Constituent ontologies and granular partitions Thomas Bittner and Barry Smith IFOMIS – Leipzig and...
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Transcript of Constituent ontologies and granular partitions Thomas Bittner and Barry Smith IFOMIS – Leipzig and...
Constituent ontologies and granular partitions
Thomas Bittner and Barry SmithIFOMIS – Leipzig
and
Department of Philosophy, SUNY Buffalo
http://ontologist.com
• User Ontologies for Adaptive Interactive Software Systems (with I. Nebel)
Adaptivity
<< Hypoglycaemia >> -- improves performance in recall and behavior
Our idea: User Ontologies, Competency Ontologies
vs. Statistical Stereotyping Methods
User-Ontology
vs. User-Profiles
To support adaptivity:
• Need for reasoning simultaneously with cross-cutting ontologies at different levels of granularity
Overview
• The method of constituent ontology• 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 features in the domain and the interrelations between them
Examples of constituent ontologies
Constituent ontologies
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Constituent ontologies
• Database tables • Category trees
Nice properties
• Very simple structure
• Correspond to the way people represent domains– In databases– Spreadsheets– Maps
Meta-level relations between constituent ontologies
Meta level (sub-ontologies)
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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 theoryObject-Level • 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 • Relations between 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
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
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
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
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Ordering relations between LGPs (2)
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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
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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
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NDSD
N
Composition of COs
IM
W=
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NDSD
N
°ND
SDW
N
Composition of COs
IM
W°
IM
W IM
W=
The End