ogies, composing ontologies via fusions, noting dependen- cies between ontologies, declaring the use of other ontolo- 3 Semantic Integration in the IFF gies , etc. The IFF takes a building blocks approach to- wards the development of object-level ontological struc- ture. This is a rather elaborate categorical approach, which Robert E. Kent uses insights and ideas from the theory of distributed logic Ontologos known as information flow (Barwise and Seligman, 1997) [email protected] and the theory of formal concept analysis (Ganter and Wille, 1999). The IFF represents metalogic, and as such operates at the structural level of ontologies. In the IFF, Abstract there is a precise boundary between the metalevel and the object level. The IEEE P1600.1 Standard Upper Ontology (SUO) The modular architecture of the IFF consists of metale- project aims to specify an upper ontology that will provide vels, namespaces and meta-ontologies. There are three me- a structure and a set of general concepts upon which do- talevels: top, upper and lower. This partition, which cor- main ontologies could be constructed. The Information responds to the set-theoretic distinction between small Flow Framework (IFF), which is being developed under (sets), large (classes) and generic collections, is permanent. the auspices of the SUO Working Group, represents the Each metalevel services the level below by providing a structural aspect of the SUO. The IFF is based on category language that is used to declare and axiomatize that level. theory. Semantic integration of object-level ontologies in * The top metalevel services the upper metalevel, the upper the IFF is represented with its fusion construction . The metalevel services the lower metalevel, and the lower meta- IFF maintains ontologies using powerful composition pri- level services the object-level. Within each metalevel, the mitives, which includes the fusion construction. terminology is partitioned into namespaces†. The number 1. The Information Flow Framework of namespaces and the content may vary over time: new namespaces may be created or old namespaces may be de- The IEEE P1600.1 Standard Upper Ontology (SUO)1 precated, and new terminology and axiomatization within project aims to specify an upper ontology that will provide any particular namespace may change. In addition, within a structure and a set of general concepts upon which ob- each level, various namespaces are collected together into ject-level domain ontologies could be constructed. These meaningful composites called meta-ontologies. At any par- object-level domain ontologies will utilize the SUO for ticular metalevel, these meta-ontologies cover all the na- “applications such as data interoperability, information mespaces at that level, but they may overlap. The number search and retrieval, automated inferencing, and natural of meta-ontologies and the content of any meta-ontology language processing”. A central purpose of the SUO may vary over time: new meta-ontologies may be created project is interoperability. or old meta-ontologies may be deprecated, and new na- The Information Flow Framework (IFF)2 is being de- mespaces within any particular meta-ontology may change veloped to represent the structural aspect of the SUO. It (new versions). aims to provide semantic interoperability among various The top IFF metalevel provides an interface between object-level ontologies. The IFF supports this interopera- the simple IFF-KIF language and the other IFF terminolo- bility by its architecture and its use of a particular branch gy. By analogy, the simple IFF-KIF language is like a ma- of mathematics known as category theory (Mac Lane, chine language and the top IFF metalevel is like an assem- 1971). A major reason that the IFF uses the architecture bly language. There is only one namespace and one meta- and formalisms that it does is to support modular ontology ontology in the top metalevel: the Top Core (meta) Ontol- development. Modularity facilitates the development, test- ogy. This meta-ontology represents generic collections. In ing, maintenance, and use of ontologies. The categorical a sense, it bootstraps the rest of the IFF into existence. The approach of the IFF provides a principled framework for single namespace, the meta-ontology and the top metalevel modular design via a structural metatheory of object-level can be identified with each other. The upper and lower IFF ontologies. Such a metatheory is a method for representing metalevels represent the structural aspect of the SUO. By the structural relationships between ontologies. analogy, the structural aspect of the SUO is like a high lev- The IFF provides mechanisms for the principled foun- dation of a metalevel ontological framework – a framework † for sharing ontologies, manipulating ontologies as objects, The IFF terminology is disambiguated via the disjoint union of local namespace terminology. A fully qualified term in the IFF is of the form relating ontologies through morphisms, partitioning ontol- “$”, where the namespace prefix label “” is a “.” separated sequence of alphabetic strings that uniquely represents an IFF namespace, and the local unqualified term “” is a unique lowercase alphanumeric-dash * Throughout this paper, we use the intuitive terminology of mathematical string within that namespace. For example: the term context, passage/construction, pair of invertible passages and fusion for “th.col.psh$coequalizer-diagram” the mathematical concepts of category, functor, adjunction and colimit, represents the coequalizer diagram underlying a pushout diagram of respectively. theories within the theory pushout namespace in the lower IFF metalevel. el programming language such as Lisp, Java, ML, etc. 2. Basic Concepts of the IFF-OO There are three permanent meta-ontologies in the upper metalevel: the Upper Core (meta) Ontology represents the The metalevel axiomatic framework for object-level on- large collections called classes; the Category Theory (me- tologies represented in first order logic and model theory is ta) Ontology represents category theory; and the Upper concentrated in the lower metalevel IFF Ontology (meta) Classification (meta) Ontology represents information flow Ontology (IFF-OO). The IFF-OO is a generic framework and formal concept analysis. There will eventually be many for the representation and manipulation of object-level on- meta-ontologies situated in the lower IFF metalevel‡. Cur- tologies. The architecture of the IFF-OO (Figure 1) con- sists of four central mathematical contexts* interconnected rently there are only four: the Lower Core (meta) Ontology * represents the small collections called sets; the Lower by five pairs of invertible passages . Each of the four con- Classification (meta) Ontology is a small and more specia- texts represents a basic concept axiomatized in the IFF- OO. These four concepts are language, theory, model and lized version of its upper counterpart; the Algebraic Theory 6 (meta) Ontology represents equational logic; and the On- logic. The context of first order logic languages sits at the tology (meta) Ontology represents first order logic and base of the IFF-OO – everything depends upon it. The model theory. All versions of these meta-ontologies are three other contexts – models, theories and logics – are listed as links in the SUO IFF site map4. situated above the language context. Models provide the interpretive semantics for object-level ontologies, theories Logic provide the formal or axiomatic semantics, and logics pro- vide the combined semantics. Any theory is based on a language, and the context of theories is connected to the Model Theory context of languages by the base passage. An object-level ontology is populated when it has instance data. Unpopu- lated object-level ontologies are represented by IFF theo- ries, whereas populated object-level ontologies are Language represented by IFF logics. This paper deals only with for- Figure 1: IFF-OO Architecture mal, axiomatic semantics for object-level ontologies. Inter- pretive semantics will be combined with this in future The IFF, which is situated at the metalevel, represents work. form. The ontologies, which are situated at the object level, The concept of an IFF language is many-sorted – the represent content§. By analogy, the content aspect of the definition follows (Enderton, 1972), generalizing the stan- SUO is like the various software applications, such as word dard notion of a single-sorted language. The IFF terminol- processors, browsers, spreadsheet software, databases, etc. ogy is somewhat different from Enderton – it uses the two The distinction between content and form is basic in the polarities of entities versus relations and instances versus general grammar of natural languages, in logic and in on- types: an IFF entity type corresponds to a sort, an IFF rela- tology. In all of these realms, but especially in logic and tion type corresponds to a predicate, and an IFF function ontology, the IFF offers a coherent principled approach to type corresponds to a function symbol. In this paper, we form. Such form is realized in the structuring, mapping and ignore function types for simplicity – these are adequately integration of ontologies. The IFF offers axiomatization handled in the IFF Algebraic Theory (meta) Ontology. and techniques for the hierarchical structuring of object- Note that an IFF language deals only with type informa- level ontologies via the lattice of theories, the mapping tion. Constants are regarded as nullary function