Semantic interoperability can be grounded in ontology reconciliation. Integrating heterogeneous resources of the web requires finding agreement between the ontologies involved: this is called ontology matching. We express these agreements (or disagreement) through an alignment: a set of relationships, e.g., equivalence or subsumption, between ontology entities.
We have contributed to the definition of alignments and their semantics and we are still working towards deeper expressivity and flexibility of ontology alignments. We also develop supporting software.
Goal: Our work on ontology alignments aims at providing a robust and wellfounded alignment infrastructure for the semantic web. 
In the context of the Knowledge web network of excellence, we are developing activities for structuring the research on ontology alignment at the European level. These activities have helped shaping the European research community on the topics of ontology alignment and matching.
This has led to numerous activities and results, including:
A large part of the work described in this section started with the activities of the Heterogeneity work package (2.2) of the Knowledge web network of excellence.
We provided a general model for ontology alignment (the result of the mapping task [Euzenat 2003i, Bouquet 2004a]).
An alignment between two ontologies o and o' is a set of correspondences ⟨e, r, e'⟩ in which
Most matching algorithms provide correspondences between named entities, more rarely between compound terms. The relationships are generally the equivalence between these entities. Some systems are able to provide subsumption relations as well as other relations in the support language (like incompatibility or instanciation). Confidence measures are usually given a value between 0 and 1 and are used for expressing preferences between two correspondences.
This format designed by Exmo is used in the Alignment API [Euzenat 2004f, David 2011a] which provides a common ground for implementing and comparing alignment algorithms and is used in benchmarking ontology matching. It is widely adopted.
In order to offer a declarative format which is both expressive and independent from concrete ontology languages we developed the Expressive Alignment Language EDOAL [Euzenat 2005h, Euzenat 2007e]. The high expressivity of the language allows the expression of complex alignments even if the ontology languages are not themselves expressive [Kovalenko 2016a]. The language independence guarantees that we can define expressive alignments between any languages and provides a declarative definition of the alignments which will be usable in various manners (ontology merging, data translation, etc.).
We defined for this language an abstract syntax (used for describing the semantics), an exchange syntax (in RDF/XML) and a more readable surface syntax. We provided a model theoretic semantics for the language which rely upon the semantics of aligned ontologies while remaining independent from their details. We also provided support for this language by combining and extending the API for the Alignment Mapping Language developed at Innsbruck and our own Alignment API.
This work has started in cooperation with Innsbruck Universität (François Scharffe) in the context of the Knowledge web project and has been pursued since the within various projects.
When dealing with alignments, it is important, both for generating them and for using them to know their interpretation. This is even more important when users are dealing with a whole network of ontologies related by alignments. Such a structure, composed of a set of ontologies, interconnected with ontology alignments, is called a distributed system or a network of ontologies. A legitimate question is: given the semantics of these ontologies, what are the consequences of a network of ontologies.
So far, alignments have been given semantics only related to a precise logical framework (e.g., firstorder logic). In the continuation of our work on categorical definition of alignment (see below), we aimed at an alignment semantics independent from the ontology semantics.
For that purpose, we have defined a parameterised family of modeltheoretic semantics for alignments and networks of ontologies. This semantics is parameterised by the interpretation of the set of relation, it uses and rely transparently on the semantics of the ontologies (which is only supposed to define the consequence relation ⊨). This means that the models, i.e., satisfying interpretations, of an alignment or a network of ontologies are defined in function of the models of the local ontologies, even when different ontologies are written in different languages.
Given the semantics of the two ontologies provided by their consequence relation, we define an interpretation of the two ontologies as a triple made of an interpretation for each ontology and an equalising function (γ) which maps the domain of each of the models to a common domain on which the relations are interpreted. Such a triple ⟨m, m', γ⟩ is a model of the aligned ontologies o and o' if and only if, for each correspondence ⟨e, e', r ⟩ ∈ A of the alignment, then m⊨o, m'⊨o' and ⟨γ(m(e)), γ(m'(e'))⟩∈r^{γ}.
This definition is extended to networks of ontologies, for which a model is a tuple of local models and an equalising function, such that each alignment is valid for the models and the equalising function involved in the tuple. In such a system, alignments play the role of model filters which select the local models which are compatible with all alignments.
We have investigated three different variations of this semantics, offering different levels of integration and supporting different paradigms. The three types of semantics use different techniques to obtain commensurate interpretations of formulas: either by constraining interpretations on a common domain, mapping the domains to a common domain or relating the entities of each pairs of domains. We studied the semantic properties of ontology alignment composition according to these three variants [Zimmermann 2006b]. It appears that only the first two types of distributed semantics are sound with respect to alignment composition, while the last one, which corresponds to the paradigm of Distributed First Order Logics (DFOL), Distributed Description Logics (DDL) and COWL, is not.
In addition, we proposed an abstraction of these semantics and applied it to networks of ontologies. This allowed for defining the notions of closure and consistency for networks of ontologies independently from the precise semantics. We also show that networks of ontologies with specific notions of morphisms define categories of networks of ontologies [Euzenat 2014d].
This work has been used for defining precisely semantic extensions of precision and recall.
It is also used for designing ontology modules. With modular ontologies, knowledge is expressed in interlinked chunks rather than large monolitic ontologies. Ontologies can be assembled from ontology modules like programme modules in software engineering. We have designed a model of modules which combines an interface and an ontology implementation, in which a module can import other modules through alignments with their interface [Bezerra 2008a, Zimmermann 2008c]. This is a very natural approach since alignments can be used to adjust the components in the ontologies. We have provided a semantics for such modules which is a combination of ontology semantics and our own alignment semantics.
The work on modules is carried out in cooperation with Frederico Freitas within the OntoCompo project and in the framework of the NeOn project.
In order to deal with uncertainty in relations between ontology entities, we have proposed to use algebra of binary relations instead of the generally used ad hoc relations. We have shown that algebras of binary relations are a natural way to represent disjunction of relations, to agregate matcher results, and to compute composition and granularity change [Euzenat 2008e].
All qualitative calculi share an implicit assumption that the universe is homogeneous, i.e., consists of objects of the same kind. For instance, the previously considered algebra of relations contains taxonomical relations between classes only. However, objects of different kinds, e.g., a concept and an individual, may also entertain relations. The problem is important because ontology matching deals with various kinds of ontological entities: concepts, individuals, properties.
We have defined the notion of heterogeneous qualitative calculus based on an algebraic construct called Schröder category. A Schröder category is to binary relations over heterogeneous universes what a relation algebra is to homogeneous ones. We have established the connection between homogeneous and heterogeneous qualitative calculi by defining two mutually inverse transition operators. We have designed an algorithm for combining two homogeneous calculi with different universes into a single calculus. This has been applied to alignment relations [Inants 2015a] combining algebras for relations between concepts and individuals. It is, first, able to deal with empty classes, and, second, incorporates all qualitative taxonomical relations that occur between individuals and concepts, including the relations ``is a'' and ``is not''. We have proved that this algebra is coherent with respect to the simple semantics of alignments.
We also introduced modularity in qualitative calculi and provided a methodology for modeling qualitative calculi with heterogeneous universes [Inants 2016a]. It is based on a special class of partition schemes which we call modular. For a qualitative calculus generated by a modular partition scheme, we can define a structure that associates each relation symbol with an abstract domain and codomain from a Boolean lattice of sorts. A module of such a qualitative calculus is a subcalculus restricted to a given sort, which is obtained through an operation called relativisation to a sort. Of a greater practical interest is the opposite operation, which allows for combining several qualitative calculi into a single calculus. We defined an operation called combination modulo glue, which combines two or more qualitative calculi over different universes, provided some glue relations between these universes. The framework is general enough to support most known qualitative spatiotemporal calculi and generalised the results we obtained in [Inants 2015a].
In addition, once we are able to ascribe a semantics to alignments [Zimmermann 2008c], it is possible to carry out approximate reasoning that does not involve ontologies but alignments alone. This can be exploited for evaluating alignments [David 2008b] or for checking consistency or preprocessing a distributed set of alignments through the computation of its compositional, symmetric and union closure. Algebra of relations are then instrumental for computing composition of alignments.
This was part of the PhD work of Armen Inants.
Alignment correspondences are often assigned a weight or confidence factor by matchers. Nonetheless, few semantic accounts have been given so far for such weights. We have proposed a formal semantics for weighted correspondences between different ontologies. It is based on a classificative interpretation of correspondences: if o and o' are two ontologies used to classify a common set X, then alignments between o and o' are interpreted as encoding how elements of X classified in the concepts of o are reclassified in the concepts of o', and weights are interpreted as measures of how precise and complete reclassifications are. This semantics is justifiable for extensional matchers. We have proven that it is a generalisation of the semantics of absolute correspondences, and we have provided properties that relate correspondence entailment with description logic constructors [Atencia 2012c].
However, this semantics introduced a discontinuity between weighted and nonweighted interpretations. Moreover, it does not provide a calculus for reasoning with weighted ontology alignments. We introduced a calculus for such alignments [Inants 2016b] provided by an infinite relationtype algebra, the elements of which are weighted taxonomic relations. In addition, it approximates the nonweighted case in a continuous manner.
This work has been made in cooperation with Alexander Borgida (Rutgers University) and Chiara Ghidini and Luciano Serafini (Fondazione Bruno Kessler).
We have designed a theoretical framework to define ontology alignments and operations that be applied to them. This framework, based on category theory, considers ontologies and ontology alignments as first order objects independently of the representation language, as opposed to entity based definitions. Indeed, if ontologies are objects in a category, and morphisms correspond to ontology refinements, then an ontology alignment is a categorical relation between two ontologies, i.e. a pair of morphisms with the same domain. So, if f: A→o and f': A→o' are both ontology refinements, then ⟨A, f, f'⟩ is an alignment of o and o', A approximates ontologies o and o' and it describes a part of the knowledge that is common to them. We call this structure a Valignment. With such a characterization, an algebra has been designed which describes ontology merging, alignment comparison, composition, union and intersection using well known categorical constructions.
Concrete categories of ontologies exist in the literature, but fail to express complex alignments, e.g. alignments expressing subsumption relation between concepts of two ontologies. To solve this problem, we investigated two approaches: defining more complex categories or improving the structure of our categorical alignments. On the one hand, more complex categories enhance the expressivity of the alignments, but the categories miss some interesting properties such as the existence of the merge for any Valignment and pair of ontologies. On the other hand, simpler categories, such as the category of theories and theory morphisms in institution theory, can describe complex alignments if the alignment structure is more elaborate. In collaboration with Markus Krötzsch and Pascal Hitzler (University of Karlsruhe), we introduced Walignments: a structure having an additional ontology containing bridge axioms relating to o and o' by two Valignments [Hitzler 2005a, Zimmermann 2006a]. This raises the expressivity of alignments while keeping desirable properties of the category. However, the resulting algebra is less natural (composition is not associative).
This was part of the PhD work of Antoine Zimmermann.
Category theory has also been applied to networks of alignments in order to generalise the global algebraic properties of such networks [Euzenat 2014d]. This work is used for defining revision of networks of ontologies.
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