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The IFF Approach to Semantic Integration

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

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### 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

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 InfluencesIFF Semantic Integration ~ 7 Nov 2002

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

Table of Contents:Category Theory

- Category Theory
- The Category Manifesto
- Examples: Categories

IFF 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.

IFF Semantic Integration ~ 7 Nov 2002

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

FG

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”

- Two special cases:
- 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

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

Table of Contents: The Information Flow Framework (IFF)

- The IFF Architecture
- The Lower Metalevel
- The Category Design Principle

IFF Semantic Integration ~ 7 Nov 2002

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 ArchitectureIFF Semantic Integration ~ 7 Nov 2002

“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

The Categorical Design Principle 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.”

- 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

Table of Contents 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.” : 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

Central contexts 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.”

Language

Theory

Model

Logic

Other contexts

Language╧

Theory╧

Prologic

Central generalized inverses

λ = logthTheory╧ Logic “integration coreflection”

μ = init-modmax-thTheory╧ Model “semantics adjunction”

ω = typ Language╧ Model“free model coreflection”

Other generalized inverses

υ = prologthTheory╧ Prologic “free prologic coreflection”

ρ = restrictinclPrologic Logic “restriction reflection”

π = modlogLogic Model “theory augmentation reflection”

κ = ⊤baseLanguage Theory “empty theory coreflection”

Logic

π

λ

Model

Theory╧

μ

ω

Theory

κ╧

Language╧

κ

Language

The IFF-ONT Architecture – overviewIFF Semantic Integration ~ 7 Nov 2002

type language 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.” 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: LanguageIFF Semantic Integration ~ 7 Nov 2002

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.” 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

- sets

IFF Semantic Integration ~ 7 Nov 2002

The reference quartet 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.”

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 :LL
- 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

- source/target languages
- Note:
- Type language morphisms preserve signatures
“on the nose”!

- Example: If (, ) rel(L),rel(f)() = , (, ) rel(L), then ent(f)() = , and ent(f)() = .

- Type language morphisms preserve signatures

IFF Semantic Integration ~ 7 Nov 2002

Theory 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.” 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: TheoryIFF Semantic Integration ~ 7 Nov 2002

Mathematical Context: Theory (continued) 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.”

- Theory morphism g :TT
- 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)

- source/target theories

IFF Semantic Integration ~ 7 Nov 2002

Model 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.” 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: ModelIFF Semantic Integration ~ 7 Nov 2002

typ 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.” (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)

- source/target models

IFF Semantic Integration ~ 7 Nov 2002

clo 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.” (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))

- component model

IFF Semantic Integration ~ 7 Nov 2002

Mathematical Context: Logic (continuation) 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.”

- 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))

- source/target logics

IFF Semantic Integration ~ 7 Nov 2002

Table of Contents 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.” : 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

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.”

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 - detailsIFF Semantic Integration ~ 7 Nov 2002

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.”

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 AdjunctionsIFF Semantic Integration ~ 7 Nov 2002

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.”

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 AdjunctionsIFF Semantic Integration ~ 7 Nov 2002

th 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.”

prolog

Prologic

Theory╧

Prologic

base

mod

mod

Model

Language╧

Model

typ

Map of Coreflectionsfree prologic adjunction

free model adjunction

- Compares/connect adjunctions
υ = prologth 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

id 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.”

Logic

Logic

Logic

incl

restrict

incl

restrict

id

Prologic

Prologic

Prologic

Prologic

id

th

prolog

prolog

th

Theory╧

Theory╧

Theory╧

id

Composition of Adjunctionsintegration coreflection

free logic coreflection

restriction reflection

- λ = υ ρ
λ = logth Theory╧ Logic

υ = prologth Theory╧ Prologic

ρ = restrictincl 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

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.” AT = idAinclA(T)

: AT⇄log(mod(AT)) = Amax-th(A)

AT = intentT(A)idT

: init-mod(T)T = log(th(AT)) ⇄AT

A, Tth component of unit of adjunction

π = modlogidLogic Model

A, Tth component of counit of adjunction

λ = logthidTheory╧ Logic

A, T

Logic

intentT(A) :init-mod(T) ⇄A

inclA(T) : Tmax-th(A)

Model Infomorphism

in model(T)

Theory Morphism

in theory(A)

Logic PresentationsIFF Semantic Integration ~ 7 Nov 2002

expr 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.” (L)

expr(K)

cloth(L)

expr(f)

entail(K)

entail(L)

cloexprf

cloth(L)

cloth(K)

join(L)

meet(L)

L

K

exprf

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 TheoriesIFF Semantic Integration ~ 7 Nov 2002

T 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.” 1

f

T2

T0

clo(T0)

Ť1

Ť2

Closed

Theory

Theory

base

L1

f

L2

L

Language

Context of Theories vs. Lattice of TheoriesIFF Semantic Integration ~ 7 Nov 2002

Table of Contents 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.” : 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

Community 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.” 2

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

Glossary for 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.” Semantic 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

g 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.”

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

Alignment Diagram – abstract 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.”

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

- formalization: portals and portal links

IFF Semantic Integration ~ 7 Nov 2002

L 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.” 1

L2

K

k1

k2

p1

p2

P1

P2

L1

L2

1

2

f1

f2

L

P1KP2

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

- Formalization:
- 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

- Formalization:

IFF Semantic Integration ~ 7 Nov 2002

L 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.” 2

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

P1KP2

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

- Components
- 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

Summary 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.”

- 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

Future Work 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.”

- 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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