Finite Choice, Convex Choice and Sorting

Abstract

We study the Weihrauch degrees of closed choice for finite sets, closed choice for convex sets and sorting infinite sequences over finite alphabets. Our main result is that choice for finite sets of cardinality \(i + 1\) is reducible to choice for convex sets in dimension j, which in turn is reducible to sorting infinite sequences over an alphabet of size \(k + 1\), iff \(i \le j \le k\). Our proofs invoke Kleene’s recursion theorem, and we describe in some detail how Kleene’s recursion theorem gives rise to a technique for proving separations of Weihrauch degrees.