Reverse Mathematics of Matroids

Chapter
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10010)

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 \(\varSigma ^0_2\) formulas.

Keywords

Reverse mathematics Matroid Induction Graph Connected component 

MSC Subject Class (2000)

03B30 03F35 05B35 

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Appalachian State UniversityBooneUSA
  2. 2.Marshall UniversityHuntingtonUSA

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