LISTIC/ESIA ~ Oct 2005

The Information Flow Framework (IFF)

Robert E. Kent

Henri Matisse. The Joy of Life.

The Information Flow Framework (IFF) is a descriptive category metatheory for building/maintaining object-level ontologies.

LISTIC/ESIA ~ Oct 2005

Peirce Quotation

“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

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Table of Contents

Part 0. Introduction to the IFF– What is an Ontology?

(definition) (dictionaries) (distinctions)

– The SUO-IFF Project– Origins and Influences– The Categorical Manifesto– IFF Design Guidelines– IFF Development Phases

Principle of Conceptual Warrant Principle of Categorical Design

Part I. The IFF Metatheory– The IFF Architecture

(old version) (new version) The IFF Metastack Four Fundamental Relations

– The IFF Code The IFF (meta) Ontologies Some Example IFF Terms The Basic IFF Terms

– The IFF Metalanguages (old version) (new version)

Part II. The IFF Institutional Application– Two IFF Efforts to represent FOL

IFF-ONT (old version) (new version)

IFF-FOL– Logical Environments

Languages Language Expressions Language Colimits Models Theories The Category of Theories Theory Colimits (Fusion) Truth Classification & Truth Concept Latti

ce Truth Concept Lattice Functionality

– Institutions The Grothendieck Construction The Truthful Connection

Summary Appendix

– What is an Ontology?1. (other views)2. (other views)

– Logical Environments Language Morphisms Language Sums Language Endorelations Language Quotients Theory Morphisms Theory Sums Theory Endorelations and Quotients

LISTIC/ESIA ~ Oct 2005

Part 0. Introduction to the IFF

What is an Ontology?– (definition)– (dictionaries)– (distinctions)

The SUO-IFF Project Origins and Influences The Categorical Manifesto IFF Design Guidelines IFF Development Phases

– Principle of Conceptual Warrant – Principle of Categorical Design

object-level

metalevel

01

23

n

…

… CORE

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What is an Ontology? (definition)

Example. Consider the e-commerce example (Figure 1) of a schema (ontology).

– concepts are represented as nodes,– relationships are represented as edges,– constraints are represented as parallel pairs

of edge paths, limits and coproducts.

Definition. An ontology is a formal, explicit specification of a shared conceptualization. It is an abstract model of some phenomena in the world, explicitly represented as concepts, relationships and constraints, which is machine-readable and incorporates the consensual knowledge of some community.

Location

PhysicalProduct

PhysicalLocation

VirtualProduct

Price

Delivery

Order

CustomerProduct

has has

isa

isa

isa of to

according to

Figure 1: e-commerce ontology**Michael Johnson and C.N.G. Dampney. "On Category Theory as a (meta) Ontology for Information Systems Research". FOIS'01, October 17-19, 2001, Ogunquit, Maine, USA.

Semantic conceptualization Formal and

explicit

Shared and relative

Logic-oriented

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What is an Ontology? (dictionaries)

Dictionaries and Ontologies.An ontology is similar to a dictionary, but has greater detail and structure. In the IFF there is a strong analogy between dictionaries and meta-ontologies.– Builders. There is a correspondence between the builders of dictionaries and those of

meta-ontologies. Corresponding to the lexicographers, who create dictionaries, are the ontologicians (mathematicians, particularly category-theorists) who create meta-ontologies.

– Source material. There is a correspondence between the source material used by dictionaries and that used by meta-ontologies. The entries placed and described in dictionaries have three sources:

1. terms borrowed from other dictionaries, 2. new terms used to express concepts in works of literature, and 3. new terms used to express concepts in everyday speech.

By analogy, the concepts axiomatized in meta-ontologies originate from three sources:1. terms borrowed from other meta-ontologies, 2. new metalevel terms used to express concepts in object-level ontologies, and 3. new metalevel terms used to express the conceptual structure of a community.

For both dictionaries and ontologies, the second source is most important. – Source creators. There is a correspondence between the source creators used by

dictionaries and those used by meta-ontologies. Corresponding to the literary figures who originate new terms in works of literature are the knowledge engineers who originate and use new metalevel terms in object-level ontologies.

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What is an Ontology? (distinctions)

The object/meta distinction.– An object-level ontology represents some aspect of the “real world”. We distinguish

between populated and unpopulated ontologies. Unpopulated ontologies have only type information [aka schemas or theories]. Populated ontologies have both type and instance information, plus the classification relationship

between these two kinds of things [aka databases or logics]. – A meta-level ontology is an ontology about ontologies. It represents some aspect of the

organization of object-level ontologies. The prescriptive/descriptive distinction.

– Both dictionaries and ontologies come in two basic philosophies: prescriptive or descriptive. A descriptive dictionary or ontology describes actual usage. Most modern dictionaries are descriptive. The IFF is a descriptive metatheory, since it uses the constraint called “conceptual warrant”.

The monolithic/modular distinction.– The monolithic-modular distinction is important for the maintenance of object-level

ontologies. A monolithic ontology is one-size-fits-all. The monolithic approach is not compatible with the need for continual revision and consistency checking. The modular approach, which advocates the lattice and category of theories constructions, is very compatible with these needs.

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The SUO-IFF Project

The IEEE P1600.1 Standard Upper Ontology (SUO) project – Aims to specify an upper ontology that will provide a structure and a set of general

concepts upon which object-level ontologies can be constructed.– An upper ontology is limited to concepts that are meta, generic, abstract and philosophical.

John Sowa’s “Lattice of Theories”– A framework is created which can support an open-ended number of theories (potentially

infinite) organized in a lattice 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.

The Information Flow Framework (IFF).– A descriptive category metatheory that represents the structural aspect of the SUO

containing meta, generic, abstract concepts.– Its institutional approach to logical semantics provides a principled framework for the

modular design of object-level ontologies; in particular, John Sowa’s “lattice-of-theories” framework.

object-level

metalevel

01

23

n

…

… CORE

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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.

Institutions: Abstract and generalize Tarski’s "semantic definition of truth". Formalize, represent, implement and translate the notion of "a logic".

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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.gzMathematical 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

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IFF Design Guidelines

In the development of the IFF, certain guidelines have proven to be very important. Goal. The IFF metatheory was designed to represent first order logic, its

languages, theories, model-theoretic structures and (local) logics, including satisfaction. This metatheory incorporates the theory of institutions of Goguen and Burstall.

Meta-guideline. Follow the intuitions of the working category-theorist. Such intuitions represent naive category theory. The meaning of naive here is not pejorative. It means primitive, natural, intuitive, first-formed, primary, not evolved or elemental.

Practice. Initially formulate any IFF module as a set-theoretic axiomatization using a first order expression. Eventually, by eliminating quantifiers and logical connectives, move, morph or transform this set-theoretic axiomatization toward a category-theoretic axiomatization.

Rule-of-thumb. Keep it simple! From the foundational standpoint, this means that we start with no assumptions at all. However, in view of the Cantor diagonal argument, as a first step we assume a slender hierarchy called the IFF metastack.

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IFF Development Phases Phase 1: (2000-2001)

– First ontology axiomatized: The IFF Category Theory (meta) Ontology (IFF-CAT). – A non-starter: a topos axiomatization. Objections due to its lack of warrant.

Warrant means evidence for or token of authorization. Conceptual warrant is an adaptation of the librarianship notion of literary warrant.

Phase 2: (2001-2003)– Designed bottom-up; Require conceptual warrant; Follow categorical design principle.– Important metalogic concept incorporated: finite limits; for example, composition of class functions requires

the pullback concept. Phase 3: (2004-2006)

– Central task: reorganization of the IFF metastack using "adjunctive axiomatization". This will provide the metastack with better structure. This will simplify the IFF-KIF meta-language core by not requiring the definite descriptive operator.

– Important metalogic concept incorporated: exponents. – Fibrations and indexed categories will also be central.– Institutions will be fully axiomatized.

Principles. During the IFF development, four concepts have eventually emerged as important. One of these concepts, the IFF metastack, is discussed elsewhere. The other three concepts are principles for IFF development. In chronological order these are

– the principle of conceptual warrant,– the principle of categorical design and– the principle of institutional logic.

Conceptual warrant restricts the introduction of upper metalevel terminology, whereas categorical design forces the introduction of this terminology, principally in the core.

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Principle: All IFF terminology should require conceptual warrant for their existence: any term that appears in (and is axiomatized by) a metalanguage should reference a concept needed in a lower metalevel or object level axiomatization.

Note: The terminology appearing in any standardization meta-ontology will exert authority. Because of this, in selecting which terminology to specify in the IFF, we utilize the notion of “conceptual warrant”. Warrant means evidence for, or token of, authorization.

Analogue: Conceptual warrant is an adaptation of the librarianship notion of literary warrant. According to the Library of Congress, its collections serve as the literary warrant (i.e., the literature on which the controlled vocabulary is based) for the Library of Congress subject headings system. In the same fashion, the object-level (n = 0) and lower metalevel (n = 1) terminology of the IFF serves as the conceptual warrant for the IFF upper metalevel (n = 2) axiomatization.

The Principle of Conceptual Warrant

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The Principle of Categorical Design

Principle: The design of a module at any particular metalevel should adhere to the property that its axiomatic representation is strictly category-theoretic:

– All axioms use terms from the metalanguage at that metalevel.– All axioms are atomic: no axioms use explicit logical notation; no variables, quantification

(‘forall’, ‘exists’) or logical connectives (‘and’, ‘or’, ‘not’, ‘implies’,‘iff’) are used. Note: The peripheral (non-core) modules in the lower IFF metalevel (n = 1) have the

tripartite form: outer category namespace, inner object and morphism namespaces. – Currently, the outer namespace follows the categorical design principle exactly.– And the inner namespaces follow it to a great extent (80–90%).

“A (level n) category can be thought of as a special kind of graph – a graph with monoidal properties. It consists of an underlying graph, a composition graph morphism and an identity graph morphism, both with an identity object function.”

((#n+1).set:set category) ((#n+1).ftn:function graph) ((#n+1).ftn:function underlying) (= underlying graph) (= ((#n+1).ftn:source graph) category)(= ((#n+1).ftn:target graph) #n.gph.obj:graph)(((#n+1).rel:parametric #n.gph.obj:multipliable) graph graph) ((#n+1).ftn:function graph-pair)(= ((#n+1).ftn:source graph-pair) category)(= ((#n+1).ftn:target graph-pair) ((#n+1).rel:extent #n.gph.obj:multipliable))(= graph-pair (((#n+1).rel:mediator #n.gph.obj:multipliable) [graph graph]))

((#n+1).ftn:function mu)(= ((#n+1).ftn:source mu) category)(= ((#n+1).ftn:target mu) #n.gph.mor:2-cell)(= ((#n+1).ftn:composition [mu #n.gph.mor:source]) ((#n+1).ftn:composition [graph-pair #n.gph.obj:multiplication]))(= ((#n+1).ftn:composition [mu #n.gph.mor:target]) graph) ((#n+1).ftn:function eta)(= ((#n+1).ftn:source eta) category)(= ((#n+1).ftn:target eta) #n.gph.mor:2-cell)(= ((#n+1).ftn:composition [eta #n.gph.mor:source]) ((#n+1).ftn:composition ((#n+1).ftn:composition [graph #n.gph.obj:object]) #n.gph.obj:unit]))(= ((#n+1).ftn:composition [eta #n.gph.mor:target]) graph)Example: Axioms for a Level n Category

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Part I. The IFF Metatheory

The IFF Architecture – (old version)– (new version)– The IFF Metastack– Four Fundamental Relations

The IFF Code– IFF (meta) Ontologies– Some Example IFF Terms– The Basic IFF Terms

The IFF Metalanguages – (old version)– (new version)

CORE

object-level

metalevel

object-level

metalevel

01

23

n

…

… CORE

OLDVERSION

NEWVERSION

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The IFF Architecture(old version)

Ur

Core

Top

Core

Upper

Institutions, Classifications,

Concept Lattices Core

Category Theory, Graph Theory,

Relations, Order Theory

Lower

1st Order Logic, Model Theory,

Common Logic, etc.

Institutions,

Classifications, Concept Lattices

Core

Category Theory, Graph Theory,

Relations, Order Theory

Consists of metalevels, namespaces and meta-ontologies. 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.

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The IFF Architecture(new version)

meta dgm lim expn gph cat func nat adj top fbr clsn info clg inst trm expr fol lang mod th

ur=

… … … … …

n

… … … … …

vlrg=3

lrg=2

sml=1

metalevel

objectlevel

0 SUO Botany Ontolingua WordNet SENSUS Holes Gene SUMO

Cyc Enterprise e-commerce Government Education HPKB SemWeb …

type IFF-CORE IFF-CAT IFF-INST IFF-FOL IFF-ONT

Metalevels, namespaces and meta-ontologies. Generic metalevels n with Ur supremum Goal: n/Ur metalevels satisfy Categorical Design Principle Upper metalevel (n = 2)

– Declare, define, axiomatize and reason about generic categories, functors, adjunctions, colimits, monads, classifications, concept lattices, etc.

Lower metalevel (n = 1)– Declare, define, axiomatize and reason about particular categories,

functors, adjunctions, colimits, monads, classifications, concept lattices, etc.

IFF-CORE forms a hierarchy of toposes: – Set1 Set2 Set3 … Setn … Set

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The IFF Metastack

There are axiomatizations for categories, finite (and general) limits/colimits, exponents and subobject classifiers.

The IFF metastack represents the chain of categories (toposes)Set1 Set2 Set3 ... Setn ... Set

The IFF metastack is the core hierarchy of the IFF. Its backbone (kernel) is pictured here.

The metastack axiomatizes sets and functions (and for convenience, binary relations) in two modules:

– a generic level n module and – an Ur level module.

ftn1 set1

10

11

10

11

rel1

ftn2 set2

20

21

20

21

rel2

ftn3 set3

30

31

30

31

rel3

… … …

ftn set

0

1

0

1

rel

function set relation

= urgeneric

3 = vlrgvery large

2 = lrglarge

1 = smlsmall

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Four Fundamental Relations The chain of categories (toposes)

– Set1 Set2 … Setn … Set are axiomatized in the IFF Core (meta) Ontology (IFF-CORE).

These inclusions represent the fact that axiomatization at level n includes specialization of the axiomatization at level n+1.

Specialization (technical term) means the exact adaptation of the core terminology and axiomatization at level n+1 to level n by use of four fundamental relations: subset, (function) (optimal-)restriction and (relation) abridgment (plus intersection).

abridgment

An Bn

fn

An+1 Bn+1

fn+1

commutativediagram

An Rn

An B

n Bn

An+1 Rn+1

An+1 B

n+1 Bn+1

rn+1

rn

multipullbackdiagram

Cn

Cn+1

An Bn

fn

An+1 Bn+1

fn+1

pullbackdiagram

optimalrestrictionrestrictionsubset

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IFF (meta) Ontologies IFF-UR“the generic”

(category axioms) [30 terms, 7 pages]http://suo.ieee.org/IFF/metastack/UR.pdf

IFF-TCO“the very large”

(collections; finite limit axioms) [300+ terms, 60 pages]http://suo.ieee.org/IFF/metastack/TCO.pdf

IFF-UCO“the large”

(classes; exponents, finite limits and colimits) [500+ terms, 100+ pages]http://suo.ieee.org/IFF/metastack/UCO.pdfhttp://suo.ieee.org/IFF/metalevel/upper/ontology/core/version20020102.pdf

IFF-CAT (basic category theory) [220+ terms, 53 pages]http://suo.ieee.org/IFF/metalevel/upper/ontology/category-theory/version20020102.pdf

IFF-CLS (basic information flow and formal concept analysis) [280+ terms, 78 pages]http://suo.ieee.org/IFF/metalevel/upper/ontology/classification/version20020102.pdf

IFF-INS (rudimentary institution theory)http://suo.ieee.org/IFF/work-in-progress/INS/institution-namespace.pdf

IFF-LCO“the small”

(sets; quartets; arbitrary limits and colimits) [many, many terms, 150+ pages]http://suo.ieee.org/IFF/metalevel/lower/ontology/core/version20020515.pdfhttp://suo.ieee.org/IFF/metalevel/lower/namespace/set/version20030402.pdf

IFF-SCL (simple common logic) [~100 terms, 36 pages]http://suo.ieee.org/IFF/metalevel/lower/namespace/scl/version20040505-obj.pdf

IFF-ONT (logic and ontology axiomatization)Language Namespace [320+ terms, 95 pages]

http://suo.ieee.org/IFF/metalevel/lower/namespace/type-language/version20021205.pdfTheory Namespace [170+terms, 49 pages]

http://suo.ieee.org/IFF/metalevel/lower/namespace/theory/version20021205.pdfModel Namespace [80+ terms, 36 pages]

http://suo.ieee.org/IFF/metalevel/lower/namespace/model/version20021205.pdfLogic Namespace [40+ terms, 14 pages]

http://suo.ieee.org/IFF/metalevel/lower/namespace/logic/version20021205.pdf

IFF-CORE Ur generic (level n) n = 3 n = 2 n = 1

IFF-CAT Ur generic (level n) n = 3 n = 2 n = 1(c

urre

nt)

(future)

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Some Example IFF Terms

Core PeripheryIFF-KIF ‘and’, ‘or’, ‘implies’, ‘iff’, ‘forall’, ‘not’, ‘forall’, ‘exists’

IFF-Ur ‘object’, ‘morphism’, ‘relation’, ‘subobject’, ‘restriction’, ‘abridgement’, ‘composition’, ‘identity’

IFF-Top ‘collection’, ‘function’, ‘relation’, ‘subcollection’, ‘restriction’, ‘abridgement’,‘composition’, ‘identity’

‘collection-pair’, ‘parallel-pair’, ‘opspan’, ‘binary-product’, ‘equalizer’, ‘pullback’

IFF-Upper ‘class’, ‘function’, ‘relation’, ‘subclass’, ‘restriction’, ‘abridgement’, ‘composition’, ‘identity’

‘graph’, ‘partial-order’, ‘total-order’, ‘equivalence-relation’‘category’, ‘functor’, ‘natural-transformation’, ‘adjunction’, ‘monad’, ‘classification’, ‘infomorphism’, ‘concept’, ‘concept-lattice’, ‘bond’

‘unary-function’, ‘binary-function’,‘curry’, ‘hom’, ‘exponent’

IFF-Lower ‘set’, ‘function’, ‘relation’, ‘subset’, ‘restriction’, ‘abridgement’, ‘composition’, ‘identity’

‘entity’, ‘relation’, ‘arity’, ‘language’, ‘language-morphism,’ ‘theory’, ‘theory-morphism’, ‘logic’, ‘logic-morphism’

‘diagram’, ‘colimit’, ‘direct-flow’, ‘inverse-flow’

Object level ‘person’, ‘company’, ‘employ’, ‘salary’, ‘date’, ‘real-number’, etc.

Object level (level 0) ontologies use lower metalevel (level 1) terminology and functionality.

– Example: Theory functionality (IFF-th) could be used to organize an E-commerce ontology.

Lower metalevel ontologies use upper level terminology and functionality.– Example: Category theory functionality (IFF-CAT) is used to organize the language

ontology (IFF-lang). Upper metalevel ontologies use top level terminology and functionality.

– Example: Core top level functionality (IFF-TCO) is used to axiomatize the category theory ontology (IFF-CAT).

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The Basic IFF Terms

'abridgment''antisymmetric''bijection''binary coproduct''binary intersection''binary product''binary union''bond''bonding pair''categorical equivalence''category''class''classification''cocone''coequalizer''colimit''colimit injection''collection‘'composition''concept lattice‘'cone'

'constant function''coproduct''currying' 'diagram''disjoint''domain''empty''epimorphism''equalizer''equivalence relation''evaluation''exponent''function''functional relation''functor‘'hom‘'hypergraph''identity''inclusion''infomorphism‘'initial'

'injection''institution''institution morphism''inverse currying''isomorphic''isomorphism''language''limit''limit projection''mediator''monad‘'monomorphism''morphism''mor2rel' 'natural transformation''nothing''null''object''opspan''one‘'order'

'packing''pair''parallel pair''partial order''partial function''pfn2ftn''pfn2rel‘'product''product with''pullback''pushout''reflexive' 'relation''restriction''set''signature''source''span''spangraph''subclass‘'subcollection'

'subordination''subset''surjection''symmetric‘'target''terminal''ternary coproduct''ternary product''thing''three‘'total order''total relation''transitive''triple''two''unique function''unique morphism''unit''unpacking''vocabulary''zero'

Most of the thousands of IFF terms are necessary, but conceptually derived.

Basic IFF terms (100+) are being defined in “A Dictionary for Basic Terms”

With the new IFF architecture, some terms will be combined: ‘set’, ‘class’ and ‘collection’ will become ‘#n.set:set’ for n = 1, 2 and 3.

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The IFFMetalanguages(old version)

Each metalevel has an associated metalanguage– It contains the metalanguage directly above it.– Its terminology is defined by the meta-ontologies at that level.– It is used to axiomatize the (meta-)ontologies at all lower levels.

Introductions– IFF-Ur – axiomatization for categories– IFF-Top – axiomatization for finite limits– IFF-Upper – axiomatization for exponents and finite colimits– IFF-Lower – axiomatization for subobjects and general limits/colimits

IFF-KIF enables lisp-like first-order expression: functions, binary relations, connectives and quantifiers. Uses restricted quantification.

IFF-UrIFF-Top

IFF-UpperIFF-Lower

IFF-KIF

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The IFFMetalanguages(new version)

Each metalevel has an associated metalanguage– It contains the metalanguage directly above it.– Its terminology is defined by the meta-ontologies at that level.– It is used to axiomatize the (meta-)ontologies at all lower levels.

IFF metashell– Logical part: enables lisp-like first-order expression: sets, unary

functions, binary relations, connectives and quantifiers. Uses restricted quantification with guards and definite description.

– Mathematical part: (IFF-META namespace) meta-terminology allowing IFF-CORE to conform to the principle of categorical design.

IFF meta-ur & IFF meta-n: (IFF-CORE ontology) introduces axiomatization for toposes; that is, for categories, finite limits/colimits, exponents, subobjects and general limits/colimits

meta-ur

meta-n

meta-1meta-2

metashell

meta-3

…

…

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Part II. The IFF Institutional Application

Two IFF Efforts to Represent FOL– The IFF-ONT

(old version) (new version)

– The IFF-FOL Logical Environments

– Languages Language Expressions Language Colimits

– Models– Theories

The Category of Theories Theory Colimits (fusion)

– Truth Classification & Truth Concept Lattice– Truth Concept Lattice Functionality

Institutions– The Grothendieck Construction– The Truthful Connection

object-level

metalevel

01

23

n

…

… CORE

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Two IFF Efforts to Represent FOL

Non-Traditional FOL– The IFF Ontology (meta) Ontology (IFF-ONT)

old version new version

Traditional FOL– The IFF First Order Logic (meta) Ontology (IFF-FOL)

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The IFF Ontology (meta) Ontology

(IFF-ONT) (old version)

Nontraditional. Home page: http://suo.ieee.org/IFF/metalevel/lower/ontology/ontology/version20021205.htm. Based on view that n-ary relations incorporate a notion of hypergraphs. Also, novel definition of models via classifications: t | r instead of r(t). Completely axiomatized: Language, Theory, Model and Logic. Language and Theory are cocomplete. Free models and logics exist. Models are nonstandard: they have a subset of abstract tuples. Problem:

– Model fiber functions must be restricted to bijective variable functions.– Must limit Language and Theory to corresponding subcategories Language≐ and Theory≐

– Language≐ (hence, Theory≐) is not cocomplete; in fact, does not have coproducts.– Analysis: reference (sort) function is too inflexible (Note: such a sort function is

recommended in Enderton's text!).

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The IFF Ontology (meta) Ontology

(IFF-ONT) (new version)

Traditional. Discussion and links at http://suo.ieee.org/IFF/work-in-progress/#IFF-OO. Still based on view that n-ary relations incorporate a notion of hypergraphs. In fact, identifies languages with hypergraphs (of a certain kind). This kind of hypergraph does not use a reference (sort) function. Languages (hence, theories) are cocomplete. Model fiber function exists. Special advantage: hypergraphs are equivalent to spangraphs.

– Spangraphs are a more flexible definition of logical language (in terms of relational arity).– Spangraphs nicely model the SCL "role-set syntax", where arguments form a set of role-

value pairs. Example: ‘(Married (role-set: (wife Jill) (husband Jack)))’

This notion has been advocated in SCL development by Pat Hayes. Position and argument order are not needed. According to Hayes, this provides some insurance against communication errors.

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The IFF First Order Logic (meta) Ontology (IFF-FOL)

Traditional Discussion and links at http://suo.ieee.org/IFF/work-in-progress/#FOL . Has a very modular architecture.

– Central bifurcation is between terms and expressions. FOL languages = Expression Languages Sets and Bijections (variables) Term Languages FOL languages with equality = FOL languages Term Languages Universal Algebra FOL languages with equality: pullback relations along functions and then along equations.

– Term Languages: Functions symbols and variables. Lawvere construction is defined here. Equations can be added giving equational languages (equational presentations) as an extension of term languages. They define a quotient of their Lawvere category.

– Expression Languages: Relation symbols and variables. Peircian existential graphs can be included here.

– FOL Languages: Function symbols, relation symbols and variables. An FOL language is a term language and an expression language that share a common set of variables.

Important submodules in the axiomatization– term/tuple fixpoint, expression/arity fixpoint, Lawvere construction, term-tuple coproducts,

term monad and expression monad.

LISTIC/ESIA ~ Oct 2005

Logical Environments

Distinctions– Potential versus actual– Formal versus interpretive

Four Fundamental Notions – Language

basic formalism– Theory

formal or axiomatic semantics– Model

actual or interpretive semantics– Logic

combined semantics

Language

Logic

ModelTheory

LISTIC/ESIA ~ Oct 2005

rel(L)

sign(L)

var(L)arity(L)

tuple(refer(L)) ent(L)

Xrefer(L)

Languages

We assume that each language has an adequate, possibly denumerable, set of variables X. This generalizes the usual case where sequences are used – the advantage for this generality is elimination of the dependency on sequences and natural numbers. Cardinal numbers represent a skeletal quotient.

A language L = rel(L), ent(L), arity(L), sign(L):– a relation type set rel(L), – an entity type set ent(L), – a reference set pair L = refer(L) = X, ent(L),

– an arity function L = arity(L) = sign(L) · tuple-arity(L) : rel(L)  X, and – a signature function L = sign(L) : rel(L)  tuple(refer(L)).

For any relation type ρ  rel(L)arity(L)(ρ) = {x, x, … xn  X is its arity with external form ρ, x, x, … xn,

L(ρ) : arity(L)(ρ)  ent(L) is its signature with external form ρ, x : ε, x : ε, … xn : εn,

where L(ρ)(xk) = εk for  < k < n.

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Language Expressions Associated with any language L = ent(L), rel(L), L, L, L is an expression language expr(L) =

ent(L), rel(expr(L)), L, expr(L), expr(L) that extends L.– The set of entity types and the reference set pair of expr(L) are the same as L. – The relation types of expr(L) are L-expressions. The set rel(expr(L)) of L-expressions is inductively defined

on relation types rel(L) using the logical quantifiers and connectives.– The arity and signature functions

expr(L) = arity(expr(L)) : rel(expr(L))  X expr(L) = sign(expr(L)) : rel(expr(L))  tuple(refer(L))

are recursively-defined extensions from the relation types of L to the expressions of L. Sentences are expressions with empty arity .

Associated with any language morphism f = rel(f), ent(f) : L  L is an expression language morphism expr(f) = rel(expr(f)), ent(f) : expr(L)  expr(L) that extends f.

– The entity type function is identity.– The relation function rel(f) : rel(L)  rel(L) is recursively extended to an expression function

rel(expr(f)) : rel(expr(L))  rel(expr(L)). There is a language morphism L = embed(L) : L  expr(L) that embeds any language into its

associated expression language. – The entity function is the identity. – The relation function rel(L) : rel(L)  rel(expr(L)) embeds relation types as expressions.

There is a language morphism L = collapse(L) : expr(expr(L))  expr(L) that collapses the expression of expression language of any language into its expression language.

– The entity function is identity.– The relation function rel(L) : rel(expr(expr(L)))  rel(expr(L)) collapses expressions.

LISTIC/ESIA ~ Oct 2005

Language Colimits

Language colimits are important: – they are the basis for theory colimits.

They involve two opposed processes: “summing” “keeping things apart” “preserving distinctness” “quotienting” “putting things together” “identification” “synonymy”

The “things” involved here are symbols:– relation type symbols– entity type symbolsand the concepts they denote.

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Models

A model M is a product of a language (= hypergraph) and a classification.– rel(M) is a relation classification; instances are abstract “tuples”.– ent(M) is an entity classification; types are “sorts”. Entity types are unary relation types.– The type aspect typ(M) = typ(rel(M)), typ(ent(M)), arity(M), sign(M) is a language.– The instance aspect inst(M) is a hypergraph.

This approach is comparable to the structure of Aristotle's ontological framework. – The two distinctions in Aristotle's ontological framework of “universal versus particular” and of “quality

versus substance” are analogous to the two distinctions in the semantic architecture of the IFF between “type versus instance” and “relation versus entity”.

There is a type functor from models to languages typ : Model Language. There is a model functor from languages to classes mod = typ : Language SET.

– For fixed language L we denote by mod(L) the class of models of that type.– For any language morphism f : L L, we can define a model fiber function

mod(f) : mod(L) mod(L).– There is a satisfaction relation

M |L between models M mod(L) and expressions   expr(L) (hence, sentences).

relation instance= tuple

entity type

entity instance

= the universe

hypergraphrelation entity

instance

classification

type relation type

(prehension)(nexus) (actuality)

universal quality

individual quality

universal substance

individual substance

(prehension)quality substance

particular

universal

(nexus) (actuality)

The Information Flow Framework– mathematical knowledge organization

Aristotle's ontological framework– philosophical knowledge organization

LISTIC/ESIA ~ Oct 2005

Theories

A theory T = base(T) axm(T) consists of an underlying language L = base(T) and a set of sentences axm(T)  sen(L) called the axioms of T. Theories represents formal or axiomatic semantics. We assume a fixed common variable set X for all languages.

– An L-model M is a model of the theory T, denoted M |LT, when M satisfies each axiom   axm(T); that is, when M |L for all axioms   axm(T).

– A theory T entails a sentence   sen(L) when M |LT implies M |L for any model M. Such a sentence is called a theorem of T. Any axiom is a theorem. The set of theorems of T is denoted by thm(T). The theory clo(T) = T = L, thm(T) is called the closure of T. A theory is closed when it equals its closure.

– Two theories T and T are compatible when they share the same type language: base(T) = base(T). A theory T is a specialization of a compatible theory T when T entails every theorem of T; that is, when thm(T) thm(T). Then we also say that T is a generalization of T, and denote this by T T. This is a preorder on theories over L – any theory T is equivalent to its closure T clo(T).

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The Category of Theories

Contexts:– Theory– Closed Theory– Language

Passages:– theories indexed by languages

base : Theory Language.– entailment closure of theories

clo : Theory Closed Theory. Facts:

– Theory morphisms in fiber equivalent to the (opposite) lattice ordering.– Fusion in the context of theories is direct image flow followed by meet in the

lattices of theories (fibers).

Theory

ClosedTheory

T1

T2

f

T0 clo(T0)

Ť1

Ť2

LanguageL

L1

L2

f

base

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Theory Colimits (Fusion)

1. Informally, identify the theories to be used in the construction.

2. Alignment: Formally, create a diagram of theories T of shape (indexing) graph G that indicates this selection. This diagram of theories is transient, since it will be used only for this computation. Other diagrams could be used for other fusion constructions.

3. Unification: Form the fusion theory T = T of this diagram of theories, with theory fusion cocone : T T.a. Compute the base diagram of languages L = base(T) with the same

shape. In more detail, L = base(T) = {Lk} + {Le : Lm Ln} = {base(Tk)} + {base(Te) : base(Tm) base(Tn)}.

b. Form the fusion language L = L of this diagram, with language fusion cocone : L L. In more detail, = {k : Lk L}, satisfying the conditions m = Le · n for G-edge e : m n.

c. Move (the individual theories {Tk} in) the diagram of theories T from the lattice of theory diagrams fiber(L) along the language morphisms in the fusion cocone : L L to the lattice of theories fiber(L) using the direct image function, getting the homogeneous diagram of theories dir()(T) with the same shape G, where each theory dir()(T)k = dir(k)(Tk) has the same base language L (the meaning of homogeneous).

d. Compute the meet (union) of the diagram dir()(T) within the lattice fiber(L) getting the fusion theory T = T = meet(L)(dir()(T)).

e. The language fusion cocone is the base of the theory fusion cocone: = base() : base(T) base(T).

Theory

Language

base

L

L

T

T

dir()(T)

LISTIC/ESIA ~ Oct 2005

Truth Classification & Truth Concept Lattice

Truth classification of 1st-order language L– Instances are L-models.– Types are L-sentences.– Incidence is satisfaction: M |= when is true in M.

Truth concept lattice of L– In the IFF approach, this is the appropriate

“lattice of ontological theories.” – Formal concept is a pair c = ext(c), int(c) where:

The int(c) is a closed theory (set of sentences). The ext(c) is the collection of all models for that theory.

– Lattice order is the opposite of theory inclusion.– The join or supremum of two theories is the intersection of the theories.– The meet or infimum of two theories is the theory of the common models. – Both L-models and L-sentences generate formal truth concepts (theories).

An object concept is the theory of a model. An attribute concept is the models of a sentence.

1st-order model

1st -order sentence

⊤= all models, true sentences

c = ext(c), int(c)

⊥ = no model, all sentences

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Truth Concept Lattice Functionality

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)

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Institutions

I = Sign, Mod, Sen, ⊨http://www.cs.ucsd.edu/users/goguen/projs/inst.html

Sign: an abstract category of signatures . Sen : Sign Set: a sentence functor indexing abstract sentences Sen() by signatures . Mod : Sign SETop: a model functor indexing abstract models Mod() by signatures . | Mod()Sen(): a signature parameterized satisfaction relation. Satisfaction Condition (expresses the invariance of truth under change of notation): For any signature

morphism f : ', any '-model M' and any -sentence e, M' |' Sen(f)(e) iff Mod(f)(M') | e.

Spec : Sign Set: a specification functor indexing abstract theories (specifications) Spec() by signatures . A -specification is a set of -sentences.

I : Sign CLSN: a classification functor is a concise definition of an institution. But CLSN CLG. Î : Sign CLG: a concept lattice functor is an alternate definition of an institution. Example institutions

– first order logic (with first order structures as models)– many sorted equational logic (with abstract algebras as models)– Horn clause logic– variants of higher order and of modal logic

LISTIC/ESIA ~ Oct 2005

The Grothendieck Construction

Objects:

(L, M), L a language, M a model in mod(L) (L, T), L a language, T a theory in th(L)

Morphisms:f : (L, M) (L, M)f : L L a language morphismM mod(f)(M) in mod(L)

f : (L, T) (L, T)f : L L a language morphismT th(f)(T) in th(L) or equivalently T th(f)(T) in th(L)

Models and theories are indexed by languagesmod : Language SET and th : Language Set

Using satisfaction, these can be lifted to ordersmod : Language ORDop and th : Language Ord

Using the Grothendieck construction, these indexed orders can be combined into a “flattened” category with an index projectiontyp : Model Language and base : Theory Language

Logic

Prologic

Language

ModelTheory

mod

Tbase

th mod

th

incl

max-th

typ

Every model has a associated theory maximal in subset order max-th(L) : mod(L) th(L). Using the Grothendieck construction, since this is natural in L, models map to theories

max-th : Model Theory

Structured institutions are known as “logical environments”. These are discussed further in the documenthttp://suo.ieee.org/IFF/metalevel/lower/metatheory/environment/version20041010.html.

LISTIC/ESIA ~ Oct 2005

The Truthful Connection

The Information Flow Framework The Theory of InstitutionsCategory of Languages: Language Category of Signatures: Sign

Language L Signature

Language Morphism f : L1 L2 Signature Morphism : 1 2

Expression Functor expr(f) : expr(L1) expr(L2) Sentence Functor Sen() : Sen(1) Sen(2)

Expression Functor: expr : Language Language Sentence Functor: Sen : Sign Set

Category of Theories: Theory Category of Specifications: Spec

Theory T = base(T), axm(T) Specification S = , A

L = base(T)axm(T) expr(L)

is a signatureA is a set of -sentences

Theory Morphism f : T1 T2 Specification Morphism : S1 S2

expr(f)(axm(T1)) thm(T2) f : L1 L2 maps S1-axioms into S2-theorems : 1 2

Model Fiber Functor: mod : Language CATop Model Functor: Mod : Sign CATop

Model M = ent(M), rel(M) Model m

Truth Framework Institution I = Sign, Mod, Sen, |=

truth classification: truth(L) = mod(L), expr(L), |=L

M |=L e means model M satisfies expression e

satisfaction relation: |= |Mod()| Sen()

truth infomorphism:truth(f) = mod(f), expr(f) : truth(L1) truth(L2)

mod(f)(M2) |=L1 e1 iff M2 |=L2 expr(f)(e1),

for all models M2 mod(L2), expressions e1 expr(L1)

satisfaction condition:

Mod()(m2) |=1 s1 iff m2 |=2 Sen()(s1),for all models m2 Mod(2), sentences s1 Sen(1)

LISTIC/ESIA ~ Oct 2005

Summary

The IFF is a descriptive category metatheory whose architecture contains nested metalanguages.

The main institutional application of the IFF is axiomatized in the Ontology (meta) Ontology (IFF-ONT) and the First Order Logic (meta) Ontology (IFF-FOL). These form flexible and general institutions for first order logic.

Institutions formally express semantic integration as an ontological fusion process. Each community represents their conceptual space in their own terms, and connects with others via morphisms that enable ontological alignment specification.

Institutions represent logical environments and institution morphisms connect logical environments. Amongst other things, the IFF is in the process of axiomatizing the theory of institutions.

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Appendix

What is an Ontology? 1. (other views)2. (other views)

Language Morphisms Language Sums Language Endorelations Language Quotients Theory Morphisms Theory Sums Theory Endorelations and Quotients

object-level

metalevel

01

23

n

…

… CORE

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What is an Ontology? (other views)

B. Chandrasekaran, Jorn R. Josephson, V. and Richard Benjamins. "Ontological analysis clarifies the structure of knowledge. Given a domain, its ontology forms the heart of any

system of knowledge representation for that domain. Without ontologies, or the conceptualizations that underlie knowledge, there cannot be a vocabulary for representing knowledge …. Second, ontologies enable knowledge sharing."

Tom Gruber. "An ontology is an explicit specification of a conceptualization. The term is borrowed from philosophy, where

an ontology is a systematic account of existence. For AI systems, what 'exists' is that which can be represented."

Jeff Heflin. "An ontology defines the terms used to describe and represent an area of knowledge. Ontologies are used

by people, databases, and applications that need to share domain information (a domain is just a specific subject area or area of knowledge, like medicine, tool manufacturing, real estate, automobile repair, financial management, etc.). Ontologies include computer-usable definitions of basic concepts in the domain and the relationships among them (note that here and throughout this document, definition is not used in the technical sense understood by logicians). They encode knowledge in a domain and also knowledge that spans domains. In this way, they make that knowledge reusable.... Ontologies are usually expressed in a logic-based language, so that detailed, accurate, consistent, sound, and meaningful distinctions can be made among the classes, properties, and relations."

AAAI Topics: Ontologies

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What is an Ontology? (other views)

Christopher Welty. "Ontology is a discipline of philosophy whose name dates back to 1613 and whose practice dates back to

Aristotle. It is the science of what is, the kinds and structures of objects, properties, events, processes, and relations in every area of reality. … [W]hat the field of ontology research attempts to capture is a notion that is common to a number of disciplines: software engineering, databases, and AI to name but a few. In each of these areas, developers are faced with the problem of building an artifact that represents some portion of the world in a fashion that can be processed by a machine."

John F. Sowa. "The subject of ontology is the study of the categories of things that exist or may exist in some domain. The

product of such a study, called an ontology, is a catalog of the types of things that are assumed to exist in a domain of interest D from the perspective of a person who uses a language L for the purpose of talking about D."

Tim Berners-Less, James Hendler, and Ora Lassila. "In philosophy, an ontology is a theory about the nature of existence, of what types of things exist; ontology

as a discipline studies such theories. Artificial-intelligence and Web researchers have co-opted the term for their own jargon, and for them an ontology is a document or file that formally defines the relations among terms. The most typical kind of ontology for the Web has a taxonomy and a set of inference rules."

SUO WG. "An ontology is similar to a dictionary or glossary, but with greater detail and structure that enables

computers to process its content. An ontology consists of a set of concepts, axioms, and relationships that describe a domain of interest. … An upper ontology is limited to concepts that are meta, generic, abstract and philosophical, and therefore are general enough to address (at a high level) a broad range of domain areas."

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Language Morphisms

A language morphism f = rel(f), ent(f) : L  L:– a relation type function rel(f) : rel(L)  rel(L), and– an entity type function ent(f) : ent(L)  ent(L).These must preserve arity and signature:preservation of arity:

rel(f) · arity(L) = arity(L)

L(rel(f)(ρ)) = var(f)(L(ρ)) for any relation type ρ  rel(L)

if ρ, x, x, … xn then rel(f)(ρ), x, x, … xn

preservation of signature:rel(f) · sign(L) = sign(L) · tuple(refer(f))

L(rel(f)(ρ)) = tuple(refer(f))(L(ρ)) for any relation ρ  rel(L)

if ρ, x : ε, x : ε, … xn : εn then rel(f)(ρ), x : ent(f)(ε), … xn : ent(f)(εn)

rel(L)

tuple(ref(L))

rel(L)

sign(L)

rel(f)

tuple(ref(L))

sign(L)

tuple(refer(f))

sign(f)

X

ent(L)

X

refer(L)

X

ent(L)

refer(L)

ent(f)

refer(f)

arity(f)

rel(L) rel(L)

arity(L)

rel(f)

arity(L)

X

X X

LISTIC/ESIA ~ Oct 2005

Language Sums – “keeping things apart”

Let L1 and L be two languages with common variable set X. The sum language L1+L:

– The relation type and entity type sets are the disjoint unions of their components rel(L1+L) = rel(L1)+rel(L) and ent(L1+L) = ent(L1)+ent(L).

– The variable set is the fixed set X, var(L1+L) = var(L1) = var(L) = X.– The arity function is the pairing

arityL1L = L1L = L1, L : rel(L1)rel(L) X with L1L(1) = L1(1) for 1  rel(L1) and L1L() = L() for   rel(L).

– The signature function is the pairing signL1L = L1L = L1, L : rel(L1)rel(L)  tuple(refer(L1L)) with L1L(1) = L1(1) for 1  rel(L1) and L1L() = L() for   rel(L).

The sum injection language morphism in(L1) = inrel(L1), inent(L1) : L1  L1L: – The relation function is the disjoint union injection

inrel(L1) : rel(L1)  rel(L1L) = rel(L1)rel(L); – the entity function is the sum injection

inent(L1) : ent(L1)  ent(L1L) = ent(L1)ent(L).  The sum L1+L is a coproduct in the category Language(X).

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Language Endorelations Let X be any set representing a fixed set of variables. Let endo(X) denote the class of

all language endorelations at X. A language endorelation is a triple J = lang(J), rel(J), ent(J) consisting of

– a language lang(J) = L with variable set X, – a binary endorelation rel(J) = R rel(L)rel(L) on relation types of L, and – a binary endorelation ent(J) = E ent(L)ent(L) on entity types of L.

These must satisfy the following constraints on arity and signature functions of L. Here R is the equivalence relation generated by rel(J) = R and E is the equivalence relation generated by ent(J) = E. For any pair of relation types ρ1, ρ2  rel(L), if (ρ, ρ)  rel(J) = R, then

arity(L)(ρ) = arity(L)(ρ) and sign(L)(ρ)  sign(L)(ρ),

where   sign(refer(L))sign(refer(L)) is the equivalence relation on tuples defined as follows: τ  τ when arity(refer(L))(τ) = arity(refer(L))(τ) and τ(x) E τ(x) for all x  arity(refer(L))(τ) = arity(refer(L))(τ).

Often, the endorelations rel(J) = R and ent(J) = E are equivalence relations on relation types and entity types. However, it is not only convenient but also very important not to require this. In particular, the endorelations defined by parallel pairs of language morphisms (coequalizer diagrams) do not have component equivalence relations. A simple inductive proof shows that the triple Ĵ = L, R, E is also a language endorelation.

LISTIC/ESIA ~ Oct 2005

Language Quotients – “putting things together”

The quotient L/J of an endorelation J = L, R, E = lang(J), rel(J), ent(J) on a variable set X is the language defined as follows:

– The set of relation types are the equivalence classes rel(L/J) = rel(L)/R.

– The set of entity types are the equivalence classes ent(L/J) = ent(L)/E.– The set of variables is var(L/J) = var(L) = X.– The arity function L/J = arity(L/J) : rel(L)/R  X is defined pointwise: L/J(ρR) = L(ρ)

for all relation types ρ  rel(L).– The signature function L/J = sign(L/J) : rel(L)/R  tuple(refer(L/J)) is defined pointwise: L/J(ρR)(x) = L(ρ)(x)E for all relation types ρ  rel(L) and variables x  L(ρ).

There is a canonical quotient language morphism J = rel(J), ent(J) : L  L/J

whose relation type function is the canonical surjection rel(J) = canon(rel(J)) = ‑R : rel(L)  rel(L)/R, and

whose entity type function is the canonical surjection ent(J) = canon(ent(J)) = ‑E : ent(L)  ent(L)/E.

The fundamental property for this language morphism is trivial, given the definition of the quotient language above.

LISTIC/ESIA ~ Oct 2005

Theory Morphisms

Theories are related through theory morphisms. A morphism of theories f : T  T is a language morphism f : L  L that maps

source axioms to target theorems: sen(f)(axm(T))  thm(T). Then, a morphism of theories maps source theorems to target theorems: sen(f)(thm(T))  thm(T).

When the source and target share the same base language (they are compatible) and the base language morphism is the identity, a morphism of theories f : T  T is equivalent to a theory ordering T T, which states that theory T generalizes theory T or theory T specializes theory T.

The composition and identities for theory morphisms are defined in terms of their base type language morphisms. Theories and theory morphisms form the category Theory.

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Theory Sums

Let T1 = base(T1), axm(T1) and T2 = base(T2), axm(T2) be two theories over languages L1 = base(T1) and L2 = base(T2) with common variable set X.

The sum theory T1+T:– The base of a sum is the sum of the bases base(T1T2) = base(T1)base(T2).

– The axiom set is the (disjoint) union A1A2 of the images A1 = expr(base())(axm(T1)) of axioms from T1 or images A2 = expr(base(2))(axm(T2)) of axioms from T2.

– The set of expressions expr(base(T1T2)) = E1E12E2 falls into three disjoint parts: E1 = images of the L1-expressions; E2 = images of the L2-expressions; and E12 = mixed expressions. Clearly, A1  E1 and A2  E2.

The sum injection theory morphism in(T1) : T1  T1T: – The base of the injections is the injection of the base: base(in(T1)) = in(base(T1)).

The sum T1+T is a coproduct in the category Theory.

LISTIC/ESIA ~ Oct 2005

Theory Endorelations and Quotients

Given a theory T = base(T), axm(T), a theory endorelation J is a language endorelation J = ent(J), rel(J) over L = base(T) . Recall that J has an associated type language endorelation expr(J) = ent(J), expr(J) over expr(L).

The entity, relation and expression endorelations generate equivalence relations E  ent(L)ent(L), R  rel(L)rel(L), and R*  expr(L)expr(L).

The set of expressions on the quotient base language is expr(T/J) = expr(L)/R*. The quotient T/J = base(T/J), axm(T/J) of an endorelation J on a theory T has

– base(T/J) = base(T)/base(J) , the quotient of the base language of the base endorelation,– axm(T/J) = φR*  φ  axm(T), the collection of equivalence classes of axioms of T.

There is a canonical quotient theory morphism J : T  T/J whose base is the canonical quotient type language morphism

base(J) = base(base(J)) : base(T)  base(T/J) = base(T)/base(J) .This morphism clearly maps axioms of T to axioms of T/J.