Advertisement

Rewriting input expressions in complex algebraic geometry provers

  • Z. KovácsEmail author
  • T. Recio
  • C. Sólyom-Gecse
Article

Abstract

We present an algorithm to help converting expressions having non-negative quantities (like distances) in Euclidean geometry theorems to be usable in a complex algebraic geometry prover. The algorithm helps in refining the output of an existing prover, therefore it supports immediate deployment in high level prover systems. We prove that the algorithm may take doubly exponential time to produce the output in polynomial form, but in many cases it is still computable and useful.

Keywords

Automatic theorem proving Automatic theorem deduction Complex algebraic geometry Elementary geometry Dynamic geometry software GeoGebra 

Mathematics Subject Classification (2010)

14Q99 13P10 68T15 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgments

The MEP formula was suggested by Bernard Parisse, inventor of Giac.

We are thankful to Predrag Janičić, Julien Narboux, Francisco Botana and the anonymous reviewers for their suggestions to improve the text of this paper.

First and second authors are partially supported by the grant MTM2017-88796-P from the Spanish MINECO (Ministerio de Economía y Competitividad) and the ERDF (European Regional Development Fund). Second author was partially supported by the grant MTM2014-54141-P.

References

  1. 1.
    Bogomolny, A.: Viviani’s 3D analogue from interactive mathematics miscellany and puzzles. Downloaded from. http://www.cut-the-knot.org/triangle/VivianiTetrahedron.shtml, accessed in April 2016
  2. 2.
    Botana, F., Hohenwarter, M., Janičić, P., Kovács, Z., Petrović, I., Recio, T., Weitzhofer, S.: Automated theorem proving in geogebra current achievements. J. Autom. Reason. 55(1), 39–59 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chou, S.-C.: Mechanical Geometry Theorem Proving. Springer Science + Business Media, Berlin (1987)CrossRefzbMATHGoogle Scholar
  4. 4.
    Cox, D., Little, J., O’Shea, D.: Ideals varieties and algorithms. Springer, New York (2007)CrossRefzbMATHGoogle Scholar
  5. 5.
    Decker, W., Greuel, G.-M., Pfister, G., Schönemann, H.: Singular 4-0-2 — A computer algebra system for polynomial computations. http://www.singular.uni-kl.de(2015)
  6. 6.
    Dolzmann, A., Sturm, T., Weispfenning, V.: A new approach for automatic theorem proving in real geometry. J. Autom. Reason. 21(3), 357–380 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Gao, X.-S.: Automated geometry diagram construction and engineering geometry. In: Automated deduction in geometry. ADG 1998. Lecture Notes in Computer Science, 1669. Springer, Berlin (1999)Google Scholar
  8. 8.
    Hoyles, C., Jones, K.: Proof in dynamic geometry contexts. In: Mammana, C., Villani, V. (eds.) Perspectives on the teaching of geometry for the 21st century, pp 121–128. Kluwer, Dordrecht (1998)Google Scholar
  9. 9.
    Kapur, D.: Using Grȯbner bases to reason about geometry problems. J. Symb. Comput. 2(4), 399–408 (1986)CrossRefzbMATHGoogle Scholar
  10. 10.
    Kovács, Z., Sólyom-Gecse, C.: GeoGebra tools with proof capabilities. arXiv:1603.01228 (2016)
  11. 11.
    Parisse, B.: About Giac’s Gröbner basis and ideal elimination computation. Presentation at the conference on Applications of Computer Algebra, Kassel. http://test.geogebra.org/kovzol/guests/BernardParisse/aca16-parisse.pdf (2016)
  12. 12.
    Petrović, I., Janičić, P.: Integration of OpenGeoProver with GeoGebra. http://argo.matf.bg.ac.rs/events/2012/fatpa2012/slides/IvanPetrovic.pdf (2012)
  13. 13.
    Recio Muñiz, T. J.: Cálculo simbólico y geométrico. Editorial Síntesis, Madrid (1998)Google Scholar
  14. 14.
    Recio, T.T., Botana, F.: Where the truth lies (in automatic theorem proving in elementary geometry). In: Laganà, A., Gavrilova, M.L., Kumar, V., Mun, Y., Tan, C.J.K., Gervasi, O. (eds.) Computational science and its applications ICCSA 2004. Lecture Notes in Computer Science 3044. Springer, Berlin (2004)Google Scholar
  15. 15.
    Recio, T., Vélez, M.P.: Automatic discovery of theorems in elementary geometry. J. Autom. Reason. 23, 63–82 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Wu, W.-T.: On the decision problem and the mechanization of theorem-proving in elementary geometry. Sci. Sinica 21, 159–172 (1978)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Ye, Z., Chou, S.-C., Gao, X.-S.: An introduction to java geometry expert. In: Automated deduction in geometry, pp. 189–195. Springer Science + Business Media (2011)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.The Private University College of Education of the Diocese of LinzLinzAustria
  2. 2.Universidad de CantabriaSantanderSpain
  3. 3.Johannes Kepler UniversityLinzAustria

Personalised recommendations