the iff approach to semantic integration boeing mini-workshop on semantic integration

The IFF Approach to Semantic Integration

Boeing Mini-Workshop on Semantic Integration

http://suo.ieee.org/IFF/

http://www.ontologos.org/IFF/OntologyOntology/Introduction.htm

“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://ofcn.org/cyber.serv/resource/bookshelf/persu11/chapter06.html

7 Nov 2002IFF Semantic Integration ~ 7 Nov 20022

Origins and Influences

Information Flow Framework (IFF)

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.

http://www.springer-ny.com/detail.tpl?cart=9815698863736245&ISBN=0387984038

http://www.springer.de/cgi-bin/search_book.pl?isbn=3-540-62771-5#english

http://www.springer-ny.com/detail.tpl?cart=9815698863736245&ISBN=0387984038

http://www.springer.de/cgi-bin/search_book.pl?isbn=3-540-62771-5

7 Nov 2002IFF Semantic Integration ~ 7 Nov 20023

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

Category TheoryIFF Semantic Integration ~ 7 Nov 20024

Table of Contents:Category Theory

5. Category Theory6. The Category Manifesto7. Examples: Categories


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.

6 IFF Semantic Integration ~ 7 Nov 2002 Category Theory

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.”

A BF

C

W

The Categorical Manifesto by 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.

A BF G

IdA = F ◦ GG ◦ F = IdB

IdA F ◦ GG ◦ F IdB

IdA F ◦ GG ◦ F IdB

http://www.cs.ucsd.edu/users/goguen/ps/manif.ps.gz


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).

The IFFIFF Semantic Integration ~ 7 Nov 20028

Table of Contents: The Information Flow Framework (IFF)

9. The IFF Architecture10. The Lower Metalevel11. The Category Design Principle


The IFF Architecture

metalevel

object level

upper

lower

top

The IFF

IFF Model Theory(meta) Ontology

IFF Core (meta) Ontology

IFF Classification(meta) Ontology

IFF Category Theory(meta) Ontology

IFF Basic KIF (meta) Ontology

Upper OntologyUpper

OntologyUpper Ontologym

�ֻ

Domain OntologyDomain

OntologyDomain Ontologyp

�ֻMiddle

OntologyMiddle OntologyMiddle

Ontologyn

�ֻ

Upper metalevel– Declare, define, axiomatize and

reason about generic categories, functors, adjunctions, colimits, monads, classifications, concept lattices, etc.

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.


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

“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 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.

ContextsIFF Semantic Integration ~ 7 Nov 200212

Table of Contents: The IFF-ONT Contexts

13. The IFF-ONT Architecture – overview14. Mathematical Context: Language15. Mathematical Context: Language (continued)16. Mathematical Context: Language (continued)17. Mathematical Context: Theory18. Mathematical Context: Theory (continued)19. Mathematical Context: Model20. Mathematical Context: Model (continued)21. Mathematical Context: Logic22. Mathematical Context: Logic (continued)


The IFF-ONT Architecture – overview

Central contextsLanguageTheoryModelLogic

Other contextsLanguage╧

Theory╧

Prologic

Central generalized inversesλ = log th Theory╧ Logic “integration coreflection”

μ = init-mod max-th Theory╧ Model “semantics adjunction”

ω = typ Language╧ Model “free model coreflection”

Other generalized inversesυ = 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

Model Theory╧

Language╧

π

κ╧

μ

λ

ω Theory

Languageκ


Mathematical Context: Language

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)

Exampleindexes = Integer+

entity types = Person, String, Natno, Realvariables = (entity type, index) pairsrelation 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

Type Language

rel(L)

L

ent(L)

var(L)L

var(L)L

sign(L)

refer-arity(L)


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

Expression Type Language

expr(L)

expr(L)

ent(L)

var(L)L

var(L)expr(L)

sign(L)

refer-arity(L)rel(L)

L

LinclL


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)() = .

The arity quartet

arity(L)

var(f)

arity(L)

rel(f)

arity(f)

rel(L)rel(L)

var(L) var(L)

var(L)

ent(L)

var(L )

refer(L)

var(f)

ent(L)

refer(L)

ent(f)

refer(f)

The reference quartet

rel(L)

sign(L)

rel(L)

sign(L)

rel(f)

sign(L)

sign(L)

sign(refer(f))

sign(f)

The signature quartet


Mathematical Context: Theory

Theory T– language

base: base(T)

– set axioms: axm(T) expr-set(base(T))

– derived set theorems: thm(T)

(semantically defined using “satisfaction” )

Examplebase: above language exampleaxioms:(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 (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)


Mathematical Context: Model

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)

Exampleunderlying type language: above language exampleuniverse of discourse: all people, natural and real numberstuple space: the usual n-tuplesEntity classification:“Mike Uschold” ⊨ Person3 ⊨ natno(“George Bush”, 56) ⊨ age

Entity Classification

typ(ent(A))

univ(A)

⊨ent(A)

Relation Classification

typ(rel(A))

tuple(A)

⊨rel(A)


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)

Entity Infomorphism

typ(ent(A))

univ(A)

⊨ent(A)

typ(ent(A))

univ(A)

⊨ent(A)

typ(ent(h))

univ(h)

Relation Infomorphism

typ(rel(A))

tuple(A)

⊨rel(A)

typ(rel(A))

tuple(A)

⊨rel(A)

typ(rel(h))

tuple(h)


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))

Prologic L

Concept Lattice for

typ(mod(L)) = base(th(L))

clo(th(L))

max-th(mod(L))


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))

The IFF-ONTIFF Semantic Integration ~ 7 Nov 200223

Table of Contents: The IFF Ontology (meta) Ontology (IFF-ONT)

24. The IFF-ONT Architecture – details25. Architectural Components: Categories, Functors and

Adjunctions26. Map of Coreflections27. Composition of Adjunctions28. Logic Presentations29. Concept Lattice of Theories30. Context of Theories vs. Lattice of Theories


The IFF-ONT Architecture - details

ωωμμ

λλ

Language

Theory

Model

Language╧

Prologic

Theory╧

Logic

log th

restrict

incl

typ

th log

prolog mod

base╧⊤╧

⊥╧

modid⊤

⊥

base⊤

⊥

max-th

init-mod

25 IFF Semantic Integration ~ 7 Nov 2002 The IFF-ONT

Architectural Components: Categories, Functors and Adjunctions

λ = log th Theory╧ Logicμ = init-mod max-th Theory╧ Modelω = typ Language╧ Model

Theoryclosure-inclusion [unit of μ = λ π] empty-inclusion [counit of κ] inclusion-full [unit of κ]special-empty-inclusion [counit of κ╧] special-inclusion-full [unit of κ╧]

Modellanguage-intent [counit of ω = κ╧ μ]theory-intent [counit of μ = λ π]

Prologicintent [counit of υ]restriction-inclusion [unit of ρ]empty-inclusion [counit of ν]

Logicinclusion [unit of π] intent [counit of λ = υ ρ]

υ = prolog th Theory╧ Prologicρ = restrict incl Prologic Logicπ = mod log Logic Modelν = ⊤ mod Model Prologicκ = ⊤ base Language Theoryκ╧ = ⊤ base ╧ Language╧ Theory╧

κ = base ⊥ Theory Languageκ╧ = base ⊥ ╧ Theory╧ Language╧

ωωμμλλ

Language

Theory

Model

Language╧

Prologic

Theory╧

Logic

log th

restrict

incl

typ

th log

prolog mod

base╧⊤╧

⊥╧

modid⊤

⊥

base⊤

⊥

max-th

init-mod


Dependencies between Adjunctions

Dependency Kind Expression υ ω [map of adjunctions] mod base╧ : υ ω

λ υ, ρ ω, ρ [composition of adjunctions] λ = υ ◦ ρ

μ λ, π ω, ρ, π [composition of adjunctions] μ = λ ◦ π

ω κ╧, μ [composition of adjunctions] ω = κ╧ ◦ μ

ω, ν κ╧, υ [composition of adjunctions] ω ν = κ╧ ◦ υ

ωωμμλλ

Language

Theory

Model

Language╧

Prologic

Theory╧

Logic

log th

restrict

incl

typ

th log

prolog mod

base╧⊤╧

⊥╧

modid⊤

⊥

base⊤

⊥

max-th

init-mod


Map of Coreflections

Compares/connect adjunctionsυ = prolog th Theory╧ Prologic ω = typ Language╧ Model

– prolog o mod = base o

– th o base = mod o typ

– prolog o th = idTheory╧

o typ = idLanguage╧

– ευ • mod = mod • εω

Model Language╧

Prologic Theory╧

basemod

typ

th prologPrologic

Model

mod

“The model component of the free prologic (of a theory) is the free

model of the base language”

“The theory component of the free prologic (of a theory) is that theory”

free prologic adjunctionfree model adjunction

“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”

“The base language of the theory component (of a prologic) is the type language of the model component”


restriction reflection λ = υ ρ

λ = log th Theory╧ Logic υ = prolog th Theory╧ Prologic ρ = restrict incl Prologic Logic

– log = prolog o restrict– th = incl o th = prolog • ρ • th λ = incl υ restrict

Composition of Adjunctions

free logic coreflection

“The theory morphism component of the prologic intent of any free prologic (of a

theory) is the identity at that theory”

“The component theory (of a logic) is the component theory of that logic regarded as a prologic”

“The free logic (of a theory) is the restriction of the free

prologic of that theory”

Prologic

Theory╧

Logic

restrict incl

thprolog

Prologic

Theory╧

id

idPrologi

c

Logic

Theory╧

restrictincl

th prolog

Prologic

Logic

id

id

“The intent morphism (of a logic) is the restriction of the intent morphism of that

logic regarded as a prologic”

integration coreflection


Logic Presentations

A, T

Logic

intentT(A) : init-mod(T) ⇄ A

Model Infomorphismin model(T)

inclA(T) : T max-th(A)

Theory Morphismin theory(A)

A T = idA inclA(T)

: A T ⇄ log(mod(A T)) = A max-th(A)

A, Tth component of unit of adjunctionπ = mod log id Logic Model

A T = intentT(A) idT

: init-mod(T) T = log(th(A T)) ⇄ A T

A, Tth component of counit of adjunctionλ = log th id Theory╧ Logic


Concept Lattice of Theories

expr(L)

mod(L)

expr(K)

mod(K)

cloth(L)

cloth(L)

cloth(K)

cloth(K)

expr(f)

mod(f)

L K

cloexprf

exprf

entail(L) entail(K)

max-th(K)max-th(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

mod(L) expr(L)extent(L)

inst-gen(L) typ-gen(L)intent(L)

cloth(L)mod(L) expr(L)max-th(L) entail(L)

cloth(L)

join(L) meet(L)


Context of Theories vs. Lattice of Theories

Theory

ClosedTheory

T1

T2

f

T0 clo(T0)

Ť1

Ť2

LanguageL

L1

L2

f

base

Semantic IntegrationIFF Semantic Integration ~ 7 Nov 200232

Table of Contents: Semantic Integration

32. Semantic Integration Process – schema33. Glossary for Semantic Integration34. Refinement35. Alignment (Partial Compatibility)36. Unification37. Semantic Integration Process – details

33 IFF Semantic Integration ~ 7 Nov 2002 Semantic Integration

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

Semantic Integration Process – schema Participant

Community2 Portal(Logic)

ParticipantCommunity1 Portal

(Logic)

Core Ontology ofCommunity Connections

(Virtual Logic)

Community1

Alignment Link(Theory Morphism,Logic Morphism)

Community1

Unification Link(Virtual Logic

Morphism)

Community2

Unification Link(Virtual Logic

Morphism)

Community2

Alignment Link(Theory Morphism,Logic Morphism)

Common GenericExtensible Ontology

(Theory, Logic)

ParticipantCommunity1 Ontology

(Logic)

ParticipantCommunity2 Ontology

(Logic)

Community2

Portal Link(Logic Morphism)

Community1

Portal Link(Logic Morphism)


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.


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

Refinement – details

function type relation type

termexpression

entity type entity type

Refinement – abstract

gT1 T2


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

Alignment or Partial Compatibility – details

Alignment Diagram – abstract

equivalentfunction types

linkedfunction types

equivalentrelation types

linkedrelation types

k1 k2

K

P1 P2

L1 L2

p1 p2


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

Aligned Unification DiagramUnaligned Unification Diagram

k1 k2

K

P1 P2

L1 L2

21P1K P2

p1 p2

f1 f2L

L1 L2


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

P1 P2

L1 L2

p1 p2

g2g1

T

th(P1) th(P2)

f1 f2

LL1 L2

L2L1

k1 k2

K

P1 P2

L1 L2

21P1 K P2

p1 p2

Unification Diagram

k1 k2

K

P1 P2

L1 L2

p1 p2

Alignment DiagramStep 2:Unification (Fusion in Logic)

Step 1.ii:Alignment

Logic Alignment (Lifting to Logic)

Step 1.i:Alignment

Portal Specification Theory Alignment

SummaryIFF Semantic Integration ~ 7 Nov 200239

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.

Future WorkIFF Semantic Integration ~ 7 Nov 200240

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

the iff approach to semantic integration boeing mini-workshop on semantic integration

Documents