Formalization and Specification of Geometric Knowledge Objects


This paper presents our work on the identification, formalization, structuring, and specification of geometric knowledge objects for the purpose of semantic representation and knowledge management. We classify geometric knowledge according to how it has been accumulated and represented in the geometric literature, formalize geometric knowledge statements by adapting the language of first-order logic, specify knowledge objects with embedded knowledge in a retrievable and extensible data structure, and organize them by modeling the hierarchic structure of relations among them. Some examples of formal specification for geometric knowledge objects are given to illustrate our approach. The underlying idea of the approach has been used successfully for automated geometric reasoning, knowledge base creation, and electronic document generation.

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


  1. 1

    Chou S., Gao X.: Automated reasoning in geometry. In: Robinson, A., Voronkov, A. (eds.) Handbook of Automated Reasoning, vol. I, pp. 707–749. Elsevier, North Holland (2001)

  2. 2

    List of Interactive Geometry Software.

  3. 3

    Kerber, M. (ed.): Management of Mathematical Knowledge. Special issue of Mathematics in Computer Science, vol. 2, no. 2. Birkhäuser, Basel (2008)

  4. 4

    MathML. Accessed 1 Nov 2013

  5. 5

    OpenMath. Accessed 1 Nov 2013

  6. 6

    OMDoc. Accessed 1 Nov 2013

  7. 7

    MathDox. Accessed 1 Nov 2013

  8. 8

    NIST Digital Library of Mathematical Functions. Accessed 1 Nov 2013

  9. 9

    Miller, B.: Three years of DLMF: web, math and search. In: Carette, J., Aspinall, D., Lange, C., Sojka, P., Windsteiger, W. (eds.) Proceedings of Conference on Intelligent Computer Mathematics, CICM ’13. LNAI, vol. 7961, pp. 288–295. Springer, Berlin (2013)

  10. 10

    Hypertextual Electronic Library of Mathematics. Accessed 1 Nov 2013

  11. 11

    MBase: A Mathematical Knowledge Base. Accessed 1 Nov 2013

  12. 12

    arXMLiv: Translating the arXiv to XML+MathML. Accessed 1 Nov 2013

  13. 13

    ActiveMath. Accessed 1 Nov 2013

  14. 14

    Wortel TUE. Accessed 1 Nov 2013

  15. 15

    Quaresma, P., Janic˘ić, P., Tomasevic, J., Vujosevic-Janicic, M., Tosic, D.: XML-based format for geometry—XML-based Format for descriptions of geometrical constructions and geometrical proofs. Chapter in Communicating Mathematics in Digital Era, pp. 183–197. A K Peters, Wellesley (2008)

  16. 16

    Janic˘ić, P.: GCLC—a tool for constructive Euclidean geometry and more than that. In: Takayama, N., Iglesias, A. (eds.) Proceedings of the 2nd International Congress on Mathematical Software, ICMS ’06, LNCS, vol. 4151, pp. 58–73. Springer, Berlin (2006)

  17. 17

    Janic˘ić P.: Geometry constructions language. J. Autom. Reason. 44(1–2), 3–24 (2010)

    Google Scholar 

  18. 18

    Quaresma, P.: An XML-format for conjectures in geometry. In: Conferences on Intelligent Computer Mathematics, CICM ’12, Work in Progress, 8–13 July 2012, Jacobs University, Bremen, Germany

  19. 19

    Quaresma, P., Janic˘ić, P.: GeoThms—a web system for Euclidean constructive geometry. In: Proceedings of the 7th Workshop on User Interfaces for Theorem Provers, UITP ’06. Electron. Notes Theor. Comput. Sci. 174(2), 35–48 (2007)

  20. 20

    Janic˘ić, P., Quaresma, P.: System description: GCLCprover + GeoThms. In: Furbach, U., Shankar, N. (eds.) Proceedings of the 3rd International Joint Conference on Automated Reasoning, IJCAR ’06. LNAI, vol. 4130, pp.145–150. Springer, Berlin (2006)

  21. 21


  22. 22

    Egido, S., Hendriks, M., Kreis, Y., Kortenkamp, U., Marquès, D.: i2g Common File Format Final Version. Tech. Rep. D3.10, The Intergeo Consortium (2010)

  23. 23

    Dynamic Geometry Software Speaking I2geo. Accessed 1 Nov 2013

  24. 24

    Intergeo. Accessed 1 Nov 2013

  25. 25

    Quaresma, P.: Thousands of geometric problems for geometric theorem provers (TGTP). In: Schreck, P., Narboux, J., Richter-Gebert, J. (eds.) Proceedings of the 8th International Workshop on Automated Deduction in Geometry, ADG ’10. LNAI 6877, pp. 169–181. Springer, Berlin (2011)

  26. 26

    TGTP. Accessed 1 Nov 2013

  27. 27

    Chen, X., Huang, Y., Wang, D.: On the design and implementation of a geometric knowledge base. In: Sturm, T., Zengler, C. (eds.) Proceedings of the 7th International Workshop on Automated Deduction in Geometry, ADG ’08. LNAI 6301, pp. 22–41. Springer, Berlin (2011)

  28. 28

    Chen, X.: Electronic geometry textbook: a geometric textbook knowledge management system. In: Autexier, S., Calmet, J., Delahaye, D., Ion, P., Rideau, L., Rioboo, R., Sexton, A. (eds.) Proceedings of Conference on Intelligent Computer Mathematics CICM ’10. LNAI, vol. 6167, pp. 278–292. Springer, Berlin (2010)

  29. 29

    Chen, X., Zhao, T., Wang, D.: GeoText: an intelligent dynamic geometry textbook. In: Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation, ISSAC ’12. ACM Commun. Comput. Algebra 46(4) Issue 182, 171–175 (2012)

  30. 30

    Chen X., Wang D.: Management of geometric knowledge in textbooks. Data Knowl. Eng. 73, 43–57 (2012)

    Article  Google Scholar 

  31. 31

    Kerber, M.: On the Representation of Mathematical Concepts and their Translation into First-Order Logic. Ph.D. thesis, Fachbereich Informatik, Universität Kaiserslautern (1992)

  32. 32

    Li, W.: Mathematical Logic: Foundations for Information Science. Birkhäuser, Basel (2010). Chinese edition: Science Press, Beijing (2007)

  33. 33

    Kerber, M., Pollet, M.: On the design of mathematical concepts. In: McKay, B., Slaney, J. (eds.) Proceedings of the 15th Australian Joint Conference on Artificial Intelligence: Advances in Artificial Intelligence, AI ’02. LNAI 2557, p. 716. Springer, Berlin (2002)

  34. 34

    GeoGebra. Accessed 1 Nov 2013

  35. 35

    Corollary. Accessed 1 Nov 2013

  36. 36

    Wang, D.: GEOTHER 1.1: handling and proving geometric theorems automatically. In: Winkler, F. (ed.) Proceedings of the 4th International Workshop on Automated Deduction in Geometry, ADG ’02. LNAI, vol. 2930, pp. 194–215. Springer, Berlin (2004)

  37. 37

    GEOTHER. Accessed 1 Nov 2013

  38. 38

    Liang T., Wang D.: The design and implementation of a geometric-object-oriented language. Frontiers Comput. Sci. China 1(2), 180–190 (2007)

    Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Xiaoyu Chen.

Additional information

This paper is a substantially revised and expanded version of a research note (by the second author with the same title) published informally in Proceedings of the 6th Asian Workshop on Foundations of Software, AWFS ’09 (Hu Z, Zhang J, eds.), National Institute of Informatics, Japan, 2009, 86–98.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Chen, X., Wang, D. Formalization and Specification of Geometric Knowledge Objects. Math.Comput.Sci. 7, 439–454 (2013).

Download citation


  • Embedded knowledge
  • Formal specification
  • Semantic representation
  • Knowledge management

Mathematics Subject Classification (2010)

  • 68T30
  • 68P05