Semantic Integration in the IFF

The IEEE P1600.1 Standard Upper Ontology (SUO) project aims to specify an upper ontology that will provide a structure and a set of general concepts upon which domain ontologies could be constructed. The Information Flow Framework (IFF), which is being developed under the auspices of the SUO Working Group, represents the structural aspect of the SUO. The IFF is based on category theory. Semantic integration of object-level ontologies in the IFF is represented with its fusion construction. The IFF maintains ontologies using powerful composition primitives, which includes the fusion construction. 1. The Information Flow Framework The IEEE P1600.1 Standard Upper Ontology (SUO) project aims to specify an upper ontology that will provide a structure and a set of general concepts upon which object-level domain ontologies could be constructed. These object-level domain ontologies will utilize the SUO for “applications such as data interoperability, information search and retrieval, automated inferencing, and natural language processing”. A central purpose of the SUO project is interoperability. The Information Flow Framework (IFF) is being developed to represent the structural aspect of the SUO. It aims to provide semantic interoperability among various object-level ontologies. The IFF supports this interoperability by its architecture and its use of a particular branch of mathematics known as category theory (Mac Lane, 1971). A major reason that the IFF uses the architecture and formalisms that it does is to support modular ontology development. Modularity facilitates the development, testing, maintenance, and use of ontologies. The categorical approach of the IFF provides a principled framework for modular design via a structural metatheory of object-level ontologies. Such a metatheory is a method for representing the structural relationships between ontologies. The IFF provides mechanisms for the principled foundation of a metalevel ontological framework – a framework for sharing ontologies, manipulating ontologies as objects, relating ontologies through morphisms, partitioning on* Throughout this paper, we use the intuitive terminology of mathematical context, passage/construction, pair of invertible passages and fusion for the mathematical concepts of category, functor, adjunction and colimit, respectively. tologies, composing ontologies via fusions, noting dependencies between ontologies, declaring the use of other ontologies, etc. The IFF takes a building blocks approach towards the development of object-level ontological structure. This is a rather elaborate categorical approach, which uses insights and ideas from the theory of distributed logic known as information flow (Barwise and Seligman, 1997) 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, there is a precise boundary between the metalevel and the object level. The modular architecture of the IFF consists of metalevels, namespaces and meta-ontologies. There are three metalevels: top, upper and lower. This partition, which corresponds to the set-theoretic distinction between small (sets), large (classes) and generic collections, is permanent. Each metalevel services the level below by providing a language that is used to declare and axiomatize that level. The top metalevel services the upper metalevel, the upper metalevel services the lower metalevel, and the lower metalevel services the object-level. Within each metalevel, the terminology is partitioned into namespaces. The number of namespaces and the content may vary over time: new namespaces may be created or old namespaces may be deprecated, and new terminology and axiomatization within any particular namespace may change. In addition, within each level, various namespaces are collected together into meaningful composites called metaontologies. At any particular metalevel, these metaontologies cover all the namespaces at that level, but they may overlap. The number of meta-ontologies and the content of any meta-ontology may vary over time: new metaontologies may be created or old meta-ontologies may be deprecated, and new namespaces within any particular meta-ontology may change (new versions). The top IFF metalevel provides an interface between the simple IFF-KIF language and the other IFF terminology. By analogy, the simple IFF-KIF language is like a machine language and the top IFF metalevel is like an assembly language. There is only one namespace and one meta-ontology in the top metalevel: the Top Core (meta) Ontology. This meta-ontology represents generic collections. In a sense, it bootstraps the rest of the IFF into existence. The single namespace, the meta-ontology and the top metalevel can be identified with each other. The upper and lower IFF metalevels represent the structural aspect of the SUO. By analogy, the structural aspect of the SUO is † The IFF terminology is disambiguated via the disjoint union of local namespace terminology. A fully qualified term in the IFF is of the form “ν$τ”, where the namespace prefix label “ν” is a “.” separated sequence of alphabetic strings that uniquely represents an IFF namespace, and the local unqualified term “τ” is a unique lowercase alphanumeric-dash string within that namespace. For example: the term “th.col.psh$coequalizer-diagram” represents the coequalizer diagram underlying a pushout diagram of theories within the theory pushout namespace in the lower IFF metalevel. like a high level programming language such as Lisp, Java, ML, etc. There are three permanent meta-ontologies in the upper metalevel: the Upper Core (meta) Ontology represents the large collections called classes; the Category Theory (meta) Ontology represents category theory; and the Upper Classification (meta) Ontology represents information flow and formal concept analysis. There will eventually be many meta-ontologies situated in the lower IFF metalevel. Currently there are only four: the Lower Core (meta) Ontology represents the small collections called sets; the Lower Classification (meta) Ontology is a small and more specialized version of its upper counterpart; the Algebraic Theory (meta) Ontology represents equational logic; and the Ontology (meta) Ontology represents first order logic and model theory. All versions of these metaontologies are listed as links in the SUO IFF site map. The IFF, which is situated at the metalevel, represents form. The ontologies, which are situated at the object level, represent content. By analogy, the content aspect of the SUO is like the various software applications, such as word processors, browsers, spreadsheet software, databases, etc. The distinction between content and form is basic in the general grammar of natural languages, in logic and in ontology. In all of these realms, but especially in logic and ontology, the IFF offers a coherent principled approach to form. Such form is realized in the structuring, mapping and integration of ontologies. The IFF offers axiomatization and techniques for the hierarchical structuring of objectlevel ontologies via the lattice of theories, the mapping between ontologies via syntax directed translation, and the semantic integration of ontologies via mediating or reference ontologies. To paraphrase John Sowa, developing the tools and methodologies for extending, refining, and sharing object-level ontologies is more important than developing the content for those ontologies. ‡ A module in the IFF lower metalevel should represent a well-researched area. In addition to the IFF-OO, which represents first order logic and model theory, other non-core lower metalevel modules are also being considered: a module for the “soft computation” of both rough sets and fuzzy logic; a module for theories of semiotics; a module for gametheoretic semantics; etc. § Many current object-level ontologies contain generic axiomatizations for notions such as binary relations, partial orders, etc. In the IFF, these are not needed, since such axiomatizations are included in the Lower Core (meta) Ontology, etc. When compliant with the IFF, object-level ontologies can concentrate on their core axiomatics. 2. Basic Concepts of the IFF-OO The metalevel axiomatic framework for object-level ontologies represented in first order logic and model theory is concentrated in the lower metalevel IFF Ontology (meta) Ontology (IFF-OO). The IFF-OO is a generic framework for the representation and manipulation of object-level ontologies. The architecture of the IFF-OO (Figure 1) consists of four central mathematical contexts interconnected by five pairs of invertible passages. Each of the four contexts represents a basic concept axiomatized in the IFFOO. These four concepts are language, theory, model and logic. The context of first order logic languages sits at the base of the IFF-OO – everything depends upon it. The three other contexts – models, theories and logics – are situated above the language context. Models provide the interpretive semantics for object-level ontologies, theories provide the formal or axiomatic semantics, and logics provide the combined semantics. Any theory is based on a language, and the context of theories is connected to the context of languages by the base passage. An object-level ontology is populated when it has instance data. Unpopulated object-level ontologies are represented by IFF theories, whereas populated object-level ontologies are represented by IFF logics. This paper deals only with formal, axiomatic semantics for object-level ontologies. Interpretive semantics will be combined with this in future work. The concept of an IFF language is many-sorted – the definition follows (Enderton, 1972), generalizing the standard notion of a single-sorted language. The IFF terminology is somewhat different from Enderton – it uses the two polarities of entities versus relations and instances versus types: an IFF entity type corresponds to a sort, an IFF relation type corresponds to a predicate, and an IFF function type corresponds to a function symbol. In this paper, we ignore function types for simplicity – these are adequately handled in the IFF Algebraic Theory (meta) Ontology. Note that an IFF language deals only with type information. Constants are regarded as nullary function types. Languages are comparable via language morphisms, and theories are comparable via theory morphisms. Any language L determines a lattice of theories fiber(L), a base passage fiber. Any language morphism f : L1 → L2 determines a function expr(f) : expr(L1) → expr(L2) by induction, and from this a lattice morphism of theories fiber(f) = 〈inv(f), dir 〉 : fiber(L2) → fiber(L1), ** The lattice of theories fiber(L) for a language L is the complete lattice of all theories with base language L using entailment order between theories: T2 ≤ T1 means that T2 is more specialized than T1 in the sense that T1 is contained in the closure of T2; or equivalently, that any theorem of T1 is entailed by the axioms of T2. †† A fiber of a passage P : C → B for fixed object b ∈ B is analogous to the inverse image of b along P, thus forming the sub-context fiberP(b) ⊆ C of all C-objects that map to b and all C-morphisms that map to the identity at b. Language Logic