Outline
Introduction
The algebra of Contextualized Entities
Formal Framework
Conclusion
Ontology and Context
I. Cafezeiro
E. H. Haeusler
A. Rademaker
April 10, 2008
I. Cafezeiro E. H. Haeusler A. Rademaker
Ontology and Context
Outline
Introduction
The algebra of Contextualized Entities
Formal Framework
Conclusion
Introduction
The algebra of Contextualized Entities
Entity Integration
Context Integration
Combined Integration
Formal Framework
Conclusion
I. Cafezeiro E. H. Haeusler A. Rademaker
Ontology and Context
Outline
Introduction
The algebra of Contextualized Entities
Formal Framework
Conclusion
Ontologies and Computer Science
I
Ontologies describe real world things. Hierarchically
organizing concepts and enriching this hierarchy with
relationships among concepts.
I
A real word entity to be represented is always related to a
context. A semantically consistent body of information in
which the entity makes sense.
I
The need of contexts? In mobile applications, where the
environment suffer dynamic re-configurations.
I. Cafezeiro E. H. Haeusler A. Rademaker
Ontology and Context
Outline
Introduction
The algebra of Contextualized Entities
Formal Framework
Conclusion
Proposal
I
I
I
I
We propose an algebra for manipulate Contextualized
Ontologies with a few basic formal concepts turn it
accessible.
We adopt: (i) an homogeneous description of entities and
contexts and; (ii) maps that consistently link entities and
contexts.
Flexibility to: (i) combine entities or contexts in several ways
and; (ii) changing and inheritance of context by an entity, and
other useful operations.
The formal approach: (i) rigorous definition of
Contextualized Ontologies; (ii) abstract enough to make
possible the replacement of ontologies by other knowledge
representation technique.
I. Cafezeiro E. H. Haeusler A. Rademaker
Ontology and Context
Outline
Introduction
The algebra of Contextualized Entities
Formal Framework
Conclusion
Contextualized Entities
I
Entities are described by three parts: the entity itself, a
context, and a link between entity and its context.
I
A triple (entity, link, context) represented by e → c, will be
named contextualized entity.
I
As both entity and context are ontologies, an entity can be
the context of other entity.
I
The context, gives general information about the entity or
about 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 call
contextualized ontology.
I. Cafezeiro E. H. Haeusler A. Rademaker
Ontology and Context
Outline
Introduction
The algebra of Contextualized Entities
Formal Framework
Conclusion
Constraints about Links
The link entity–context ensures the coherence. The context
preserves 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 to
itself, 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 called
codomain of a link;
iii links can be composed in an associative way if the codomain
of the first is the domain of the second. “◦” denotes
composition of links!
I. Cafezeiro E. H. Haeusler A. Rademaker
Ontology and Context
Outline
Introduction
The algebra of Contextualized Entities
Formal Framework
Conclusion
Entity Integration
Context Integration
Combined 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 also
attached to that context.
I
The original entities play the role of context to the produced
entity. By transitivity, C is also a context for E .
-e1
E1
0
e1
E2
-e2 0
-e1
e2
E1
e2
C
C
E2
E
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
Outline
Introduction
The algebra of Contextualized Entities
Formal Framework
Conclusion
Entity Integration
Context Integration
Combined Integration
Properties
i The diagram is commutative. E1 , E2 and E are coherent
with respect of the context.
e1
-
ii E is the more complete entity that makes the diagram
commute. All components of E1 and E2 linked to the same
element in C have a corresponding in E , and nothing more.
C e
2
E1
0
e1
0
E I. Cafezeiro E. H. Haeusler A. Rademaker
E2
e 2!
E 00
Ontology and Context
Outline
Introduction
The algebra of Contextualized Entities
Formal Framework
Conclusion
Entity Integration
Context Integration
Combined Integration
Definition: entity integration
e1
-
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 e10 : E → E1 and
e20 : E → E2 such that e1 ◦ e10 = e2 ◦ e20 , and, (ii) For any other other
entity E 00 , with links e100 : E 00 → E1 and e200 : E 00 → E2 there exists a
unique link ! : E 00 → E with e10 ◦! = e100 and e20 ◦! = e200
C e
2
E1
0
e1
0
E I. Cafezeiro E. H. Haeusler A. Rademaker
E2
e 2!
E 00
Ontology and Context
Outline
Introduction
The algebra of Contextualized Entities
Formal Framework
Conclusion
Entity Integration
Context Integration
Combined Integration
Example 1/3: E1 and C
Whole Plant Growth Stage contextualized by Life Stage:
whole plant growth stage
is-a
vegetative stage
is-a
flowering
is-a
reproductive stage
is-a
life stage
innative
is-a
fruit formation
active
is-a
fertile
is-a
reproductive
I. Cafezeiro E. H. Haeusler A. Rademaker
is-a
is-a
unproductive
is-a
is-a
mature
Ontology and Context
pre-fertile
Outline
Introduction
The algebra of Contextualized Entities
Formal Framework
Conclusion
Entity Integration
Context Integration
Combined Integration
Example 2/3: E2 and C
Human contextualized by Life Stages:
human life stage
is-a
gestative stage
is-a
is-a
pos-partum
is-a is-a
first age
life stage
second age
innative
is-a
active
is-a
is-a
third age
fertile
is-a
pre-fertile
I. Cafezeiro E. H. Haeusler A. Rademaker
is-a
unproductive
is-a
reproductive
Ontology and Context
is-a
mature
Outline
Introduction
The algebra of Contextualized Entities
Formal Framework
Conclusion
Entity Integration
Context Integration
Combined Integration
Example 3/3: E and C
The semantic intersection of E1 and E2 guided by C :
life stage
is-a
reproductive
is-a
mature
life stage
is-a
innative
is-a is-a
innative
active
is-a
fertile
is-a
reproductive
I. Cafezeiro E. H. Haeusler A. Rademaker
is-a
mature
Ontology and Context
is-a
unproductive
is-a
pre-fertile
Outline
Introduction
The algebra of Contextualized Entities
Formal Framework
Conclusion
Entity Integration
Context Integration
Combined Integration
Example comments
I
Both are contextualized by an ontology that describes life
stages.
I
The result embodies the semantic intersection of “plant” and
“human”.
I
Both plant and human ontologies could also be viewed as
context 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 component
of “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
Outline
Introduction
The algebra of Contextualized Entities
Formal Framework
Conclusion
Entity Integration
Context Integration
Combined 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 , HEC , 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′2 are 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′ := fe′ ∪ (fe2 (x2 ) ∈ CE ) 7→ x2
2
2
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′ := ge′ ∪ (ge2 (y2 ) ∈ CE ) 7→ y2
2
2
return (fe1 , ge1 ) ◦ (fe′1 , ge′1 )
I. Cafezeiro E. H. Haeusler A. Rademaker
Ontology and Context
Outline
Introduction
The algebra of Contextualized Entities
Formal Framework
Conclusion
Entity Integration
Context Integration
Combined Integration
A summation (amalgamation) of contexts
C1
C1
0
e 1-
e
E
C
e 2-
e 2-
C2
I. Cafezeiro E. H. Haeusler A. Rademaker
e0
E
2
-
I
-1
I
A single entity E can be viewed in different ways, C1 and C2 .
A new context as a result of combining and integrating given
contexts.
The resulting context must be coherent with respect to the
corresponding entity.
All components of the original contexts will be represent in C ,
resulting links have the original contexts as domain.
e
I
-1
I
C2
Ontology and Context
Outline
Introduction
The algebra of Contextualized Entities
Formal Framework
Conclusion
Entity Integration
Context Integration
Combined Integration
Properties
i The diagram commutes, so C is a coherent sum with respect
to the entity E .
ii C is the less informative context that makes the diagram
commute. All elements of C1 and C2 are represent in C , and
nothing more.
e
0
e 1-
-1
C1
C
!
- C 00
e0
e 2-
2
-
E
C2
I. Cafezeiro E. H. Haeusler A. Rademaker
Ontology and Context
Outline
Introduction
The algebra of Contextualized Entities
Formal Framework
Conclusion
Entity Integration
Context Integration
Combined 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 e10 : C1 → C
and e20 : C2 → C such that e10 ◦ e1 = e20 ◦ e2 , and, (ii) For any other other
context C 00 , with maps e100 : C1 → C 00 and e200 : C2 → C 00 there exists a
unique map ! : C → C 00 with ! ◦ e10 = e100 and ! ◦ e20 = e200 .
e
0
e 1-
-1
C1
E
!
- C 00
e0
e 2-
2
-
C
C2
I. Cafezeiro E. H. Haeusler A. Rademaker
Ontology and Context
Outline
Introduction
The algebra of Contextualized Entities
Formal Framework
Conclusion
Entity Integration
Context Integration
Combined Integration
Example 1/3: E and C1
Part of Whole Plant Growth Stage ontology:
whole plant growth stage
is-a
vegetative stage
whole plant growth stage
is-a
reproductive stage
is-a
vegetative stage
is-a
reproductive stage
is-a
germination
part-of
imbibition
part-of
seeding growth
part-of
radicle emergence
I. Cafezeiro E. H. Haeusler A. Rademaker
part-of
shoot emergence
Ontology and Context
Outline
Introduction
The algebra of Contextualized Entities
Formal Framework
Conclusion
Entity Integration
Context Integration
Combined Integration
Example 2/3: E and C2
Another part of Whole Plant Growth Stage ontology:
whole plant growth stage
is-a
vegetative stage
is-a
reproductive stage
whole plant growth stage
is-a
vegetative stage
is-a
reproductive stage
is-a
flowering
I. Cafezeiro E. H. Haeusler A. Rademaker
Ontology and Context
is-a
fruit formation
Outline
Introduction
The algebra of Contextualized Entities
Formal Framework
Conclusion
Entity Integration
Context Integration
Combined Integration
Example 3/3: E and C
Ammalgamation of entities of the ontologies:
whole plant growth stage
is-a
vegetative stage
whole plant growth stage
is-a
reproductive stage
is-a
vegetative stage
is-a
reproductive stage
is-a
germination
part-of
imbibition
is-a
flowering
fruit formation
part-of
seeding growth
part-of
radicle emergence
I. Cafezeiro E. H. Haeusler A. Rademaker
is-a
part-of
shoot emergence
Ontology and Context
Outline
Introduction
The algebra of Contextualized Entities
Formal Framework
Conclusion
Entity Integration
Context Integration
Combined Integration
Example comments
I
A plant growth stage ontology is composed by two separate
parts: vegetative stage and reproductive stage.
I
These parts can be developed in separate and glued later to
form the complete plant growth stage.
I
In the glue process part of the ontology (the glue points) must
be contextualized by the ontologies containing the new parts.
The integration of these contexts results the whole plant growth
stage ontology.
I. Cafezeiro E. H. Haeusler A. Rademaker
Ontology and Context
Outline
Introduction
The algebra of Contextualized Entities
Formal Framework
Conclusion
Entity Integration
Context Integration
Combined 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′ , fe′ are empty functions.
1
2
(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
Outline
Introduction
The algebra of Contextualized Entities
Formal Framework
Conclusion
Entity Integration
Context Integration
Combined Integration
Morphism between Contextualized Entities
How to operate contextualized entities as a whole (entity and
context)? The commutative square:
C1
c C2
6
6
e2
e1
E1
e0 E2
e
e
1
2
Given two contextualized entities E1 −→C
1 and E2 −→C2 , a pair
c
e
0
contextualized entities (C1 −→C2 , E1 −→E2 ) is a map from e1 to e2 if
c◦e
e ◦e
0
1
2
E1 −→C
2 = E1 −→C2 is a contextualized entity.
I. Cafezeiro E. H. Haeusler A. Rademaker
Ontology and Context
Outline
Introduction
The algebra of Contextualized Entities
Formal Framework
Conclusion
Entity Integration
Context Integration
Combined Integration
The Relative Intersection
I
I
I
I
Commonalities among entities with different contexts.
Coherent intersection of two given contextualized entities
with respect to a third one.
CE 0 is the more informative contextualized entity, coherent
with CE1 and CE2 with respect to CE .
All the lateral squares of the right cube commute (by def).
The bottom and top squares of the cube also commute.
m1
CE
m2
@
I
@
CE2
m20
CE1
I
m10@
@
00
CE
CE 0
I. Cafezeiro E. H. Haeusler A. Rademaker
C1
C
6@
I
@
C2
E
@
I
@
6
I 6
@
C0 @
E1
E2
6
I
@
@
E0
Ontology and Context
Outline
Introduction
The algebra of Contextualized Entities
Formal Framework
Conclusion
Entity Integration
Context Integration
Combined Integration
The Collapsing Union
I
I
I
Acts in context and entity of CE1 and CE2 . Results the union
of them, possibly collapsing some components.
Any concept in CE1 or CE2 is mapped a component in CE 0 .
The concepts of CE are mapped to the same concept of CE 0
via links through CE1 or CE2 .
CE 0 is the less informative Cont.Ent. coherent with CE1 and
CE2 .
CE 00 0 CE 0
m1
m0
I
@
@ 2
CE1
CE2
m1@
m
I
2
@
CE
I. Cafezeiro E. H. Haeusler A. Rademaker
C1
C0
6@
I
@
C2
0
E
@
I
@
6
I 6
@
E @
E1
E2
6
I
@
@
C
Ontology and Context
Outline
Introduction
The algebra of Contextualized Entities
Formal Framework
Conclusion
Entity Integration
Context Integration
Combined Integration
Examples
C0
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Computer
Science
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Prog . Lang .
Computer
Science
6
Prog .Lang .
@ 6
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Func.Ling .
E0
I. Cafezeiro E. H. Haeusler A. Rademaker
Specialization
@@ E 0 6
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Ontology and Context
Outline
Introduction
The algebra of Contextualized Entities
Formal 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
Outline
Introduction
The algebra of Contextualized Entities
Formal 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 morphisms
composition; id morphisms idA for each object A ∈ O.
I. Cafezeiro E. H. Haeusler A. Rademaker
Ontology and Context
Outline
Introduction
The algebra of Contextualized Entities
Formal 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
g
oi
- oj
-
fj
fi
o
I. Cafezeiro E. H. Haeusler A. Rademaker
Ontology and Context
Outline
Introduction
The algebra of Contextualized Entities
Formal Framework
Conclusion
Limits
A limit for a diagram D with objects oi is a cone {fi : o → oi }
such that for any other cone {fi 0 : o 0 → oi }, for D, there is a
unique morphism ! : o 0 → o for which fi ◦! = fi 0 with fi 0 : o 0 → oi .
g
oi
- oj
fj
fi
fi 0
-
6
!
-
o
o0
Special cases of D: two single objects is product; o1 → o ← o2 is
pullback (product guide by o).
I. Cafezeiro E. H. Haeusler A. Rademaker
Ontology and Context
Outline
Introduction
The algebra of Contextualized Entities
Formal 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 {fi 0 : oi → o 0 }, for D, there is a
unique morphism ! : o → o 0 for which ! ◦ fi = fi 0 with fi 0 : oi → o 0 .
g
fj
fi
-
fi 0
oj
!
o
oi ?
o0
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
Outline
Introduction
The algebra of Contextualized Entities
Formal Framework
Conclusion
An Ontology
An ontology O is a structure:
O = (C , R, H C , rel, A)
Concepts, relations, H C ⊆ C × C hierarchy of concepts (taxonomic
relation), rel : R → C × C relates concepts non-taxonomically and
Axioms.
(x1 , x0 ) ∈ H c means x1 is subconcept of x0 .
I. Cafezeiro E. H. Haeusler A. Rademaker
Ontology and Context
Outline
Introduction
The algebra of Contextualized Entities
Formal 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 between
them.
alignment is the task of establishing a collection of binary relations
between 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 the
semantic differences and collapses the semantic intersection.
matching finding commonalities between ontologies.
hiding erasing a concept/relation preserving hierarchy, conceptual
relations and semantic relations.
I. Cafezeiro E. H. Haeusler A. Rademaker
Ontology and Context
Outline
Introduction
The algebra of Contextualized Entities
Formal Framework
Conclusion
Ontologies in a Categorical view
The category Ont of ontologies:
I
I
objects are ontology structures;
morphisms pairs of functions (f , g ) : O → O 0 where
0
O = (C , R, H C , rel) and O 0 = (C 0 , R 0 , H C , rel 0 ) and
0
0
f : C → C and g : R → R such that:
0
i if (c1 , c2 ) ∈ H C then (f (c1 ), f (c2 )) ∈ H c , and
ii if (c1 , c2 ) ∈ rel(r ) then (f (c1 ), f (c2 )) ∈ rel 0 (g (r )).
(f , g ) are links!
I. Cafezeiro E. H. Haeusler A. Rademaker
Ontology and Context
Outline
Introduction
The algebra of Contextualized Entities
Formal Framework
Conclusion
Ontologies in a Categorical view
I
A contextualized entity is an object of Ont → where objects
are morphisms of Ont, morphisms are pairs of Ont morphisms
(m, m0 );
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
Outline
Introduction
The algebra of Contextualized Entities
Formal Framework
Conclusion
Conclusion
I
Contexts are essential to clarify the meaning of entities;
I
Uniform representation of entities and contexts and the
compositional definitions of operations give us abstraction,
modularity and reuse;
I
The role of an object (entity or context) is given by the net of
links from or to it;
I
Expansive, specificity, explicit, separated and transparent
(from Roman, Julien & Payton“Formal Treatment of Context
Awareness”, 2004);
I
. . . Interoperability of heterogeneous descriptions (different
kinds of categories).
I. Cafezeiro E. H. Haeusler A. Rademaker
Ontology and Context
Outline
Introduction
The algebra of Contextualized Entities
Formal Framework
Conclusion
Future works
Contextualizing web queries: a query is a morphism in Ont where
dom ontology is the information to be search (O) and cod is the
context (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 the
operations on the structural level.
I. Cafezeiro E. H. Haeusler A. Rademaker
Ontology and Context