Abstract
We show that a comprehensive Boolean Gröbner basis of an ideal I in a Boolean polynomial ring B \((\bar A,\bar X)\) with main variables \(\bar X\) and parameters \(\bar A\) can be obtained by simply computing a usual Boolean Gröbner basis of I regarding both \(\bar X\) and \(\bar A\) as variables with a certain block term order such that \(\bar X \gg \bar A\). The result together with a fact that a finite Boolean ring is isomorphic to a direct product of the Galois field \(\mathbb{GF}_2\) enables us to compute a comprehensive Boolean Gröbner basis by only computing corresponding Gröbner bases in a polynomial ring over \(\mathbb{GF}_2\). Our implementation in a computer algebra system Risa/Asir shows that our method is extremely efficient comparing with existing computation algorithms of comprehensive Boolean Gröbner bases.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Kapur, D.: An Approach for Solving Systems of Parametric Polynomial Equations. In: Saraswat, Van Hentenryck (eds.) Principles and Practices of Constraint Programming, pp. 217–244. MIT Press, Cambridge (1995)
Manubens, M., Montes, A.: Improving DISPGB algorithm using the discriminant ideal. J. Symb. Comp. 41, 1245–1263 (2006)
Montes, A.: A new algorithm for discussing Gröbner bases with parameters. J. Symb. Comp. 33(2), 183–208 (2002)
Noro, M., et al.: A Computer Algebra System Risa/Asir (2009), http://www.math.kobe-u.ac.jp/Asir/asir.html
Sakai, K., Sato, Y.: Boolean Gröbner bases. ICOT Technical Momorandum 488 (1988), http://www.icot.or.jp/ARCHIVE/Museum/TRTM/tm-list-E.html
Sakai, K., Sato, Y., Menju, S.: Boolean Gröbner bases(revised). ICOT Technical Report 613. (1991), http://www.icot.or.jp/ARCHIVE/Museum/TRTM/tr-list-E.html
Sato, Y., et al.: Set Constrains Solvers(Prolog version) (1996), http://www.icot.or.jp/ARCHIVE/Museum/FUNDING/funding-96-E.html
Sato, Y.: A new type of canonical Gröbner bases in polynomial rings over Von Neumann regular rings. In: Proceedings of ISSAC 1998, pp. 317–332. ACM Press, New York (1998)
Sato, Y., et al.: Set Constrains Solvers(Klic version) (1998), http://www.icot.or.jp/ARCHIVE/Museum/FUNDING/funding-98-E.html
Sato, Y., Inoue, S.: On the Construction of Comprehensive Boolean Gröbner Bases. In: Proceedings of the Seventh Asian Symposium on Computer Mathematics (ASCM 2005), pp. 145–148 (2005)
Sato, Y., Inoue, S., Suzuki, A., Nabeshima, K.: Boolean Gröbner Bases and Sudoku (submitted for publication)
Sato, Y., Nagai, A., Inoue, S.: On the Computation of Elimination Ideals of Boolean Polynomial Rings. In: Kapur, D. (ed.) ASCM 2007. LNCS (LNAI), vol. 5081, pp. 334–348. Springer, Heidelberg (2008)
Suzuki, A., Sato, Y.: An Alternative approach to Comprehensive Gröbner Bases. J. Symb. Comp. 36(3-4), 649–667 (2003)
Suzuki, A., Sato,Y.: A Simple Algorithm to Compute Comprehensive Gröbner Bases Using Gröbner Bases. In: International Symposium on Symbolic and Algebraic Computation (ISSAC 2006), Proceedings, pp. 326–331 (2006)
Weispfenning, V.: Gröbner bases in polynomial ideals over commutative regular rings. In: Davenport, J.H. (ed.) ISSAC 1987 and EUROCAL 1987. LNCS, vol. 378, pp. 336–347. Springer, Heidelberg (1989)
Weispfenning, V.: Comprehensive Gröbner bases. J. Symb. Comp. 14(1), 1–29 (1992)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Inoue, S. (2009). On the Computation of Comprehensive Boolean Gröbner Bases. In: Gerdt, V.P., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2009. Lecture Notes in Computer Science, vol 5743. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04103-7_13
Download citation
DOI: https://doi.org/10.1007/978-3-642-04103-7_13
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-04102-0
Online ISBN: 978-3-642-04103-7
eBook Packages: Computer ScienceComputer Science (R0)