Papers by Bruno Buchberger
Gröbner Bases in Non-Commutative Reduction Rings
Cambridge University Press eBooks, Feb 26, 1998
Grobner Bases in Non-Commutative Reduction Rings Klaus Madlener and Birgit Reinert1 Fachbereich I... more Grobner Bases in Non-Commutative Reduction Rings Klaus Madlener and Birgit Reinert1 Fachbereich Informatik, Universitat Kaiserslautern 67663 Kaiserslautern, Germany {madlener, reinert}@ informatik. uni-kl. de Abstract Grobner bases of ideals in polynomial rings can be ...
A Formal Knowledge Base for Groebner Bases Theory
数式処理, Feb 1, 2005
Algorithms of Computeralgebra (Dagstuhl Seminar 9151)
Proceedings of the Third Joint International Conference on Vector and Parallel Processing: Parallel Processing
Invited Lectures from the European Conference on Computer Algebra-Volume I - Volume I
RISC: Innovation – global und regional
Gröbner bases computation by triangularizing Macaulay matrices
Advanced studies in pure mathematics, Mar 7, 2019
An Algorithmic Criterion for the Solvability of a System of Algebraic Equations (translated by Michael Abramson and Robert Lumbert)
Methodische Analyse der Fallstudie
Springer eBooks, 1980
Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics): Preface
Algebraic Biology, 2008
Gröbner Bases Computation and Macaulay Matrices
In my PhD thesis 1965 and the subsequent publication 1970 in Aequationes Mathematicae, I introduc... more In my PhD thesis 1965 and the subsequent publication 1970 in Aequationes Mathematicae, I introduced the notion of Grobner bases and proved a characterization theorem for Grobner bases on which an algorithm for constructing Grobner bases can be based. The main idea for the theorem and the algorithm was the notion of "�S-polynomials"�. Most of the subsequent work on the algorithmic theory of Grobner bases, including the implementation of the Grobner bases technology in mathematical software systems like Mathematica, Maple, etc. was based on this approach.
Annals of Mathematics and Artificial Intelligence
Annals of Mathematics and Artificial Intelligence, Sep 30, 2023
In this note, I present my personal view on the interaction of the three areas Automated Programm... more In this note, I present my personal view on the interaction of the three areas Automated Programming, Symbolic Computation, and Machine Learning. Programming is the activity of finding a (hopefully) correct program (algorithm) for a given problem. Programming is central to automation in all areas and is considered one of the most creative human activities. However, already very early in the history of programming, people started to "jump to the meta-level" of programming, i.e., started to develop procedures that automate, or semiautomate, (various aspects or parts of) the process of programming. This area has various names like "Automated Programming", "Automated Algorithm Synthesis", etc. Developing compilers can be considered an early example of a problem in automated programming. Automated reasoners for proving the correctness of programs with respect to a specification is an advanced example of a topic in automated programming. ChatGPT producing (amazingly good) programs from problem specifications in natural language is a recent example of automated programming. Programming tends to become the most important activity as the level of technological sophistication increases. Therefore, automating programming is maybe the most exciting and relevant technological endeavor today. It also will have enormous impact on the global job market in the software industry. Roughly, I see two main approaches to automated programming: • symbolic computation • and machine learning. In this note, I explain how the two approaches work and that they are fundamentally different because they address two completely different ways of how problems are specified. Together, the two approaches constitute (part of) what some people like to call "artificial intelligence". In my analysis, both approaches are just part of (algorithmic) mathematics. The approaches, like all non-trivial mathematical methods, need quite some intelligence on the side of the human inventors of the methods. However, applying the methods is just "machine execution" of algorithms. It is misleading to call the application "machine intelligence" or "artificial intelligence". The analysis of the two approaches to automated programming also suggests that the two approaches, in the future, should be combined to achieve even higher levels of sophistication. At the end of this note, I propose some research questions for this new direction.
Proceedings of the Second International Conference on Mathematical Knowledge Management
THEOREMA: a formal frame for Mathematics
Parallel processing : CONPAR 94-VAPP VI : Third Joint International Conference on Vector and Parallel Processing, Linz, Austria, September 6-8, 1994 : proceedings
This volume presents the proceedings of the Third Joint International Conference on Vector and Pa... more This volume presents the proceedings of the Third Joint International Conference on Vector and Parallel Processing (CONPAR 94 - VAPP VI), held in Linz, Austria in September 1994. The 76 papers contained were carefully selected from a wealth of submissions and address the most important aspects of parallel processing research. The volume is organized into sections on performance analysis and monitoring, parallel program development, parallel algorithms and complexity models, parallel architectures and abstract machines, parallel languages and compiler technology, networks and routing, and scheduling in distributed memory systems.
Journal of Symbolic Computation, Mar 1, 2006
Comments on the translation of my PhD thesis The editor of the JSC, Hoon Hong, and the editor of ... more Comments on the translation of my PhD thesis The editor of the JSC, Hoon Hong, and the editor of this special issue, Deepak Kapur, proposed including an English translation of my PhD thesis Buchberger (1965) in which I initiated the theory and algorithmic construction of Gröbner bases. In fact, Michael Abramson had prepared an English translation of both my thesis and (with co-translator Robert Lumbert) my subsequent journal publication Buchberger (1970) already in 1997 and we included the English translation of the journal publication in the book Buchberger and Winkler (1998). The translation of the thesis will now be included in this special issue. I am grateful to H. Hong, D. Kapur, and M. Abramson for their concern for making my early papers on Gröbner bases available to a broad readership who might be interested in seeing how the ideas evolved. Special thanks go to M. Abramson for his enormous patience and effort in doing the translation. While the journal publication Buchberger (1970) already concentrated exclusively on the notion of Gröbner bases, some of their properties, and their algorithmic construction, my thesis started from the problem and the ideas for a procedure for solving the problem that my thesis advisor, Professor Wolfgang Gröbner (1899-1980), gave me. Only in the second part of the thesis did I introduce the notion of S-polynomials and my own algorithm for constructing Gröbner bases which is based on S-polynomials. Thus, when reading my thesis, it may be somewhat hard to understand what I was driving at in the first few sections of the thesis. Just consider these sections as a tribute to my advisor: He had posed the problem of finding a linearly independent basis of the associative algebra constituted by the residue class ring of a (zero-dimensional) polynomial ideal in his seminar in 1964. Also in this seminar, he presented a procedure for solving this problem: Consider all the power products and reduce them in all possible ways. If, for a given power product, you find different normal forms, add the difference to the basis of the ideal. In the first sections of my thesis (up to Lemmata (5.8) and (5.13)), I described this procedure and tried to prove a few observations on this procedure. Lemma (5.13), which refers to (5.8), is then the crucial lemma in which I introduced the concept of S-polynomials and proved their fundamental role in the construction of Gröbner bases. (In (5.13) I did not yet use the name "S-polynomials"; I introduced this name only in the journal publication Buchberger (1970).) After having S-polynomials and the lemma on S-polynomials, one could forget the ideas that led to this lemma. However, in the first part of my thesis (before (5.13)), I gave an account of how I was led to (5.13) starting from the ideas of my advisor. Only in the second part (starting from (5.13)), which could be read independently of the first part, did I then develop the essential ingredients of "Gröbner bases theory":
Methodische Analyse der Fallstudie
Informatik-Fachberichte, 1980
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Papers by Bruno Buchberger