url email. dateSubmission. proverId description category. userId bibref. teoId. teoId code. drawerId figure. userId bibref. figureId. teoId code. proverId proof status.
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
MKM 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
MKM 2006
2 / 19
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) /
GeoThms - Browsing Recent work Conclusions Future Work
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
4 / 19
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
MKM 2006
1
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
MKM 2006
1
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.
GeoThms - Browsing Recent work Conclusions Future Work
MKM 2006
synthetic proof style Synthetic methods attempt to automate traditional geometry proof methods that produce human-readable proofs.
7 / 19
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
MKM 2006
very efficient for many conjectures
8 / 19
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
MKM 2006
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
9 / 19
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
MKM 2006
GeoThms is accessible from http://hilbert.mat.uc.pt/ geothms
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10 / 19
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
contributers regular users
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
GeoThms - Browsing Recent work Conclusions Future Work
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
MKM 2006
contributers regular users
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
GeoThms - Browsing Recent work Conclusions Future Work
D
2
MKM 2006
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
MKM 2006
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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
MKM 2006
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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
MKM 2006
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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)
“
Conclusions Future Work
(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
MKM 2006
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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.
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
MKM 2006
16 / 19
Recent work Introduction
XML and SVG support.
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
MKM 2006
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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
Conclusions Future Work
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.
MKM 2006
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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
19 / 19