2014 | OriginalPaper | Buchkapitel
Groebner Bases in Theorema
verfasst von : Bruno Buchberger, Alexander Maletzky
Erschienen in: Mathematical Software – ICMS 2014
Verlag: Springer Berlin Heidelberg
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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.