## Section: Scientific Foundations

### Ontology alignment

When different representations are used, it is necessary to identify their correspondences. This task is called ontology matching and its result is an alignment. It can be described as follows: given two ontologies, each describing a set of discrete entities (which can be classes, properties, rules, predicates, etc.), find the relationships, e.g., equivalence or subsumption, if any, that hold between these entities.

An alignment between two ontologies o and is a set of correspondences in which

e and are the entities between which a relation is asserted by the correspondence, e.g., formulas, terms, classes, individuals;

r is the relation asserted to hold between e and . This relation can be any relation applying to these entities, e.g., equivalence, subsumption.

n is a degree of confidence in this particular correspondence (which will be omitted here).

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 ( ) which maps the domain of each of the models to a common domain on which the relations are interpreted. Such a triple is a model of the aligned ontologies o and if and only if, for each correspondence of the alignment A , mo , and .

This definition is extended to a system of networked ontologies which is a set of ontologies and associated alignments. A model of such a system of networked ontologies 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 will select the local models which are compatible with all alignments.

So, given a system of networked ontologies, it is possible to interpret it. However, given a set of ontologies, it is necessary to find the alignments between them and the semantics does not tell which ones they are. Ontology matching aims at finding these alignments. A variety of methods is used for this task. They perform pair-wise comparisons of entities from each of the ontologies and select the most similar pairs. Most matching algorithms provide correspondences between named entities, more rarely between compound terms. The relationships are generally 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.