Armen Inants, Qualitative calculi with heterogeneous universes, Thèse d'informatique, Université de Grenoble, Grenoble (FR), April 2016

Qualitative representation and reasoning operate with non-numerical relations holding between objects of some universe. The general formalisms developed in this field are based on various kinds of algebras of relations, such as Tarskian relation algebras. All these formalisms, which are called qualitative calculi, share an implicit assumption that the universe is homogeneous, i.e., consists of objectsof the same kind. However, objects of different kinds may also entertain relations. The state of the art of qualitative reasoning does not offer a general combination operation of qualitative calculi for different kinds of objects into a single calculus. Many applications discriminate between different kinds of objects. For example, some spatial models discriminate between regions, lines and points, and different relations are used for each kind of objects. In ontology matching, qualitative calculi were shown useful for expressing alignments between only one kind of entities, such as concepts or individuals. However, relations between individuals and concepts, which impose additional constraints, are not exploited. This dissertation introduces modularity in qualitative calculi and provides a methodology for modeling qualitative calculi with heterogeneous universes. Our central contribution is a framework based on a special class of partition schemes which we call modular. For a qualitative calculus generated by a modular partition scheme, we 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 sub-calculus restricted to a given sort, which is obtained through an operation called relativization to a sort. Of a greater practical interest is the opposite operation, which allows for combining several qualitative calculi into a single calculus. We define 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 spatio-temporal calculi.

Qualitative calculus, Schröder category, Relation algebra, Ontology alignment