Ontology and Context - GitHub Pagesarademaker.github.io/files/comorea2008-slides.pdf · Introduction The algebra of Contextualized Entities Formal Framework Conclusion Constraints

OutlineIntroduction

The algebra of Contextualized EntitiesFormal Framework

Conclusion

Ontology and Context

I. CafezeiroE. H. HaeuslerA. Rademaker

April 10, 2008

I. Cafezeiro E. H. Haeusler A. Rademaker Ontology and Context

OutlineIntroduction


Conclusion

Introduction

The algebra of Contextualized EntitiesEntity IntegrationContext IntegrationCombined Integration

Formal Framework

Conclusion


OutlineIntroduction


Conclusion

Ontologies and Computer Science

I Ontologies describe real world things. Hierarchicallyorganizing concepts and enriching this hierarchy withrelationships among concepts.

I A real word entity to be represented is always related to acontext. A semantically consistent body of information inwhich the entity makes sense.

I The need of contexts? In mobile applications, where theenvironment suffer dynamic re-configurations.


OutlineIntroduction


Conclusion

Proposal

I We propose an algebra for manipulate ContextualizedOntologies with a few basic formal concepts turn itaccessible.

I We adopt: (i) an homogeneous description of entities andcontexts and; (ii) maps that consistently link entities andcontexts.

I Flexibility to: (i) combine entities or contexts in several waysand; (ii) changing and inheritance of context by an entity, andother useful operations.

I The formal approach: (i) rigorous definition ofContextualized Ontologies; (ii) abstract enough to makepossible the replacement of ontologies by other knowledgerepresentation technique.


OutlineIntroduction


Conclusion

Contextualized Entities

I Entities are described by three parts: the entity itself, acontext, and a link between entity and its context.

I A triple (entity, link, context) represented by e → c , will benamed contextualized entity.

I As both entity and context are ontologies, an entity can bethe context of other entity.

I The context, gives general information about the entity orabout the environment wherein the entity operates.

I Any context can be linked to a (meta)context.

I If the entity, or context, is represented by ontologies, we callcontextualized ontology.


OutlineIntroduction


Conclusion

Constraints about Links

The link entity–context ensures the coherence. The contextpreserves the nature of the entity.

F : E → C such that F (f (e1, e2)) = F (f )[F (e1),F (e2)]

Constrains:

i any entity must have an identity link, that maps the entity toitself, and thus the entity may be viewed as a(non-informative) context of itself;

ii an entity is called domain of a link, while a context is calledcodomain of a link;

iii links can be composed in an associative way if the codomainof the first is the domain of the second. “◦” denotescomposition of links!


OutlineIntroduction


Conclusion

Entity IntegrationContext IntegrationCombined Integration

Integrating entities that share the same context

I A semantic intersection of contextualized entities.

I Is guided by the context (C ) and results E that is alsoattached to that context.

I The original entities play the role of context to the producedentity. By transitivity, C is also a context for E .

C

E1

e 1-

E2

�e2

C

E1

e 1-

E2

�e2

E

e′2-

�e ′1

If E1 and E2 give different approaches about a subject C ,then E express their agreement with respect to C .


OutlineIntroduction


Conclusion


Properties

i The diagram is commutative. E1,E2 and E are coherentwith respect of the context.

ii E is the more complete entity that makes the diagramcommute. All components of E1 and E2 linked to the sameelement in C have a corresponding in E , and nothing more.

C

E1

e 1-

E2

� e2

E �!

e′2-

� e ′1

E ′′


OutlineIntroduction


Conclusion


Definition: entity integration

Given two contextualized entities sharing the same context e1 : E1 → C

and e2 : E2 → C , the integration of E1 and E2 with respect to C is the

contextualized entity E → C , such that, (i) There exists e′1 : E → E1 and

e′2 : E → E2 such that e1 ◦ e′1 = e2 ◦ e′2, and, (ii) For any other other

entity E ′′, with links e′′1 : E ′′ → E1 and e′′2 : E ′′ → E2 there exists a

unique link ! : E ′′ → E with e′1◦! = e′′1 and e′2◦! = e′′2

C

E1

e 1-

E2

� e2

E �!

e′2-

� e ′1

E ′′


OutlineIntroduction


Conclusion


Example 1/3: E1 and C

Whole Plant Growth Stage contextualized by Life Stage:

whole plant growth stage

vegetative stage

is-a

reproductive stage

is-a

life stage

innative

flowering

is-a

fruit formation

is-a

fertile

reproductive mature

is-a

active

is-a

is-a

unproductive

is-a

pre-fertile

is-ais-a is-a


OutlineIntroduction


Conclusion


Example 2/3: E2 and C

Human contextualized by Life Stages:

human life stage

gestative stage

is-a

pos-partum

is-a

life stage

innative

first age

is-a

second age

is-a

third age

is-a

active

pre-fertile reproductive mature

is-a is-a

fertile

is-a

unproductive

is-a

is-a is-a is-a


OutlineIntroduction


Conclusion


Example 3/3: E and C

The semantic intersection of E1 and E2 guided by C :

life stage

innative

is-a

reproductive

is-a

mature

is-a

life stage

innative

reproductive mature

is-a

active

is-a

fertile

is-a

unproductive

is-a

pre-fertile

is-ais-a is-a


OutlineIntroduction


Conclusion


Example comments

I Both are contextualized by an ontology that describes lifestages.

I The result embodies the semantic intersection of “plant” and“human”.

I Both plant and human ontologies could also be viewed ascontext to the resulting entity.

I By (i), components of the entity E correspond to those of“plant” and “human” that are linked to the same componentof “life stage”.

I By (ii), all the components that satisfies (i) are present in E .


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Conclusion


The algorithm

Algorithm. (Entity Integration)

Input: e1 : E1 → C and e2 : E2 → C Output: E → C

Notation: xi are variables for concepts of entities and yi are variables for relations of

entities. (CE, RE , HCE , relE) identify the components of an entity E. fe is component

f of a link e and ge is component g of a link e. The symbol 7→ denotes the association

by a function of the element at the left to the element at the right of the symbol 7→.

Initial conditions: CE , RE are empty sets and fe′1, fe′

2, ge′

1, ge′

2are empty functions.

For all x1 ∈ CE1

If there is x2 ∈ CE2 with fe1 (x1) = fe2(x2)

CE := CE ∪ fe1(x1)

fe′1

:= fe′1∪ (fe1(x1) ∈ CE) 7→ x1

fe′2

:= fe′2∪ (fe2(x2) ∈ CE) 7→ x2

For all y1 ∈ RE1

If there is y2 ∈ RE2 with ge1(y1) = ge2(y2)

RE := RE ∪ ge1(y1)

ge′1

:= ge′1∪ (ge1(y1) ∈ CE) 7→ y1

ge′2

:= ge′2∪ (ge2(y2) ∈ CE) 7→ y2

return (fe1 , ge1) ◦ (fe′1, ge′

1)


OutlineIntroduction


Conclusion


A summation (amalgamation) of contexts

I A single entity E can be viewed in different ways, C1 and C2.

I A new context as a result of combining and integrating givencontexts.

I The resulting context must be coherent with respect to thecorresponding entity.

I All components of the original contexts will be represent in C ,resulting links have the original contexts as domain.

C1

E

e 1-

C2

e2-

C1

E

e 1-

C

e ′1-

C2

e′2-

e2-


OutlineIntroduction


Conclusion


Properties

i The diagram commutes, so C is a coherent sum with respectto the entity E .

ii C is the less informative context that makes the diagramcommute. All elements of C1 and C2 are represent in C , andnothing more.

C1

E

e 1-

C!-

e ′1-

C ′′

C2

e′2-

e2-


OutlineIntroduction


Conclusion


Definition: context integration

Given two contextualizations of the same entity e1 : E → C1 and

e2 : E → C2, the context integration of C1 and C2 with respect to E is

the contextualized entity E → C , such that, (i) There exists e′1 : C1 → C

and e′2 : C2 → C such that e′1 ◦ e1 = e′2 ◦ e2, and, (ii) For any other other

context C ′′, with maps e′′1 : C1 → C ′′ and e′′2 : C2 → C ′′ there exists a

unique map ! : C → C ′′ with ! ◦ e′1 = e′′1 and ! ◦ e′2 = e′′2 .

C1

E

e 1-

C!-

e ′1-

C ′′

C2

e′2-

e2-


OutlineIntroduction


Conclusion


Example 1/3: E and C1

Part of Whole Plant Growth Stage ontology:


vegetative stage

is-a

reproductive stage

is-a


vegetative stage reproductive stage

is-a is-a

germination

is-a

imbibition

part-of

seeding growth

part-of

radicle emergence

part-of

shoot emergence

part-of


OutlineIntroduction


Conclusion


Example 2/3: E and C2

Another part of Whole Plant Growth Stage ontology:


vegetative stage

is-a

reproductive stage

is-a



is-a is-a

flowering

is-a

fruit formation

is-a


OutlineIntroduction


Conclusion


Example 3/3: E and C

Ammalgamation of entities of the ontologies:


vegetative stage

is-a

reproductive stage

is-a



is-a is-a

germination

is-a

flowering

is-a

fruit formation

is-a

imbibition

part-of

seeding growth

part-of

radicle emergence

part-of

shoot emergence

part-of


OutlineIntroduction


Conclusion


Example comments

I A plant growth stage ontology is composed by two separateparts: vegetative stage and reproductive stage.

I These parts can be developed in separate and glued later toform the complete plant growth stage.

I In the glue process part of the ontology (the glue points) mustbe contextualized by the ontologies containing the new parts.

The integration of these contexts results the whole plant growthstage ontology.


OutlineIntroduction


Conclusion


The algorithm

Algorithm. (Context Integration)

Input: e1 : E → C1 and e2 : E → C2 Output: E → C

Notation: similar of Entity Integration.

Initial conditions: CE is the empty set and fe′1, fe′

2are empty functions.

(i) For all x ∈ CE

CC := CC ∪ x

fe′1

:= fe′1∪ (fe1(x) ∈ CC1) 7→ (x ∈ CC)

fe′2

:= fe′2∪ (fe2(x) ∈ CC2) 7→ (x ∈ CC)

(ii) For all x ∈ CE1 that is not in the image of fe1

CC := CC ∪ x

fe′1

:= fe′1∪ (x ∈ CC1) 7→ (x ∈ CC)

(iii) For all x ∈ CE2 that is not in the image of fe2

CC := CC ∪ x

fe′2

:= fe′2∪ (x ∈ CC2) 7→ (x ∈ CC)

return fe′1◦ fe1


OutlineIntroduction


Conclusion


Morphism between Contextualized Entities

How to operate contextualized entities as a whole (entity andcontext)? The commutative square:

C1c- C2

E1

e1

6

e ′- E2

e2

6

Given two contextualized entities E1e1−→C1 and E2

e2−→C2, a pair

contextualized entities (C1c−→C2,E1

e′−→E2) is a map from e1 to e2 if

E1c◦e1−→C2 = E1

e2◦e′−→C2 is a contextualized entity.


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Conclusion


The Relative Intersection

I Commonalities among entities with different contexts.I Coherent intersection of two given contextualized entities

with respect to a third one.I CE ′ is the more informative contextualized entity, coherent

with CE1 and CE2 with respect to CE .I All the lateral squares of the right cube commute (by def).

The bottom and top squares of the cube also commute.

CE

CE2CE1

@@Im2

��m1

CE ′@@I m′2��m′1-CE ′′

E

E2E1

@@I��

E ′@@I ��

C

C2C1

@@I��

C ′@@I ��6 6

6

6


OutlineIntroduction


Conclusion


The Collapsing Union

I Acts in context and entity of CE1 and CE2. Results the unionof them, possibly collapsing some components.

I Any concept in CE1 or CE2 is mapped a component in CE ′.The concepts of CE are mapped to the same concept of CE ′

via links through CE1 or CE2.I CE ′ is the less informative Cont.Ent. coherent with CE1 and

CE2.

CE ′

CE2CE1

@@Im′2��

m′1

CE@@I m2��m1

�CE ′′

E ′

E2E1

@@I��

C@@I ��

C ′

C2C1

@@I��

E@@I ��6 6

6

6


OutlineIntroduction


Conclusion


Examples

UFF

Prog . Lang .ComputerScience

@@I

��

E ′@@I

��

University

SpecializationGraduation

@@I��

C ′@@I ��6 6

6

6

E ′

Prog .Lang .ComputerScience

@@I

��

Func.Ling .

@@I

��

C ′

SpecializationGraduation

@@I��

FirstYear

@@I ��6 6

6

6


OutlineIntroduction


Conclusion

Category Theory: what is it? Why use it?

I The presented algebra is an application of Category Theory;

I “Thing” (objects) described abstractly by their interactions;Focus in the relationship (morphisms);

I Functors relates categories, co-existence of heterogeneous“things”;

I Successfully used where interoperability is a crucial point;


OutlineIntroduction


Conclusion

A Category

A category C is a structure:

C = (O,M,Dom,Cod , ◦, id)

O collection of objects; M morphisms f : A→ B where A,B ∈ O;Dom,Cod : M → O; ◦ associative operation of morphismscomposition; id morphisms idA for each object A ∈ O.


OutlineIntroduction


Conclusion

Diagrams and Cones

I A category can be pictured as graphs (diagrams) for reasoning.

I Some definitions can be easily obtained by just reversing“arrows”.

Cone {fi : o → oi}: for any g : oi → oj we have g ◦ fi = fj

oig

- oj

o

f j -�

fi


OutlineIntroduction


Conclusion

Limits

A limit for a diagram D with objects oi is a cone {fi : o → oi}such that for any other cone {f ′i : o ′ → oi}, for D, there is aunique morphism ! : o ′ → o for which fi◦! = f ′i with f ′i : o ′ → oi .

oig

- oj

o

f j -�

fi

o ′

f ′i

6

! -

Special cases of D: two single objects is product; o1 → o ← o2 ispullback (product guide by o).


OutlineIntroduction


Conclusion

Colimits: duality of limits

A colimit for a diagram D with objects oi is a cocone {fi : oi → o}such that for any other cocone {f ′i : oi → o ′}, for D, there is aunique morphism ! : o → o ′ for which ! ◦ fi = f ′i with f ′i : oi → o ′.

oi�

goj

o�

f jfi-

o ′

f ′i

?�

!

Special cases of D: two single objects is coproduct; o1 ← o → o2

is pushout (sum of o1 and o2 possibly collapsing according to o).I. Cafezeiro E. H. Haeusler A. Rademaker Ontology and Context

OutlineIntroduction


Conclusion

An Ontology

An ontology O is a structure:

O = (C ,R,HC , rel ,A)

Concepts, relations, HC ⊆ C ×C hierarchy of concepts (taxonomicrelation), rel : R → C × C relates concepts non-taxonomically andAxioms.(x1, x0) ∈ Hc means x1 is subconcept of x0.


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Conclusion

Ontology operations

Given two ontologies o1 and o2:

mapping total mapping between o1 and o2 which preserves hierarchy,conceptual relations and specify semantic overlap betweenthem.

alignment is the task of establishing a collection of binary relationsbetween vocabularies of o1 and o2. A pair of total functions(ontology mappings) from a intermediate o.

merging unification of o1 and o2 into a new one that embodies thesemantic differences and collapses the semantic intersection.

matching finding commonalities between ontologies.

hiding erasing a concept/relation preserving hierarchy, conceptualrelations and semantic relations.


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Conclusion

Ontologies in a Categorical view

The category Ont of ontologies:

I objects are ontology structures;I morphisms pairs of functions (f , g) : O → O ′ where

O = (C ,R,HC , rel) and O ′ = (C ′,R ′,HC ′, rel ′) and

f : C → C ′ and g : R → R ′ such that:

i if (c1, c2) ∈ HC then (f (c1), f (c2)) ∈ Hc′, and

ii if (c1, c2) ∈ rel(r) then (f (c1), f (c2)) ∈ rel ′(g(r)).

(f , g) are links!


OutlineIntroduction


Conclusion

Ontologies in a Categorical view

I A contextualized entity is an object of Ont→ where objectsare morphisms of Ont, morphisms are pairs of Ont morphisms(m,m′);

I Entity integration is a pullback in Ont (matching);

I Context integration is a pushout in Ont (merging);

I Combined operations are performed in Ont→.

I Proofs that operations presented are well defined;


OutlineIntroduction


Conclusion

Conclusion

I Contexts are essential to clarify the meaning of entities;

I Uniform representation of entities and contexts and thecompositional definitions of operations give us abstraction,modularity and reuse;

I The role of an object (entity or context) is given by the net oflinks from or to it;

I Expansive, specificity, explicit, separated and transparent(from Roman, Julien & Payton“Formal Treatment of ContextAwareness”, 2004);

I . . . Interoperability of heterogeneous descriptions (differentkinds of categories).


OutlineIntroduction


Conclusion

Future works

Contextualizing web queries: a query is a morphism in Ont wheredom ontology is the information to be search (O) and cod is thecontext (O1).

for all O2 ∈ search(O)if ι : O → O2 ∈ Morphisms(Ont)

Results = Results ∪ pushout(O1 ← O ↪→ O2)return Results

Institutions: model-theoretical and syntactic mechanism from theoperations on the structural level.


Documents

Ontology and Context - GitHub Pagesarademaker.github.io/files/comorea2008-slides.pdf · Introduction The algebra of Contextualized Entities Formal Framework Conclusion Constraints