integrating dynamic geometry software, deduction systems, and

MKM 2006 1 / 19

Integrating Dynamic Geometry Software, DeductionSystems, and Theorem Repositories

Pedro QuaresmaCISUC/Mathematics Department

University of CoimbraPortugal

Predrag JanicicFaculty of MathematicsUniversity of Belgrade

Serbia

August 10-12, 2006

Introduction

Introduction

Computers & Geometry

GeoThms Framework

Dynamic Geometry Software

GCLC & Eukleides

ATP in Geometry

GCLCprover

GeoDB - ERD

GeoThms

GeoThms - by Example

Describe the Construction

Testing the Conjecture

The Proof - Area Method

Adding a New Theorem to theDatabase

GeoThms - Browsing

Recent work

Conclusions

Future Work

MKM 2006 2 / 19

The axiomatic presentation of geometry fills the gapbetween formal logic and our spatial intuition.

The study of geometry is, and will always be, veryimportant for a mathematical practitioner.

GeoThms framework provides an environment suitable for new ways of studying andteaching geometry at different levels and for storing geometrical knowledge:descriptions of construction; geometrical conjectures; geometrical proofs


Introduction


GeoThms Framework


GCLC & Eukleides

ATP in Geometry

GCLCprover

GeoDB - ERD

GeoThms






GeoThms - Browsing

Recent work

Conclusions

Future Work

MKM 2006 3 / 19

Computer technologies give new ways for studying geometry

Dynamic Geometry Software Visualise/Explore/Test Conjectures

Geometric Automated Theorem Proving synthetic proofs (human-readable) /algebraic proofs (efficiency).

Problems Repositories browse through the existing knowledge.

GeoThms integrates all these features bringing new forms in communicatingmathematics.

GeoThms Framework

Introduction


GeoThms Framework


GCLC & Eukleides

ATP in Geometry

GCLCprover

GeoDB - ERD

GeoThms






GeoThms - Browsing

Recent work

Conclusions

Future Work

MKM 2006 4 / 19

GeoThms integrates DGSs, ATPs, and a repository of constructive geometry theoremsin one single tool.

Dynamic Geometry Software GCLC & Eukleides

Geometric Automated Theorem Proving GCLCprover (implements the areamethod).

Problems Repositories geoDB - geometric theorems, illustrations and proofsdatabase.

GeoThms provides an environment suitable for new ways of studying and teachinggeometry at different levels, and for storing geometrical knowledge: descriptions ofconstruction; geometrical conjectures; geometrical proofs


Introduction


GeoThms Framework


GCLC & Eukleides

ATP in Geometry

GCLCprover

GeoDB - ERD

GeoThms






GeoThms - Browsing

Recent work

Conclusions

Future Work

MKM 2006 5 / 19

Dynamic geometry software visualise geometric objects and link formal, axiomaticnature of geometry (most often — Euclidean) with its standard models (e.g., Cartesianmodel) and corresponding illustrations.

GCLC & Eukleides - two DGSs designed to be close to the traditional language ofelementary Euclidean geometry.

� they provide support for primitive constructions based on ruler and compass

� transformations, labelling components of figures, interactive work, animations, etc.

� graphical user interface.

By using the set of primitive constructions, one can define more complex constructions.

GCLC & Eukleides

Introduction


GeoThms Framework


GCLC & Eukleides

ATP in Geometry

GCLCprover

GeoDB - ERD

GeoThms






GeoThms - Browsing

Recent work

Conclusions

Future Work

MKM 2006 6 / 19

GCLC1 is a tool for teaching and studying mathematics, especially geometry andgeometric constructions, and also for storing descriptions of mathematical figures andproducing digital illustrations of high quality.

1Predrag Janicic, www.matf.bg.ac.yu/~janicic/gclc/

GCLC & Eukleides

Introduction


GeoThms Framework


GCLC & Eukleides

ATP in Geometry

GCLCprover

GeoDB - ERD

GeoThms






GeoThms - Browsing

Recent work

Conclusions

Future Work

MKM 2006 6 / 19

Eukleides1 is an Euclidean geometry drawing language (with localised versions).

� eukleides is a compiler for typesetting geometric figures within a (La)TeXdocument.

� xeukleides is a GUI front-end for creating interactive geometric figures.

1Christian Obrecht; EukleidesPT (Pedro Quaresma) gentzen.mat.uc.pt/~EukleidesPT/

ATP in Geometry

Introduction


GeoThms Framework


GCLC & Eukleides

ATP in Geometry

GCLCprover

GeoDB - ERD

GeoThms






GeoThms - Browsing

Recent work

Conclusions

Future Work

MKM 2006 7 / 19

Automated theorem proving in geometry has two major lines of research:

algebraic proof style Algebraic proof style methods are based on reducinggeometry properties to algebraic properties expressed in terms of Cartesiancoordinates. These methods are usually very efficient, but the proofs theyproduce do not reflect the geometry nature of the problem and they give only ayes/no conclusion.

synthetic proof style Synthetic methods attempt to automate traditional geometryproof methods that produce human-readable proofs.

GCLCprover

Introduction


GeoThms Framework


GCLC & Eukleides

ATP in Geometry

GCLCprover

GeoDB - ERD

GeoThms






GeoThms - Browsing

Recent work

Conclusions

Future Work

MKM 2006 8 / 19

GCLCprover - synthetic geometric ATP (area method)

� implements the area method

� simple and tight integration with GCLC and Eukleides

� human-readable proofs

� very efficient for many conjectures

GeoDB - ERD

Introduction


GeoThms Framework


GCLC & Eukleides

ATP in Geometry

GCLCprover

GeoDB - ERD

GeoThms






GeoThms - Browsing

Recent work

Conclusions

Future Work

MKM 2006 9 / 19

provers

nameversiondescriptionurlemaildateSubmission

proverId

descriptioncategoryuserIdbibref

teoIdteoIdcodedrawerIdfigureuserIdbibref

figureIdteoIdcodeproverIdproofstatususerIdbibref

demIdteoId teoId

dateSubmissiondateSubmission

dateSubmission

teoName

dateSubmissionemailurldescriptionversionnamedrawerId

drawers

drawerId

authordrawer

authorIddrawerId dateSubmission

emailurlaffiliationnameauthorId

authors

authorprover

authorIdproverId

proverId

bibrefIduserId

users

userIdnameusernamepasswdtype

affiliationurlemaildateSubmission

bibrefs

bibrefIdbibtexEntry

figures theorems proofs

GeoThms

Introduction


GeoThms Framework


GCLC & Eukleides

ATP in Geometry

GCLCprover

GeoDB - ERD

GeoThms






GeoThms - Browsing

Recent work

Conclusions

Future Work

MKM 2006 10 / 19

GeoThms2, is a framework that links dynamic geometry software (GCLC, Eukleides),geometry theorem provers (GCLCprover), and a repository of geometry problems(geoDB).

Interactionmodule

statement

LaTeX+

auxiliary tools

statements

constructiongeometric

geometricconstruction

withconjecture

contributersregular users

Web

Interface

Repository

contributers

Reports

(listings/technical reports)

(GCLC,Eukleides,...)

DGS


figures

(GCLCprover,...)

ATP

proofs

(provers/drawers/...)

Forms

(add/update data)

2GeoThms is accessible from http://hilbert.mat.uc.pt/~geothms

GeoThms

Introduction


GeoThms Framework


GCLC & Eukleides

ATP in Geometry

GCLCprover

GeoDB - ERD

GeoThms






GeoThms - Browsing

Recent work

Conclusions

Future Work

MKM 2006 10 / 19

GeoThms, is a framework that links dynamic geometry software (GCLC, Eukleides),geometry theorem provers (GCLCprover), and a repository of geometry problems(geoDB).

Interactionmodule

statement

LaTeX+

auxiliary tools

statements

constructiongeometric

geometricconstruction

withconjecture


Web

Interface

Repository

contributers

Reports

(listings/technical reports)

(GCLC,Eukleides,...)

DGS


figures

(GCLCprover,...)

ATP

proofs

(provers/drawers/...)

Forms

(add/update data)

DGS code

orGCLC

Eukleides

ATP code

GCLCprover

Inputvia

HTML forms

HTML files

Outputvia

LaTeXformat

PDFformat

Figures inJPEGformat

Proofs inPDF

format

GCLC code + conjecture

Eukleides code + conjecture(via a conversion tool)

or

Outputvia

HTML files


Introduction


GeoThms Framework


GCLC & Eukleides

ATP in Geometry

GCLCprover

GeoDB - ERD

GeoThms






GeoThms - Browsing

Recent work

Conclusions

Future Work

MKM 2006 11 / 19

Theorem 1 (Gramy P1432) Given a parallelogram ABCD, a point N , obtained bythe intersection of a line parallel to AC passing through B, and a line perpendicular toAC passing through D, then the point P , which is given by the intersection of AN

and BC , is the midpoint of QB, where Q is the intersection of BC and DN .

A

BC

D

N

PQ

2P143 of “Gramy: A Geometry Theorem Prover Capable of Construction” by Matsuda and Vanlehn.


Introduction


GeoThms Framework


GCLC & Eukleides

ATP in Geometry

GCLCprover

GeoDB - ERD

GeoThms






GeoThms - Browsing

Recent work

Conclusions

Future Work

MKM 2006 12 / 19

We begin by specifying the construction in the DGSs language.


Introduction


GeoThms Framework


GCLC & Eukleides

ATP in Geometry

GCLCprover

GeoDB - ERD

GeoThms






GeoThms - Browsing

Recent work

Conclusions

Future Work

MKM 2006 13 / 19

Having described the construction of the figure, now we have to add the conjecture, P

is the midpoint of QB.


Introduction


GeoThms Framework


GCLC & Eukleides

ATP in Geometry

GCLCprover

GeoDB - ERD

GeoThms






GeoThms - Browsing

Recent work

Conclusions

Future Work

MKM 2006 13 / 19

All the commands used in the construction of the figure are internally (within theprover) transformed into primitive constructions of the area method.


Introduction


GeoThms Framework


GCLC & Eukleides

ATP in Geometry

GCLCprover

GeoDB - ERD

GeoThms






GeoThms - Browsing

Recent work

Conclusions

Future Work

MKM 2006 14 / 19

(1)−→

QP−→

PB= 1 , by the statement

(2)

−1 ·

−→

PQ−→

PB

!

= 1 , by geometric simplifications

(3)

−1 ·

SPDF3

dn

SPDBF3

dn

!

= 1 ,by Lemma 37 , second case — points P , B,and C are collinear (point Q eliminated)

(4)

0

@−1 ·

SDF3

dnP

“

SDBP + SBF3

dnP

”

1

A = 1 , by geometric simplifications

(5)

“

−1 · SDF3

dnP

”

“

SDBP + SBF3

dnP

” = 1 , by algebraic simplifications

(6)

−1 ·

„„

SBAN ·SDF3

dnC

«

+

„

−1·

„

SCAN ·SDF3

dnB

«««

SBACN

!

“

SDBP + SBF3

dnP

” = 1 , by Lemma 30 (point P eliminated)

(7)

““

−1 ·

“

SBAN · SDF3

dnC

””

+“

SCAN · SDF3

dnB

””

“

(SBACN · SDBP ) +“

SBACN · SBF3

dnP

”” = 1 , by algebraic simplifications

2

Adding a New Theorem to the Database

Introduction


GeoThms Framework


GCLC & Eukleides

ATP in Geometry

GCLCprover

GeoDB - ERD

GeoThms






GeoThms - Browsing

Recent work

Conclusions

Future Work

MKM 2006 15 / 19

The user (with the status of contributer) can select the “Forms” section in order to adda statement for the new result and the corresponding figure and proof.

GeoThms - Browsing

Introduction


GeoThms Framework


GCLC & Eukleides

ATP in Geometry

GCLCprover

GeoDB - ERD

GeoThms






GeoThms - Browsing

Recent work

Conclusions

Future Work

MKM 2006 16 / 19

The user has many other options for browsing the database.

Recent work

Introduction


GeoThms Framework


GCLC & Eukleides

ATP in Geometry

GCLCprover

GeoDB - ERD

GeoThms






GeoThms - Browsing

Recent work

Conclusions

Future Work

MKM 2006 17 / 19

XML and SVG support.

Recent work

Introduction


GeoThms Framework


GCLC & Eukleides

ATP in Geometry

GCLCprover

GeoDB - ERD

GeoThms






GeoThms - Browsing

Recent work

Conclusions

Future Work

MKM 2006 17 / 19

XML and SVG support.

� geometrical constructions stored in strictly structured files; easy to parse, process,and convert into different forms and formats

� input/output tasks will be supported by generic, external tools and differentgeometry tools will communicate easily

� growing corpora of geometrical constructions will be unified and accessible tousers of different geometry tools

� easier communication and exchange of material with the rest of mathematical andcomputer science community

� there is a wide and growing support for XML

� different sorts of presentation (text form, LATEX form, HTML) easily enabled

� strict content validation of documents with respect to given restrictions.

Conclusions

Introduction


GeoThms Framework


GCLC & Eukleides

ATP in Geometry

GCLCprover

GeoDB - ERD

GeoThms






GeoThms - Browsing

Recent work

Conclusions

Future Work

MKM 2006 18 / 19

GeoThms:

� DGSs (GCLC and Eukleides)� ATP (GCLCprover)� Database - GeoDB

All accessible through a Web interface. GeoThms system is, as far as we know, theonly system that integrates DGSs, ATPs, and a database of geometric problemsin a Web interface.

This framework provides:

� an environment suitable for new ways of studying and teaching geometry atdifferent levels.

� an environment for storing mathematical knowledge (in explicit, declarative way) —about geometrical constructions, proofs, and illustrations.

We hope that GeoThms would contribute to a modern mathematical education.

Future Work

Introduction


GeoThms Framework


GCLC & Eukleides

ATP in Geometry

GCLCprover

GeoDB - ERD

GeoThms






GeoThms - Browsing

Recent work

Conclusions

Future Work

MKM 2006 19 / 19

We hope that with the support from interested parties GeoThms can grow and becamea widely used repository. We would try to make GeoThms a major Internet resource forgeometrical constructions.

We will also work on the following tasks:

� To implement a e-Learning module for the study of Euclidean geometry athigh-school and university level.

� To implement a module for proof visualisation and for moving through thegenerated proofs

� To improve the search mechanism

� To further develop the XML based interchange format (and the corresponding XMLsuite) that can link most of the current geometrical software.

� To implement/develop additional proving methods, primarily synthetic ones (e.g.angle method).

� To link additional geometry programs and additional theorem provers to ourframework and to further develop the Web interface.

integrating dynamic geometry software, deduction systems, and

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