A Generic and Executable Formalization of Signature-Based Gr\"obner Basis Algorithms

TL;DR

This paper formalizes signature-based Gr"obner basis algorithms in Isabelle/HOL, ensuring correctness, efficiency, and minimality detection, and is the first such formalization to date.

Contribution

It provides the first formal, executable, and verified formalization of signature-based Gr"obner basis algorithms in a proof assistant, generalizing existing variants.

Findings

Algorithms detect and avoid useless reductions to zero under certain conditions.

The formalization is effectively executable for concrete inputs.

It proves the algorithms return minimal signature Gr"obner bases.

Abstract

We present a generic and executable formalization of signature-based algorithms (such as Faug\`ere's $F_5$) for computing Gr\"obner bases, as well as their mathematical background, in the Isabelle/HOL proof assistant. Said algorithms are currently the best known algorithms for computing Gr\"obner bases in terms of computational efficiency. The formal development attempts to be as generic as possible, generalizing most known variants of signature-based algorithms, but at the same time the implemented functions are effectively executable on concrete input for efficiently computing mechanically verified Gr\"obner bases. Besides correctness the formalization also proves that under certain conditions the algorithms a-priori detect and avoid all useless reductions to zero, and return minimal signature Gr\"obner bases. To the best of our knowledge, the formalization presented here is the only formalization of signature-based Gr\"obner basis algorithms in existence so far.