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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
X. Chen ([email protected]) ICMS2014, Seoul August 6, 2014 2 / 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 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?
X. Chen ([email protected]) ICMS2014, Seoul August 6, 2014 3 / 30
Motivation
A geometric knowledge base is a special database for storing andmanaging geometric knowledge data.
X. Chen ([email protected]) ICMS2014, Seoul August 6, 2014 4 / 30
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)
X. Chen ([email protected]) ICMS2014, Seoul August 6, 2014 5 / 30
Motivation
http://geo.cc4cm.org/geodata/
X. Chen ([email protected]) ICMS2014, Seoul August 6, 2014 6 / 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 ([email protected]) ICMS2014, Seoul August 6, 2014 7 / 30
Geometric knowledge base: design methodology
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)
X. Chen ([email protected]) ICMS2014, Seoul August 6, 2014 8 / 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 ([email protected]) ICMS2014, Seoul August 6, 2014 9 / 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 ([email protected]) ICMS2014, Seoul August 6, 2014 10 / 30
Geometric knowledge base: design methodology
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.
X. Chen ([email protected]) ICMS2014, Seoul August 6, 2014 11 / 30
Geometric knowledge base: design methodology
Definition class
X. Chen ([email protected]) ICMS2014, Seoul August 6, 2014 12 / 30
Geometric knowledge base: design methodology
Other classes
X. Chen ([email protected]) ICMS2014, Seoul August 6, 2014 13 / 30
Geometric knowledge base: design methodology
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.
X. Chen ([email protected]) ICMS2014, Seoul August 6, 2014 14 / 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 are concurrent
D2: Definition of incenter of a triangle
D3: Another definition of incenter ofa triangle
X. Chen ([email protected]) ICMS2014, Seoul August 6, 2014 15 / 30
Geometric knowledge base: design methodology
Knowledge graph: inheritance relations between concepts
X. Chen ([email protected]) ICMS2014, Seoul August 6, 2014 16 / 30
Geometric knowledge base: design methodology
Knowledge graph: inheritance relations between concepts
X. Chen ([email protected]) ICMS2014, Seoul August 6, 2014 17 / 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→decide BA→introduce BA→remarkOn BA→complicate BA→solve BA→exerciseOf B
Association
A→justify BA→applyOn BA→exampleOf BA↔associate BA↔equal B
X. Chen ([email protected]) ICMS2014, Seoul August 6, 2014 18 / 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 ([email protected]) ICMS2014, Seoul August 6, 2014 19 / 30
OpenGeo: an enhanced version of GeoData
OpenGeo is an enhanced version of GeoData, which is equipped with
web-based interfaces,
new management facilities, and
made open online.
X. Chen ([email protected]) ICMS2014, Seoul August 6, 2014 20 / 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) 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.
X. Chen ([email protected]) ICMS2014, Seoul August 6, 2014 21 / 30
OpenGeo: an enhanced version of GeoData
Implementation techniques: meta-knowledgerepresentation
We adopt ontology (OWL) to formally specify geometric knowledgeobjects and relations among them.
X. Chen ([email protected]) ICMS2014, Seoul August 6, 2014 22 / 30
OpenGeo: an enhanced version of GeoData
Implementation techniques: meta-knowledgerepresentation
knowledge object7→ ontology instanceknowledge class7→ ontology class
X. Chen ([email protected]) ICMS2014, Seoul August 6, 2014 23 / 30
OpenGeo: an enhanced version of GeoData
Implementation techniques: meta-knowledgerepresentation
knowledge class structure7→ ontology attributeknowledge graph7→ ontology relation
X. Chen ([email protected]) ICMS2014, Seoul August 6, 2014 24 / 30
OpenGeo: an enhanced version of GeoData
Implementation techniques: database schema
Database schema (relational data tables) can be automatically generatedfrom the ontologies.
−→
X. Chen ([email protected]) ICMS2014, Seoul August 6, 2014 25 / 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 ([email protected]) ICMS2014, Seoul August 6, 2014 26 / 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 ([email protected]) ICMS2014, Seoul August 6, 2014 27 / 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, 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.
X. Chen ([email protected]) ICMS2014, Seoul August 6, 2014 28 / 30
Conclusion and future work
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.
X. Chen ([email protected]) ICMS2014, Seoul August 6, 2014 29 / 30
Conclusion and future work
Thanks
X. Chen ([email protected]) ICMS2014, Seoul August 6, 2014 30 / 30
MotivationGeometric knowledge base: design methodologyOpenGeo: an enhanced version of GeoDataConclusion and future work