Abstract
The main proposal in this paper is the merging of two techniques that have been recently developed. On the one hand, we consider a new approach for computing some specializable Gröbner basis, the so called Minimal Canonical Comprehensive Gröbner Systems (MCCGS) that is -roughly speaking- a computational procedure yielding “good” bases for ideals of polynomials over a field, depending on several parameters, that specialize “well”, for instance, regarding the number of solutions for the given ideal, for different values of the parameters. The second ingredient is related to automatic theorem discovery in elementary geometry. Automatic discovery aims to obtain complementary (equality and inequality type) hypotheses for a (generally false) geometric statement to become true. The paper shows how to use MCCGS for automatic discovering of theorems and gives relevant examples.
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References
Beltrán, C., Dalzotto, G., Recio, T.: The moment of truth in automatic theorem proving in elementary geometry. In: Botana, F., Roanes-Lozano, E. (eds.) Proceedings ADG 2006 (extended abstracts), Universidad de Vigo (2006)
Botana, F., Recio, T.: Towards solving the dynamic geometry bottleneck via a symbolic approach. In: Hong, H., Wang, D. (eds.) ADG 2004. LNCS (LNAI), vol. 3763, pp. 92–111. Springer, Heidelberg (2006)
Chen, X.F., Li, P., Lin, L., Wang, D.K.: Proving geometric theorems by partitioned-parametric Gröbner bases. In: Hong, H., Wang, D. (eds.) ADG 2004. LNCS (LNAI), vol. 3763, pp. 34–44. Springer, Heidelberg (2005)
Chen, X.F., Wang, D.K.: The projection of a quasivariety and its application on geometry theorem proving and formula deduction. In: Winkler, F. (ed.) ADG 2002. LNCS (LNAI), vol. 2930, pp. 21–30. Springer, Heidelberg (2004)
Chou, S.-C.: Proving Elementary Geometry Theorems Using Wu’s Algorithm Contemporary Mathematics. Automated Theorem Proving: After 25 Years, American Mathematical Society, Providence, Rhode Island 29, 243–286 (1984)
Chou, S.-C.: Proving and discovering theorems in elementary geometries using Wu’s method Ph.D. Thesis, Department of Mathematics, University of Texas, Austin (1985)
Chou, S.C.: A Method for Mechanical Derivation of Formulas in Elementary Geometry. Journal of Automated Reasoning 3, 291–299 (1987)
Chou, S.-C.: Mechanical Geometry Theorem Proving. Mathematics and its Applications, D. Reidel Publ. Comp. (1988)
Chou, S.-C., Gao, X.-S.: Methods for Mechanical Geometry Formula Deriving. In: Proceedings of International Symposium on Symbolic and Algebraic Computation, pp. 265–270. ACM Press, New York (1990)
Capani, A., Niesi, G., Robbiano, L.: CoCoA, a System for Doing Computations in Commutative Algebra. The version 4.6 is available at the web site, http://cocoa.dima.unige.it
Dalzotto, G., Recio, T.: On protocols for the automated discovery of theorems in elementary geometry. J. Automated Reasoning (submitted)
Dolzmann, A., Gilch, L.: Generic Hermitian Quantifier Elimination. In: Buchberger, B., Campbell, J.A. (eds.) AISC 2004. LNCS (LNAI), vol. 3249, pp. 80–93. Springer, Heidelberg (2004)
Dolzmann, A., Sturm, T., Weispfenning, V.: A new approach for automatic theorem proving in real geometry. Journal of Automated Reasoning 21(3), 357–380 (1998)
Kapur, D.: Using Gröbner basis to reason about geometry problems. J. Symbolic Computation 2(4), 399–408 (1986)
Kapur, D.: Wu’s method and its application to perspective viewing. In: Kapur, D., Mundy, J.L. (eds.) Geometric Reasoning, The MIT press, Cambridge (1989)
Kapur, D.: An approach to solving systems of parametric polynomial equations. In: Saraswat, Van Hentenryck (eds.) Principles and Practice of Constraint Programming, MIT Press, Cambridge (1995)
Kreuzer, M., Robbiano, L.: Computational Commutative Algebra 1. Springer, Heidelberg (2000)
Koepf, W.: Gröbner bases and triangles. The International Journal of Computer Algebra in Mathematics Education 4(4), 371–386 (1998)
Manubens, M., Montes, A.: Improving DISPGB Algorithm Using the Discriminant Ideal. Jour. Symb. Comp. 41, 1245–1263 (2006)
Manubens, M., Montes, A.: Minimal Canonical Comprehensive Groebner System. arXiv: math.AC/0611948. (2006)
Montes, A.: New Algorithm for Discussing Gröbner Bases with Parameters. Jour. Symb. Comp. 33(1-2), 183–208 (2002)
Montes, A.: About the canonical discussion of polynomial systems with parameters. arXiv: math.AC/0601674 (2006)
Recio, T.: Cálculo simbólico y geométrico. Editorial Síntesis, Madrid (1998)
Recio, T., Pilar, M., Pilar Vélez, M.: Automatic Discovery of Theorems in Elementary Geometry. J. Automat. Reason. 23, 63–82 (1999)
Recio, T., Botana, F.: Where the truth lies (in automatic theorem proving in elementary geometry). In: Laganà, A., Gavrilova, M., Kumar, V., Mun, Y., Tan, C.J.K., Gervasi, O. (eds.) ICCSA 2004. LNCS, vol. 3044, pp. 761–771. Springer, Heidelberg (2004)
Richard, P.: Raisonnement et stratégies de preuve dans l’enseignement des mathématiques. Peter Lang Editorial, Berne (2004)
Seidl, A., Sturm, T.: A generic projection operator for partial cylindrical algebraic decomposition. In: Sendra, R. (ed.) ISSAC 2003. Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation, Philadelphia, Pennsylvania, pp. 240–247. ACM Press, New York (2003)
Sturm, T.: Real Quantifier Elimination in Geometry. Doctoral dissertation, Department of Mathematics and Computer Science. University of Passau, Germany, D-94030 Passau, Germany (December 1999)
Wang, D.: Gröbner Bases Applied to Geometric Theorem Proving and Discovering. In: Buchberger, B., Winkler, F. (eds.) Gröbner Bases and Applications, 251th edn. London Mathematical Society Lecture Notes Series, vol. 251, pp. 281–301. Cambridge University Press, Cambridge (1998)
Weispfenning, V.: Comprehensive Grobner bases. Journal of Symbolic Computation 14(1), 1–29 (1992)
Wibmer, M.: Gröbner bases for families of affine or projective schemes. arXiv: math.AC/0608019. (2006)
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Montes, A., Recio, T. (2007). Automatic Discovery of Geometry Theorems Using Minimal Canonical Comprehensive Gröbner Systems. In: Botana, F., Recio, T. (eds) Automated Deduction in Geometry. ADG 2006. Lecture Notes in Computer Science(), vol 4869. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77356-6_8
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DOI: https://doi.org/10.1007/978-3-540-77356-6_8
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