Reverse Mathematics of Matroids

TL;DR

This paper investigates the logical foundations of matroid theory using reverse mathematics and Weihrauch reducibility, establishing equivalences between basis theorems, combinatorial principles, and induction schemes.

Contribution

It applies reverse mathematics and Weihrauch analysis to matroids, linking basis theorems to logical and combinatorial principles, and formalizing reductions for foundational insights.

Findings

Existence of bases for bounded dimension vector spaces is equivalent to $ ext{I} ext{-} ext{Sigma}^0_2$ induction.

Reverse mathematics characterizes the strength of matroid basis theorems.

Weihrauch reducibility relates basis theorems to combinatorial choice principles.

Abstract

Matroids generalize the familiar notion of linear dependence from linear algebra. Following a brief discussion of founding work in computability and matroids, we use the techniques of reverse mathematics to determine the logical strength of some basis theorems for matroids and enumerated matroids. Next, using Weihrauch reducibility, we relate the basis results to combinatorial choice principles and statements about vector spaces. Finally, we formalize some of the Weihrauch reductions to extract related reverse mathematics results. In particular, we show that the existence of bases for vector spaces of bounded dimension is equivalent to the induction scheme for $\Sigma^0_2$ formulas.