Integrating Dynamic Geometry Software, Deduction Systems, and ...

Integrating Dynamic Geometry Software, Deduction Systems, and Theorem Repositories

Pedro Quaresma CISUC/Mathematics Department University of Coimbra Portugal

Predrag Janiˇci´c Faculty of Mathematics University of Belgrade Serbia August 10-12, 2006

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Introduction Introduction Computers & Geometry GeoThms Framework Dynamic Geometry Software GCLC & Eukleides ATP in Geometry GCLCprover GeoDB - ERD

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

GeoThms GeoThms - by Example Describe the Construction Testing the Conjecture The Proof - Area Method Adding a New Theorem to the Database

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

GeoThms - Browsing Recent work Conclusions Future Work

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

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Computers & Geometry Introduction

Computer technologies give new ways for studying geometry

Computers & Geometry GeoThms Framework Dynamic Geometry Software GCLC & Eukleides ATP in Geometry GCLCprover GeoDB - ERD

Dynamic Geometry Software

Visualise/Explore/Test Conjectures

GeoThms GeoThms - by Example Describe the Construction Testing the Conjecture The Proof - Area Method Adding a New Theorem to the Database

Geometric Automated Theorem Proving algebraic proofs (efficiency).

synthetic proofs (human-readable) /


Problems Repositories

browse through the existing knowledge.

GeoThms integrates all these features bringing new forms in communicating mathematics. MKM 2006

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GeoThms Framework Introduction Computers & Geometry GeoThms Framework

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

Dynamic Geometry Software GCLC & Eukleides ATP in Geometry GCLCprover

Dynamic Geometry Software

GCLC & Eukleides

GeoDB - ERD GeoThms GeoThms - by Example Describe the Construction Testing the Conjecture The Proof - Area Method

Geometric Automated Theorem Proving method).

GCLCprover (implements the area

Adding a New Theorem to the Database GeoThms - Browsing Recent work Conclusions Future Work

Problems Repositories database.

geoDB - geometric theorems, illustrations and proofs

GeoThms provides an environment suitable for new ways of studying and teaching geometry at different levels, and for storing geometrical knowledge: descriptions of construction; geometrical conjectures; geometrical proofs MKM 2006

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Dynamic Geometry Software Introduction Computers & Geometry GeoThms Framework Dynamic Geometry Software GCLC & Eukleides ATP in Geometry GCLCprover

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

GeoDB - ERD GeoThms GeoThms - by Example Describe the Construction Testing the Conjecture

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

The Proof - Area Method Adding a New Theorem to the Database GeoThms - Browsing Recent work Conclusions Future Work

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. MKM 2006

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GCLC & Eukleides Introduction Computers & Geometry GeoThms Framework Dynamic Geometry Software GCLC & Eukleides

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

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 the Database GeoThms - Browsing Recent work Conclusions Future Work

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Predrag Janiˇci´c, www.matf.bg.ac.yu/ janicic/gclc/

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GCLC & Eukleides Introduction Computers & Geometry

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

GeoThms Framework Dynamic Geometry Software

eukleides is a compiler for typesetting geometric figures within a (La)TeX

GCLC & Eukleides

document.

ATP in Geometry GCLCprover

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

GeoDB - ERD GeoThms GeoThms - by Example Describe the Construction Testing the Conjecture The Proof - Area Method Adding a New Theorem to the Database GeoThms - Browsing Recent work Conclusions Future Work

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

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ATP in Geometry Introduction

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

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 the Database

algebraic proof style Algebraic proof style methods are based on reducing geometry properties to algebraic properties expressed in terms of Cartesian coordinates. These methods are usually very efficient, but the proofs they produce do not reflect the geometry nature of the problem and they give only a yes/no conclusion.


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

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GCLCprover Introduction

GCLCprover - synthetic geometric ATP (area method)

Computers & Geometry GeoThms Framework Dynamic Geometry Software GCLC & Eukleides ATP in Geometry GCLCprover GeoDB - ERD GeoThms GeoThms - by Example

implements the area method

Describe the Construction Testing the Conjecture The Proof - Area Method

simple and tight integration with GCLC and Eukleides

Adding a New Theorem to the Database GeoThms - Browsing Recent work

human-readable proofs

Conclusions Future Work

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very efficient for many conjectures

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GeoDB - ERD Introduction

users

Computers & Geometry

userId name username passwd type affiliation url email dateSubmission

GeoThms Framework Dynamic Geometry Software GCLC & Eukleides ATP in Geometry GCLCprover GeoDB - ERD

bibrefs userId

bibrefId bibtexEntry

bibrefId

GeoThms GeoThms - by Example

figures

theorems

proofs

Describe the Construction Testing the Conjecture The Proof - Area Method Adding a New Theorem to the Database GeoThms - Browsing Recent work

figureId teoId code drawerId figure userId bibref dateSubmission

teoId

teoId teoName description category userId bibref dateSubmission

teoId

demId teoId code proverId proof status userId bibref dateSubmission

Conclusions

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proverId

drawerId

Future Work

drawers

authors

provers

drawerId name version description url email dateSubmission

authorId name affiliation url email dateSubmission

proverId name version description url email dateSubmission

authordrawer authorId drawerId

authorprover authorId proverId

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GeoThms Introduction Computers & Geometry GeoThms Framework Dynamic Geometry Software

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

GCLC & Eukleides ATP in Geometry

contributers regular users

contributers

GCLCprover Forms

Interaction module

GeoDB - ERD (add/update data)

GeoThms GeoThms - by Example Repository

W e b

Describe the Construction Testing the Conjecture The Proof - Area Method

I n t e r f a c e

Adding a New Theorem to the Database GeoThms - Browsing Recent work Conclusions Future Work

statement

LaTeX + auxiliary tools

geometric construction

DGS

ATP

(GCLC,Eukleides,...)

(GCLCprover,...)

figures

statements (provers/drawers/...)

geometric construction with conjecture

proofs

Reports (listings/technical reports) contributers regular users

2

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

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GeoThms Introduction Computers & Geometry GeoThms Framework Dynamic Geometry Software

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

GCLC & Eukleides ATP in Geometry GCLCprover GeoDB - ERD

contributers

GeoThms

Input via HTML forms


Forms

GeoThms - by Example

Output via HTML files DGS code

Interaction module

(add/update data)

Describe the Construction Testing the Conjecture The Proof - Area Method Adding a New Theorem to the Database

LaTeX format


Repository

W e b I n t e r f a c e

GCLC or Eukleides ATP code GCLCprover

statement

LaTeX + auxiliary tools

geometric construction

DGS

ATP

(GCLC,Eukleides,...)

(GCLCprover,...)

figures

statements (provers/drawers/...)

geometric construction with conjecture

proofs

Reports

GCLC code + conjecture or Eukleides code + conjecture (via a conversion tool)

Proofs in PDF format

(listings/technical reports)

PDF format

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Output via HTML files

Figures in JPEG format

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GeoThms - by Example Introduction Computers & Geometry GeoThms Framework Dynamic Geometry Software GCLC & Eukleides ATP in Geometry

Theorem 1 (Gramy P1432 ) Given a parallelogram ABCD , a point N , obtained by the intersection of a line parallel to AC passing through B , and a line perpendicular to AC 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 .

GCLCprover GeoDB - ERD

N

GeoThms GeoThms - by Example Describe the Construction Testing the Conjecture The Proof - Area Method

C

Adding a New Theorem to the Database

Q

B

P


D

2

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A

P143 of “Gramy: A Geometry Theorem Prover Capable of Construction” by Matsuda and Vanlehn. 11 / 19

Describe the Construction Introduction

We begin by specifying the construction in the DGSs language.

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 the Database GeoThms - Browsing Recent work Conclusions Future Work

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Describe the Construction Introduction

We begin by specifying the construction in the DGSs language.


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Testing the Conjecture Introduction Computers & Geometry GeoThms Framework

Having described the construction of the figure, now we have to add the conjecture, P is the midpoint of QB .

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 the Database GeoThms - Browsing Recent work Conclusions Future Work

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Testing the Conjecture 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 the Database GeoThms - Browsing Recent work Conclusions Future Work

All the commands used in the construction of the figure are internally (within the prover) transformed into primitive constructions of the area method. MKM 2006

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The Proof - Area Method Introduction Computers & Geometry GeoThms Framework Dynamic Geometry Software GCLC & Eukleides

−→ QP −→ = 1 PB

(1)

ATP in Geometry GCLCprover

−→ ! PQ −1 · −→ = 1 PB

(2)

GeoDB - ERD

,

by the statement

,

by geometric simplifications

,

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

,

by geometric simplifications

,

by algebraic simplifications

,

by Lemma 30 (point P eliminated)

,

by algebraic simplifications

GeoThms GeoThms - by Example

SP DF 3

(3)

−1 ·

Describe the Construction

The Proof - Area Method

0

(4)

2

@−1 · “

(5)

“


(6)

−1 ·

(7)

=1

“

P dn

SDBP + SBF 3

P dn

−1 · SDF 3

P dn

SDBP + SBF 3

”

dn

P

1

”A = 1

” =1

««« ! « „ „ „„ SBAN ·SDF 3 C + −1· SCAN ·SDF 3 B dn dn SBACN

“ ““

!

SDF 3

GeoThms - Browsing Recent work

SP DBF 3

dn

Testing the Conjecture

Adding a New Theorem to the Database

dn

SDBP + SBF 3

P dn

”

=1

“ ”” “ ”” −1 · SBAN · SDF 3 C + SCAN · SDF 3 B dn dn “ “ ”” =1 (SBACN · SDBP ) + SBACN · SBF 3 P dn

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Adding a New Theorem to the Database Introduction Computers & Geometry GeoThms Framework

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

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 the Database GeoThms - Browsing Recent work Conclusions Future Work

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GeoThms - Browsing Introduction

The user has many other options for browsing the database.


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Recent work Introduction

XML and SVG support.


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Recent work Introduction

XML and SVG support.

Computers & Geometry GeoThms Framework Dynamic Geometry Software GCLC & Eukleides ATP in Geometry

geometrical constructions stored in strictly structured files; easy to parse, process,

and convert into different forms and formats

GCLCprover GeoDB - ERD GeoThms

input/output tasks will be supported by generic, external tools and different

geometry tools will communicate easily

GeoThms - by Example Describe the Construction Testing the Conjecture The Proof - Area Method Adding a New Theorem to the Database GeoThms - Browsing Recent work

growing corpora of geometrical constructions will be unified and accessible to

users of different geometry tools easier communication and exchange of material with the rest of mathematical and

computer 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.

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Conclusions Introduction

GeoThms:

Computers & Geometry GeoThms Framework Dynamic Geometry Software GCLC & Eukleides ATP in Geometry GCLCprover GeoDB - ERD

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

GeoThms GeoThms - by Example Describe the Construction Testing the Conjecture The Proof - Area Method

All accessible through a Web interface. GeoThms system is, as far as we know, the only system that integrates DGSs, ATPs, and a database of geometric problems in a Web interface.

Adding a New Theorem to the Database GeoThms - Browsing

This framework provides:

Recent work Conclusions Future Work

an environment suitable for new ways of studying and teaching geometry at

different 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. MKM 2006

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Future Work Introduction Computers & Geometry GeoThms Framework Dynamic Geometry Software

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

GCLC & Eukleides ATP in Geometry

We will also work on the following tasks:

GCLCprover GeoDB - ERD GeoThms GeoThms - by Example Describe the Construction Testing the Conjecture The Proof - Area Method Adding a New Theorem to the Database GeoThms - Browsing

To implement a e-Learning module for the study of Euclidean geometry at

high-school and university level. To implement a module for proof visualisation and for moving through the

generated proofs To improve the search mechanism

Recent work Conclusions Future Work

To further develop the XML based interchange format (and the corresponding XML

suite) 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 our

framework and to further develop the Web interface. MKM 2006

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