A network of alignments can then be combined to an integrated alignment ontology via a so-called colimit. Reasoning in a network of aligned ontologies is then the same as reasoning in the combined ontology. Thus, in order to implement a reasoner, it is in principle sucient to dene the relativisation procedure for the local logics and the alignment transformation for each kinds of semantics. 3
Networks of ontologies and their semantics
In this section we recall networks of ontologies and their semantics introduced in [
A semantics of networks of ontologies is given in terms of local interpretation of the ontologies and alignments it consists of. To be able to give such a semantics, one needs to give an interpretation of the relations between entities that are expressed in the correspondences. In the following three subsections let S = f(Oi)i2I ; (Aij )i;j2I g be a NeO over a set of indexes I.
Smiamntpicles, sthemeaasnstuimcsptiIonn tihsethsaimtpallle osne-- O1 O2 : : : On tologies are interpreted over the same m1 m2 mn domain (or universe of interpretation) !( D w D. The relations in R are interpreted as relations over D, and we denote the interpretation of R 2 R by RD.
If O1, O2 are two ontologies and c = (e1; e2; R) is a correspondence between O1 and O2, we say that c is satised by interpretations m1, m2 of O1, O2 i m1(e1) RD m2(e2). This is written m1; m2 j=S c. A model of an alignment A between ontologies O1 and O2 is then a pair m1, m2 of interpretations of O1, O2 such that for all c 2 A, m1; m2 j=S c. We denote this by m1; m2 j=S A. An interpretation of S is a family (mi)i2I of models mi of Oi. A simple interpretation of S is an interpretation (mi)i2I of S over the same domain D.
Denition 1. [
Integrated Semantics Another pos- O1 O2 : : : On sibility is to consider that the domain of interpretation of the ontologies of a m m2 mn NeO is not constrained, and a global D1 D2 : : : Dn gdeotmhearinwoitfhinatfearmprielytaotifonequUaliesxinisgtsf,untco-- 1 2 n tions i : Di ! U , where Di is the do- !( U w main of Oi, for each i 2 I. A relation R in R is interpreted as a relation RU on the global domain. Satisfaction of a correspondence c = (e1; e2; R) by two models m1 of O1 and m2 of O2 means that i(mi(e1))RU j (mj (e2)). We denote this by m1; m2 j=I1; 2 c and by m1; m2 j=I1; 2 A we denote that m1; m2 j=I1; 2 c for each c 2 A.
An integrated interpretation of S is then f(mi)i2I ; ( i)i2I g where (mi)i2I is an
interpretation of S and i : Di ! U is a function to a common global domain U for
each i 2 I. We here assume that the i are inclusions.2
Denition 2. [
Contextualised Semantics The func- O1 O2 : : : On
tional notion of contextualised
semantics in [
The idea is to relate the domains of the ontologies by a family of relations r = (rij )i;j2I . The relations R in R are interpreted in each domain of the ontologies in the NeO. Satisfaction of a correspondence c = (e1; e2; R) by two models m1 of O1 and m2 of O2 means that mi(e1)Rirji(mj (e2)), where Ri is the interpretation of R in Di. We denote it by m1; m2 j=rC c, and extend this to alignments, denoted m1; m2 j=rC A if all correspondences of the alignment are satised by m1; m2 w.r.t. r.
A contextualised interpretation of S is a pair f(mi)i2I ; (rij )i;j2I g where (mi)i2I is an interpretation of S and (rij )i;j2I is a family of domain relations such that rij relates the domain of mi to the domain of mj and rii is the identity (diagonal) relation. Further assumptions about domain relations can be added, thus restricting more the class of interpretations of a NeO.
Denition 3. A contextualised model of the NeO S is a contextualised interpretation ((mi)i2I ; (rij )i;j2I ) of S such that for each i; j 2 I, mi; mj j=rC Aij . We denote by M odcon(S) the class of all contextualised models of a NeO S. 2 The theory also works for injections without much change. Arbitrary, i.e. possibly noninjective maps, are conceptually not necessary: a local model can be quotiented by the kernel of a non-injective such map, and then be replaced by the quotient, leading to an injective map again.
In this section we start by introducing the DOL concepts necessary for giving semantics of alignments. We then introduce the syntax of alignments in DOL and illustrate with the help of an example involving OWL ontologies how the semantics of alignments can be given using diagrams and colimits. We then present the main result of the paper, showing how the categorical semantics of DOL alignments captures the three semantics of networks of ontologies. 4.1
DOL Diagrams and Combinations
graph D = D1; : : : ; Dm; O1; : : : ; On; M1; : : : ; Mp; A1; : : : ; Ak where Di are (sub-)diagrams, Oi are ontologies, Mi are morphisms and Ai are alignments. The user species a diagram D formed with the subgraphs given by diagrams Di, extended with ontologies Oi and the morphisms Mi and the subdiagrams of the alignments Ai
DOL also provides means for combining a diagram of ontologies into a new ontology, such that the symbols related in the diagram are identied. The syntax of combinations is ontology O = combine D , where D is a diagram, named or specied as above. The semantics of a combination O is the class of models of the colimit ontology of the diagram specied in the combination. Under rather mild technical assumptions, this model class captures exactly the models of the diagram. 4.2
Syntax of DOL Alignments
DOL represents the general alignment format in a similar way to the Alignment API
[
Before starting to analyse the three semantics for NeOs in our setting, we can rst dene the diagram of a NeO in terms of the diagrams of its parts. Denition 4. The diagram of a NeO S = f(Oi)i2I ; (Aij )i;j2I g is obtained by putting together the diagrams of all alignments Aij it consists of.
The gap to be lled is the construction of the diagram associated with a single assignment, in all three possible assumptions about the semantics. Once this has been given, we can dene the semantics of a NeO as the colimit ontology of its associated diagram.
Example 1. We illustrate the three approaches to semantics with the help of a simple example. Let us consider the following two ontologies: ontology S = Class: Person
Individual: alex Types: Person
Class: Child ontology T = Class: HumanBeing
Class: Male SubClassOf: HumanBeing
Class: Employee together with the following correspondences: S:Person = T:HumanBeing, S:alex 2 T:Male and S:Child v : T: Employee.
Using the AlignmentAPI syntax, we can write this alignment as alignment A : S to T = Person = HumanBeing, alex 2 Male,
Child < : Employee
The assumption about the domains of S and T, which determines which of the three semantics is used, is left to be added in the specication of A.
In all three cases, the semantics of the alignment is the class of models of the colimit of the diagram of the alignment, which can be specied in DOL by writing ontology C = combine A . 4.3
Simple Semantics
In this simplest case, we simply turn the correspondences into OWL sentences to
generate the bridge ontology. Moreover, for each entity occuring in an alignment
we want to use both its axiomatisation in the original ontology as well as the bridge
axioms introduced by the alignment. For this reason, we keep track of the dependency
between the symbols of the bridge ontology and the ontology they have origin from
by adding a common source in the diagram for these two occurences. This is a
wellknown construction, see [
Denition 5. Let A be an alignment (using the notations of Sec. 4.2). The diagram
of the alignment is of the following shape (a W-alignment in the sense of [
O2
B
Bridge O1'
O2' Its constituents are obtained as follows. The ontologies O10 and O20 collect, respectively, all the symbols s1 and s2 that appear in a correspondence s1REL s2 in A, and have no sentences. The morphisms i from Oi0 to Oi, where i = 1; 2, are inclusions. The ontology B is constructed by turning the correspondences of the alignment into OWL axioms. The morphisms 1 and 2 map the symbols occurring in correspondences to their counterpart in B. The alignment is ill-formed when it contains an equivalence between symbols of dierent kinds, or if B fails to be a well-formed ontology. Example 2. We start by adding the assumption that we have a shared domain for the ontologies in the alignment of Ex. 1: alignment A : S to T = : : :
assuming SingleDomain where S0 consists of the concepts Person and Child and the individual alex and T 0 consists of the concepts HumanBeing, Employee and Male, 1 and 2 are inclusions and 1 and 2 map, respectively, Person and HumanBeing to Person_HumanBeing and all other concepts and/or individuals identically.
The bridge ontology B is: ontology B = Class: Person_HumanBeing
Class: Employee Class: Male Class: Child SubClassOf: : Employee
Individual: alex Types: Male The colimit ontology of the diagram of A is: ontology C = Class: Person_HumanBeing
Class: Employee Class: Male SubClassOf: Person_HumanBeing Class: Child SubClassOf: : Employee
Individual: alex Types: Male ; Person_HumanBeing 4.4
Integrated Semantics
Capturing integrated semantics in DOL using families of models compatible with a
diagram is more dicult, as compatibility with the diagram implies uniqueness of
the domain. To remedy this, we use relativisation of an ontology where the universal
concept becomes a new concept and thus can be interpreted as a subset of the
relativised domain. Relativisations have previously been used in dening Common Logic
modules [
Denition 6. Let O be an OWL ontology. We dene the relativisation of O, denoted O~, as follows. The concepts of O~ are the concepts of O together with a new concept, denoted >O. The roles and individuals of O~ are the same as in O. O~ contains axioms stating that each concept C of O is subsumed by >O, each individual i of O is an instance of >O, each role r has its domain and range, if present, intersected with >O, otherwise they are >O. and the axioms of O where the following replacement of concepts is made: each occurence of > is replaced by >O, and each concept :C is replaced by >O u :C each concept 8R:C is replaced by >O u 8R:C.
Example 3. We add the assumption that we have a global domain where the domains of the ontologies in our alignment are included: alignment A : S to T = : : :
assuming GlobalDomain
~ S _ ~ ? B _ 1
1 S0 2
2 ~ ? T where S0 consists of the concepts T hingS , Person and Child and the individual alex and T 0 consists of the concepts T hingT , HumanBeing, Employee and Male, 1 and 2 are inclusions and 1 and 2 map Person and respectively HumanBeing to Person_HumanBeing and all other concepts and/or individuals identically.
The relativisations S~ and T~ of the ontologies S and T are ontology S~ = Class: T hingS
Class: Person SubClassOf: T hingS Individual: alex Types: Person, T hingS
Class: Child SubClassOf: T hingS ontology T~ = Class: T hingT
Class: HumanBeing SubClassOf: T hingT Class: Male SubClassOf: HumanBeing, T hingT
Class: Employee SubClassOf: T hingT
The relativised bridge ontology of an alignment is built by relativising the axioms that result from translating the correspondences of A to OWL sentences. Since we made the assumption that equalising functions are all inclusions, there is no need to introduce explicit symbols for them in the bridge ontology. In our case, the bridge ontology of A is ontology B~ = Class: T hingS Class: T hingT
Class: Person_HumanBeing SubClassOf: T hingS, T hingT Class: Male Class: Employee Class: Child SubClassOf: T hingT and : Employee
Individual: alex Types: Male
The colimit ontology of the relativised diagram of the alignment in Ex. 1 is: ontology C = Class: ThingS
Class: ThingT Class: Person_HumanBeing SubClassOf: ThingS, ThingC Class: Male SubClassOf: Person_HumanBeing Class: Employee SubClassOf: ThingT Class: Child SubClassOf: ThingS Class: Child SubClassOf: ThingT and : Employee
Individual: alex Types: Male, Person_HumanBeing 4.5
Contextualised Semantics Here we need to introduce explicitly the relations between the domains in the language of the bridge ontology. The diagram of the alignment has thus the same shape as in Def. 5, but now the bridge ontology is computed dierently and, as in the previous section, the ontologies are relativised. We denote the bridge ontology by B and dene it to modify B as follows: rji is added to B as a role with domain >T and range >S the correspondences are translated to axioms involving these roles:
Ci = Cj becomes Ci 9rji Cj ai = aj becomes ai rji aj ai 2 Cj becomes ai 2 9rji Cj Ci < Cj becomes Ci v 9rji Cj
Ci%Cj becomes Ci u 9rji Cj = ; the properties of the rji are added as axioms in B.
Here we assume that the alignment Aij contains no correspondence (ri; rj ; R), where ri and rj are roles. Having such correspondences leads to sentences that cannot be expressed in OWL.
Example 4. We add the assumption that we have dierent domains for the ontologies, which are related by domain relations: alignment A : S to T = : : :
assuming ContextualisedDomain The diagram of A is then ~ S _
? B _ 1
1 S0 2
2 ~ ? T where the constituents of the diagram, except B, are as dened in Ex. 3. The bridge ontology of A now becomes: ontology B = Class: ThingS
Class: ThingT ObjectPropery: rT S Domain: ThingT Range: ThingS Class: Person EquivalentTo: rT S some HumanBeing Class: Employee Class: Male Class: Child SubClassOf: rT S some : Employee
Individual: alex Types: rT S some Male The colimit ontology of this diagram is: ontology C = Class: ThingS
Class: ThingT ObjectPropery: rT S Domain: ThingT Range: ThingS Class: Person EquivalentTo: rT S some HumanBeing Class: Male SubClassOf: Person_HumanBeing Class: Employee Class: Child SubClassOf: rT S some : Employee
Individual: alex Types: rT S some Male, Person 4.6
The three semantics in DOL In this section let S = ((Oi)i2I ; (Aij )i;j2I ) be a network of OWL ontologies. We denote C(S) the colimit ontology of the diagram associated to S, regardless if the assumption about the alignments in S is that they use single, integrated or contextualised semantics. The model class of C(S) is denoted JC(S)K.
Theorem 1. 1. If the alignments of S use SingleDomain and the diagram of S is connected, then JC(S)K is in bijection with M odsim(S). 2. If the alignments of S use GlobalDomain , then JC(S)K is in bijection with the class M odint(S) of integrated models ((mi); ( i)) of S where i are inclusions. 3. IwfitthheMaolidgcnomn(eSn)ts. of S use ContextualisedDomain , then JC(S)K is in bijection
DOL is supported by Ontohub ( https://ontohub.org ), a Web-based repository
engine for managing distributed heterogenous ontologies. The back-end of Ontohub
is the Heterogeneous Tool Set HETS [
Our theoretical contributions to the foundations of ontology alignment and combination have a potentially large impact on future alignment practices and reasoning. Regardless of the semantic paradigm employed, ‘reasoning’ with alignments involves at least three levels: (1) the nding/discovery of alignments (often based heavily on statistical methods), (2) the construction of the aligned ontology (the ‘colimit’), and (3) reasoning over the aligned result, respectively debugging and repair, closing the loop to (1). Our contributions in this paper address levels (2) and (3).
Regarding (2), platforms such as Bioportal (with hundred thousands of
mappings) illustrate that mappings between ontologies, ontology modules, and the
concepts and denitions living in them, are of great importance to support re-use. The
importance of alignment has also been well demonstrated for foundational
ontologies in the repository ROMULUS [
Regarding (3), alignment tools such as LogMap [
The approach presented here provides an integration of the major paradigms of
ontology alignment in one coherent framework. This includes standard alignment
relations, DDLs, PD-L, IDDL, and E -connections [
Future work includes the combination of dierent alignment paradigms within
one network (as principally enabled by our unifying framework) and an integration
of techniques for the revision of NeOs [
Acknowledgements We gratefully acknowledge the nancial support of the Future and Emerging Technologies (FET) programme within the Seventh Framework Programme for Research of the European Commission, under FET-Open Grant number: 611553, project COINVENT.