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The Information Flow Foundation for Conceptual Knowledge Organization. Robert E. Kent. Intuitive. Mathematical. context. category. passage. functor. invertible. adjoint. sum. coproduct. quotient. —. fusion. pushout. Table 1: Definition of Terminology. Focus and Presentation.

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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**Intuitive**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**Logic**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)) Ontologies ISKO 6, July 2000**chasing**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 SharingDiagram ISKO 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**Step 1**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**Ontology SharingPrinciples**• 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**typ(A)**⊨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(f)**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**Type Hierarchy**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**Thing**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**Type Hierarchy**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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