Convex choice, finite choice and sorting

05/02/2019

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by   Takayuki Kihara, et al.

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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 results are: One, 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 ≤ j ≤ k. Two, that convex choice in dimension two is not reducible to the product of convex choice in dimension one with itself. Three, that sequential composition of one-dimensional convex choice is not reducible to convex choice in any dimension. The latter solves an open question raised at a Dagstuhl seminar on Weihrauch reducibility in 2015. 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.