THEOREMAOMEGAULTRA

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COMPUTER-BASED MATHEMATICS
Mathematische Assistenzsysteme
Jörg Siekmann
http//www.dfki.de/siekmann

• Universität des Saarlandes
• und
• Deutsches Forschungszentrum
• für Künstliche Intelligenz GmbH
• DFKI

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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 partiell der
  Textfragmente!
• Hilfsmittel und Tools Computer-Algebra Systeme
• Hilsmittel und Tools Beweis Systeme
• Hilfsmittel und Tools Proofchecker

http//www.activemath.org
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..... to set the stage.
1954
Martin Davis
Theorem The Sum of two even numbers is again
even Proof (Presburger Arithmetic)
1956
Alan Newell, Herb Simon
Theorems from Principia Mathematica Proof
Logic Theorist
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3 Paradigms
1. Classical Automated Theorem Proving

• Resolution
• Tableaux-Methods
• Matrix and Connection Method

2. Tactical Theorem Proving

• Automath
• NUPRL
• IMPS
• ISABELLE etc.

3. Human oriented Theorem Proving

• Natural Deduction
• Woody Bledsoe
• Proof Planing OYSTER-CLAM, OMEGA

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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 hunt through 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
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Can we do better
?
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Knowledge based Proof Planning
AI-PLANNING IN THE BLOCKS WORLD

• Initial State

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

• Goal

on_table(B)
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Methods in Proof Planning
Declarations
Premises
Specification
Declarative Part
Constraints
Conclusions
Declarative Content
Tactic
Procedural Part
Procedural Content
Alan Bundy (1989) A Science Of Reasoning

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Methods An Example
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Knowledge based Proof Planning
Method 1
Method 2
Method 3

• Classical Proof Planning
• Knowledge based Proof Planning

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Mathematical Control Knowledge
Global mathematical control

• Prove a lt b directly or via auxiliary
  variabels ? prove a lt b by Solveltb, Solve or
  ... LimHeuristic.

• Use important parts of assumptions to
  introduce auxiliary variabels/inequalities e.g.
  LimHeuristic requires

• Focus
• UNWRAPHYP
• REmoveFocus
• MP-b

Source Erica Melis
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Control knowledge represented as rules
(control-rule attack-inequality (IF
(goal-matches (?goal (?x lt ?y)))) (THEN
(prefer((Solvelt ?goal) (Solve
?goal) (ComplexEstimate ?goal)
(Simplify ?goal))))
(control-rule case-analysis-intro (IF
(last-method (Rewrite (?C -gt ?R))) AND
(failure-condition (trivial ?C)))
(THEN (select (CaseSplit (?C or not ?C)))))
Source Erica Melis
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Peter Deussen Semigroups and Automata,
Springer Verlag, 1971
Theorem 4.8 Let s and t be two equivalence
relations. Then (s ? t) t is also an
equivalence relation.
Proof (Idea)
To be shown

• Symmetry
• Reflexivity
• Transitivity

of (s ? t) t
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More Examples epsilon-delta Proofs
Woody Bledsoe Challenges
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Method for Limit Theorems
Source Erica Melis
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Construction of mathematical Objects
CONSTRAINT SOLVING
Collecting constraints and check for consitency
Final constraint store for LIM
Source Erica Melis
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Proof Presentation to the User
Verbalisation of
Source Erica Melis
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PDS Representation of (partial) Proofs
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OMEGA -ANTS Combining ATP with Proof Planning

• concurrency and ressource
  adaptive behaviour
• anytime algorithms
• flexible integration of
• natural deduction
• tactics and methods
• external systems

Chris Benzmüller, Volker Sorge
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The OMEGA SYSTEM
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Proof Planning A Screen Shot
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Zwei Entwicklungsrichtungen

• Verifikationswerkzeuge z.B. VSE am DFKI
• 2. Mathematische Tutorsysteme
• z.B. ActiveMath
  am DFKI
• __________________________________________________
  ___
• anwendungsorientierte
• Grundlagenforschung
  !

CHALLENGE Ein integriertes mathematisches
Assistenzsystem
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Knowledge Representation for Mathematics

• XML-Representation
• Semantics (OpenMath) extended by meta data (publ,
  mathematical, and pedagogical)

• Formal content for
• Calling external systems
• Intelligent search functionalities

Mathematical Ontology
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Knowledge the building blocks in OMDoc
ltdefinition id"c6s1p4_Th2_def_monoid"
for"c6s1p4_monoid"gt ltmetadatagt
ltdepends-ongt ltref theory"cp1_Th3"
name"structure" /gt lt/depends-ongt ltTitle
xmllang"en"gtDefinition of a monoidlt/Titlegt
lt/metadatagt ltCMP xmllang"en" format"omtext"gt
A monoid is a ltref xref"cp1_Th3_def_structure"gt
structure lt/refgt ltOMOBJgt ltOMS cd"elementary"
name"ordered-triple"/gt ltOMV name"M"/gt
ltOMS cd"cp4_Th2" name"times"/gt ltOMS
cd"cp4_Th2" name"unit"/gt lt/OMOBJgt in
which ltOMOBJgt ltOMS cd"elementary"
name"ordered-pair"/gt ltOMV name"M"/gt ltOMS
cd"cp4_Th2" name"times"/gt lt/OMOBJgt is a
semi-group with ltref xref"c6s1p3_Th2_def_unit"gtun
itlt/refgt ltOMOBJ xmlns"http//www.openmath.or
g/OpenMath"gt ltOMS cd"cp4_Th2"
name"unit"/gt lt/OMOBJgt. lt/CMPgt ltFMPgtltOMOBJgt
... lt/OMOBJgtlt/FMPgt lt/definitiongt
ltdefinition id"c6s1p4_Th2_def_monoid"
for"c6s1p4_monoid"gt A monoid is a
structure M times
unit in which M times is a
semi-group with unit e lt/definitiongt
ltdefinition id"c6s1p4_Th2_def_monoid"
for"c6s1p4_monoid"gt ltCMP xmllang"en"
format"omtext"gt A monoid is a
structure M times unit in
which M times is a semi-group with
unit e lt/CMPgt ltFMPgtltOMOBJgt ...
lt/OMOBJgtlt/FMPgt lt/definitiongt
ltdefinition id"c6s1p4_Th2_def_monoid"
for"c6s1p4_monoid"gt ltmetadatagt
ltdepends-ongt ltref theory"cp1_Th3"
name"structure" /gt lt/depends-ongt ltTitle
xmllang"en"gtDefinition of a monoidlt/Titlegt
lt/metadatagt ltCMP xmllang"en" format"omtext"gt
A monoid is a
structure M times unit in which M
times is a semi-group with unit e lt/CMPgt ltFMPgt
ltOMOBJgt ... lt/OMOBJgtlt/FMPgt lt/definitiongt
A monoid is a
structure M times unit in which M
times is a semi-group with unit e
An Example A MONOID
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Ein mathematisches Assistenzsystem
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Computer Supported Mathematics !!
Schickard Die erste mechanische Rechenmaschine
der Welt.
. . . . . . Zuse die erste elektronische
Rechenmaschine.
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