Glossary

² ontology: In philosophy, ontology is a branch of metaphysics concerned with the nature and relations of being. In knowledge representation, ontology is the study of the categories of things in some domain of interest to a community of agents. The types in the ontology represent the entities and relations of the domain. Ontologies are distinguished along a spectrum of formality. An informal ontology contains a list of types that are either undefined or defined only by statements in a natural language. A formal ontology or taxonomy is specified by a set of types with constraints for subtyping, covering and partition. An axiomatized ontology is a formal ontology that uses first order logic for a richer expression of the constraints between the entity and relation types.

² category theory: Category theory is a general mathematical theory of structures and systems of structures. It explicates how structures of different kinds are related to one another. It also describes the universal objects of a family of structures of a given kind. Category theory is regarded by many to be an alternative to set theory as a foundation for mathematics. Category theory provides a unifying and economic mathematical modeling language. Category theory extracts and generalizes elementary and essential notions and constructions from many mathematical disciplines. A category is a mathematical context consisting of a class of objects and a class of morphisms (arrows) between those objects, which satisfy some intuitively clear constraints. Examples of categories include Set, the category of sets and functions, Grp, the category of groups and group homomorphisms, and Top, the category of topologies and continuous maps. A functor is a passage between mathematical contexts, which satisfies some elementary constraints concerning its behavior on the objects and morphisms of the source category. Some of the other basic notions of category theory include natural transformations, adjoint functors and (co)limits.

² classification: A classification is essentially just a binary relation. It consists of a collection of instances, tokens or formal objects, a collection of types or formal attributes and an incidence relation between instances and types. This is the central notion of the theory of Information Flow (see the references for Barwise and Seligman). Concept lattices and classifications are equivalent notions: any classification generates a concept lattice, and any concept lattice has an underlying classification. For any first order language L, the associated truth classification cls(L) has the models (model-theoretic structures) of L as instances, the expressions (= formulas) of L as types, and satisfaction between models and expressions as incidence.

² lattice of theories: A partial order consists of an underlying collection of elements plus an binary order relation, which is transitive, reflexive and antisymmetric. A complete lattice is a partial order, which has meets and joins for arbitrary subcollections of elements. A concept lattice is a complete lattice whose elements are called formal concepts. A concept lattice possesses two collections of generators or atoms. At one pole these is a set of join atoms (called formal objects) in the sense that any formal concept is the (possibly infinite) join of a subset of join atoms. At the other pole these is a set of meet atoms (called formal attributes) in the sense that any formal concept is the (possibly infinite) meet of a subset of meet atoms. For any first order language L, the associated truth concept lattice lat(L) of theories over L is the concept lattice of the truth classification of L. Its formal concepts correspond to the closed theories of L, whose order is entailment (or equivalently reverse subset order on theories – the smaller the theory the more general it is and the higher up it is in the truth concept lattice).

² semantic integration: Two agents are semantically integrated when they can successfully communicate with each other (Uschold and Gruninger). The agents involved in semantic integration can only access the meaning of the other agent’s terms via the axioms of that agent’s ontology. The IFF approach (see the references for Kent) for the semantic integration of ontologies is the two-step process of alignment and unification. Ontological alignment consists of the sharing of common terminology and semantics through a mediating or reference ontology. Ontological unification, concentrated in a virtual ontology of community connections, is fusion of the alignment diagram of participant community ontologies – the quotient of the sum of the participant portals modulo the ontological alignment structure.