how description logic ontologies benefit from formal concept analysis barış sertkaya sap research...

How Description Logic Ontologies Benefit fromFormal Concept Analysis

Barış SertkayaSAP Research Center DresdenGermany

© 2010 SAP AG. All rights reserved. /

FCA and DLs, what are they?

Formal Concept Analysis (FCA)

field of mathematics based on

lattice theory

analyze data and derive 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


FCA vs. DLs

Formal Concept Analysis (FCA)

data

algorithms

formal concepts:

concept lattice


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


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)


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, …



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


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:


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


Example of an interpretation

Interpretation domainConcept names

BodyOfWater

Sea

Ocean

Individual names

Mediterranean

Atlantic

Country

OceanCountry

Roles

hasBorderTo

hasNeighbour

Pacific

Spain

Portugal

Austria

LandlockedCountry

Inte

rpre

tatio

n fu

nctio

n

Interpretation function


Example of an interpretation

Interpretation domain

OceanAtlantic

Country

hasBorderTo

hasNeighbour

Spain

Portugal


Reasoning

Main reasoning task:

Concept subsumption: Is subsumed by ? (written )

(Does hold for all )

Concept subsumption for computing the subsumption hierarchy (classification)

BodyOfWater

OceanSea

LandMass

Country

OceanCountry LandLockedCountryReasoner


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


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]


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


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


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


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


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?


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?


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?


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!


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


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


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


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, …

http://www.aletheia-projekt.de/

Thank you


Early Days of KR

PieceOfLand

Ocean

OceanCountryCountry

BodyOfWater

IslandCountry

is a

is a

is a

is ahasBorderTo

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

how description logic ontologies benefit from formal concept analysis barış sertkaya sap research...

Documents

sap ag

order logic slide

domain role names

welldefined formal semantics

conceptual knowledge

dls dls intensional

concepts negation

roles relations