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Towards a Certified and Efficient Computing of Gröbner Bases

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3643))

Abstract

In this paper, we present an example of the implementation and verification of a functional program. We expose an experience in developing an application in the area of symbolic computation: the computing of Gröbner basis of a set of multivariate polynomials. Our aim is the formal certification of several aspects of the program written in the functional language Caml. In addition, efficient computing of the algorithm is another issue to take into account.

Supported by MCyT TIC2002-02859 and Xunta de Galicia PGIDIT03PXIC10502PN.

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References

  1. Buchberger, B.: An Algorithm for Finding a Basis for the Residue Class Ring of a Zero-Dimensional Polynomial Ideal. PhD thesis, Univ. of Innsbruck, Austria (1965)

    Google Scholar 

  2. Bird, R., Wadler, P.: Introduction to Functional Programming. Prentice-Hall, Englewood Cliffs (1988)

    Google Scholar 

  3. Hudak, P.: Conception, evolution, and application of functional programming languages. ACM Computing Surveys 21 (1989)

    Google Scholar 

  4. Paulson, L.C.: ML for the Working Programmer, 2nd edn. Cambridge University Press, Cambridge (1996)

    MATH  Google Scholar 

  5. Jorge, J.S.: Estudio de la verificación de propíedades de programas funcionales: de las pruebas manuales al uso de asistentes de pruebas. PhD thesis, University of A Coruña, Spain (2004)

    Google Scholar 

  6. Weis, P., Leroy, X.: Le langage Caml, 2nd edn. Dunod (1999)

    Google Scholar 

  7. Leroy, X., et al.: The Objective Caml system: Documentation and User’s Manual, Release 3.08. INRIA (2004), http://caml.inria.fr

  8. The Coq Development Team: The Coq Proof Assistant Reference Manual, Version 7.3. INRIA (2002), http://coq.inria.fr

  9. Bertot, Y., Casteran, P.: Interactive Theorem Proving and Program Development, Coq’Art: The Calculus of Inductive Constructions. Springer, Heidelberg (2004)

    MATH  Google Scholar 

  10. Théry, L.: A machine-checked implementation of Buchberger’s algorithm. Journal of Automated Reasoning 26 (2001)

    Google Scholar 

  11. Medina-Bulo, I., Palomo-Lozano, F., Alonso-Jiménez, J.A., Ruiz-Reina, J.-L.: Verified computer algebra in ACL2 (Gröbner bases computation). In: Buchberger, B., Campbell, J. (eds.) AISC 2004. LNCS (LNAI), vol. 3249, pp. 171–184. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  12. Pérez, G.: Bases de Gröbner: Desarrollo formal en Coq. PhD thesis, University of A Coruña, Spain (2005)

    Google Scholar 

  13. Coquand, T., Huet, G.: The calculus of constructions. Information and Computation 76 (1988)

    Google Scholar 

  14. Barja, J.M., Pérez, G.: Demostración en implementaciones concretas de anillos de polinomios. RSME (2000)

    Google Scholar 

  15. Medina-Bulo, I., Alonso-Jiménez, J.A., Palomo-Lozano, F.: Automatic verification of polynomial rings fundamental properties in ACL2. In: 2nd International Workshop on the ACL2 Theorem Prover and Its Applications (2000)

    Google Scholar 

  16. Paulson, L.C.: Constructing recursion operators in intuitionistic type theory. Journal of Symbolic Computation 2 (1986)

    Google Scholar 

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© 2005 Springer-Verlag Berlin Heidelberg

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Jorge, J.S., Gulías, V.M., Freire, J.L., Sánchez, J.J. (2005). Towards a Certified and Efficient Computing of Gröbner Bases. In: Moreno Díaz, R., Pichler, F., Quesada Arencibia, A. (eds) Computer Aided Systems Theory – EUROCAST 2005. EUROCAST 2005. Lecture Notes in Computer Science, vol 3643. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11556985_16

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  • DOI: https://doi.org/10.1007/11556985_16

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-29002-5

  • Online ISBN: 978-3-540-31829-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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