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

We present a generic and executable formalization of signature-based algorithms (such as Faugère'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...

Ausführliche Beschreibung

Gespeichert in:

Bibliographische Detailangaben
Veröffentlicht in:	arXiv.org
1. Verfasser:	Maletzky, Alexander
Format:	Paper
Sprache:	Englisch
Veröffentlicht:	Ithaca Cornell University Library, arXiv.org 03.12.2020
Schlagworte:	Algorithms
ISSN:	2331-8422
Online-Zugang:	Volltext
Tags:	Tag hinzufügen Keine Tags, Fügen Sie den ersten Tag hinzu!

Beschreibung
Zusammenfassung:	We present a generic and executable formalization of signature-based algorithms (such as Faugère'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.
Bibliographie:	SourceType-Working Papers-1 ObjectType-Working Paper/Pre-Print-1 content type line 50
ISSN:	2331-8422
DOI:	10.48550/arxiv.2012.02239