OpenGeo: An Open Geometric Knowledge Basesites.nlsde.buaa.edu.cn/~chenxiaoyu/slides/icms2014.pdfOpenGeo: An Open Geometric Knowledge Base Dongming Wang, Xiaoyu Chen, Wenya An, Lei - [PDF Document] (2024)

OpenGeo: An Open Geometric Knowledge Base

Dongming Wang, Xiaoyu Chen, Wenya An, Lei Jiang, and DanSong

Beihang University, China

August 6, 2014

X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 20141 / 30

Motivation

Outline

1 Motivation

2 Geometric knowledge base: design methodology

3 OpenGeo: an enhanced version of GeoData

4 Conclusion and future work

X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 20142 / 30

Motivation

Geometric knowledge

Geometric knowledge is

rich in content: definitions, axioms, theorems, proofs,problems,solutions, and algorithms;

sophisticated in structure: from basic concepts to derivedconcepts,from simple diagrams to complicated configurations.

Problem

How to digitalize geometric knowledge and make it easilyaccessible,presentable, interoperable, and processable on advancedcomputingmachines and communication devices?

X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 20143 / 30

Motivation

A geometric knowledge base is a special database for storingandmanaging geometric knowledge data.

X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 20144 / 30

Motivation

GeoData: a geometric knowledge base

Resourcesµ

H. S. M. Coxeter and S. L. Greitzer. Geometry Revisited. TheMathematicalAssociation of America, Washington D.C., 1967

S. Chou. Mechanical Geometry Theorem Proving. Reidel, Dordrecht,1988

J. Hadamard. Lessons in Geometry: I. Plane Geometry. AmericanMathematicalSociety, Providence, 2008

GeoData currently includes

- 849 Euclidean plane geometric theorems

- 104 definitions of geometric concepts

- introductions to the historical background of somewell-knowntheorems (e.g., Simson’s theorem)

X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 20145 / 30

Motivation

http://geo.cc4cm.org/geodata/

X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 20146 / 30

http://localhost/geodata

Geometric knowledge base: design methodology

Outline

1 Motivation

2 Geometric knowledge base: design methodology

3 OpenGeo: an enhanced version of GeoData

4 Conclusion and future work

X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 20147 / 30

Geometric knowledge base: design methodology

Geometric knowledge base

The following aspects are needed to be studied for constructingageometric knowledge base.

Geometric knowledge representation

Meta-knowledge representation(the knowledge about geometricknowledge)

X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 20148 / 30

Geometric knowledge base: design methodology

Represent geometric knowledge: multiple forms

Natural languageµa circle with center O andradius r

Algebraic expressionµ

(x, y)|x2 + y2 = r2 or

x = r · 1− t

2

1 + t2

y = r · 2t1 + t2

Drawing instructionµCircle[O, r]

Degeneracy conditionµr = 0

Image:

X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 20149 / 30

Geometric knowledge base: design methodology

Represent geometric knowledge: multiple forms (cont.)

Formalization:- Definition(intersection(l::Line,m::Line),[A::Point where and(incident(A, l),

incident(A,m))], not(parallel(l,m)))

- Theorem([A:=point(), B:=point(), C:=point(), D:=point(),incident(D,

circumcircle(triangle(A,B,C)))], [collinear(foot(D,line(A, B)),foot(D,line(A,

C)), foot(D, line(B, C)))])

Dynamic diagram:

Multimedia: video, audio

X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 201410 / 30

Geometric knowledge base: design methodology

Represent the meta-knowledge: encapsulationandclassification

A knowledge object is individual knowledge unit that can berecognized,differentiated, understood, and manipulated in theprocess of management.

Knowledge objects are used to encapsulate interrelatedgeometricknowledge data.

Knowledge classes are used to define the internal structureofknowledge objects.

- Definition, Axiom, Lemma, Theorem, Corollary, Conjecture,Problem,Example, Exercise, Proof, Solution, Algorithm,Introduction, Remark.

X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 201411 / 30

Geometric knowledge base: design methodology

Definition class

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Geometric knowledge base: design methodology

Other classes

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Geometric knowledge base: design methodology

Organize knowledge objects

Catalog is used to describe how knowledge objects areclustered.

Chapter: Points and Lines Connected with a Triangle

Section: Points of interest

Definition of orthocenter

Knowledge graph is used to describe how knowledge objectsarerelated.

X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 201414 / 30

Geometric knowledge base: design methodology

Knowledge graph: Section 1.5 from ”Geometry Revisited”

C: Points and Lines Connected with aTriangle

T1: Steiner-Lehmus theorem

P1: Steiner-Lehmus theorem’s proof

L1, L2: Lemma used in P1

E1, E2: Exercise for T1

S1, S2: Solution to the exercises

I1, R1: Introduction and remark on T1

D1: Definition of bisector

T2: Theorem: the three innerbisectors of a triangle areconcurrent

D2: Definition of incenter of a triangle

D3: Another definition of incenter ofa triangle

X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 201415 / 30

Geometric knowledge base: design methodology

Knowledge graph: inheritance relations between concepts

X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 201416 / 30

Geometric knowledge base: design methodology

Knowledge graph: inheritance relations between concepts

X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 201417 / 30

Geometric knowledge base: design methodology

Types of relations

Inclusion

A→include B

Inheritance

A→inherit B

Dependance

A→contextOf BA→deriveFrom BA→imply BA→hasProperty BA→decideBA→introduce BA→remarkOn BA→complicate BA→solve BA→exerciseOf B

Association

A→justify BA→applyOn BA→exampleOf BA↔associate BA↔equal B

X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 201418 / 30

OpenGeo: an enhanced version of GeoData

Outline

1 Motivation

2 Geometric knowledge base: design methodology

3 OpenGeo: an enhanced version of GeoData

4 Conclusion and future work

X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 201419 / 30

OpenGeo: an enhanced version of GeoData

OpenGeo is an enhanced version of GeoData, which is equippedwith

web-based interfaces,

new management facilities, and

made open online.

X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 201420 / 30

OpenGeo: an enhanced version of GeoData

Open to users

knowledge objects can be edited or deleted;

meta-information (e.g., language, format, and keyword) canbeannotated for organizing and classifying knowledge objects;

revisions of knowledge objects can be recorded;

knowledge objects can be retrieved in meta-information-basedways;

knowledge objects can be rated and commented forscreeninghigh-quality versions;

new knowledge objects can be created and added to OpenGeo.

*Creative Commons Attribution-ShareAlike license is adopted asits main content license.

X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 201421 / 30

OpenGeo: an enhanced version of GeoData

Implementation techniques: meta-knowledgerepresentation

We adopt ontology (OWL) to formally specify geometricknowledgeobjects and relations among them.

X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 201422 / 30

OpenGeo: an enhanced version of GeoData

Implementation techniques: meta-knowledgerepresentation

knowledge object7→ ontology instanceknowledge class7→ ontologyclass

X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 201423 / 30

OpenGeo: an enhanced version of GeoData

Implementation techniques: meta-knowledgerepresentation

knowledge class structure7→ ontology attributeknowledge graph7→ontology relation

X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 201424 / 30

OpenGeo: an enhanced version of GeoData

Implementation techniques: database schema

Database schema (relational data tables) can be automaticallygeneratedfrom the ontologies.

−→

X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 201425 / 30

OpenGeo: an enhanced version of GeoData

Implementation techniques: user interface

The LAMP (Linux Apache MySQL PHP/Perl/Python) framework

MathEdit: editing formatted formulas in a WISIWIG style

Sketchometry: drawing and exporting dynamic diagrams

GeoGebra: constructing and rendering dynamic diagrams

X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 201426 / 30

http://localhost:8000

Conclusion and future work

Outline

1 Motivation

2 Geometric knowledge base: design methodology

3 OpenGeo: an enhanced version of GeoData

4 Conclusion and future work

X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 201427 / 30

Conclusion and future work

Conclusion

OpenGeo is created for the purpose of research and education,and mayserve as

a public resource for users to test, for instance, geometrictheoremprovers and problem solvers; and

an infrastructure for developing new educational applications(e.g.,generation of textbooks and courses) in online learningenvironments.

We are

formalizing geometric theorems in the OpenGeo collection and

developing semantic querying tools based on images ofdiagrams.

We expect to complete these tasks and release a preliminaryversion ofOpenGeo in early 2015.

X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 201428 / 30

Conclusion and future work

Automated knowledge acquisition

Input Output

If the points A,B, and C are arbitrary, the point D is onthecircumcircle of the triangle ABC, F is theperpendicular foot of theline AC to the line DF , G isthe perpendicular foot of the line BCto the line DG,and E is the perpendicular foot of the line BA totheline DE, then the point F is on the line EG.

X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 201429 / 30

Conclusion and future work

Thanks

X. Chen ([emailprotected]) ICMS2014, Seoul August 6, 201430 / 30

MotivationGeometric knowledge base: design methodologyOpenGeo:an enhanced version of GeoDataConclusion and future work