Advertisement

Proving Formally the Implementation of an Efficient gcd Algorithm for Polynomials

  • Assia Mahboubi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4130)

Abstract

We describe here a formal proof in the Coq system of the structure theorem for subresultants, which allows to prove formally the correctness of our implementation of the subresultants algorithm. Up to our knowledge it is the first mechanized proof of this result.

Keywords

Computer Algebra Formal Proof Computer Algebra System Correctness Proof Proof Assistant 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Basu, S., Pollack, R., Roy, M.-F.: Algorithms in Real Algebraic Geometry. In: Algorithms and Computation in Mathematics. Algorithms and Computation in Mathematics, vol. 10, Springer, Heidelberg (2003), draft for snd edition available at http://name.math.univ-rennes1.fr/marie-francoise.roy/bpr-posted1.html Google Scholar
  2. 2.
    Bertot, Y., Casteran, P.: Interactive Theorem Proving and Program Development. Coq’Art: the Calculus of Inductive Constructions. Texts in Theoretical Computer Science. Springer, Heidelberg (2004)Google Scholar
  3. 3.
    Boulmé, S.: Vers la spécification formelle d’un algorithme non trivial de calcul formel: le calcul de pgcd de deux polynômes par la chaîne de pseudo-restes de sous-résultants. Master’s thesis, SPI team, Paris VI University (September 1997)Google Scholar
  4. 4.
    Brown, W.S., Traub, J.F.: On Euclid’s Algorithm and the The Theory of Subresultants. Journal of the ACM 18(4), 505–514 (1971)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Collins, G.E.: Subresultant and Reduced Polynomial Remainder Sequences. Journal of the ACM 14, 128–142 (1967)MATHCrossRefGoogle Scholar
  6. 6.
    Delahaye, D., Mayero, M.: Quantifier Elimination over Algebraically Closed Fields in a Proof Assistant using a Computer Algebra System. In: Proceedings of Calculemus 2005 (2005)Google Scholar
  7. 7.
    Geddes, K., Czapor, S.R., Labahn, G.: Algorithms for Computer Algebra. Kluwer Academic Publishers, Dordrecht (1992)MATHCrossRefGoogle Scholar
  8. 8.
    Gilles Barthe, V.C., Pons, O.: Setoids in type theory. Journal of Functional Programming 13(2), 261–293 (2003)MATHMathSciNetGoogle Scholar
  9. 9.
    Grégoire, B., Leroy, X.: A Compiled Implementation of Strong Reduction. In: International Conference on Functional Programming 2002, pp. 235–246. ACM Press, New York (2002)Google Scholar
  10. 10.
    Grégoire, B., Mahboubi, A.: Proving Ring Equalities Done Right in Coq. In: Hurd, J., Melham, T. (eds.) TPHOLs 2005. LNCS, vol. 3603, Springer, Heidelberg (2005)CrossRefGoogle Scholar
  11. 11.
    Grégoire, B., Théry, L.: A Purely Functional Library for Modular Arithmetic and its Application for Certifying Large Prime Numbers. In: Furbach, U., Shankar, N. (eds.) IJCAR 2006. LNCS (LNAI), vol. 4130, Springer, Heidelberg (2006)CrossRefGoogle Scholar
  12. 12.
    Harrison, J.: The HOL-Light System 2.20. University of Cambrige, DSTO, SRI International (May 2006), http://www.cl.cam.ac.uk/jrh/hol-light/
  13. 13.
    Harrison, J., Théry, L.: A Skeptic’s Approach to Combining HOL and Maple. Journal of Automated Reasoning 21, 279–294 (1998)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Jacobi, C.: De Eliminatione Variablilis e Duabus Aequationibus. J. Reine Angew. Math 15, 101–124 (1836)Google Scholar
  15. 15.
    Knuth, D.: The Art of Computer Programming, Semi-numerical Algorithms, vol. 2. Addison-Wesley, Reading (1998)Google Scholar
  16. 16.
    Liao, H.-C., Fateman, R.J.: Evaluation of the Heuristic Polynomial gcd. In: ISSAC 1995: Proceedings of the 1995 international symposium on Symbolic and algebraic computation, pp. 240–247. ACM Press, New York (1995)CrossRefGoogle Scholar
  17. 17.
    Mahboubi, A.: Programming and Certifying a CAD Algorithm in the Coq system. In: Mathematics, Algorithms, Proofs, number 05021 in Dagstuhl Seminar Proceedings. IBFI, Schloss Dagstuhl, Germany (2006)Google Scholar
  18. 18.
    Nipkow, T., Paulson, L.C., Wenzel, M.: Isabelle/HOL. LNCS, vol. 2283. Springer, Heidelberg (2002)MATHGoogle Scholar
  19. 19.
    Sacerdoti, C.: A Semi-reflexive Tactic for (Sub-)Equational Reasoning. In: Filliâtre, J.-C., Paulin-Mohring, C., Werner, B. (eds.) TYPES 2004. LNCS, vol. 3839, pp. 98–114. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  20. 20.
    The Coq Development Team. The Coq Proof Assistant Reference Manual – Version V8.0 (April 2004), http://coq.inria.fr
  21. 21.
    Théry, L.: A Machine-Checked Implementation of Buchberger’s Algorithm. Journal of Automated Reasoning 26, 107–137 (2001)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    von zur Gathen, J., Lücking, T.: Subresultants Revisited. Theoretical Computer Science 297(1-3), 199–239 (2003)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Yap, C.K.: Fundamental Problems of Algorithmic Algebra. Oxford University Press, Oxford (2000)MATHGoogle Scholar
  24. 24.
    Zippel, R.: Effective Polynomial Computation. Kluwer Academic Publishers, Dordrecht (1993)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Assia Mahboubi
    • 1
  1. 1.INRIA Sophia-AntipolisSophia Antipolis CedexFrance

Personalised recommendations