Conservativity in Structured and Heterogeneous Ontologies

Two notions are becoming increasingly important in ontological engineering: modularity and heterogeneity. Whilst heterogeneity allows ontologies to be specified in different logical formalisms and thus calls for sophisticated integration and combination techniques, various notions of modularity have been envisaged in the literature in order to support maintenance and reuse of (parts of) ontologies. Can these demands be brought to a common basis? We propose to use the language of category theory, in particular diagrams and their colimits, for answering this question. We outline a general approach for representing (heterogeneous) combinations of logical theories, or ontologies, through interfaces of various kinds, based on diagrams and the theory of institutions. In particular, we cover theory interpretations, (definitional) language extensions, symbol identifications, and conservative extensions. We study the problem of inheriting conservativity between sub-theories in a diagram to its colimit ontology. Finally, we apply this to the problem of localisation of reasoning in ‘modular ontology languages’ such as DDLs or E -connections.