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Proving Formally the Implementation of an Efficient gcd Algorithm for Polynomials

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Automated Reasoning (IJCAR 2006)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 4130))

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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.

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References

  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. 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. 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. 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)

    Article  MATH  MathSciNet  Google Scholar 

  5. Collins, G.E.: Subresultant and Reduced Polynomial Remainder Sequences. Journal of the ACM 14, 128–142 (1967)

    Article  MATH  Google Scholar 

  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. Geddes, K., Czapor, S.R., Labahn, G.: Algorithms for Computer Algebra. Kluwer Academic Publishers, Dordrecht (1992)

    Book  MATH  Google Scholar 

  8. Gilles Barthe, V.C., Pons, O.: Setoids in type theory. Journal of Functional Programming 13(2), 261–293 (2003)

    MATH  MathSciNet  Google Scholar 

  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. 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)

    Chapter  Google Scholar 

  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)

    Chapter  Google Scholar 

  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. Harrison, J., Théry, L.: A Skeptic’s Approach to Combining HOL and Maple. Journal of Automated Reasoning 21, 279–294 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  14. Jacobi, C.: De Eliminatione Variablilis e Duabus Aequationibus. J. Reine Angew. Math 15, 101–124 (1836)

    Google Scholar 

  15. Knuth, D.: The Art of Computer Programming, Semi-numerical Algorithms, vol. 2. Addison-Wesley, Reading (1998)

    Google Scholar 

  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)

    Chapter  Google Scholar 

  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. Nipkow, T., Paulson, L.C., Wenzel, M.: Isabelle/HOL. LNCS, vol. 2283. Springer, Heidelberg (2002)

    MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  20. The Coq Development Team. The Coq Proof Assistant Reference Manual – Version V8.0 (April 2004), http://coq.inria.fr

  21. Théry, L.: A Machine-Checked Implementation of Buchberger’s Algorithm. Journal of Automated Reasoning 26, 107–137 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  22. von zur Gathen, J., Lücking, T.: Subresultants Revisited. Theoretical Computer Science 297(1-3), 199–239 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  23. Yap, C.K.: Fundamental Problems of Algorithmic Algebra. Oxford University Press, Oxford (2000)

    MATH  Google Scholar 

  24. Zippel, R.: Effective Polynomial Computation. Kluwer Academic Publishers, Dordrecht (1993)

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. INRIA Sophia-Antipolis, 2204, routes des Lucioles, B.P. 93, 06902, Sophia Antipolis Cedex, France

    Assia Mahboubi

Authors
  1. Assia Mahboubi
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Editor information

Editors and Affiliations

  1. Department of Computer Science, Artificial Intelligence Research Group, University of Koblenz-Landau, Universitätsstr. 1, 56070, Koblenz

    Ulrich Furbach

  2. SRI International, MS EL256,, 333 Ravenswood Avenue, 94025-3493, Menlo Park, CA, USA

    Natarajan Shankar

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Mahboubi, A. (2006). Proving Formally the Implementation of an Efficient gcd Algorithm for Polynomials. In: Furbach, U., Shankar, N. (eds) Automated Reasoning. IJCAR 2006. Lecture Notes in Computer Science(), vol 4130. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11814771_37

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  • DOI: https://doi.org/10.1007/11814771_37

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-37187-8

  • Online ISBN: 978-3-540-37188-5

  • eBook Packages: Computer ScienceComputer Science (R0)

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