Source: Erica Melis Universität des Saarlandes Saarbrücken den 16. 2. 2005 Ringvorlesung: Perspektiven der Informatik Prof. Dr. Jörg H. Siekmann Saarbrücken

Ringvorlesung:

Perspektiven der Informatik

Universität des SaarlandesSaarbrücken den 16. 2. 2005

Prof. Dr. Jörg H. Siekmann

Saarbrücken 16th of February, 20052005

German Research Center for Artificial Intelligence, DFKI

GmbH Stuhlsatzenhausweg 3

66123 Saarbruecken, Germanyphone: (+49 681) 302-5276e-mail: [email protected]:http://www.dfki.de/~siekmann

German Research Centre for Artificial Intelligence

DFKIPerspektiven der Informatik 2005

Ringvorlesung:



The Structure of DFKI Key Directors

Prof. W. Wahlster (CEO)Dr. W. Olthoff (CFO)

Administration and ServicesDr. H.-J. Bürckert

PR and Corporate CommunicationR. Karger, M.A.

Competence Centers Transfer Centers

Institute for Information

Systems (IWi)

Prof. A.-W. Scheer

Research LabKnowledge

Management

Prof. A. Dengel

Research Lab Intelligent

Visualizationand Simulation

Systems

Prof. H. Hagen

Research Lab Deduction

andMultiagentSystems

Prof. J. Siekmann

Research Lab Language

Technology

Prof. H. Uszkoreit

Research Lab Intelligent

UserInterfaces

Prof. W. Wahlster

Research LabImage Understanding Pattern RecognitionProf. Th. Breuel

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Research Lab: Deduction and Multi Agent Systems

Safety and SecurityLab

Dr W.Stephan, Dr D.Hutter

Multi Agent Lab

Dr K.Fischer, Dr M.Klusch

CompetenceCentre

E-LearningJ.Burgard, K.P.Jantke,

E.Melis

MathematicalAssistent Systems

Dr S.Autexier, Dr Ch.BenzmüllerUniversity

DeMAS

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DFKI Covers the Complete Innovation Cycle

DFKI

Labs at theUniversity

Application-oriented

BasicResearch

Applied Research

and Development

TransferProjects

‘Blue Sky‘Basic Research

DFKI projectsfor

externalclients and

shareholders

Spin-offCompanieswith DFKIequity

Commercialization/Exploitation

Shareholders

ExternalClients

DFKIprojects

forfederal

government,EU

DFKIprojects

forstate

governments,clients and

shareholders

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DFKI is a Joint Venture of:

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Es existieren mehrere Kandidaten für weitere Gesellschafteranteile am DFKI, die ihr Interesse für 2005 nach Vertragsprüfung bekundet haben.

Hybride, personalisierte Navigationssysteme, Ferndiagnose

Ontologie-basiertes Matchmaking, Virtuelles Arbeitsamt

Multilinguale Autorensysteme

Beratungsagenten für UMTSBill Gates: MicroSoft

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Staff Development from 1988 - 2001

0

20

40

60

80

100

120

140

160

180

200

1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001

2005: ~ 250

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DFKI has created >600 new Jobs in IT Industry

and 33 Start-up and Spin-Off Companies

InnoPwww.innop.de

springwww.spring.de

Plansoftwww.plansoft.de

ProCAEsswww.procaess.de

tec:innowww.tecinno.de

TNMwww.tnmsoft.com

SIEDAwww.sieda.com

TransCatwww.transcat.de

EYELEDwww.eyeled.co

mXtraMindwww.xtramind.com

Insiders Information Management GmbHwww.im-insiders.de

sonicsonwww.sonicson.com

DACOSwww.dacos.de

Imagowww.imago.de

DeepWebhttp://www.deep-web.de

acrolinxhttp://www.acrolinx.de/

Erster Deutscher SPIN-OFF PREIS

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Universität des SaarlandesSaarbrücken den 16. 2. 2005© 2002 DFKI Design by R.O.

Thank You for

Your Attention

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Universität des Saarlandesund

Deutsches Forschungszentrum für Künstliche Intelligenz GmbH

DFKI

COMPUTER-BASED MATHEMATICS

http://www.dfki.de/~siekmann

Jörg Siekmann

Mathematische Assistenzsysteme

Ringvorlesung:



Was ist ein mathematisches Assistenzsystem?

• Textverarbeitung a la LATEX: TeXmacs• Information Retrieval aus einer mathematischen Datenbank ( MBase, Open Math, OmDoc,... )• Theorie Kontext• Semantik: der Definitionen, Theoreme, Beweise und Lemmata . . . und part iell der Textfragmente!• Hilfsmittel und Tools: Computer-Algebra Systeme• Hilsmittel und Tools: Beweis Systeme• Hilfsmittel und Tools: Proofchecker

• interakt ive Übungen durch integrierte Service-Systeme (Computeralgebra etc)• web-basiert, verteilte Architektur• semantische XML-Repräsentation

– Wiederverwendung des Inhalts

http://www.activemath.org

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..... to set the stage.1954:

1956:

Martin DavisTheorem: The Sum of two even numbers is again evenProof: (Presburger Arithmetic)Alan Newell, Herb Simon

Theorems: from Principia MathematicaProof: Logic Theorist

Dartmouth Conference

Psychology Scruffy

Logic Theorist GPSNewellWoody Bledsoe

LogicNeat

WangResolution

Matrix

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3 Paradigms:

• Natural Deduction• Woody Bledsoe

• Proof Planing: OYSTER-CLAM, ΩMEGA

• Resolution• Tableaux-Methods• Matrix and Connection Method

• Automath• NUPRL• IMPS• ISABELLE etc.

1. Classical Automated Theorem Proving

2. Tactical Theorem Proving

3. Human oriented Theorem Proving

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Deduction Systems

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Automated theorem proving is not the beautiful process we know asmathematics. This is „cover your eyes with blinders and huntthrough a cornfield for a diamond-shaped grain of corn“ ...

Mathematicians have given us a great deal of direction over the last two or three millennia. Let us pay attention to it.

Woody Bledsoe, 1986

Ringvorlesung:



Can we do better ?

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Knowledge based Proof Planning

• Initial State

• Goal

on(A,B), on(B,C), on_table(C),on_table(D), free(A), ...

on_table(B)

• Operators PUTDOWN(X):precondition: holding (X)effect: (+) on_table(X), hand_empty

(-) holding(X)

• Planpick(A), putdown(A), pick(B), putdown(B)

AI-PLANNING IN THE BLOCKS WORLD

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Methods: An Example

Ringvorlesung:



Method 1 Method 2 Method 3

Knowledge based Proof Planning

•Classical Proof Planning

•Knowledge based Proof Planning

•Knowledge

•Time

•User interaction

Ressources

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Mathematical Control Knowledge

Global mathematical control:

•Prove |a| < b directly or via auxiliary variabels

prove |a| < b by Solve<b, Solve* or

... LimHeuristic.

•Use important parts of assumptions to introduce

auxiliary variabels/inequalities:

e.g. LimHeuristic requires:•Focus•UNWRAPHYP•REmoveFocus•MP-bSource: Erica Melis

Ringvorlesung:



Theorem 4.8: Let σ and τ be two equivalence

relations. Then (σ τ) t is also an

equivalence relation.Proof: (Idea)

•Symmetry•Reflexivity•Transitivity

Peter Deussen: Semigroups and Automata,

Springer Verlag, 1971

To be shown:

of (σ τ) t

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More Examples: epsilon-delta Proofs

Woody Bledsoe: “Challenges“

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Method for Limit Theorems

Source: Erica Melis

Ringvorlesung:



Construction of mathematical Objects

Final constraint store for LIM+

Collecting constraints and check for consitency

CONSTRAINT SOLVING:

Source: Erica Melis

Ringvorlesung:



Proof Presentation to the User

Verbalisation of

Source: Erica Melis

Ringvorlesung:



Mathematical Assistance Systems

Integrated MathematicalAssistant Environment

‘Pen-and-Paper’Mathematicsvs.

Applications Join of resources necessaryMathematics reserachMathematics educationFormal methods

System level: Coq, NuPrl,Isabelle/HOL, PVS,

Theorema, MEGA, Clam, ...

Research Networks: Calculemus, MKM, Monet, MoWGLI

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The OMEGA SYSTEM

Ringvorlesung:



Proof Planning: A Screen Shot

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Proof Verbalization

• lifting of proofs in the PDS to assertion level

• macro-planning text structure

• micro-planning sentence structure and linguistic realization

• generation of natural language representation

• pre-required: linguistic knowledge

• user-adaptive proof explanation

P.REX (successor of PROVERB):

Ringvorlesung:



Zwei Entwicklungsrichtungen:

1. Verifikationswerkzeuge: z.B. VSE am DFKI

2. Mathematische Tutorsysteme: z.B. ActiveMath am DFKI_____________________________________________________

anwendungsorientierte Grundlagenforschung!

CHALLENGE:

Ein integriertes mathematisches Assistenzsystem

Ringvorlesung:



Knowledge Representation for Mathematics

– XML-Representation– Semantics (OpenMath) extended by meta

data (publ, mathematical, and pedagogical)

Mathematical Ontology

– Formal content for

•Calling external systems

•Intelligent search functionalities

Ringvorlesung:



Knowledge: the building blocks in OMDoc

A monoid is a structure [M times unit]

in which[M times]

is a semi-group

with unite

<definition id="c6s1p4_Th2_def_monoid" for="c6s1p4_monoid">

A monoid is a structure [M times unit]

in which[M times]

is a semi-group

with unite

</definition>

<definition id="c6s1p4_Th2_def_monoid" for="c6s1p4_monoid">

<CMP xml:lang="en" format="omtext"> A monoid is a structure [M times unit]

in which[M times]

is a semi-group

with unite

</CMP><FMP><OMOBJ> ... </OMOBJ></FMP></definition>

<definition id="c6s1p4_Th2_def_monoid" for="c6s1p4_monoid"><metadata> <depends-on> <ref theory="cp1_Th3" name="structure" /></depends-on> <Title xml:lang="en">Definition of a monoid</Title> </metadata><CMP xml:lang="en" format="omtext"> A monoid is a structure [M times unit]

in which[M times]

is a semi-group

with unite

</CMP><FMP><OMOBJ> ... </OMOBJ></FMP></definition>

<definition id="c6s1p4_Th2_def_monoid" for="c6s1p4_monoid"><metadata> <depends-on> <ref theory="cp1_Th3" name="structure" /> </depends-on> <Title xml:lang="en">Definition of a monoid</Title> </metadata><CMP xml:lang="en" format="omtext"> A monoid is a <ref xref="cp1_Th3_def_structure"> structure </ref> <OMOBJ> <OMS cd="elementary" name="ordered-triple"/> <OMV name="M"/> <OMS cd="cp4_Th2" name="times"/> <OMS cd="cp4_Th2" name="unit"/></OMOBJ>in which<OMOBJ> <OMS cd="elementary" name="ordered-pair"/> <OMV name="M"/> <OMS cd="cp4_Th2" name="times"/></OMOBJ> is a semi-groupwith <ref xref="c6s1p3_Th2_def_unit">unit</ref> <OMOBJ xmlns="http://www.openmath.org/OpenMath"> <OMS cd="cp4_Th2" name="unit"/> </OMOBJ>. </CMP><FMP><OMOBJ> ... </OMOBJ></FMP></definition>

An Example:

A MONOID

Ringvorlesung:



Ein mathematisches Assistenzsystem

Ringvorlesung:



Computer Supported Mathematics !!

Schickard:Die erste mechanischeRechenmaschine der Welt.

. . . . . . Zuse: die erste elektronische Rechenmaschine.

Documents

Source: Erica Melis Universität des Saarlandes Saarbrücken den 16. 2. 2005 Ringvorlesung: Perspektiven der Informatik Prof. Dr. Jörg H. Siekmann Saarbrücken