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IFF Semantic Integration ~ 7 Nov Sections Category Theory (3 slides) Category Theory The IFF (3 slides) The IFF The IFF-ONT Contexts (10 slides) The IFF-ONT Contexts The IFF-ONT (7 slides) The IFF-ONT Semantic Integration (6 slides) Semantic Integration Summary & Future Work SummaryFuture Work
Citation preview
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
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.
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 TheoryIFF Semantic Integration ~ 7 Nov 2002
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
Category TheoryIFF Semantic Integration ~ 7 Nov 20027
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 IFFIFF Semantic Integration ~ 7 Nov 20029
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 IFFIFF Semantic Integration ~ 7 Nov 200210
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 IFFIFF Semantic Integration ~ 7 Nov 200211
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)
ContextsIFF Semantic Integration ~ 7 Nov 200213
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
ContextsIFF Semantic Integration ~ 7 Nov 200214
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)
ContextsIFF Semantic Integration ~ 7 Nov 200215
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
ContextsIFF Semantic Integration ~ 7 Nov 200216
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
ContextsIFF Semantic Integration ~ 7 Nov 200217
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.
ContextsIFF Semantic Integration ~ 7 Nov 200218
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)
ContextsIFF Semantic Integration ~ 7 Nov 200219
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)
ContextsIFF Semantic Integration ~ 7 Nov 200220
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)
ContextsIFF Semantic Integration ~ 7 Nov 200221
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))
ContextsIFF Semantic Integration ~ 7 Nov 200222
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-ONTIFF Semantic Integration ~ 7 Nov 200224
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
26 IFF Semantic Integration ~ 7 Nov 2002 The IFF-ONT
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
The IFF-ONTIFF Semantic Integration ~ 7 Nov 200227
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”
The IFF-ONTIFF Semantic Integration ~ 7 Nov 200228
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
The IFF-ONTIFF Semantic Integration ~ 7 Nov 200229
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
30 IFF Semantic Integration ~ 7 Nov 2002 The IFF-ONT
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)
31 IFF Semantic Integration ~ 7 Nov 2002 The IFF-ONT
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)
34 IFF Semantic Integration ~ 7 Nov 2002 Semantic Integration
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.
Semantic IntegrationIFF Semantic Integration ~ 7 Nov 200235
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
Semantic IntegrationIFF Semantic Integration ~ 7 Nov 200236
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
Semantic IntegrationIFF Semantic Integration ~ 7 Nov 200237
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
38 IFF Semantic Integration ~ 7 Nov 2002 Semantic Integration
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