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On the Computation of Comprehensive Boolean Gröbner Bases

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Computer Algebra in Scientific Computing (CASC 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5743))

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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.

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Authors and Affiliations

  1. Tokyo University of Science, 1-3, Kagurazaka, Shinjuku-ku, Tokyo, Japan

    Shutaro Inoue

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  1. Shutaro Inoue
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Editors and Affiliations

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

    Vladimir P. Gerdt

  2. Technische Universität München‘, Institut für Informatik, Garching, Germany

    Ernst W. Mayr

  3. Institute of Theoretical and Applied Mechanics, Russian Academy of Sciences, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

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© 2009 Springer-Verlag Berlin Heidelberg

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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

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  • 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)

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