Skip to main content
SpringerLink
Log in

On computation of Boolean involutive bases

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

Abstract

Gröbner bases in Boolean rings can be calculated by an involutive algorithm based on the Janet or Pommaret division. The Pommaret division allows calculations immediately in the Boolean ring, whereas the Janet division implies use of a polynomial ring over field \( \mathbb{F}_2 \). In this paper, both divisions are considered, and distributive and recursive representations of Boolean polynomials are compared from the point of view of calculation effectiveness. Results of computer experiments with both representations for an algorithm based on the Pommaret division and for lexicographical monomial order are presented.

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. Sakai, K. and Sato, Y., Boolean Gröbner Bases, ICOT Tech. Memorandum 448, 1988. Sakai, K., Sato, Y., and Menju, S., Boolean Gröbner Bases (Revised), ICOT Tech. Report 613, 1991. http://www.icot.or.jp/ARCHIVE/Museum/TRTM/tr-list-E.html.

  2. Faugère, J.-C. and Joux, A., Algebraic Cryptanalysis of Hidden Field Equations (HFE) Using Gröbner Bases, Lecture Notes in Computer Science, Springer, 2003, vol. 2729, pp. 44–60.

    Article  Google Scholar 

  3. Faugère, J.-C., Gianni, P., Lazard, D., and Mora, T., Efficient Computation of Zero-Dimensional Gröbner Bases by Change of Ordering, J. Symbol. Computations, 1993, vol. 16, no. 4, pp. 329–344.

    Article  MATH  Google Scholar 

  4. von zur Ghaten, J. and Gerhard, J., Modern Computer Algebra, Cambridge: Cambridge Univ. Press, 2003, 2nd edition.

    Google Scholar 

  5. Chistov, A.L., Two Times Exponential Lower Bound on the Degree of System of Generators for a Polynomial Prime Ideal, Algebra I Analiz, 2008, vol. 20, no. 6, pp. 186–213.

    Google Scholar 

  6. Garey, M.R. and Johnson, D.S., Computers and Intractability: A Guide to the Theory of NP-completeness, New York: Freeman, 1978. Translated under the title Vychislitel’nye mashiny i trudnoreshaemye zadachi, Moscow: Mir, 1982.

    Google Scholar 

  7. Papadimitriou, C.H., Computational Complexity, Reading, Mass.: Addison-Wesley, 1994.

    MATH  Google Scholar 

  8. 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.

  9. Condrat, C. and Kalla, P., A Gröbner Basis Approach to CNF-Formulae Preprocessing. Tools and Algorithms for the Construction and Analysis of Systems, Lecture Notes in Computer Science, Springer, 2007, vol. 4424, pp. 618–631.

    Article  Google Scholar 

  10. 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.

  11. Gerdt, V.P. and Zinin, M.V., Involutive Method for Computing Gröbner Bases over F 2, Programmirovanie, 2008, no. 4, pp. 8–24 [Programming Comput. Software (Engl. Transl.), 2008, vol. 34, no. 4, pp. 112–123].

  12. Gerdt, V.P. and Zinin, M.V., A Pommaret Division Algorithm for Computing Gröbner Bases in Boolean Rings, Proc. of ISSAC 2008 (Hagenberg, 2008), ACM, 2008, pp. 191–203.

  13. Tran, Q.-N., A P-SPACE Algorithm for Gröbner Bases Computation in Boolean Rings, PWASET, 2008, vol. 35, pp. 495–501.

    Google Scholar 

  14. Sato, Y., Nagai, A., and Inoue, S., On the Computation of Elimination Ideals of Boolean Polynomial Rings, Lecture Notes in Artificial Intelligence, Springer, 2008, vol. 5081, pp. 334–348.

    Google Scholar 

  15. 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 

  16. 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 

  17. Dawson, C.M., Haselgrove, H.L., Hines, A.P., Mortimer, D., Nielsen, M.A., and Osborne, T.J., Quantum Computing and Polynomial Equations over the Finite Field Z2. arXiv:quant-ph/0408129.

  18. Seiler, W.M., A Combinatorial Approach to Involution and Delta-Regularity I: Involutive Bases in Polynomial Algebras of Solvable Type; II: Structure Analysis of Polynomial Modules with Pommaret Bases, Preprints of Universität Kassel, 2007.

  19. Plesken, W. and Robertz, D., Janet’s Approach to Presentations and Resolutions for Polynomials and Linear Pdes, Arch. Math., 2005, vol. 84, pp. 22–37.

    Article  MATH  MathSciNet  Google Scholar 

  20. 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 

  21. Gerdt, V.P., On the Relation between Pommaret and Janet Bases, Proc. of Conf. “Computer Algebra in Scientific Computing” (CASC 2000), (2000), Berlin: Springer, 2000, pp. 164–171. arXiv:math.AC/0004100.

    Google Scholar 

  22. Apel, J., Theory of Involutive Division and an Application to Hilbert Function, J. Symb. Computations, 1998, vol. 25, pp. 683–704.

    Article  MATH  MathSciNet  Google Scholar 

  23. Cox, D., Little, J., and O’shea, D., Ideals, Varieties, and Algorithms, New York: Springer, 2007, 3d edition.

    MATH  Google Scholar 

  24. 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 

  25. Gerdt, V.P. and Blinkov, Yu.A., Specialized Computer Algebra System GINV, Programmirovanie, 2008, no. 2, pp. 67–80 [Programming Comput. Software (Engl. Transl.), 2008, vol. 34, no. 2, pp. 112–123].

  26. http://www-sop.inria.fr/saga/POL; http://www.math. uic.edu/~jan/demo.html.

  27. Kornyak, V.V., On Compatibility of Discrete Relations, Lecture Notes in Computer Science, Springer, 2005, vol. 3718, pp. 272–284, arXiv:math-ph/0504048.

    Article  MathSciNet  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

  2. Department of Mathematics and Mechanics, Saratov State University, Saratov, 410071, Russia

    Yu. A. Blinkov

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

  3. Yu. A. Blinkov
    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, Yu.A. Blinkov, 2010, published in Programmirovanie, 2010, Vol. 36, No. 2.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gerdt, V.P., Zinin, M.V. & Blinkov, Y.A. On computation of Boolean involutive bases. Program Comput Soft 36, 117–123 (2010). https://doi.org/10.1134/S0361768810020106

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0361768810020106

Keywords

Search

Navigation