OpenGeo: An Open Geometric Knowledge Base Dongming Wang, Xiaoyu - PowerPoint PPT Presentation

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 (franknewchen@gmail.com) ICMS 2014, Seoul August 6, 2014 1 / 30

Motivation Outline Motivation 1 Geometric knowledge base: design methodology 2 OpenGeo: an enhanced version of GeoData 3 Conclusion and future work 4 X. Chen (franknewchen@gmail.com) ICMS 2014, 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 computing machines and communication devices? X. Chen (franknewchen@gmail.com) ICMS 2014, Seoul August 6, 2014 3 / 30

Motivation A geometric knowledge base is a special database for storing and managing geometric knowledge data. X. Chen (franknewchen@gmail.com) ICMS 2014, Seoul August 6, 2014 4 / 30

Motivation GeoData: a geometric knowledge base Resources µ H. S. M. Coxeter and S. L. Greitzer. Geometry Revisited. The Mathematical Association of America, Washington D.C., 1967 S. Chou. Mechanical Geometry Theorem Proving. Reidel, Dordrecht, 1988 J. Hadamard. Lessons in Geometry: I. Plane Geometry. American Mathematical Society, Providence, 2008 GeoData currently includes - 849 Euclidean plane geometric theorems - 104 definitions of geometric concepts - introductions to the historical background of some well-known theorems (e.g., Simson’s theorem) X. Chen (franknewchen@gmail.com) ICMS 2014, Seoul August 6, 2014 5 / 30

Motivation http://geo.cc4cm.org/geodata/ X. Chen (franknewchen@gmail.com) ICMS 2014, Seoul August 6, 2014 6 / 30

Geometric knowledge base: design methodology Outline Motivation 1 Geometric knowledge base: design methodology 2 OpenGeo: an enhanced version of GeoData 3 Conclusion and future work 4 X. Chen (franknewchen@gmail.com) ICMS 2014, Seoul August 6, 2014 7 / 30

Geometric knowledge base: design methodology Geometric knowledge base The following aspects are needed to be studied for constructing a geometric knowledge base. Geometric knowledge representation Meta-knowledge representation (the knowledge about geometric knowledge) X. Chen (franknewchen@gmail.com) ICMS 2014, Seoul August 6, 2014 8 / 30

Geometric knowledge base: design methodology Represent geometric knowledge: multiple forms Natural language µ a circle with center O and radius r Image: Algebraic expression µ  x = r · 1 − t 2   ( x, y ) | x 2 + y 2 = r 2 or  1 + t 2 2 t y = r ·    1 + t 2 Drawing instruction µ Circle[ O , r ] Degeneracy condition µ r = 0 X. Chen (franknewchen@gmail.com) ICMS 2014, 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 (franknewchen@gmail.com) ICMS 2014, Seoul August 6, 2014 10 / 30

Geometric knowledge base: design methodology Represent the meta-knowledge: encapsulation and classification 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 geometric knowledge data. Knowledge classes are used to define the internal structure of knowledge objects. - Definition, Axiom, Lemma, Theorem, Corollary, Conjecture, Problem, Example, Exercise, Proof, Solution, Algorithm, Introduction, Remark. X. Chen (franknewchen@gmail.com) ICMS 2014, Seoul August 6, 2014 11 / 30

Geometric knowledge base: design methodology Definition class X. Chen (franknewchen@gmail.com) ICMS 2014, Seoul August 6, 2014 12 / 30

Geometric knowledge base: design methodology Other classes X. Chen (franknewchen@gmail.com) ICMS 2014, 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 are related. X. Chen (franknewchen@gmail.com) ICMS 2014, 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 a Triangle T 1 : Steiner-Lehmus theorem P 1 : Steiner-Lehmus theorem’s proof L 1 , L 2 : Lemma used in P 1 E 1 , E 2 : Exercise for T 1 S 1 , S 2 : Solution to the exercises I 1 , R 1 : Introduction and remark on T 1 D 1 : Definition of bisector T 2 : Theorem: the three inner bisectors of a triangle are concurrent D 2 : Definition of incenter of a triangle D 3 : Another definition of incenter of a triangle X. Chen (franknewchen@gmail.com) ICMS 2014, Seoul August 6, 2014 15 / 30

Geometric knowledge base: design methodology Knowledge graph: inheritance relations between concepts X. Chen (franknewchen@gmail.com) ICMS 2014, Seoul August 6, 2014 16 / 30

Geometric knowledge base: design methodology Knowledge graph: inheritance relations between concepts X. Chen (franknewchen@gmail.com) ICMS 2014, Seoul August 6, 2014 17 / 30

Geometric knowledge base: design methodology Types of relations Dependance A → contextOf B A → deriveFrom B Association Inclusion A → imply B A → justify B A → include B A → hasProperty B A → applyOn B A → decide B A → exampleOf B Inheritance A → introduce B A ↔ associate B A → remarkOn B A → inherit B A ↔ equal B A → complicate B A → solve B A → exerciseOf B X. Chen (franknewchen@gmail.com) ICMS 2014, Seoul August 6, 2014 18 / 30

OpenGeo: an enhanced version of GeoData Outline Motivation 1 Geometric knowledge base: design methodology 2 OpenGeo: an enhanced version of GeoData 3 Conclusion and future work 4 X. Chen (franknewchen@gmail.com) ICMS 2014, 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 (franknewchen@gmail.com) ICMS 2014, 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 be annotated 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 screening high-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 (franknewchen@gmail.com) ICMS 2014, Seoul August 6, 2014 21 / 30

OpenGeo: an enhanced version of GeoData Implementation techniques: meta-knowledge representation We adopt ontology (OWL) to formally specify geometric knowledge objects and relations among them. X. Chen (franknewchen@gmail.com) ICMS 2014, Seoul August 6, 2014 22 / 30

OpenGeo: an enhanced version of GeoData Implementation techniques: meta-knowledge representation knowledge object �→ ontology instance knowledge class �→ ontology class X. Chen (franknewchen@gmail.com) ICMS 2014, Seoul August 6, 2014 23 / 30

OpenGeo: an enhanced version of GeoData Implementation techniques: meta-knowledge representation knowledge class structure �→ ontology attribute knowledge graph �→ ontology relation X. Chen (franknewchen@gmail.com) ICMS 2014, Seoul August 6, 2014 24 / 30

OpenGeo: an enhanced version of GeoData Implementation techniques: database schema Database schema (relational data tables) can be automatically generated from the ontologies. − → X. Chen (franknewchen@gmail.com) ICMS 2014, 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 (franknewchen@gmail.com) ICMS 2014, Seoul August 6, 2014 26 / 30

Conclusion and future work Outline Motivation 1 Geometric knowledge base: design methodology 2 OpenGeo: an enhanced version of GeoData 3 Conclusion and future work 4 X. Chen (franknewchen@gmail.com) ICMS 2014, Seoul August 6, 2014 27 / 30