Complexity Analysis of the Bivariate Buchberger Algorithm in Theorema

  • Alexander Maletzky
  • Bruno Buchberger
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8592)


In this talk we present the formalization and formal verification of the complexity analysis of Buchberger’s algorithm in the bivariate case in the computer system Theorema as a case study for using the system in mathematical theory exploration.

We describe how Buchberger’s original complexity proof for Groebner bases can be carried out within the Theorema system. As in the original proof, the whole setting is transferred from rings of bivariate polynomials over fields to the discrete space of pairs of natural numbers by mapping each polynomial to the exponent vector of its leading monomial. The complexity analysis is then carried out in the discrete space, mostly by means of combinatorial methods that require many tedious case distinctions, making this proof a natural candidate for automated theorem proving. However, following our Theorema philosophy, we do not expect general theorem provers (like resolution provers) to carry out this task in a natural and efficient way. Rather, we designed and implemented a special prover for such proofs. We show how the Theorema philosophy of working in parallel both on the meta level (designing and implementing special provers) and on the object level (design of the notions and theorems) of a theory can lead to a new quality and style of mathematical research.


Groebner basis Buchberger algorithm mathematical theory exploration complexity analysis Theorema 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Buchberger, B.: Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal (An Algorithm for Finding the Basis Elements in the Residue Class Ring Modulo a Zero Dimensional Polynomial Ideal). PhD thesis, Mathematical Institute, University of Innsbruck, Austria 1965: English translation in J. of Symbolic Computation, Special Issue on Logic, Mathematics, and Computer Science: Interactions 41(3-4), 475–511 (2006)Google Scholar
  2. 2.
    Buchberger, B.: A Criterion for Detecting Unnecessary Reductions in the Construction of Groebner Bases. In: Ng, K.W. (ed.) EUROSAM 1979 and ISSAC 1979. LNCS, vol. 72, pp. 3–21. Springer, Heidelberg (1979)CrossRefGoogle Scholar
  3. 3.
    Buchberger, B., Winkler, F.: Miscellaneous Results on the Construction of Groebner-Bases for Polynomial Ideals. Technical Report 137, Johannes Kepler University Linz, Technisch-Naturwissenschaftliche Fakultaet, Insitut fuer Mathematik (June 1979)Google Scholar
  4. 4.
    Buchberger, B.: A Note on the Complexity of Constructing Groebner-Bases. In: van Hulzen, J.A. (ed.) ISSAC 1983 and EUROCAL 1983. LNCS, vol. 162, pp. 137–145. Springer, Heidelberg (1983)CrossRefGoogle Scholar
  5. 5.
    Buchberger, B.: Miscellaneous Results on Groebner-Bases for Polynomial Ideals II. Technical Report 83-1, Department of Computer and Information Sciences, University of Delaware (1983)Google Scholar
  6. 6.
    Buchberger, B.: Mathematica as a Rewrite Language. In: Ida, T., Ohori, A., Takeichi, M. (eds.) Functional and Logic Programming (Proceedings of the 2nd Fuji International Workshop on Functional and Logic Programming, Shonan Village Center, November 1-4), pp. 1–4. World Scientific, Singapore (1996)Google Scholar
  7. 7.
    Buchberger, B.: Introduction to Groebner Bases. London Mathematical Society Lecture Notes Series, vol. 251. Cambridge University Press (April 1998)Google Scholar
  8. 8.
    Buchberger, B.: Groebner Rings in Theorema: A Case Study in Functors and Categories. Technical Report 2003-49, Johannes Kepler University Linz, Spezialforschungsbereich F013 (November 2003)Google Scholar
  9. 9.
    Buchberger, B.: Proving by First and Intermediate Principles, November 1-2, Invited Talk at Workshop on Types for “Mathematics/Libraries of Formal Mathematics”, University of Nijmegen, The Netherlands (2004)Google Scholar
  10. 10.
    Buchberger, B., Crǎciun, A., Jebelean, T., Kovács, L., Kutsia, T., Nakagawa, K., Piroi, F., Popov, N., Robu, J., Rosenkranz, M., Windsteiger, W.: Theorema: Towards Computer-Aided Mathematical Theory Exploration. Journal of Applied Logic 4(4), 470–504 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Giese, M., Buchberger, B.: Towards Practical Reflection for Formal Mathematics. RISC Report Series 07-05, Research Institute for Symbolic Computation (RISC), University of Linz, Schloss Hagenberg, 4232 Hagenberg, Austria (2007)Google Scholar
  12. 12.
    Giusti, M.: Some effectivity problems in polynomial ideal theory. In: Fitch, J. (ed.) EUROSAM 1984 and ISSAC 1984. LNCS, vol. 174, pp. 159–171. Springer, Heidelberg (1984)CrossRefGoogle Scholar
  13. 13.
  14. 14.
    Mayr, E.W., Meyer, A.R.: The complexity of the word problems for commutative semigroups and polynomial ideals. Advances in Mathematics 46(3), 305–329 (1982)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Michael Moeller, H., Mora, F.: Upper and lower bounds for the degree of Groebner bases. In: Fitch, J. (ed.) EUROSAM 1984 and ISSAC 1984. LNCS, vol. 174, pp. 172–183. Springer, Heidelberg (1984)Google Scholar
  16. 16.
  17. 17.
    Windsteiger, W.: Building Up Hierarchical Mathematical Domains Using Functors in THEOREMA. In: Armando, A., Jebelean, T. (eds.) Electronic Notes in Theoretical Computer Science. ENTCS, vol. 23, pp. 401–419. Elsevier (1999)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Alexander Maletzky
    • 1
  • Bruno Buchberger
    • 2
  1. 1.Doctoral College “Computational Mathematics” and RISCJohannes Kepler UniversityLinzAustria
  2. 2.RISCJohannes Kepler UniversityLinzAustria

Personalised recommendations