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The IFF Approach to Semantic Integration

The IFF Approach to Semantic Integration. Boeing Mini-Workshop on Semantic Integration.

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The IFF Approach to Semantic Integration

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  1. The IFF Approach to Semantic Integration Boeing Mini-Workshop on Semantic Integration “She acknowledged it to be very fitting, that every little social commonwealth should dictate its own matters of discourse; and hoped, ere long, to become a not unworthy member of the one she was now transplanted into. With the prospect of spending at least two months at —, it was highly incumbent on her to clothe her imagination, her memory, and all her ideas in as much of — as possible.” Persuasion, Chapter 6, (1818). Jane Austen http://suo.ieee.org/IFF/ http://www.ontologos.org/IFF/OntologyOntology/Introduction.htm

  2. Category Theory: the study of structures and structure morphisms; starts with the observation that many properties of mathematical systems can be unified and simplified by a presentation with diagrams of arrows. Information Flow: the logic of distributed systems; a mathematically rigorous, philosophically sound foundation for a science of information. Formal Concept Analysis: advocates methods and instruments of conceptual knowledge processing that support people in their rational thinking, judgments and actions. Information Flow Framework (IFF) Origins and Influences IFF Semantic Integration ~ 7 Nov 2002

  3. Sections • Category Theory (3 slides) • The IFF (3 slides) • The IFF-ONT Contexts (10 slides) • The IFF-ONT (7 slides) • Semantic Integration (6 slides) • Summary & Future Work IFF Semantic Integration ~ 7 Nov 2002

  4. Table of Contents:Category Theory • Category Theory • The Category Manifesto • Examples: Categories IFF Semantic Integration ~ 7 Nov 2002

  5. Category Theory • Started in 1945 with Eilenberg & Mac Lane’s paper entitled "General Theory of Natural Equivalences." • It is a general mathematical theory of structures and systems of structures. • Reveals how structures of different kinds are related to one another (morphisms), as well as the universal components of a family of structures of a given kind (limits/colimits). • It is considered by many as being an alternative to set theory as a foundation for mathematics. IFF Semantic Integration ~ 7 Nov 2002

  6. The Categorical Manifestoby Joseph Goguen (1989) <http://www.cs.ucsd.edu/users/goguen/ps/manif.ps.gz> Mathematical Structures in Computer Science, Volume 1, Number 1, March 1991, pages 49–67. Four “dogmas” for categories, functors, adjunctions and colimits – all concepts of central importance in the structure of IFF meta-ontologies. Dogma (M-W): something held as an established opinion, especially a definite authoritative tenet. The intended meaning is not the pejorative sense of the word. IdA= F ◦ G G ◦ F = IdB IdA F ◦ G G ◦ F  IdB F IdA F ◦ G G ◦ F  IdB A B W FG A B The Categorical Manifesto • Mathematical Context (~ Category) “To each species of mathematical structure, there corresponds a category whose objects have that structure, and whose morphisms preserve it.” • Passage (Construction) between Contexts (~ Functor) “To any natural construction on structures of one species, yielding structures of another species, there corresponds a functor from the category of the first species to the category of the second.” • Generalized Inverse (~ Adjunction) “To any canonical construction from one species of structure to another corresponds an adjunction between the corresponding categories.” • Two special cases: • Reflection: G◦F=IdB “G is rali to F” “B reflective subcategory A” • Coreflection: IdA=F◦G “G is rari to F” “A coreflective subcategory B” • Sums, Quotients and Fusions (~ Colimit) “Given a species of structure, say widgets, then the result of interconnecting a system of widgets to form a super-widget corresponds to taking the colimit of the diagram of widgets in which the morphisms show how they are interconnected.” C IFF Semantic Integration ~ 7 Nov 2002

  7. Examples: Categories • Almost every known example of a mathematical structure with the appropriate structure preserving map yields a category. • Sets with functions between them. • Groups with group homomorphisms. • Topological spaces with continuous maps. • Vector spaces and linear transformations. • Any class itself is a category with only identity morphisms. • Any monoid is a one-object category with elements being morphisms. • Any preordered class is a category with morphisms being pair orderings. • Classifications and infomorphisms (or bonds, or bonding pairs). • Hypergraphs and their morphisms; first order type languages and their morphisms; • Theories and theory morphisms; • Models and model infomorphisms; Logics and logic infomorphisms. • Concept lattices and concept morphisms; • Complete lattices and adjoint pairs (or complete homomorphisms). IFF Semantic Integration ~ 7 Nov 2002

  8. Table of Contents: The Information Flow Framework (IFF) • The IFF Architecture • The Lower Metalevel • The Category Design Principle IFF Semantic Integration ~ 7 Nov 2002

  9. Lower metalevel Declare, define, axiomatize and reason about particular categories, functors, adjunctions, colimits, monads, classifications,concept lattices, etc. Categories include Hypergraph, Language, Theory, Model, Logic, etc. Functors include typ, , init-mod, max-th, log, th, etc. Upper metalevel Declare, define, axiomatize and reason about generic categories, functors, adjunctions, colimits, monads, classifications, concept lattices, etc. The IFF IFF Basic KIF (meta) Ontology top IFF Model Theory (meta) Ontology IFF Classification (meta) Ontology IFF Core (meta) Ontology metalevel upper IFF Category Theory (meta) Ontology lower ֻ object level ֻ ֻ Middle Ontology Middle Ontologyn Middle Ontology Upper Ontology Domain Ontology Domain Ontology Upper Ontology Domain Ontologyp Upper Ontologym The IFF Architecture IFF Semantic Integration ~ 7 Nov 2002

  10. “Philosophy cannot become scientifically healthy without an immense technical vocabulary. We can hardly imagine our great-grandsons turning over the leaves of this dictionary without amusement over the paucity of words with which their grandsires attempted to handle metaphysics and logic. Long before that day, it will have become indispensably requisite, too, that each of these terms should be confined to a single meaning which, however broad, must be free from all vagueness. This will involve a revolution in terminology; for in its present condition a philosophical thought of any precision can seldom be expressed without lengthy explanations.” – Charles Sanders Peirce, Collected Papers 8:169 The IFF Lower Metalevel • The lower metalevel of the IFF makes heavy use of the upper metalevel for both representation and reasoning. • The following modules will be located on the lower metalevel: • IFF Model Theory Ontology • IFF Algebraic Theory Ontology • IFF Ontology Ontology • Nonaligned versions of languages, models and logics • Elaboration of span graphs and span models (akin to RDF triples?) • Other possible modules on the lower metalevel include the following: • Module for categorical model theory • Modules for modal, tense and linear logic • Modules for rough and fuzzy sets • Module for semiotics • etcetera IFF Semantic Integration ~ 7 Nov 2002

  11. The Categorical Design Principle • Principle: A central goal in modeling the lower metalevel is to abide by the following categorical property. • [strictly category-theoretic] all axioms are expressed in terms of category-theoretic notions, such as the composition and identity of functions or the pullback of diagrams of classes and functions. • [no KIF] no axioms use explicit KIF connectives or quantification. • [no basic KIF ontology] no axioms use terms from the basic KIF ontology. • This principle is an ideal that has proven very useful in the design of the IFF-MT, the IFF-AT and the IFF-ONT. All modules that satisfy this property should (i) be easier to design and (ii) provide the basis for simpler proof techniques. • This design principle would seem to extend to all ontologies for true categories (not quasi-categories) – those categories whose object and morphism collections are classes (not generic collections). All ontologies that reside at the lower metalevel will be centered on true categories. IFF Semantic Integration ~ 7 Nov 2002

  12. Table of Contents: The IFF-ONT Contexts • The IFF-ONT Architecture – overview • Mathematical Context: Language • Mathematical Context: Language (continued) • Mathematical Context: Language (continued) • Mathematical Context: Theory • Mathematical Context: Theory (continued) • Mathematical Context: Model • Mathematical Context: Model (continued) • Mathematical Context: Logic • Mathematical Context: Logic (continued) IFF Semantic Integration ~ 7 Nov 2002

  13. Central contexts Language Theory Model Logic Other contexts Language╧ Theory╧ Prologic Central generalized inverses λ = logthTheory╧ Logic “integration coreflection” μ = init-modmax-thTheory╧ Model “semantics adjunction” ω = typ Language╧ Model“free model coreflection” Other generalized inverses υ = prologthTheory╧ Prologic “free prologic coreflection” ρ = restrictinclPrologic  Logic “restriction reflection” π = modlogLogic  Model “theory augmentation reflection” κ = ⊤baseLanguage  Theory “empty theory coreflection” Logic π λ Model Theory╧ μ ω Theory κ╧ Language╧ κ Language The IFF-ONT Architecture – overview IFF Semantic Integration ~ 7 Nov 2002

  14. type language L sets variables: var(L) entity types: ent(L) (~ hypergraph nodes) relation types: rel(L) (~ hyperedges) functions reference: refer(L) = L : var(L)ent(L) arity : arity(L) = L : rel(L)var(L) signature : sign(L) = L : rel(L)sign(L), where L () : L()ent(L) The abbreviated notation (, )  rel(L) means L() =  and L() = . kind of aligned hypergraph, aligned along its reference function (not true for hypergraphs) Example indexes = Integer+ entity types = Person, String, Natno, Real variables = (entity type, index) pairs relation types = name(Person, String), spouse(Person, Person), age(Person, Natno), height(Unit, Person, Real), leq(Natno, Natno) reference = projection from variables to entity types arity, signature contained in above description L var(L) rel(L) var(L) L refer-arity(L) L sign(L) ent(L) Type Language Mathematical Context: Language IFF Semantic Integration ~ 7 Nov 2002

  15. expr(L) var(L) expr(L) var(L) inclL L L refer-arity(L) expr(L) rel(L) sign(L) ent(L) L Expression Type Language Mathematical Context: Language (continued) • expression type language expr(L) • sets • variables: same as L • entity types: same as L • relation types: expressions of expr-set(L) • functions • reference: same as L • expression arity : arity(expr(L)) = expr(L) : expr-set(L)var(L) • expression signature : sign(expr(L)) = expr(L) : expr-set(L)sign(L) • expression set, arity and signature defined by induction IFF Semantic Integration ~ 7 Nov 2002

  16. The reference quartet The arity quartet var(f) rel(f) The signature quartet var(L) var(L) rel(L) rel(L) refer(f) refer(L) refer(L) sign(f) sign(L) sign(L) ent(L) ent(L) sign(L) sign(L) ent(f) sign(refer(f)) rel(f) rel(L) rel(L) arity(f) arity(L) arity(L) var(L) var(L) var(f) Mathematical Context: Language (continued) • type language morphism f :LL • source/target languages • src(f) = L and tgt(f) = L • functions • variable function: var(f) :var(L) var(L) • entity type function: ent(f) :ent(L) ent(L) • relation type function: rel(f) :rel(L) rel(L) • preservation constraints • refer(L) · ent(f) = var(f) · refer(L) “preserves reference” • arity(L) · var(f) = rel(f) · arity(L) “preserves arity” • sign(L) · sign(refer(f)) = rel(f) · sign(L) “preserves signature” • kind of aligned hypergraph morphism • Note: • Type language morphisms preserve signatures “on the nose”! • Example: If (, )  rel(L),rel(f)() = , (, )  rel(L), then ent(f)() = , and ent(f)() = . IFF Semantic Integration ~ 7 Nov 2002

  17. Theory T language base: base(T) set axioms: axm(T) expr-set(base(T)) derived set theorems: thm(T) (semantically defined using “satisfaction” ) Example base: above language example axioms: (x:Person) (y:Person) (spouse(x,y) →spouse(y,x)) (x:Person)(n:Natno) (age(x,n) →leq(n,1000)) (n:Natno)(m:Natno) (p:Natno) ((leq(n,m)(leq(m.p)) →leq(n,p)) “A framework is created which can support an open-ended number of theories (potentially infinite) organized in a lattice [category] together with systematic metalevel techniques for moving from one to another, for testing their adequacy for any given problem, and for mixing, matching, combining, and transforming them to whatever form is appropriate for whatever problem anyone is trying to solve.” – John Sowa The context of theories can adequately play this role. The lattice of theories is a somewhat derivative notion. Mathematical Context: Theory IFF Semantic Integration ~ 7 Nov 2002

  18. Mathematical Context: Theory (continued) • Theory morphism g :TT • source/target theories • src(g) = T and tgt(g) = T • type language morphism • base: base(g) :base(L) base( L) • preservation property • base(g)(axm(T))  thm(T) IFF Semantic Integration ~ 7 Nov 2002

  19. Model A type language underlying type language: typ(A) sets variables: var(A) universe of discourse: univ(A) entity types: typ(ent(A)) tuple space: tuple(A) “abstract tuples” relation types: typ(rel(A)) functions reference: typ(refer(A)) = typ(A) :var(A) typ(ent(A)) type arity: 1 = typ(arity(A))= typ(A) :typ(rel(A)) var(A) type signature: 1 = typ(sign(A))= typ(A) :typ(rel(A)) sign(typ(refer(A))) instance arity: 0 = inst(arity(A))= inst(A) :tuple(A) var(A) instance signature: 0 = inst(sign(A))= inst(A) :tuple(A) tuple(refer(A)) classifications entity: ent(A) = univ(A), typ(ent(A)), ⊨ent(A) relation: rel(A) = tuple(A), typ(rel(A)), ⊨rel(A) Example underlying type language: above language example universe of discourse: all people, natural and real numbers tuple space: the usual n-tuples Entity classification: “Mike Uschold” ⊨ Person 3 ⊨ natno (“George Bush”, 56) ⊨ age typ(ent(A)) typ(rel(A)) ⊨ent(A) ⊨rel(A) univ(A) tuple(A) Entity Classification Relation Classification Mathematical Context: Model IFF Semantic Integration ~ 7 Nov 2002

  20. typ(ent(h)) typ(ent(A)) typ(ent(A)) ⊨ent(A) ⊨ent(A) univ(A) univ(A) univ(h) Entity Infomorphism typ(rel(h)) typ(rel(A)) typ(rel(A)) ⊨rel(A) ⊨rel(A) tuple(A) tuple(A) tuple(h) Relation Infomorphism Mathematical Context: Model (continued) • Model infomorphism h : A⇄A • source/target models • src(h) = A and tgt(h) = A • type language morphism • underlying type language morphism: typ(h) : typ(A) typ(A) • functions • variable function: var(h) :var(A) var(A) • universe of discourse function: univ(h) :univ(A) univ(A) • entity type function: typ(ent(h)) :typ(ent(A)) typ(ent(A)) • tuple space function: tuple(h) :tuple(A) tuple(A) • relation type function: typ(rel(h)) :typ(rel(A)) typ(rel(A)) • classification infomorphisms • entity infomorphism: ent(h) :ent(A) ⇄ent(A) • relation infomorphism: rel(h) :rel(A) ⇄rel(A) • variable invertible pair: var(h) :var(A) ⇄var(A) IFF Semantic Integration ~ 7 Nov 2002

  21. clo(th(L)) Prologic L max-th(mod(L)) Concept Lattice for typ(mod(L)) = base(th(L)) Mathematical Context: Logic • Logic L • component model • mod(L) • component theory • th(L) • compatibility constraint • typ(mod(L)) = base(th(L)) • satisfaction constraint (several equivalent statements) • mod(L) satisfies th(L) • mod(L) satisfies  for all expressions  th(L), • r⊨expr((mod(L)) for all expressions  th(L) and all tuples r tuple(mod(L)) • Any theorem of th(L) is a theorem of the maximal theory of mod(L): th(L)  max-th(mod(L)) IFF Semantic Integration ~ 7 Nov 2002

  22. Mathematical Context: Logic (continuation) • Logic Infomorphism h : L⇄L • source/target logics • src(h) = L and tgt(h) = L • component model infomorphism • mod(h) : mod(L) ⇄mod(L) • component theory morphism • th(h) :th(L) th(L) • compatibility constraint • typ(mod(h)) = base(th(h)) IFF Semantic Integration ~ 7 Nov 2002

  23. Table of Contents: The IFF Ontology (meta) Ontology (IFF-ONT) • The IFF-ONT Architecture – details • Architectural Components: Categories, Functors and Adjunctions • Map of Coreflections • Composition of Adjunctions • Logic Presentations • Concept Lattice of Theories • Context of Theories vs. Lattice of Theories IFF Semantic Integration ~ 7 Nov 2002

  24.  restrict λ  Prologic Logic incl mod th log  id   μ   th log ⊥ ⊤ prolog mod max-th ω  Theory╧ Model init-mod base╧ base typ ⊤╧ ⊤       ⊥╧ ⊥ Theory Language╧ Language The IFF-ONT Architecture - details IFF Semantic Integration ~ 7 Nov 2002

  25.  restrict λ  Prologic Logic incl mod th log  id   μ   th log ⊥ ⊤ prolog mod max-th ω  Theory╧ Model init-mod base╧ base typ ⊤╧ ⊤       ⊥╧ ⊥ Theory Language╧ Language Architectural Components: Categories, Functors and Adjunctions IFF Semantic Integration ~ 7 Nov 2002

  26.  restrict λ  Prologic Logic incl mod th log  id   μ   th log ⊥ ⊤ prolog mod max-th ω  Theory╧ Model init-mod base╧ base typ ⊤╧ ⊤       ⊥╧ ⊥ Theory Language╧ Language Dependencies between Adjunctions IFF Semantic Integration ~ 7 Nov 2002

  27. th prolog Prologic Theory╧ Prologic base mod mod Model Language╧ Model  typ Map of Coreflections free prologic adjunction free model adjunction • Compares/connect adjunctions υ = prologth Theory╧ Prologic ω = typ Language╧ Model • prologomod = baseo • thobase = modotyp • prologoth = idTheory╧ • otyp = idLanguage╧ • ευ•mod = mod•εω “The model component of the free prologic (of a theory) is the free model of the base language” “The base language of the theory component (of a prologic) is the type language of the model component” “The theory component of the free prologic (of a theory) is that theory” “The type language component of the free model (of a language) is that language” “The model component of the prologic intent infomorphism (for any prologic) is the language intent infomorphism of the model component of that prologic” IFF Semantic Integration ~ 7 Nov 2002

  28. id Logic Logic  Logic  incl restrict incl restrict id   Prologic Prologic  Prologic Prologic  id th prolog prolog th  Theory╧  Theory╧ Theory╧ id Composition of Adjunctions integration coreflection free logic coreflection restriction reflection • λ = υ  ρ λ = logth Theory╧ Logic υ = prologth Theory╧ Prologic ρ = restrictincl Prologic  Logic • log = prologorestrict • th = incloth •  = prolog• ρ•th • λ = incl υ restrict “The free logic (of a theory) is the restriction of the free prologic of that theory” “The component theory (of a logic) is the component theory of that logic regarded as a prologic” “The theory morphism component of the prologic intent of any free prologic (of a theory) is the identity at that theory” “The intent morphism (of a logic) is the restriction of the intent morphism of that logic regarded as a prologic” IFF Semantic Integration ~ 7 Nov 2002

  29. AT = idAinclA(T) : AT⇄log(mod(AT)) = Amax-th(A) AT = intentT(A)idT : init-mod(T)T = log(th(AT)) ⇄AT A, Tth component of unit of adjunction π = modlogidLogic  Model A, Tth component of counit of adjunction λ = logthidTheory╧ Logic A, T Logic intentT(A) :init-mod(T) ⇄A inclA(T) : Tmax-th(A) Model Infomorphism in model(T) Theory Morphism in theory(A) Logic Presentations IFF Semantic Integration ~ 7 Nov 2002

  30. expr(L) expr(K) cloth(L) expr(f) entail(K) entail(L) cloexprf cloth(L) cloth(K) join(L) meet(L) L K exprf entail(L) max-th(L) max-th(K) max-th(L) mod(L) expr(L) cloth(L) mod(f) cloth(K) cloth(L) Functionality for the concept lattice of theories morphism over a type language morphism f : L K Functionality, truth classes and functions, for the concept lattice of theories over a type language L intent(L) inst-gen(L) typ-gen(L) mod(L) extent(L) mod(K) mod(L) expr(L) Concept Lattice of Theories IFF Semantic Integration ~ 7 Nov 2002

  31. T1 f T2  T0 clo(T0) Ť1  Ť2 Closed Theory Theory base L1 f L2 L Language Context of Theories vs. Lattice of Theories IFF Semantic Integration ~ 7 Nov 2002

  32. Table of Contents: Semantic Integration • Semantic Integration Process – schema • Glossary for Semantic Integration • Refinement • Alignment (Partial Compatibility) • Unification • Semantic Integration Process – details IFF Semantic Integration ~ 7 Nov 2002

  33. Community2 Alignment Link (Theory Morphism, Logic Morphism) Community1 Alignment Link (Theory Morphism, Logic Morphism) Community1 Portal Link (Logic Morphism) Community2 Portal Link (Logic Morphism) Community1 Unification Link (Virtual Logic Morphism) Community2 Unification Link (Virtual Logic Morphism) Common Generic Extensible Ontology (Theory, Logic) Participant Community2 Ontology (Logic) Participant Community1 Ontology (Logic) Participant Community2 Portal (Logic) Participant Community1 Portal (Logic) Core Ontology of Community Connections (Virtual Logic) Semantic Integration Process – schema • Participant community ontologies • terminology and semantics of a community’s knowledge • formalizable as a local logic (types, constraints, instances, classifications) • Common mediating ontology; Alignment links • common generic extensible ontology • component alignment link: from common ontology to participating community ontology • Ontology of community connections • quotient of participants connected through common ontology • specified as dual invariant IFF Semantic Integration ~ 7 Nov 2002

  34. Glossary forSemantic Integration • Ontology  IFF theory • An ontology is a specification of a conceptualization (Gruber). It is a description or formal specification of the concepts and relationships that can exist for a community. All notions here are types. This is a formal or axiomatized semantics. • Populated Ontology  IFF logic (= IFF model IFF Language IFF theory) • A community's ontology augmented with its instances and linked through its classification structures. Both instances and types. This is a combined semantics, both an axiomatized semantics and an interpretative semantics. • Refinement  IFF morphism (language, theory, logic) • A mapping of the categories and relations of one ontology to the categories and relations of another ontology. Refinements can be composed. Isomorphic ontologies are refinements of each other. • Integration  1st: alignment, 2nd: unification • The process of finding commonalities between community ontologies and the derivation of a new ontology that facilitates interoperability. • Alignment  span of IFF theory morphisms • A mapping of some of the types between two ontologies that preserves signatures and constraints. Mapped items (categories, functions or relations) are regarded as equivalent. • Portals  IFF logic morphism • The alignment mapping may be partial – many types in one ontology may have no equivalents in the other ontology. First, it may be necessary to introduce new types of concepts or relations in order to provide suitable targets for alignment. • Unification  IFF pushout of alignment span • A unification is the complete alignment of refinements of two ontologies. The IFF uses only an aligned version of unification – unify modulo an alignment diagram. IFF Semantic Integration ~ 7 Nov 2002

  35. g T1 T2 Refinement – abstract Refinement – details  entity type entity type  function type term  expression relation type Refinement • A key representation – alignment and unification are expressed in terms of refinement. • primitive type  IFF type (entity, function or relation type) • composite type  IFF term or IFF expression • generalizes a primitive type • terms generalize function types and expressions generalize relation types. • entity types are not composite. • The general notion of refinement • maps the entity types of the first ontology to entity types of the second ontology • maps the function or relation types of the first ontology to terms and expressions of the second ontology • ontology  IFF theory • refinement  IFF theory morphism • populated ontology  IFF logic • extended refinement  IFF logic infomorphism IFF Semantic Integration ~ 7 Nov 2002

  36. Alignment Diagram – abstract Alignment or Partial Compatibility – details equivalent function types linked function types   L1 L2 K k1 k2 p1 p2 equivalent relation types linked relation types P2 P1   Alignment (partial compatibility) • Basic alignment • intent of alignment: mapped categories are equivalent. • formalization: mediating ontology, alignment links • represent an equivalence pair of types as a single type in a mediating ontology • two projective mappings from this new type back to the participant community types. • structure • alignment is represent as a span of theory morphisms • mediating ontology represents both the equivalenced categories and the axiomatization needed for the degree of compatibility, partial or complete. • since the theoretical alignment links preserve axiomatization, compatibility is enforced • Extended alignment • formalization: portals and portal links • New types may needed in order to provide suitable targets for alignment • New community instances needed for interaction • structure • 'W'-shaped diagram of logic morphisms • logical portal links connect participant community ontology with portal ontology • direction of the portal links is compatible with unification diagram IFF Semantic Integration ~ 7 Nov 2002

  37. L1 L2 K k1 k2 p1 p2 P1 P2 L1 L2 1 2 f1 f2 L P1KP2 Unaligned Unification Diagram Aligned Unification Diagram Unification • Unaligned approach: • Formalization: • refinements from two participant community ontologies to refined ontologies, where the latter are isomorphic • because of isomorphism, replace two refined ontologies with single ontology • Structure: • an opspan of IFF logic infomorphisms • that is, two logic infomorphisms with a common target logic • Aligned unification: • Formalization: • unaligned opspan representation is too loose – it is not aligned • Structure: • to tighten, assume that opspan is the fusion of an alignment span of logic morphisms IFF Semantic Integration ~ 7 Nov 2002

  38. L2 L1 Portal Specification   Theory Alignment Step 1.i: Alignment L1 L2 T g1 g2 p1 p2 th(P2) P1 th(P1) P2 Logic Alignment     (Lifting to Logic) Step 1.ii: Alignment L1 L2 K k1 k2 p1 p2 P2 P1 Alignment Diagram Step 2: Unification (Fusion in Logic) L1 K L2 k1 k2 p1 p2 P2 P1 2 1 P1KP2 Unification Diagram f1 f2 L2 L1 L Semantic Integration Process - details • Specification diagrams • Components • Community ontology • Community portal and portal link • Where do we want to interact with the other community? Where is the locus of integration? The question of "where" refers to the local portal, the logic we use for interaction. • How is this place of interaction related to our community ontology? The question of "how" refers to the portal link for our community. • Common mediating ontology and alignment link • What do we want to say? What common meaning do we want to express? The question about "what" refers to the mediating ontology – what is the language and theory of the mediating ontology? • How do we say it in our own terms? How does our community formalize the common semantics? The "how" question refers to how we specify the alignment link, the theory interpretation from the theory of the mediating logic to the theory of our community logic? • Contexts • Theory • Logic • Process result diagram • Alignment diagram • Community connections ontology • Steps • Lifting from Theory context to Logic context • Fusion in Logic context IFF Semantic Integration ~ 7 Nov 2002

  39. Summary • The Information Flow Framework (IFF) • The IFF is a metalogic – it can represent the metalevel structure of the SUO. • The IFF is founded on category theory – more strongly, the Category Theory Ontology (IFF-CAT) in the upper level of the IFF represents Category Theory. • The IFF Architecture. • The three levels of the IFF represent the generic/large/small distinction. • The upper metalevel consists of the Category Theory Ontology and the Upper Classification Ontology (IFF-UCLS) anchored at the Upper Core Ontology. • The lower metalevel consists of the Model Theory Ontology (IFF-MT), the Algebraic Theory Ontology (IFF-AT) and the Ontology Ontology (IFF-ONT). It adheres as closely as possible to the category design principle. • The object level is the location for ordinary ontologies (upper ontologies, domain ontologies, etc.) • The metalevel and the object level of the IFF have a distinct and obvious boundary. • The Ontology (meta) Ontology (IFF-ONT) • This ontology provides a metalogic for semantics – both an interpretative semantics and a formal or axiomatic semantics. • The concept “lattice of theories” has been axiomatized in the IFF-ONT as base-fibers within the theory context. • The semantic integration requirements. • Colimits should exist for both theories and logics. The theory and logic contexts should be cocomplete. • There should exist free logics over theories. IFF Semantic Integration ~ 7 Nov 2002

  40. Future Work • Mathematical background and theoretical foundations • Submit IFF Ontology Ontology • Complete and submit IFF Algebraic Theory Ontology (currently 4/5 done). • Develop nonaligned versions of languages, models and logics (adapt 3 yr old paper on onto logic and make use of aligned versions). • Elaborate span graphs and span models (check kinship to RDF triples) • Applications, examples and tutorials • Develop IFF interface: control and I/O portals (CG, CycL, Teknowledge KIF, OWL, etc.) • Check connection with Kestrel Institute’s Specware (ontologies as specifications) • Standards • Assist with standards documents development for the SUO IFF Semantic Integration ~ 7 Nov 2002

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