OpenGeo: An Open Geometric Knowledge Base · OpenGeo: an enhanced version of GeoData OpenGeois an...

OpenGeo: An Open Geometric Knowledge Base

Dongming Wang, Xiaoyu Chen, Wenya An, Lei Jiang, and Dan Song

Beihang University, China

August 6, 2014

X. Chen ([email protected]) ICMS2014, Seoul August 6, 2014 1 / 30

Motivation

Outline

1 Motivation

2 Geometric knowledge base: design methodology

3 OpenGeo: an enhanced version of GeoData

4 Conclusion and future work


Motivation

Geometric knowledge

Geometric knowledge is

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

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

Problem

How to digitalize geometric knowledge and make it easily accessible,presentable, interoperable, and processable on advanced computingmachines and communication devices?


Motivation

A geometric knowledge base is a special database for storing andmanaging geometric knowledge data.


Motivation

GeoData: a geometric knowledge base

Resourcesµ

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

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

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

GeoData currently includes

- 849 Euclidean plane geometric theorems

- 104 definitions of geometric concepts

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


Motivation

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


http://localhost/geodata

Geometric knowledge base: design methodology

Outline

1 Motivation






Geometric knowledge base

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

Geometric knowledge representation

Meta-knowledge representation(the knowledge about geometric knowledge)



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:



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



Represent the meta-knowledge: encapsulation andclassification

A knowledge object is individual knowledge unit that can be recognized,differentiated, understood, and manipulated in the process of management.

Knowledge objects are used to encapsulate interrelated geometricknowledge data.

Knowledge classes are used to define the internal structure ofknowledge objects.

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



Definition class



Other classes



Organize knowledge objects

Catalog is used to describe how knowledge objects are clustered.

Chapter: Points and Lines Connected with a Triangle

Section: Points of interest

Definition of orthocenter

Knowledge graph is used to describe how knowledge objects arerelated.



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 are concurrent

D2: Definition of incenter of a triangle

D3: Another definition of incenter ofa triangle



Knowledge graph: inheritance relations between concepts



Types of relations

Inclusion

A→include B

Inheritance

A→inherit B

Dependance

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

Association

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


OpenGeo: an enhanced version of GeoData

Outline

1 Motivation






OpenGeo is an enhanced version of GeoData, which is equipped with

web-based interfaces,

new management facilities, and

made open online.



Open to users

knowledge objects can be edited or deleted;

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

revisions of knowledge objects can be recorded;

knowledge objects can be retrieved in meta-information-based ways;

knowledge objects can be rated and commented for screeninghigh-quality versions;

new knowledge objects can be created and added to OpenGeo.

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



Implementation techniques: meta-knowledgerepresentation

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




knowledge object7→ ontology instanceknowledge class7→ ontology class




knowledge class structure7→ ontology attributeknowledge graph7→ ontology relation



Implementation techniques: database schema

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

−→



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


http://localhost:8000

Conclusion and future work

Outline

1 Motivation






Conclusion

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

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

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

We are

formalizing geometric theorems in the OpenGeo collection and

developing semantic querying tools based on images of diagrams.

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



Automated knowledge acquisition

Input Output

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



Thanks


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

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