How Description Logic Ontologies Benefit from Formal Concept Analysis

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7 Αυγ 2012 (πριν από 4 χρόνια και 8 μήνες)

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How Description Logic
Ontologies

Benefit from

Formal Concept Analysis


Bar
ış Sertkaya

SAP Research Center Dresden

Germany

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2010 SAP AG. All rights reserved. / Page
2

FCA and DLs, what are they?

Formal
Concept

Anal
ysis

(FCA)



field of mathematics based on

lattice theory



anal
yze data

and der
ive

a

conceptual

structuring



medicine, psychology,
ontologies
,

linguistic databases, software

engineering, musicology, …

Description Logics (DLs)



logical languages that are fragments

of First Order Logic



represent

conceptual

knowledge

of an application domain



semantic web,
ontologies
, life

sciences, bio
-
medical computer

science, software engineering, …

Concept: collection of objects sharing certain properties

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2010 SAP AG. All rights reserved. / Page
3

FCA vs. DLs

Formal
Concept

Anal
ysis

(FCA)




data





algorithms




formal concepts:





concept lattice

Description Logics (DLs)




atomic concepts, roles:

logical constructors:




concept descriptions:




classification algorithm




subsumption


hierarchy

a

b

c

1

X

2

X

X

3

X

X

4

X

X

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2010 SAP AG. All rights reserved. / Page
4

FCA vs. DLs


DLs



intensional

definition

of a concept given
independent of a specific domain



rich language

for describing concepts

(negation, exists,
forall
, number
restrictions…)


individuals
partially
described


(open world semantics)



FCA


intensional

knowledge derived from the
extensional knowledge



concept definitions are
conjunctions

of

atomic concepts (attributes)



objects fully described

(closed world semantics)


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2010 SAP AG. All rights reserved. / Page
5

Knowledge Representation (KR)




Develop formalisms



for representing conceptual knowledge of an application domain,



that have a well
-
defined
syntax
,


formal,
unambigious

semantics
,


and practical
methods for reasoning
/ efficient implementations.



Conceptual Knowledge


Classes
:
country, ocean
-
country, …


Relations
:
has border to, has neighbor, …


Individuals
:
Spain, Mediterranean, Atlantic, …

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2010 SAP AG. All rights reserved. / Page
6

Description Logics (DLs)



family of
logic
-
based knowledge representation formalisms


describe an application domain in terms of


concepts
(classes): like
Country
,
Ocean
, …


roles
(relations): like
hasBorderTo
,
hasNeighbour
, …


individuals
like
Spain
,
Atlantic
, …


logical constructors:



well
-
defined formal semantics,

decidable fragments of First Order Logic


















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2010 SAP AG. All rights reserved. / Page
7

The DL

: The smallest propositionally closed description logic



atomic concepts:
A
,
B
, …
(unary predicates)



atomic roles:
r
,
s
, …
(binary predicates)



constructors:



(negation)



(conjunction)



(disjunction)



(existential restriction)



(value restriction)

Examples:










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2010 SAP AG. All rights reserved. / Page
8

Semantics of

Based on
interpretation

consisting of:


a
domain

(a non
-
empty set), and an
interpretation function



Concept and role names:



(
concept names interpreted as subsets of the domain)



(role names interpreted as binary relations)



Complex concept descriptions:














is a
model

of if

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2010 SAP AG. All rights reserved. / Page
9

Example

of

an
interpretation

Interpretation
domain

Concept

names

BodyOfWater

Sea

Ocean

Individual
names

Mediterranean

Atlantic

Country

OceanCountry

Roles

hasBorderTo

hasNeighbour

Pacific

Spain

Portugal

Austria

LandlockedCountry

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2010 SAP AG. All rights reserved. / Page
10

Example

of

an
interpretation

Interpretation
domain

Ocean

Atlantic

Country

hasBorderTo

hasNeighbour

Spain

Portugal

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2010 SAP AG. All rights reserved. / Page
11

Reasoning

Main reasoning task:


Concept
subsumption
: Is subsumed by ? (written )

(Does hold for all )


Concept
subsumption

for computing the
subsumption

hierarchy (classification)

BodyOfWater

Ocean

Sea

LandMass

Country

OceanCountry

LandLockedCountry

Reasoner

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2010 SAP AG. All rights reserved. / Page
12







DL
Knowledge

Bases (
Ontologies
)



DL Knowledge Base (Ontology)
=
TBox

+
ABox



TBox

defines the terminology of the application domain



ABox

states facts about a specific world




TBox
: a set of


concept definitions





ABox
:
concept
and



role assertions






General
TBox
:

General concept inclusion


axioms

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2010 SAP AG. All rights reserved. / Page
13

Bridging the gap between FCA and DLs

Existing work mainly under 2 categories:

1.

enriching FCA by borrowing constructors from DLs



theory
-
driven logical scaling
[Prediger,Stumme’99]



terminological attribute logic
[Prediger’00]



relational concept analysis
[Rouane,Huchard,Napoli,Valtchev’07]



logical concept analysis
[
Ferré
, Ridoux’01]

2.

employing FCA methods in DL knowledge bases


Computation of an extended
subsumption

hierarchy
[Baader’95]



Subsumption

hierarchy of conjunctions and disjunctions of DL concepts
[Stumme’96]



Subsumption

hierarchy of least common
subsumers

[Baader,Molitor’00]



Relational exploration
[Rudolph’04,06]



Supporting bottom
-
up construction of DL knowledge bases
[Baader,Turhan,Sertkaya’07]



Knowledge Base Completion
[Baader,Ganter,Sattler,Sertkaya’07]


Role assertion analysis
[
Coulet
,
Smail
-
Tabbone
, Napoli, Devignes’08]


Exploring finite models
[Baader,Distel’08,09]

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2010 SAP AG. All rights reserved. / Page
14

Extended
Subsumption

Hierarchy of DL
Concepts



traditional
TBox

classification:
subsumption

hierarchy of concepts



not sufficient in some settings
: interaction between defined concepts not visible




consider the concepts , , and



no
subsumption

relation between these three concepts



but, subsumed by



not visible from the
subsumption

hierarchy!






hierarchy of conjunctions of defined concepts enables
faster inferences
.



precompute

and store it.




how?
Using attribute exploration





define a formal context whose concept lattice
represents this hierarchy

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2010 SAP AG. All rights reserved. / Page
15

Extended
Subsumption

Hierarchy of DL
Concepts

Formal context
s.t
. the concept lattice is isomorphic to the hierarchy of conjunctions
of DL concepts
[Baader’95]
:







X


X














and , but , which is not visible in the usual hierarchy



implication questions are
subsumption

tests



a DL
reasoner

can act as an expert



a modified DL
reasoner

is needed for providing
countexamples

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2010 SAP AG. All rights reserved. / Page
16

Contributions to bridging the gap:

1) supporting bottom
-
up construction of KBs



traditional way of creating
ontologies
: (top
-
down manner)

1.

define concepts

2.

specify properties of individuals using them



not always adequate



which concepts are relevant?



how to define them correctly?



alternative: bottom
-
up construction of
ontologies

ABox

1.
User selects similar
ABox

individuals

2.
Individuals automatically generalized into
concept descriptions (MSC computation)

3.
Commonalities automatically extracted (LCS
computation)

4.
The LCS inspected/modified by the ontology
engineer and added to the ontology




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2010 SAP AG. All rights reserved. / Page
17

Supporting bottom
-
up construction of KBs



subsumption

hierarchy of conjunctions of concept names and their negations needed for
computing LCS



requires
subsumption

tests

for a
TBox

containing concept names



each
subsumption

test computationally
expensive



computing the hierarchy smartly without checking all pairs?



using
attribute exploration


Again define an
appropriate formal context


DL
reasoner

can answer implication questions


Use
background knowledge






implies



implies on the FCA side



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2010 SAP AG. All rights reserved. / Page
18

Bridging the gap:

2) Ontology completion

Existing

ontology

tools

support
:

1.
Detecting inconsistencies

2.
Inferring consequences

3.
Finding reasons for them

Quality
dimesion

of
soundness

What about
completeness
?



are there



missing relations
between classes?



missing individuals
?



if so how to
extend

the ontology appropriately?

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2010 SAP AG. All rights reserved. / Page
19



Ontology Completion

ABox

Asian

EUmember

European

Mediterranean

Russia

+

?

?

?

China

+

-

-

?

Montenegro

?

?

+

?

Germany

-

+

+

-

Italy

-

+

+

+

TBox



All European countries EU members?



All EU members that have a border to Mediterranean have territories in Europe?

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2010 SAP AG. All rights reserved. / Page
20

The
Phosphatese

Ontology

OWL Ontology for human protein
phosphatese

family
[
Wolstencroft
, Brass,
Horrocks
, Lord, Sattler,
Turi
, &
Stevens (2005)]




developed based on peer
-
reviewed publications



detailed knowledge about different classes of such proteins



TBox
:
classes of proteins, relations
among these classes



ABox
: large
set of human
phospthateses

identified and documented by expert biologists


Given this ontology, the biologist wants to know:

1.

Are there relations that hold in the real world, but that do not follow from the
TBox
?

2.

Are there
phospthateses

that are not represented in the
ABox
, or even that have


not yet been identified?

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2010 SAP AG. All rights reserved. / Page
21

When is an ontology (formally) complete?


is complete
w.r.t
. the intended application domain if these are equivalent:


( and are sets of concept names)

1.

is satisfied by

2.

follows from

3.

does not contain a counterexample to


Cannot be achieved by an automated tool alone, a domain expert needed!



questions ( the number of concept names)


Many of them redundant


Do not bother the expert unnecessarily


A smart way to get answers to these questions:
attribute exploration!

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2010 SAP AG. All rights reserved. / Page
22

Attribute Exploration for DL
Ontologies


Extension for open
-
world semantics of DL
ABoxes


Attribute exploration for
partial/incomplete formal contexts


Already
existing approaches

[
Burmeister

&
Holzer

2005]


the resulting knowledge is incomplete (certain implications, uncertain implications)


In contrast we want to have
complete knowledge

at the end


Our expert has / can access to complete knowledge


But he should be able to give
partial descriptions
of objects during exploration


Proved termination, correctness, minimum number of questions


An
ABox

is a partial context


Integrated a
DL
reasoner

for avoiding questions


Improved usability. The expert can:


Skip questions


Stop exploration, see previous answers, undo previous actions,


See why an implication automatically was accepted

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2010 SAP AG. All rights reserved. / Page
23

Ontology Completion

When a question is asked:



first check if it follows from the ontology



if not ask the expert



if the expert confirms, add a
new axiom

to the
TBox



if the expert rejects, get a
new
ABox


assertion

as counterexample

©
2010 SAP AG. All rights reserved. / Page
24

Summary:

How DLs benefit from FCA?



Mainly 2 categories:




using concept lattice to detect implicit relations between classes


Extended
subsumption

hierarchy (of conjunctions of concepts)


Subsumption

hierarchy of least common
subsumers


Supporting bottom
-
up construction



using attribute exploration to complete knowledge


Knowledge base completion

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2010 SAP AG. All rights reserved. / Page
25

FCA
at

SAP Research


The
Aletheia

Project


Obtaining product information through the use of
semantic technologies


FCA used for requirement analysis


sponsored by the Federal Ministry of Education and Research (BMBF)


Partners: SAP AG, ABB, BMW Group, Deutsche Post,
OntoPrise
, Otto, TU Dresden, FU
Berlin, HU Berlin,
Frauenhofer

IIS,
TecO
,
Giesecke

&
Devrient
,
Eurolog
,


http://www.aletheia
-
projekt.de




New project
CUBIST

(Combining and Uniting
Business Intelligence

with
Semantic Technologies
)



FCA

used for visual analytics on top of business intelligence


Partners: SAP AG, Sheffield
Halam

University,
Heriot
-
Watt University,
Innovantage
,
Ontotext

Lab
,
Centrale

Rechereche

S.A. (CRSA)


Laboratoire

MAS, Space
Applications

Services NV




Academic articles at ICCS, ICFCA on Role Based Access Control for
Ontologies
, …



Thank you

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2010 SAP AG. All rights reserved. / Page
27



Early Days of KR

PieceOfLand

Ocean

OceanCountry

Country

BodyOfWater

IslandCountry

is a

is a

is a

is a

hasBorderTo

hasBorderTo

Semantic Networks
[
Quilian

1967]



nodes

represent
classes



links

represent
relations



hasBorderTo
: does it mean
there exists

a border
,
or
for all
borders?



ambigious

semantics
!


KL
-
ONE
[
Brachman

& Levesque 1985]



logic
-
based semantics