C-OWL: contextualizing ontologies

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C
-
OWL: contextualizing
ontologies

Fausto Giunchiglia

October 22, 2003



Paolo Bouquet, Fausto Giunchiglia, Frank van
Harmelen, Luciano Serafini, and Heiner Stuckenschmidt


The Talk



Ontologies vs. Contexts


A (restated) global semantics for OWL


Intuitions


Three motivating examples


A (new) local models semantics for OWL


Intuitions


C
-
OWL: extending OWL with (context)
mappings


Ontologies vs. Contexts


An Ontology is a model of some domain which is
supposed to encode a view common to a set of
different parties



An ontology is built to be

shared
;



A Context is a model of some domain which is
supposed to encode a view of a party



A context is built to be

kept local

(where
local


implies
not shared
)



A context and an ontology of the
same

domain are
likely to be
very different

(different goals, different
approach, …)

Pro’s and Contra’s


Ontologies


Strengths


“easy” exchange of information


Weaknesses


consensus must be reached about their contents


maintenance may become arbitrarily hard


Contexts


Strengths


“easy” to define and to maintain


can be constructed with no consensus with the other parties


Weaknesses


Exchange of information by constructing explicit mappings
among the elements of the contexts of the involved parties

Contextual Ontologies

Contextual ontology = Ontology + Context mappings

Key idea (in two steps):

1.
Share as much as possible (
OWL import construct
)

2.
Keep it local whenever sharing does not work (
C
-
OWL
context mappings
)


Notes:

1.
In many (most in the Web?) cases
sharing does not work

and produces undesired results (semantic heterogeneity)

2.
Using context allows for
incremental
, piece
-
wise
construction of the Semantic Web (bottom up vs. top
down approach).


The Talk



Contexts vs. Ontologies


A (restated) global semantics for OWL


Intuitions


Three motivating examples


A (new) local models semantics for OWL


Intuitions


C
-
OWL: extending OWL with (context)
mappings


A Global Semantics for OWL

Index

OWL Ontologies:
<i, O
i
>
and their languages
(e.g.,
i:C, j:E, i:

r.C
)


(
Local language
).
A
local concept (role, individual)
,
C
i
(
R
i,

O
i
) is an element of
C
that appears in
O
i

either
without indexes or with index equal to
i
.


(
Foreign language
): … Anything (concept, role,
individual) which is not local



(
OWL space
).
An
OWL space
is a family of ontologies
{<i, O
i
>}

such that the language of every
O
i

contains all the other foreign languages

A Global Semantics for OWL (cont’ed)

(
OWL interpretation
).
An
OWL interpretation
for the OWL space
{<i, O
i
>}

is a pair I
=
<

I
,
(.)
I
>
, such that


I(i, C)




I
for any
i


I

and
C



C
i
;


I(i, r)




I

x

I

for any
i


I

and
r



R
i
;


I(i, o)



I

for any
i


I

and
o



O
i
;

With ∆
I

domain of interpretation and


(.)
I

interpretation function

Note: a global interpretation!


A Global Semantics for OWL (cont’ed)

(
OWL axiom and fact satisfiability
).
I

satisfies a fact
or an axiom
ø

of
O
i
according to the rules defined in



[*]
P.F. Patel
-
Schneider, P. Hayes, and I. Horrocks. Web
Ontology Language (OWL) Abstract Syntax and
Semantics. Technical report, W3C, February 2003.


An OWL interpretation
I

satisfies an OWL space
{<i,
O
i
>}
, if
I

satisfies each axiom and fact of
O
i
, for any
i


The Talk



Contexts vs. Ontologies


A (restated) global semantics for OWL


Intuitions


Three motivating examples


A (new) local models semantics for OWL


Intuitions


C
-
OWL: extending OWL with (context)
mappings


Example 1: directionality

Need to keep track of source and target ontology


Example:


Construct
O
2

by importing
O
1

and adding it some
new axiom


Want that axioms added to
O
2

do not affect
O
1


O
1

contains axioms
A B

and
C D


O
2

contains also axiom
1:
B 1:C


In new semantics, we want 1:
A 1:D

in
O
2
, but not in
O
1
.

Example 1 (cont’ed): directionality


We want to avoid propagation of inconsistency


Example:


O
1

contains axioms
A B

and
C D


O
2

contains also axiom 1
:B 1:C


We want to derive
1:
A 1:D

in
O
2

but not in
O
1





O2
contains also 1:
A(a)

and
1: not D(a)


O
2

is inconsistent


In new semantics, we want to keep
O
1

consistent



Example 2: local domains

Need to give up hypothesis that of single global domain of interpretation


Example:

Car manufacturing ontology

O
WCM

with domain of interpretation the totality of cars


individual constants
Diesel

and
Petrol

for Diesel engine and petrol engine


Axiom: a car has only one engine which is either Diesel or petrol

Car

(

1) hasEngine.{Diesel, Petrol}


Diesel


Petrol


Ferrari ontology,
O
Ferrari

describing Ferrari’s production


Imports
O
WCM

standard


Axiom: engine of a Ferrari is either an F23 or and F34i

Ferrari

(WCM:car (

1) (WCM:hasEngine).{F23, F34i}



F23

F34i




In new semantics, we want to avoid
(F23)
IFerrari

= (Diesel)
IWCM

since Ferrari
produces only petrol engines


Example 3: context mappings

Need to state that two elements of two ontologies, though
being extensionally different, are
contextually
related

Example:


O
FIAT

describes cars from manufacturer point of view


O
Sale

describes cars from car vendor point of view


O
FIAT

and
O
Sale

are largely independent and different


Two concepts of car defined in
O
FIAT

and
O
Sale
, (i.e.
Sale:Car

and
FIAT:Car
) may be very different, still
describing same real world object (
different viewpoints
)


Not possible to state relation between two concepts with
OWL syntax


The Talk



Contexts vs. Ontologies


A (restated) global semantics for OWL


Intuitions


Three motivating examples


A (new) local models semantics for OWL


Intuitions


C
-
OWL: extending OWL with (context)
mappings


Exampe 1: Directionality


Consider all (
local
) ontologies as part of a OWL space



Split global interpretation into a family of
local

interpretations,

one for each ontology



Allow for an ontology to be
locally
inconsistent

(
i.e.,

not
to have a local interpretation)





Technically:

Associate inconsistent ontologies to a special
“interpretation”, called a
hole
, that verifies any set of axioms

Example 2: Local Domains


Associate to each ontology a
local
domain



Local domains may overlap (two ontologies
may refer to the same object)




Technically:

An
OWL interpretation with local domains
for the OWL
space
{<i, O
i
>}

is a family
I = {I
i
}
, where each
I
i

=
<

Ii
, (.)
Ii
>
, called
the
local interpretation
of
O
i
, is either an interpretation of
L
i

on

Ii
, or
a hole

The Talk



Contexts vs. Ontologies


A (restated) global semantics for OWL


Intuitions


Three motivating examples


A (new) local models semantics for OWL


Intuitions


C
-
OWL: extending OWL with (context)
mappings


Example 3: adding context mappings to
syntax


(
Bridge rules
). A bridge rule from i to j is a statement of
one of the four following forms,


where x and y are concepts, or individuals, or roles of the
languages L
i
and L
j



(
Context mapping
). Given a OWL space {<i, O
i
>} a
mapping M
ij
from O
i

to O
j

is a set of bridge rules from O
i

to O
j.

Context mappings (cont’ed)


(
Contextual ontology
): It is a
local ontology

plus a set of
bridge rules (
context mappings
). We sometimes write
context

meaning
contextual ontology.


(
Context space
).
A context space is the pair

1.
OWL space {<i, O
i
>}
(
of
local ontologies
)

2.
family {M
ij
} of
(context) mappings

from i to j, for any
pair i,j


(Interpretation for context spaces
). It is
the pair

1.
I,
where

I is an OWL interpretation with holes and
local
domains

and

2.
r
ij
, the
domain relation

from i to j, is a subset of

Ii

x

Ii


Examples: Context mappings

From example 3:

Sale:Car

and
FIAT:car

describe the
same
set of objects from two different viewpoints:


(**)


Domain relation satisfying (**):

r
ij
(
Car
I
Sale
)= Car
I
FIAT

From example 2:


(*)



Domain relation satisfying (*):

r
WCM, Ferrari
(
Petrol)
I
WCM



{F23
I
Ferrari
, F34i
I
Ferrari
}

Context OWL (C
-
OWL)

A
contextual ontology

is a pair:


OWL ontology


a set of context mappings


where a mapping is a set of
bridge rules with the same
target ontology


A
context mapping

is a 4
-
tuple:


A mapping identifier (URI)


A source context containing an
OWL ontology


A target context containing an
OWL ontology


A set of bridge rules from the
local language of the source
ontology to the local language
of the target ontology


NOTE: mappings are objects (!!)

Conclusions


Ontologies
: share knowledge


Contexts
: keep knowledge local (not shared)


Contextual ontologies
: share as much as possible, keep
local whenever necessary


C
-
OWL (Context OWL)
:


OWL +


Local models semantics +


context mappings (limited, explicitly defined, visibility
from outside)

Will C
-
OWL be of any use?


How often in the Web we will import ontologies and
how often we will define context mappings (diversity
as a defect, or diversity as a feature)?



Shouldn’t the Semantic Web be a Web of
Semantic
links

(e.g., context mappings)? Context mappings
useful for: maintaining alignment, propagating info,
(semantics driven) navigation, …



Shouldn’t discovering context mappings (e.g.,
Semantic matching)
be one of the core issues in
building the Semantic Web?


Context mappings (cont’ed)








(
Satisfiability of bridge rules
)








A interpretation for a context space is a model for it if all the bridge rules are satisfied