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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
The algebra of Contextualized EntitiesFormal Framework
Conclusion
Introduction
The algebra of Contextualized EntitiesEntity IntegrationContext IntegrationCombined Integration
Formal Framework
Conclusion
I. Cafezeiro E. H. Haeusler A. Rademaker Ontology and Context
OutlineIntroduction
The algebra of Contextualized EntitiesFormal Framework
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.
I. Cafezeiro E. H. Haeusler A. Rademaker Ontology and Context
OutlineIntroduction
The algebra of Contextualized EntitiesFormal Framework
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.
I. Cafezeiro E. H. Haeusler A. Rademaker Ontology and Context
OutlineIntroduction
The algebra of Contextualized EntitiesFormal Framework
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.
I. Cafezeiro E. H. Haeusler A. Rademaker Ontology and Context
OutlineIntroduction
The algebra of Contextualized EntitiesFormal Framework
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!
I. Cafezeiro E. H. Haeusler A. Rademaker Ontology and Context
OutlineIntroduction
The algebra of Contextualized EntitiesFormal Framework
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 .
I. Cafezeiro E. H. Haeusler A. Rademaker Ontology and Context
OutlineIntroduction
The algebra of Contextualized EntitiesFormal Framework
Conclusion
Entity IntegrationContext IntegrationCombined Integration
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 ′′
I. Cafezeiro E. H. Haeusler A. Rademaker Ontology and Context
OutlineIntroduction
The algebra of Contextualized EntitiesFormal Framework
Conclusion
Entity IntegrationContext IntegrationCombined Integration
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 ′′
I. Cafezeiro E. H. Haeusler A. Rademaker Ontology and Context
OutlineIntroduction
The algebra of Contextualized EntitiesFormal Framework
Conclusion
Entity IntegrationContext IntegrationCombined Integration
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
I. Cafezeiro E. H. Haeusler A. Rademaker Ontology and Context
OutlineIntroduction
The algebra of Contextualized EntitiesFormal Framework
Conclusion
Entity IntegrationContext IntegrationCombined Integration
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
I. Cafezeiro E. H. Haeusler A. Rademaker Ontology and Context
OutlineIntroduction
The algebra of Contextualized EntitiesFormal Framework
Conclusion
Entity IntegrationContext IntegrationCombined Integration
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
I. Cafezeiro E. H. Haeusler A. Rademaker Ontology and Context
OutlineIntroduction
The algebra of Contextualized EntitiesFormal Framework
Conclusion
Entity IntegrationContext IntegrationCombined Integration
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 .
I. Cafezeiro E. H. Haeusler A. Rademaker Ontology and Context
OutlineIntroduction
The algebra of Contextualized EntitiesFormal Framework
Conclusion
Entity IntegrationContext IntegrationCombined Integration
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)
I. Cafezeiro E. H. Haeusler A. Rademaker Ontology and Context
OutlineIntroduction
The algebra of Contextualized EntitiesFormal Framework
Conclusion
Entity IntegrationContext IntegrationCombined Integration
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-
I. Cafezeiro E. H. Haeusler A. Rademaker Ontology and Context
OutlineIntroduction
The algebra of Contextualized EntitiesFormal Framework
Conclusion
Entity IntegrationContext IntegrationCombined Integration
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-
I. Cafezeiro E. H. Haeusler A. Rademaker Ontology and Context
OutlineIntroduction
The algebra of Contextualized EntitiesFormal Framework
Conclusion
Entity IntegrationContext IntegrationCombined Integration
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-
I. Cafezeiro E. H. Haeusler A. Rademaker Ontology and Context
OutlineIntroduction
The algebra of Contextualized EntitiesFormal Framework
Conclusion
Entity IntegrationContext IntegrationCombined Integration
Example 1/3: E and C1
Part of Whole Plant Growth Stage ontology:
whole plant growth stage
vegetative stage
is-a
reproductive stage
is-a
whole plant growth stage
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
I. Cafezeiro E. H. Haeusler A. Rademaker Ontology and Context
OutlineIntroduction
The algebra of Contextualized EntitiesFormal Framework
Conclusion
Entity IntegrationContext IntegrationCombined Integration
Example 2/3: E and C2
Another part of Whole Plant Growth Stage ontology:
whole plant growth stage
vegetative stage
is-a
reproductive stage
is-a
whole plant growth stage
vegetative stage reproductive stage
is-a is-a
flowering
is-a
fruit formation
is-a
I. Cafezeiro E. H. Haeusler A. Rademaker Ontology and Context
OutlineIntroduction
The algebra of Contextualized EntitiesFormal Framework
Conclusion
Entity IntegrationContext IntegrationCombined Integration
Example 3/3: E and C
Ammalgamation of entities of the ontologies:
whole plant growth stage
vegetative stage
is-a
reproductive stage
is-a
whole plant growth stage
vegetative stage reproductive stage
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
I. Cafezeiro E. H. Haeusler A. Rademaker Ontology and Context
OutlineIntroduction
The algebra of Contextualized EntitiesFormal Framework
Conclusion
Entity IntegrationContext IntegrationCombined Integration
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.
I. Cafezeiro E. H. Haeusler A. Rademaker Ontology and Context
OutlineIntroduction
The algebra of Contextualized EntitiesFormal Framework
Conclusion
Entity IntegrationContext IntegrationCombined Integration
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
I. Cafezeiro E. H. Haeusler A. Rademaker Ontology and Context
OutlineIntroduction
The algebra of Contextualized EntitiesFormal Framework
Conclusion
Entity IntegrationContext IntegrationCombined Integration
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.
I. Cafezeiro E. H. Haeusler A. Rademaker Ontology and Context
OutlineIntroduction
The algebra of Contextualized EntitiesFormal Framework
Conclusion
Entity IntegrationContext IntegrationCombined Integration
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
I. Cafezeiro E. H. Haeusler A. Rademaker Ontology and Context
OutlineIntroduction
The algebra of Contextualized EntitiesFormal Framework
Conclusion
Entity IntegrationContext IntegrationCombined Integration
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
I. Cafezeiro E. H. Haeusler A. Rademaker Ontology and Context
OutlineIntroduction
The algebra of Contextualized EntitiesFormal Framework
Conclusion
Entity IntegrationContext IntegrationCombined Integration
Examples
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I. Cafezeiro E. H. Haeusler A. Rademaker Ontology and Context
OutlineIntroduction
The algebra of Contextualized EntitiesFormal Framework
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;
I. Cafezeiro E. H. Haeusler A. Rademaker Ontology and Context
OutlineIntroduction
The algebra of Contextualized EntitiesFormal Framework
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.
I. Cafezeiro E. H. Haeusler A. Rademaker Ontology and Context
OutlineIntroduction
The algebra of Contextualized EntitiesFormal Framework
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
I. Cafezeiro E. H. Haeusler A. Rademaker Ontology and Context
OutlineIntroduction
The algebra of Contextualized EntitiesFormal Framework
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).
I. Cafezeiro E. H. Haeusler A. Rademaker Ontology and Context
OutlineIntroduction
The algebra of Contextualized EntitiesFormal Framework
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
The algebra of Contextualized EntitiesFormal Framework
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.
I. Cafezeiro E. H. Haeusler A. Rademaker Ontology and Context
OutlineIntroduction
The algebra of Contextualized EntitiesFormal Framework
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.
I. Cafezeiro E. H. Haeusler A. Rademaker Ontology and Context
OutlineIntroduction
The algebra of Contextualized EntitiesFormal Framework
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!
I. Cafezeiro E. H. Haeusler A. Rademaker Ontology and Context
OutlineIntroduction
The algebra of Contextualized EntitiesFormal Framework
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;
I. Cafezeiro E. H. Haeusler A. Rademaker Ontology and Context
OutlineIntroduction
The algebra of Contextualized EntitiesFormal Framework
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).
I. Cafezeiro E. H. Haeusler A. Rademaker Ontology and Context
OutlineIntroduction
The algebra of Contextualized EntitiesFormal Framework
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.
I. Cafezeiro E. H. Haeusler A. Rademaker Ontology and Context