Representation and automated transformation of geometric statements

Abstract

The emergence of a large quantity of digital resources in geometry, various geometric automated theorem proving systems, and kinds of dynamic geometry software systems has made geometric computation, reasoning, drawing, and knowledge management dynamic, automatic or interactive on computer. Integration of electronic contents and different systems is desired to enhance their accessibility and exploitability. This paper proposes an equivalent transformation framework for manipulating geometric statements available in the literature by using geometry software systems. Such a framework works based on a newly designed geometry description language (GDL), in which geometric statements can be represented naturally and easily. The author discusses and presents key procedures of automatically transforming GDL statements into target system-native representations for manipulation. The author also demonstrates the framework by illustrating equivalent transformation processes and interfaces for compiling the transformation results into executable formats that can be interpreted by the target geometry software systems for automated theorem proving and dynamic diagram drawing.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    Wu W, Basic principles of mechanical theorem proving in elementary geometries, Journal of Systems Science and Mathematical Sciences, 1984, 4(3): 207–235.

    MathSciNet  Google Scholar 

  2. [2]

    Chou S and Gao X, Automated reasoning in geometry, Handbook of Automated Reasoning (Volume I), Elsevier, North Holland, 2001.

    Google Scholar 

  3. [3]

    List of Interactive Geometry Software, http://en.wikipedia.org/wiki/List of interactive geometry software.

  4. [4]

    Chen X and Wang D, Towards an electronic geometry textbook, Proceedings of the 6th International Workshop on Automated Deduction in Geometry (eds. by Botana F and Recio T), Lecture Notes in Artificial Intelligence 4869, Pontevedra, 2006.

    Google Scholar 

  5. [5]

    Chen X, Electronic geometry textbook: A geometric textbook knowledge management system, Proceedings of the 9th International Conference on Mathematical Knowledge Management (eds. by Autexier S, Calmet J, Delahaye D, Ion P, Rideau L, Rioboo R, and Sexton A), Lecture Notes in Artificial Intelligence 6167, Paris, 2010.

    Google Scholar 

  6. [6]

    Chen X and Wang D, Management of geometric knowledge in textbooks, Data & Knowledge Engineering, 2012, 73: 43–57.

    Article  Google Scholar 

  7. [7]

    Quaresma P, Janicić P, Tomasevic J, Vujosevic-Janicic M, and Tosic D, XML-based format for geometry — XML-based format for descriptions of geometrical constructions and geometrical proofs, Communicating Mathematics in Digital Era, Peters A K, Ltd. Wellesley, MA, USA, 2008.

    Google Scholar 

  8. [8]

    GCLC, http://poincare.matf.bg.ac.rs/janicic//gclc/.

  9. [9]

    Janicić P, GCLC — A tool for constructive euclidean geometry and more than that, Proceedings of the 2nd International Congress on Mathematical Software (ed. by Iglesias A and Takayama N), Lecture Notes in Computer Science 4151, Castro Urdiales, 2006.

    Google Scholar 

  10. [10]

    Janicić P, Geometry constructions language, Journal of Automated Reasoning, 2010, 44(1–2): 3–24.

    Article  MATH  MathSciNet  Google Scholar 

  11. [11]

    GeoThms, http://hilbert.mat.uc.pt/geothms/.

  12. [12]

    Quaresma P and Janicić P, Geothms — A web system for Euclidean constructive geometry, Electronic Notes in Theoretical Computer Science, 2007, 174(2): 35–48.

    Article  Google Scholar 

  13. [13]

    Janicić P and Quaresma P, System description: GCLCprover + Geothms, Proceedings of the 3rd International Joint Conference on Automated Reasoning (eds. by Furbach U and Shankar N), Lecture Notes in Artificial Intelligence 4130, Seattle, 2006.

    Google Scholar 

  14. [14]

    Egido S, Hendriks M, Kreis Y, Kortenkamp U, and Marquès D, i2g Common File Format Final Version, Technical Report D3.10, The Intergeo Consortium, 2010.

    Google Scholar 

  15. [15]

    Dynamic Geometry Software Speaking I2geo, http://i2geo.net/xwiki/bin/view/Softwares/.

  16. [16]

    Intergeo, http://i2geo.net/.

  17. [17]

    OpenMath, http://www.openmath.org/cd/.

  18. [18]

    TGTP, http://hilbert.mat.uc.pt/TGTP/.

  19. [19]

    Quaresma P, Thousands of Geometric problems for geometric Theorem Provers (TGTP), Proceedings of the 8th International Workshop on Automated Deduction in Geometry (eds. by Schreck P, Narboux J, and Richter-Gebert J), Lecture Notes in Artificial Intelligence 6877, Munich, 2010.

    Google Scholar 

  20. [20]

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

    MATH  Google Scholar 

  21. [21]

    GeoGebra, http://www.geogebra.org/cms/.

  22. [22]

    GEOTHER, http://www-calfor.lip6.fr/wang/GEOTHER/.

  23. [23]

    Coxeter H S M and Greitzer S L, Geometry Revisited, The Mathematical Association of America, Washington DC, 1967.

    MATH  Google Scholar 

  24. [24]

    Cinderella, http://www.cinderella.de/.

  25. [25]

    Gräbe H G, The symbolicData GEO records — A public repository of geometry theorem proof schemes, Proceedings of the 4th International Workshop on Automated Deduction in Geometry (ed. by Winkler F), Lecture Notes in Artificial Intelligence 2930, Linz, 2002.

    Google Scholar 

  26. [26]

    GeoProver, http://www.reduce-algebra.com/docs/geoprover.html.

  27. [27]

    Wang D, GEOTHER 1.1: Handling and proving geometric theorems automatically, Proceedings of the 4th International Workshop on Automated Deduction in Geometry (ed. by Winkler F), Lecture Notes in Artificial Intelligence 2930, Linz, 2002.

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Xiaoyu Chen.

Additional information

This work has been supported partially by the SKLSDE Open Fund (SKLSDE-2011KF-02), the Natural Science Foundation of China under Grant No. 61003139, and the MOE-Intel Joint Research Fund (MOE-INTEL-11-03).

This paper was recommended for publication by Editor LI Ziming.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Chen, X. Representation and automated transformation of geometric statements. J Syst Sci Complex 27, 382–412 (2014). https://doi.org/10.1007/s11424-014-0316-0

Download citation

Keywords

  • Automated diagram drawing
  • equivalent transformation
  • formalized geometric statements
  • geometric automated theorem proving