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.
Jeffry L. Hirst and Carl Mummert. Reverse mathematics of matroids, In Computability and Complexity: Essays Dedicated to Rodney G. Downey on the Occasion of His 60th Birthday, Day, Fellows, Greenberg, Khoussainov, Melnikov, and Rosamond (Eds.), 2017, pp. 143–159. Preprint: http://arxiv.org/abs/1604.04912
This is the author’s manuscript of a chapter published in Computability and Complexity: Essays Dedicated to Rodney G. Downey on the Occasion of His 60th Birthday. The version of record is available at https://www.springer.com/gp/book/9783319500614.
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