Abstract
In this talk we show how the theory of Groebner bases can be represented in the computer system Theorema, a system initiated by Bruno Buchberger in the mid-nineties. The main purpose of Theorema is to serve mathematical theory exploration and, in particular, automated reasoning. However, it is also an essential aspect of the Theorema philosophy that the system also provides good facilities for carrying out computations. The main difference between Theorema and ordinary computer algebra systems is that in Theorema one can both program (and, hence, compute) and prove (generate and verify proofs of theorems and algorithms). In fact, algorithms / programs in Theorema are just equational (recursive) statements in predicate logic and their application to data is just a special case of simplification w.r.t. equational logic as part of predicate logic.
We present one representation of Groebner bases theory among many possible “views” on the theory. In this representation, we use functors to construct hierarchies of domains (e. g. for power products, monomials, polynomials, etc.) in a nicely structured way, which is meant to be a model for gradually more efficient implementations based on more refined and powerful theorems or at least programming tricks, data structures, etc.
This research was funded by the Austrian Science Fund (FWF): grant no. W1214-N15, project DK1.
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
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)
Buchberger, B.: A Critical-Pair/Completion Algorithm in Reduction Rings. RISC Report Series 83-21, Research Institute for Symbolic Computation (RISC), University of Linz, Schloss Hagenberg, 4232 Hagenberg, Austria (1983)
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. 1–13. World Scientific, Singapore (1996)
Buchberger, B.: Introduction to Groebner Bases. London Mathematical Society Lectures Notes Series, vol. 251. Cambridge University Press (April 1998)
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)
Buchberger, B.: Towards the Automated Synthesis of a Groebner Bases Algorithm. RACSAM - Revista de la Real Academia de Ciencias (Review of the Spanish Royal Academy of Science), Serie A: Mathematicas 98(1), 65–75 (2004)
Buchberger, B., Crǎciun, A., Jebelean, T., Kovcs, 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)
CoCoA system, cocoa.dima.unige.it
Magma Computational Algebra System, magma.maths.usyd.edu.au/magma/
Maple system, www.maplesoft.com/products/Maple/
Wolfram Mathematica, http://www.wolfram.com/mathematica/
Sage system, http://www.sagemath.org
Decker, W., Greuel, G.-M., Pfister, G., Schönemann, H.: Singular 3-1-6 — A computer algebra system for polynomial computations (2012), www.singular.uni-kl.de
Stifter, S.: A Generalization of Reduction Rings. Journal of Symbolic Computation 4(3), 351–364 (1988)
Stifter, S.: The Reduction Ring Property is Hereditary. Journal of Algebra 140(89-18), 399–414 (1991)
Theorema system, http://www.risc.jku.at/research/theorema/description/
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)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Buchberger, B., Maletzky, A. (2014). Groebner Bases in Theorema . In: Hong, H., Yap, C. (eds) Mathematical Software – ICMS 2014. ICMS 2014. Lecture Notes in Computer Science, vol 8592. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44199-2_58
Download citation
DOI: https://doi.org/10.1007/978-3-662-44199-2_58
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-44198-5
Online ISBN: 978-3-662-44199-2
eBook Packages: Computer ScienceComputer Science (R0)