Convex choice, finite choice and sorting

Takayuki Kihara; Arno Pauly

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arXiv · 1905.03190

Convex choice, finite choice and sorting

Takayuki Kihara·Arno Pauly

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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 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 $i \leq j \leq k$. 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.

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BibTeX RIS

Takayuki Kihara, Arno Pauly. 2019-05-02. Convex choice, finite choice and sorting. https://arxiv.org/abs/1905.03190

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