Skip to main content
SpringerLink
Log in

Role of involutive criteria in computing Boolean Gröbner bases

  • Published:
Programming and Computer Software Aims and scope Submit manuscript

Abstract

In this paper, effectiveness of using four criteria in an involutive algorithm based on the Pommaret division for construction of Boolean Gröbner bases is studied. One of the results of this study is the observation that the role of the criteria in computations in Boolean rings is much less than that in computations in an ordinary ring of polynomials over the field of integers. Another conclusion of this study is that the efficiency of the second and/or third criteria is higher than that of the two others.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Faugère, J.-C. and Joux, A., Algebraic Cryptanalysis of Hidden Field Equations (HFE) Using Gröbner Bases, LNCS, Springer, 2003, vol. 2729, pp. 44–60.

    Google Scholar 

  2. Faugère, J.-C., A New Efficient Algorithm for Computing Gröbner Bases without Reduction to Zero (\( \mathbb{F} \) 5), Proc. of ISSAC 2002, New York: ACM, 2002, pp. 75–83.

    Chapter  Google Scholar 

  3. http://magma.maths.usyd.edu.au/users/allan/gb/.

  4. Faugère, J.-C., A New Efficient Algorithm for Computing Gröbner Bases (\( \mathbb{F} \) 4), Pure Applied Algebra, 1999, vol. 139, nos. 1–3, pp. 61–68.

    Article  MATH  Google Scholar 

  5. Bardet, M., Faugère, J.-C., and Salvy, B., Complexity of Gröbner Basis Computation for Semi-regular Overdetermined Sequences over \( \mathbb{F} \) 2 with solutions in \( \mathbb{F} \) 2, INRIA report RR-5049, 2003.

  6. Brickenstein, M., Dreyer, A., Greuel, G.-M., and Wienand, O., New Developments in the Theory of Gröbner Bases and Applications to Formal Verification. arXiv:math.AC/0801.1177.

  7. Gerdt, V.P. and Blinkov, Yu.A., Involutive Bases of Polynomial Ideals, Math. Comput. Simulation, 1998, vol. 45, pp. 519–542. arXiv:math.AC/9912027; Minimal Involutive Bases. Ibid. pp. 543–560. arXiv:math.AC/9912029.

    Article  MATH  MathSciNet  Google Scholar 

  8. Gerdt, V.P., Blinkov, Yu.A., and Yanovich, D.A., Construction of Janet Bases: I. Monomial Bases and II. Polynomial Bases, Proc. of the Conf. “Computer Algebra in Scientific Computing” (CASC’01), (2001), Ganzha, V.G., Mayr, E.W., and Vorozhtsov, E.V., Eds., Berlin: Springer, 2001, pp. 249–263.

    Google Scholar 

  9. Gerdt, V.P., Involutive Algorithms for Computing Gröbner Bases, Computational Commutative and Non-Commutative Algebraic Geometry, Cojocaru, S., Pfister, G., and Ufnarovski, V., Eds., Amsterdam: IOS, 2005, pp. 199–225. arXiv:math.AC/0501111.

    Google Scholar 

  10. Gerdt, V.P. and Zinin, M.V., A Pommaret Division Algorithm for Computing Gröbner Bases in Boolean Rings, Proc. of ISSAC 2008 (RISC), Hagenberg, 2008. To appear.

  11. Apel, J. and Hemmecke, R., Detecting Unnecessary Reductions in an Involutive Basis Computation, J. Symbolic Computation, 2005, vol. 40, nos. 4–5, pp. 1131–1149.

    Article  MATH  MathSciNet  Google Scholar 

  12. Buchberger, B., Gröbner Bases: An Algorithmic Method in Polynomial Ideal Theory, Recent Trends in Multidimensional System Theory, Bose, N.K., Ed., Dordrecht: Reidel, 1985, pp. 184–232.

    Google Scholar 

  13. Gerdt, V.P. and Yanovich, D.A., Effectiveness of Involutive Criteria in Computation of Polynomial Janet Bases, Programmirovanie, 2006, no. 3, pp. 17–21 [Programming Comput. Software (Engl. Transl.), 2006, vol. 32, no. 3, pp. 134–138].

  14. Cox, D., Little, J., and O’shea, D., Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra, New York: Springer, 1998. Translated under the title Idealy, mnogoobraziya i algoritmy, Moscow: Mir, 2000.

    Google Scholar 

  15. http://www-sop.inria.fr/saga/POL; http://www.math.uic.edu/:_jan/demo.html.

  16. Kornyak, V.V., On Compatibility of Discrete Relations, LNCS, Springer, 2005, vol. 3718, pp. 272–284, arXiv:math-ph/0504048.

    MathSciNet  Google Scholar 

  17. Semenov, A., On Connection between Constructive Involutive Divisions and Monomial Orderings, Lecture Notes in Computer Science, Berlin: Springer, 2006, vol. 4194, pp. 261–278.

    Google Scholar 

  18. Apel, J., A Gröbner Approach to Involutive Bases, J. Symbolic Computation, 1995, vol. 19, no. 5, pp. 441–458.

    Article  MATH  MathSciNet  Google Scholar 

  19. Becker, T., Weispfenning, V., and Kredel, H., Gröbner Bases. A Computational Approach to Commutative Algebra, Graduate Texts in Mathematics, vol. 141, New York: Springer, 1993.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratory of Information Technologies, Joint Institute for Nuclear Research, Dubna, Moscow oblast, 141980, Russia

    V. P. Gerdt & M. V. Zinin

Authors
  1. V. P. Gerdt
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. V. Zinin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. P. Gerdt.

Additional information

Original Russian Text © V.P. Gerdt, M.V. Zinin, 2009, published in Programmirovanie, 2009, Vol. 35, No. 2.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gerdt, V.P., Zinin, M.V. Role of involutive criteria in computing Boolean Gröbner bases. Program Comput Soft 35, 90–97 (2009). https://doi.org/10.1134/S0361768809020042

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0361768809020042

Keywords

Search

Navigation