Automated Mathematical Invention: Would Gröbner Need a PhD Student Today?
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Eurocast 2019 - 30 years 17th International Comference on Computer Aided Systems Theory
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Wolfgang Gröbner (1899 ? 1980) was my PhD advisor back in 1964 at the University of Innsbruck, Austria. The problem he posed to me had been formulated, in a slightly different form, by Paul Gordan in 1899 and was still open in 1964. Roughly, the problem asks for an algorithmic canonical simplifier for the congruence relations w.r.t. multivariate polynomial ideals. I solved the problem in 1965 in my PhD thesis by introducing what I later called the theory and method of ?Gröbner bases?. The theory and method found numerous applications both inside mathematics and in basically all areas of science and technology in which non-linear polynomial systems play a role (e.g. robotics, cryptography, computer-aided design, software verification, systems theory, etc.) In this talk I will, first, give an easy and practical introduction to the theory and method of Gröbner bases for those with no or only little background in this area. My main emphasis, however, will be on my recent research on automating mathematical invention. For this, I will take the theory of Gröbner bases as my main example. I will show how, by recent progress in automated reasoning and, in particular, my method of ?Lazy Thinking? for the automated invention and proof of mathematical theorems and algorithms, my theory and method of Gröbner bases today could be ?invented? completely automatically. In other words, cum grano salis, Professor Gröbner today would not need any more a PhD student for solving his problem. From this, I will draw some conclusions on the future of mathematics.
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