Revista Matemática Complutense

, Volume 32, Issue 2, pp 451–474 | Cite as

Detecting truth, just on parts

  • Zoltán Kovács
  • Tomás Recio
  • M. Pilar VélezEmail author


We introduce, through a computational algebraic geometry approach, the automatic reasoning handling of propositions that are simultaneously true over some relevant collections of instances and false over some relevant collections of instances. A rigorous, algorithmic criterion is presented for detecting such cases, and its performance is exemplified through the implementation of this test within the dynamic geometry program GeoGebra. Our framework has some significant differences regarding some alternative, recent formulation of the “true on components” idea; differences and similarities between both approaches are discussed here.


Automatic deduction in geometry, automatic geometry theorem proving Automatic geometry theorem discovery Elementary geometry Gröbner basis Zero divisor True on parts, false on parts True on components Dynamic geometry 

Mathematics Subject Classification

13A15 13F20 14Q99 51-04 68W30 


  1. 1.
    Botana, F., Hohenwarter, M., Janičić, J., Kovács, Z., Petrović, I., Recio, T., Weitzhofer, S.: Automated theorem proving in GeoGebra: current achievements. J. Autom. Reason. 55, 39–59 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Botana, F., Recio, T.: On the unavoidable uncertainty of truth in dynamic geometry proving. Math. Comput. Sci. 10(1), 5–25 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chou, S.C.: Mechanical Geometry Theorem Proving. Mathematics and its Applications, vol. 41, D. Reidel Publishing Co., Dordrecht (1988)Google Scholar
  4. 4.
    Cox, D.A., Little, J., O’Shea, D.: Ideals, varieties, and algorithms. An introduction to computational algebraic geometry and commutative algebra. 4th revised ed. Undergraduate Texts in Mathematics. Springer, Cham (2015)Google Scholar
  5. 5.
    Dalzotto, G., Recio, T.: On protocols for the automated discovery of theorems in elementary geometry. J. Autom. Reason. 43, 203–236 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Kovács, Z., Recio, T., Vélez, M.P.: GeoGebra automated reasoning tools: a tutorial. (2018). Accessed 23 March 2018
  7. 7.
    Montes, A., Recio, T.: Automatic discovery of geometry theorems using minimal canonical comprehensive Gröbner systems. In: Botana, F., Recio, T. (eds.) Proceedings Automated Deduction in Geometry 2006. Lecture Notes on Artificial Intelligence, vol. 4869, pp. 13–138. Springer, Berlin (2007)Google Scholar
  8. 8.
    Ladra,M., Páez-Guillán, M.P., Recio, T.: Two ways of using Rabinowitsch trick for imposing non-degeneracy conditions. In: Narboux, J., Schreck, P., Streinu I. (eds). Proceedings of ADG 2016: Eleventh International Workshop on Automated Deduction in Geometry, pp. 144–152. Strasbourg (2016). Accessed 12 July 2018
  9. 9.
    Losada, R., Recio, T., Valcarce, J.L.: On the automatic discovery of Steiner–Lehmus generalizations (extended abstract). In: Richter-Gebert, J., Schreck, P. (eds.) Proceedings Automated Deduction in Geometry 2010, pp. 171–174. Munich (2010)Google Scholar
  10. 10.
    Recio, T.: Cálculo Simbólico y Geométrico. Editorial Síntesis, Madrid (1998)Google Scholar
  11. 11.
    Recio, T., Sterk, H., Vélez, M.P.: Automatic geometry theorem proving. In: Cohen, A.M., Cuipers, H., Sterk, H. (eds.) Some Tapas of Computer Algebra, Algorithms and Computations in Mathematics 4, pp. 276–291. Springer, Berlin (1998)Google Scholar
  12. 12.
    Recio, T., Vélez, M.P.: Automatic discovery of theorems in elementary geometry. J. Autom. Reason. 23, 63–82 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Winkler, F.: Gröbner Bases – Theory and Applications. 5th SCIEnce Training School in Symbolic Computation. RISC, Univ. Linz, 28.6.–9.7.2010. (2010). Accessed 23 July 2018
  14. 14.
    Zariski, O., Samuel, P.: Commutative Algebra, Vol. 2. Graduate Text in Mathematics 29. Springer, Berlin (1960)Google Scholar
  15. 15.
    Zhou, J., Wang, D., Sun, Y.: Automated reducible geometric theorem proving and discovery by Gröbner basis method. J. Autom. Reason. 59(3), 331–344 (2017)CrossRefzbMATHGoogle Scholar

Copyright information

© Universidad Complutense de Madrid 2018

Authors and Affiliations

  1. 1.Private Pädagogische Hochschule der Diözese LinzLinzAustria
  2. 2.Dpto. Matemáticas, Estadística y ComputaciónUniversidad de CantabriaSantanderSpain
  3. 3.Dpto. Ingeniería IndustrialUniversidad Antonio de NebrijaMadridSpain

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