Gröbner Bases of Modules and Faugère’s $$F_4$$F4 Algorithm in Isabelle/HOL

Abstract

We present an elegant, generic and extensive formalization of Gröbner bases in Isabelle/HOL. The formalization covers all of the essentials of the theory (polynomial reduction, S-polynomials, Buchberger's algorithm, Buchberger's criteria for avoiding useless pairs), but also includes more advanced features like reduced Gröbner bases. Particular highlights are the first-time formalization of Faugère's matrix-based F4 algorithm and the fact that the entire theory is formulated for modules and submodules rather than rings and ideals. All formalized algorithms can be translated into executable code operating on concrete data structures, enabling the certified computation of (reduced) Gröbner bases and syzygy modules.

A Generic and Executable Formalization of Signature-Based Gröbner Basis Algorithms

“…How exactly power-products are represented is not important, since they essentially only have to form a cancellative commutative monoid and a lattice w. r. t. divisibility. For more details see Maletzky and Immler (2018b).…”

“…Besides simple examples as the one shown above, gb-sig can also be tested on common benchmark problems and compared to other implementations of Gröbner bases. Table 1 shows such a comparison to a formally verified implementation of Buchberger's algorithm with product-and chain-criterion in Isabelle/HOL, called gb and described in Maletzky and Immler (2018a), and to function GroebnerBasis in Mathematica 11.3. Since this article is not meant as an exhaustive survey on the efficiency of different Gröbner basis algorithms, we confine ourselves here to present results of computations over the rationals w. r. t. the POT extension of the degree-reverse-lexicographic ordering and rewrite order rat .…”

“…This approach not only s-reduces one polynomial at a time, but several polynomials simultaneously by row-reducing certain matrices. Incidentally, the F 4 algorithm and corresponding F 4 -style reduction are part of the formalization by Immler and Maletzky (2016) (described in Maletzky and Immler (2018a)), and therefore could be incorporated into the formalization presented here with only moderate effort. The main reason why we have not done so as of yet is that no increase in performance can be expected from it in this concrete case: matrices are represented densely as immutable arrays in Isabelle/HOL, but F 4 -style reductions only make sense if sparse matrices are stored efficiently, possibly even involving some sort of compression.…”