Constituent ontologies and granular partitions

Thomas Bittner and Barry SmithIFOMIS – Leipzig

and

Department of Philosophy, SUNY Buffalo

Overview

• Constituent ontologies• Levels of ontological theory• The hierarchical structure of constituent

ontologies• The projective relation of constituent

ontologies and reality• Relations between constituent ontologies• Types of constituent ontologies

The method of constituent ontology:

• to study a domain ontologically – is to establish the parts and moments of

the domain and

– then to establish the interrelations between them

Examples of constituent ontologies

Constituent ontologies

I

M

W

ND

SD

N

I

M

W

Constituent ontologies

• Database tables • Category trees

Nice properties

• Very simple structure

• Very simple reasoning

• Corresponds to the way people represent domains– In databases– Spreadsheets– Maps

Meta-level relations between constituent ontologies

Meta level (sub-ontologies)

I

M

W I

M

W

ND

SD

N

x yx is sub-constituent-ontology of y

Meta-level (granularity)

Meta-level (granularity)

• Alabama• Alaska• Arkansas• Arizona• …• Wyoming

• West• Midwest• Northeast• South

Levels of granularity

• Alabama• Alaska• Arkansas• Arizona• …• Wyoming

• West• Midwest• Northeast• South

• USA

Coarse Intermediate Fine

Meta-level (themes)

USA physical• Mountains • Rivers• Planes

Meta-level (themes)

USA physical• Mountains • Rivers• Planes

USA political• Federal states

Levels of ontological theory

Constituent ontology1

Constituent ontology2

Constituent ontologyn

Levels of ontological theoryLevel of foundation• Formal relations: mereology, topology, location• Space and time• Basic categories: entities, regions, perdurants, endurants, …

Constituent ontology1

Constituent ontology2

Constituent ontologyn

Levels of ontological theoryObject-level (Taxonomies, partonomies)• Formal relations: mereology, topology, location• Space and time• Basic categories: entities, regions, perdurants, endurants, …

Meta-level• Granularity and selectivity (Theory of granular partitions)• Relations between constituent ontologies• Negation, Modality

Constituent ontology1

Constituent ontology2

Constituent ontologyn

Object-level

Levels of ontological theoryObject-level

• Formal relations: mereology, topology, location

• Space and time

• Basic categories: entities, regions, perdurants, endurants

Meta-level• Granularity and selectivity (Theory of granular partitions)• Relations between constituent ontologies

Constituent ontology1

Constituent ontology2

Constituent ontologyn

Formal relations

• Mereology (part-of) -- Partonomy• Mereotopology (is-connected-to)• Location (is-located-at)• Dependence (depends-on)• Subsumption (is-a) -- Taxonomy

Constituent ontologies

• A constituent ontology is an abstract entity

• Has constituents as parts

• Constituents are abstract entities that project onto something that is not a constituent itself

Constituent ontologies as

granular partitions

Levels of ontological theoryLevel of foundation• Formal relations: mereology, topology, location• Space and time• Basic categories: entities, regions, perdurants, endurants, …

Constituent ontology1

Constituent ontology2

Constituent ontologyn

Meta-level• Granularity and selectivity (Theory of granular partitions)

Constituent ontologies have a simple hierarchical structure

Database tables Category trees

Maps

Granular partitions: Theory A

Cell structures as Venn diagrams and trees

Animal

Bird Fish

Canary

Ostrich

Shark

Salmon

Constituent structures (1)

• minimal cells: H, He, …• non-minimal cells:

orange area, green area,yellow area (noble gases)...

• one maximal cell: the periodic table (PT)

Cell structures (2)

• - subcell relation• He noble_gases (NG) • NG PT• Partial ordering

Remember:Constituent ontologies

• A constituent ontology is an abstract entity

• Has constituents as parts

• Constituents are abstract entities that project onto something that is not a constituent itself

Granular partitions: Theory B

Projective relation to reality

Constituents project like a

flashlight onto reality

P(c, bug)

A constituent ontology is like an array of spotlights

Pets in your kitchen

Bug 1 Bug 2 Bug 3 Bug 4

Constituent 1 Constituent 2 Constituent 3 Constituent 4

Pets in your kitchen

Constituent 1

Constituent 2

Constituent 3

Constituent 4

Constituent ontology

RealityProjection

Bug 1

Bug 2

Bug 3

Bug 4

Projection of constituents

constituent ontology

Targets in reality

Hydrogen

Lithium

Projection

Projection of constituents (2)

…

Wyoming

Idaho

Montana

…

Constituent ontology

North AmericaProjection

I shall now use the notions cell and constituent synonymously!

I shall also use the notions constituent ontology and granular partition synonymously!

Projection and location

Location

L(bug,c) Being located islike being in the spotlight

Projection does not necessarily succeed

John is not located in the spotlight!L(John, c)

P(c, John)

John

Projection does not necessarily succeed

Mary is located in the spotlight! L(Mary, c)

P(c, John)

JohnMary

Misprojection

…

Idaho

Montana

Wyoming

…

P(‘Idaho’,Montana) but NOT L(Montana,’Idaho’)

Location is what results when projection succeeds

Transparency

Transparency: L(x, c) P(c, x)

P(c1, Mary) P(c2, John)

L(Mary, c1) L(John, c2)

Projection and location

Hum ans A pes U n ico rns

M am m a ls

Humans Apes

Dogs

Mammals

),Humans''( HumansP

lysuccessfulproject

NOT does Unicorn'' cell The

???),'Unicorn(' P

recognized

NOT is species The

???)L(Dogs,

Dog

)Humans'',(HumansL

Humans'' cell by the recognized

is species The Human

Functionality constraints (1)

Location is functional: If an object is located in two cells then these cells are identical, i.e., L(o,z1) and L(o,z2) z1 = z2

VenusEvening Star

Morning Star

Two cells projecting onto the same object

Functionality constraints (2)

China

Republic of China

People’s Republic of China

The same cell (name) for the two different things:

Projection is functional: If two objects are targeted by the same cell then they are identical, i.e., P(z,o1) and P(z,o2) o1 = o2

Preserve mereological structure

Helium

Noble gases

Neon

EmptyNeonHelium

gasesNobleNeon

gasesNobleHelium

EmptyNeHe

NGNe

NGHe

Potential of preserving mereological structure

Partitions should not distort mereological structure

HumansMammal

Humans''Mammal'' M am m als A p es

H um an s

Humans Apes

Dogs

Mammals

distortion

If a cell is a subcell of another cell then the object targetedby the first is a part of the object targeted by the second.

Mereological monotony

…

Helium

Noble gases

Neon

…

Helium

Noble gases

Neon

Projection does not distort mereological structure

21212,21,1 o and )( and )( zzoozPozP Projection ignores mereological structure

Well-formed constituent ontologies are granular partitions

which are such that:

• Projection and location are functions

• Location is the inverse of projection wherever defined

• Projection is order preservingIf x y then p(x) p(y)

If p(x) p(y) then x y

Mathematical Models for COs: (Z, P, )

FTM• Partial order• Unique root• Finite chain of immediate

subcells between every cell and the root

GEM

• Partial order

• Summation principle

• Extensionality

P: Z • x y P(x) P(y)

• (P(x) P(y) x y))

Constituent ontologies are mappings

Object-level

Meta-level• Granularity and selectivity (Theory of granular

partitions)

Constituent ontology1

Constituent ontology2

Constituent ontologyn

Relations between constituent ontologies (COs)

Relations between constituent ontologies

Object-level

Meta-level• Relations between constituent ontologies

Constituent ontology1

Constituent ontology2

Constituent ontologyn

Ordering relations between COs

• P1 << P2

• << is sub-partition-of• << is reflexive, transitive, antisymmetric

I

M

W I

M

W

ND

SD

N

I

M

W

ND

SD

N

Ordering relations between LGPs (2)

I

M

W

Z1

Z2

P1 P2

ff is

• one-one• into• order preserving

• if x y then f(x) f(y)• (if f(x) f(y) then x y)

P1 << P2

P2 is an extension of P1

I

M

W I

M

W

ND

SD

N

P2 is a refinement P1

<<Z1

Z2

P1 P2

f

Composition of COs

composition operation• P1 P2 = P3 iff

– P1 << P3 and – P2 << P3

IM

W°

ND

SD

N

=I

M

W

NDSD

N

Composition of COs

IM

W=

I

M

W

NDSD

N

°ND

SDW

N

Composition of COs

IM

W°

IM

W IM

W=

The End