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Finite Choice, Convex Choice and Sorting

  • Takayuki Kihara
  • Arno PaulyEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11436)

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.

Keywords

Computable analysis Weihrauch reducibility Closed choice 

Notes

Acknowledgement

We are grateful to Stéphane Le Roux for a fruitful discussion leading up to Theorems 2 and 3.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematical InformaticsNagoya UniversityNagoyaJapan
  2. 2.Department of Computer ScienceSwansea UniversitySwanseaUK
  3. 3.Department of Computer ScienceUniversity of BirminghamBirminghamUK

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