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The Information Flow Foundation for Conceptual Knowledge Organization

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### The Information Flow Foundation for Conceptual Knowledge Organization

Robert E. Kent

Mathematical

context

category

passage

functor

invertible

adjoint

sum

coproduct

quotient

—

fusion

pushout

Table 1: Definition of Terminology

Focus and Presentation- Focus: Knowledge Bases of Communities
- Presentation: on three levels
- Intuitive
- Model for ontologies
- Construction process of ontology sharing
- Mathematical (standard)
- At the level of standard mathematics
- Discussion of Information Flow primitives
- Mathematical (category-theoretic)
- Located in the endnotes
- New theorems for Information Flow
- Representation Theorem
- Factorization Theorem

ISKO 6, July 2000

Principled Approach

- The approach to ontology sharing described in the paper is a "principled approach“ – it has a well-defined foundation, the distributed (hard) logic of Information Flow.
- These principles and this approach can be compared and contrasted with the architecture of the Semantic Web.

"Semantic Web Road map"

by Tim Berners-Lee

http://www.w3.org/DesignIssues/Semantic.html

ISKO 6, July 2000

th

Log

cla

Logh

Th

Theory

Classification

Cla

Figure 2: Conceptual Knowledge Organization

Dynamism and Stability of Conceptual Structures- Stability: Theory Context
- Types and constraints specified in ontology
- Represented as an Information Flow theory
- Dynamism: Logic Context
- Instance collections; classification relations
- Links between ontologies
- Specified by
- ontological extension
- synonymy
- Represented as an Information Flow Logic

ISKO 6, July 2000

An ontology is a catalog of the types of things that are assumed to exist in a domain of interest. The types in the ontology represent entities (classes) and relations.

Onto logic (logic + ontology) represents relationships about the entities in the domain of interest.

In an ontology entity and relation types are organized in a partial ordering by the type-subtype relation. Many type collections are disjoint.

An ontology distinguishes types by axioms and definitions stated in a logical language.

Example:

Carnivore and Herbivore are subtypes of Animal.

Carnivore is disjoint from Herbivore.

Carnivores only eatAnimals.

A Lion is an Animal that only eats Herbivores.

Carnivore⊢Animal; Herbivore⊢Animal

Carnivore, Herbivore⊢

(x) (Carnivore(x)eat(x,y)) Animal(y))

Lion ⊢(Animal(x)eat(x,y)) Herbivore(y))

OntologiesISKO 6, July 2000

Cat

Mouse

agent

theme

Cat

Chase

Mouse

Chase:

Agent: ?x

Theme: ?y

Cat: *x

Mouse: *y

has

has

Relational Reification- Usually, ontologies are composed of two types of first-class constructs: entities and relations.
- We assume that relations are reified as entities – how things relate to one another is just another thing.Relations are special entities.
- Effected by relational reification, a reflective mechanism that decomposes a relation type into its constituent parts and introduces an entity type (the reificator) upon which reasoning about the relation (annotation by qualifying expressions) can be based.

ISKO 6, July 2000

Participant ontologies

terminology and semantics of a community’s knowledge

formalizable as a local logic (types, constraints, instances, classification)

Common ontology; Component links

common extensible ontology

component link: from the common ontology to the participating community ontology

Ontology of community connections

dual quotient of participants connected through common ontology

specified as dual invariant

Common Generic

Extensible Ontology

(Theory,

Logic)

Community1

Specification Link

(Theory Interpretation,

Logic Infomorphism)

Community2

Specification Link

(Theory Interpretation,

Logic Infomorphism)

Participant

Community1 Ontology

(Logic)

Participant

Community2 Ontology

(Logic)

Community1

Participation Link

(Virtual Logic

Infomorphism)

Community2

Participation Link

(Virtual Logic

Infomorphism)

Core Ontology of

Community Connections

(Virtual Logic)

Figure 1: Ontology Sharing between Communities

Ontology SharingDiagramISKO 6, July 2000

Ontology SharingComments

- Observation
- The sharing of ontologies between diverse communities of discourse allows them to compare their own information structures with that of other communities that share a common terminology and semantics.
- Participant logics
- terminology and semantics of a community’s knowledge
- specified in an ontology
- realized within the various instance collections of that community
- formalizable as a local logic (types, constraints, instances, classification)
- a classification relation between the community’s instances and types
- constraints within a theory modeling the community’s consensus semantics
- Common ontology & Component links
- common extensible ontology
- common terminology and semantics shared by diverse communities
- formalized as a theory (types plus constraints) with neither a priori instances nor classification relation.
- formal instances are added in the passage to (construction of) a local logic.
- component link: from the common ontology to the participating community ontology
- includes by ontological extension the types and constraints of the common ontology
- records any synonymy (type equivalence) prescribed by the participant
- Virtual logic of community connections
- dual quotient of participants connected through common ontology
- specified as dual invariant
- one community instance is connected to another community instance when they agree on the common inherited types
- use the types in the community participant ontologies, identifying types through the a common ontology

ISKO 6, July 2000

Lifting to Logic

Step 2

Fusion in Logic

T

Figure 3: Two-step Process

th(L1)

th(L1)

g1

g2

f1, g1

f2, g2

Log(T)

Log(T)

f1, g1

f2, g2

L1/E1

L1/E1

L2/E2

L2/E2

f1, g1

f2, g2

L1/E1Log(T)L2/E2

Ontology SharingProcess

- Specification diagrams
- Components
- Community ontologies
- Common ontology
- Community links
- Contexts
- Theory
- Logic
- Process result diagram
- Community connections ontology
- Steps
- Lifting from Theory context to Logic context
- Fusion in Logic context

ISKO 6, July 2000

- Principle 1. A community owns its collection of instances.
- It controls updates to the collection.
- It can enforce soundness.
- It controls access rights to the collection.
- Principle 2. Instances are linked through their types.
- To compare instances of two specific ontologies, we must use the free logic of the generic ontology containing all of its formal instances.

ISKO 6, July 2000

Information Flow: The Logic of Distributed Systems

- Goal: The definition and place of information in society
- Observation: Information flow is possible only within a connected distribution system
- A mathematically rigorous, philosophically sound foundation for a science of information
- Elements
- Classifications and infomorphisms
- Theories and interpretations
- Local logics and logic infomorphisms
- Colimits: sums, dual invariants/quotients, pushouts

ISKO 6, July 2000

⊨A

inst(A)

Classification- A = inst(A), typ(A), ⊨A
- inst(A), things to be classified, the instances of A
- typ(A), things used to classify the instances, the types of A
- a binary relation,⊨A, between inst(A) and typ(A)
- Examples
- Given a natural language, the dictionary classification has words as instances, parts of speech (Noun, Verb, Adjective, Adverb, etc.) as types, with functional labels as the classification relation.
- Given any ontology, the ontological classification has the classes of an ontology as types, the individuals/classes of the ontology as instances, with “instanceOf” as the classification relation.
- Given a first-order language L, the truth classification of L has L-models as instances, expressions of L as types, and satisfaction as classification relation: M⊨φ if and only if φ is true in M.
- Given any set A (of instances), the powerset classificationA = A,A,A associated with A has elements of A as instances and subsets of A as types with the membership relation serving as the classification relation. This is the free classification w.r.t. the underlying instance set functor.
- Given a set Σ of types, a partition (or truth value assignment) of Σ is a pair Γ,Δ of disjoint subsets that cover Σ. The full partition classification⊤(Σ)associated with Σ has partitions as instances, elements of Σ as types, and classification relation defined as Γ,Δ⊨α if and only if α Γ. This is the free classification w.r.t. the underlying type set functor.

notation a⊨A reads

“instance a is of type “

ISKO 6, July 2000

typ(A)

typ(B)

inst(f)

⊨A

⊨B

inst(A)

inst(B)

Infomorphism- f:A⇄B
- a contravariant pair of functions f = inst(f), typ(f)

from A = inst(A), typ(A), ⊨A to B = inst(B), typ(B), ⊨B

- inst(f):inst(A) inst(B), instance function, in the reverse direction
- typ(f):typ(A) typ(B), type function, in the forward direction
- satisfying the fundamental property of infomorphisms:

typ(f)(b) ⊨A α iff b ⊨Binst(f)(α)

- “move” information back and forth between classifications
- Examples
- The inverse image infomorphismf = f, f: A⇄B
- any function f:AB as instance function
- its inverse image function f:AB as type function
- the fundamental property of infomorphisms: f(b) A iff b Bf()
- The full partition infomorphism⊤(f) :⊤(Φ) ⊤(Ψ)
- any function f: Φ Ψ as type function
- the inverse image f(Γ, Δ) = f(Γ), f(Δ) as instance function
- the fundamental property of infomorphisms: φ f(Γ) iff f (φ) Γ

ISKO 6, July 2000

Classification Constructions

- sumA+B
- inst(A+B) = inst(A)inst(B)
- pairs (a,b) of instances, ainst(A) and binst(B)
- typ(A+B) = typ(A)+typ(B)
- tagged types, either (0,) for typ(A) or (1,) for typ(B)
- classification relation ⊨A+B
- (a,b) ⊨A+B(0,α) iff a⊨A
- (a,b) ⊨A+B(1,) iff b⊨A
- two infomorphisms A:A⇄A+B and B:B⇄A+B
- dual invariantJ = (A,R) on classification A
- a set A inst(A) of instances
- a binary relation R on types
- constraint: if R, then for each aA, a⊨A if and only if a⊨A
- dual quotientA/J
- instances A
- types are R-equivalence classes of types of A
- classification is a ⊨A/J [α] if and only if a ⊨Aα
- canonical quotient infomorphism : A ⇄ A/J

ISKO 6, July 2000

classify

IF Local Logic

Instance Collection

IF Theory

types, sequents

IF Classification

instances, types, classification

Theory- T = typ(T), ⊢
- a sequent is a pair Γ,Δ of subsets of typ(T), logically denoted by "G$D
- ⊢ is a set of sequents called constraints of the theory T, denoted by Γ ⊢ Δ
- Examples
- Special sequents
- Partition: types and partition type when
- ⊢ and ⊢ (subtypes)
- ⊢, (cover)
- , ⊢(disjoint)
- Th(A) = typ(A), ⊢A
- theory generated by classification A
- ⊢ when every instance a of A satisfies the implication:

“if instance a is of every type in Γ, then it is of some type in Δ”

validating the notation

"G$D

ISKO 6, July 2000

Intangible

Individual

Mathematical

Or

Computational

Thing

Relationship

Intangible

Individual

Microtheory

Formula

Collection

Variable

Predicate

Logical

Connective

Quantifier

TheoryExamples- Heraclitus distinction (Sowa 1999)
- Types: {Abstract, Physical}
- Constraints: (partition) Abstract, Physical ⊢; ⊢ Abstract, Physical
- Peirce distinction (Sowa 1999)
- Types: {Independent, Relative, Mediating}
- Constraints: (partition) Independent, Relative ⊢; …
- Cyc Top
- Types: {…, Individual, …, Collection , …}
- Constraints:

…; Individual, Collection ⊢;

…; Microtheory⊢ Individual;

…

ISKO 6, July 2000

classify

IF Local Logic

Instance Collection

IF Theory

types, sequents

IF Classification

instances, types, classification

TheoryExternalization Form- inheritance
- contradiction

ISKO 6, July 2000

Summary and Future Work

- Summary: The ontology sharing process has been described in terms of ideas from Information Flow.
- A new paper “The Model Theory of Onto Logic”
- better model for ontologies
- better representation for relations
- full representation of typed logic
- uses ideas from Information Flow
- Introducing a new XML application, knowledge model (and API)
- knowledge model is the theory of Information Flow
- built around new idea of ontological model
- based upon 5 years of language design

experience (OML/CKML)

- spare and parsimonious in design

iff

information flow framework

ISKO 6, July 2000

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