Reverse Mathematics of Matroids

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 Σ 0 2 formulas.

Foundations of Online Structure Theory II: The Operator Approach

“…In this particular case, we also note that a polynomial time algorithm for finding a basis of a polynomial time vector space was proven to be equivalent to P = N P suggesting intriguing connections with complexity theory. There is some relevant work by Hirst [52], who have proved that finding the basis of a vector space has the Weihrauch complexity of lim, i.e., is on the second level of the Borel hierarchy.…”

Combinatorial principles equivalent to weak induction

“…Our final result links minECT to principles considered by Hirst and Mummert [7]. The principle C # max takes as inputs a size n and the enumeration of the complement of a collection of finite subsets of N, each of size at most n, and outputs an element of the collection of maximum cardinality.…”

Comparing Representations for Function Spaces in Computable Analysis

“…2 The Weihrauch degree corresponding to C N has received significant attention, e.g. in [3,4,6,7,[15][16][17][18][19]. In particular, as shown in [23], a function between computable Polish spaces is Weihrauch reducible to C N if and only if it is piecewise computable or equivalently is effectively 0 2 -measurable.…”