Mathematical Theory Exploration in Theorema: Reduction Rings

  • Alexander MaletzkyEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9791)


In this paper we present the first-ever computer formalization of the theory of Gröbner bases in reduction rings in Theorema. Not only the formalization, but also the formal verification of all key results has already been fully completed by now; this, in particular, includes the generic implementation and correctness proof of Buchberger’s algorithm in reduction rings. Thanks to the seamless integration of proving and computing in Theorema, this implementation can now be used to compute Gröbner bases in various different domains directly within the system. Moreover, a substantial part of our formalization is made up solely by “elementary theories” such as sets, numbers and tuples that are themselves independent of reduction rings and may therefore be used as the foundations of future theory explorations in Theorema.

In addition, we also report on two general-purpose Theorema tools we developed for efficiently exploring mathematical theories: an interactive proving strategy and a “theory analyzer” that already proved extremely useful when creating large structured knowledge bases.


Gröbner bases Reduction rings Computer-supported theory exploration Automated reasoning Theorema 



I thank the anonymous referees for their valuable remarks and suggestions.

This research was funded by the Austrian Science Fund (FWF): grant no. W1214-N15, project DK1.


  1. 1.
    Adams, W.W., Loustaunau, P.: An Introduction to Gröbner Bases. Graduate Studies in Mathematics, vol. 3. American Mathematical Society, Providence (1994). doi: 10.1090/gsm/003. ISSN: 1065-7339, ISBN: 0-8218-3804-0zbMATHGoogle Scholar
  2. 2.
    Buchberger, B.: A criterion for detecting unnecessary reductions in the construction of Gröbner bases. In: Ng, E.W. (ed.) EUROSAM 1979. LNCS, vol. 72, pp. 3–21. Springer, Heidelberg (1979)CrossRefGoogle Scholar
  3. 3.
    Buchberger, B.: A critical-pair/completion algorithm for finitely generated ideals in rings. In: Börger, E., Hasenjaeger, G., Rödding, D. (eds.) Logic and Machines: Decision Problems and Complexity. LNCS, vol. 171, pp. 137–161. Springer, Heidelberg (1984)CrossRefGoogle Scholar
  4. 4.
    Buchberger, B.: Gröbner Rings in Theorema: A Case Study in Functors and Categories. Technical report 2003–49, Johannes Kepler University Linz, Spezialforschungsbereich F013, November 2003Google Scholar
  5. 5.
    Buchberger, B., Jebelean, T., Kutsia, T., Maletzky, A., Windsteiger, W.: Theorema 2.0 computer-assisted natural-style mathematics. J. Formalized Reasoning 9(1), 149–185 (2016)MathSciNetGoogle Scholar
  6. 6.
    Jorge, J.S., Guilas, V.M., Freire, J.L.: Certifying properties of an efficient functional program for computing Gröbner bases. J. Symbolic Comput. 44(5), 571–582 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Maletzky, A.: Exploring reduction ring theory in theorema. Technical report 2016–06, Doctoral Program “Computational Mathematics”, Johannes Kepler University Linz, Austria, July 2015Google Scholar
  8. 8.
    Maletzky, A.: Verifying Buchberger’s algorithm in reduction rings. In: Jebelean, T., Wang, D. (eds.) Proceedings of PAS 2015 (Program Verification, Automated Debugging and Symbolic Computation, Beijing, China, 21–23 October 2015. arXiv:1604.08736
  9. 9.
    Medina-Bulo, I., Palomo-Lozano, F., Ruiz-Reina, J.L.: A verified common lisp implementation of Buchberger’s algorithm in ACL2. J. Symbolic Comput. 45(1), 96–123 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Piroi, F., Kutsia, T.: The Theorema environment for interactive proof development. In: Sutcliffe, G., Voronkov, A. (eds.) Logic for Programming, Artificial Intelligence, and Reasoning. LNCS, vol. 3835, pp. 261–275. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  11. 11.
    Robbiano, L.: Term orderings on the polynomial ring. In: Caviness, B.F. (ed.) EUROCAL 1985. LNCS, vol. 204, pp. 513–517. Springer, Heidelberg (1985)Google Scholar
  12. 12.
    Schwarzweller, C.: Gröbner bases — theory refinement in the mizar system. In: Kohlhase, M. (ed.) MKM 2005. LNCS (LNAI), vol. 3863, pp. 299–314. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  13. 13.
    Stifter, S.: A generalization of reduction rings. J. Symbolic Comput. 4(3), 351–364 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Stifter, S.: The reduction ring property is hereditary. J. algebra 140(89–18), 399–414 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Thery, L.: A machine-checked implementation of Buchberger’s algorithm. J. Autom. Reasoning 26, 107–137 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Wiesinger-Widi, M.: Gröbner Bases and Generalized Sylvester Matrices. Ph.D. Thesis, Johannes Kepler University Linz (2015).
  17. 17.
    Windsteiger, W.: Building up hierarchical mathematical domains using functors in theorema. In: Armando, A., Jebelean, T. (eds.) Proceedings of Calculemus 1999, Trento, Italy. ENTCS, vol. 23, pp. 401–419. Elsevier, Amsterdam (1999)Google Scholar
  18. 18.
    Windsteiger, W.: Theorema 2.0: a system for mathematical theory exploration. In: Hong, H., Yap, C. (eds.) ICMS 2014. LNCS, vol. 8592, pp. 49–52. Springer, Heidelberg (2014)Google Scholar
  19. 19.
    Winkler, F., Buchberger, B.: A criterion for eliminating unnecessary reductions in the Knuth-Bendix algorithm. In: Colloqium on Algebra, Combinatorics and Logic in Computer Science, pp. 849–869 (1983)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Doctoral Program “Computational Mathematics” and RISCJohannes Kepler UniversityLinzAustria

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