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)


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.


Computable analysis Weihrauch reducibility Closed choice 



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


  1. 1.
    Blum, L., Cucker, F., Shub, M., Smale, S.: Complexity and Real Computation. Springer, New York (1998). Scholar
  2. 2.
    Brattka, V., de Brecht, M., Pauly, A.: Closed choice and a uniform low basis theorem. Ann. Pure Appl. Log. 163(8), 968–1008 (2012). Scholar
  3. 3.
    Brattka, V., Gherardi, G.: Effective choice and boundedness principles in computable analysis. Bull. Symb. Log. 17, 73–117 (2011). arXiv:0905.4685MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brattka, V., Gherardi, G.: Weihrauch degrees, omniscience principles and weak computability. J. Symb. Log. 76, 143–176 (2011). arXiv:0905.4679MathSciNetCrossRefGoogle Scholar
  5. 5.
    Brattka, V., Gherardi, G., Hölzl, R.: Probabilistic computability and choice. Inf. Comput. 242, 249–286 (2015). Scholar
  6. 6.
    Brattka, V., Gherardi, G., Hölzl, R., Pauly, A.: The vitali covering theorem in the Weihrauch lattice. In: Day, A., Fellows, M., Greenberg, N., Khoussainov, B., Melnikov, A., Rosamond, F. (eds.) Computability and Complexity. LNCS, vol. 10010, pp. 188–200. Springer, Cham (2017). Scholar
  7. 7.
    Brattka, V., Gherardi, G., Pauly, A.: Weihrauch complexity in computable analysis. arXiv:1707.03202 (2017)
  8. 8.
    Brattka, V., Hertling, P., Weihrauch, K.: A tutorial on computable analysis. In: Cooper, S.B., Löwe, B., Sorbi, A. (eds.) New Computational Paradigms: Changing Conceptions of What is Computable, pp. 425–491. Springer, New York (2008). Scholar
  9. 9.
    Brattka, V., Hölzl, R., Kuyper, R.: Monte Carlo computability. In: Vollmer, H., Vallée, B. (eds.) 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), vol. 66, pp. 17:1–17:14. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany (2017).
  10. 10.
    Brattka, V., Miller, J., Le Roux, S., Pauly, A.: Connected choice and Brouwer’s fixed point theorem. J. Math. Log. (20XX, accepted for publication). arXiv:1206.4809
  11. 11.
    Brattka, V., Pauly, A.: Computation with advice. Electron. Proc. Theor. Comput. Sci. 24, 41–55 (2010)., cCA 2010
  12. 12.
    Brattka, V., Pauly, A.: On the algebraic structure of Weihrauch degrees. Log. Methods Comput. Sci. 14(4) (2018).
  13. 13.
    Chadzelek, T., Hotz, G.: Analytic machines. Theor. Comput. Sci. 219, 151–167 (1999)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Dorais, F.G., Dzhafarov, D.D., Hirst, J.L., Mileti, J.R., Shafer, P.: On uniform relationships between combinatorial problems. Trans. AMS 368, 1321–1359 (2016). arXiv:1212.0157MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gärtner, T., Hotz, G.: Computability of analytic functions with analytic machines. In: Ambos-Spies, K., Löwe, B., Merkle, W. (eds.) CiE 2009. LNCS, vol. 5635, pp. 250–259. Springer, Heidelberg (2009). Scholar
  16. 16.
    Gherardi, G., Marcone, A.: How incomputable is the separable Hahn-Banach theorem? Notre Dame J. Form. Log. 50(4), 393–425 (2009). Scholar
  17. 17.
    Gherardi, G., Marcone, A., Pauly, A.: Projection operators in the Weihrauch lattice. Computability (20XX, accepted for publication). arXiv:1805.12026
  18. 18.
    Gura, K., Hirst, J.L., Mummert, C.: On the existence of a connected component of a graph. Computability 4(2), 103–117 (2015). Scholar
  19. 19.
    Kihara, T., Pauly, A.: Dividing by zero - how bad is it, really? In: Faliszewski, P., Muscholl, A., Niedermeier, R. (eds.) 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), vol. 58, pp. 58:1–58:14. Schloss Dagstuhl (2016).
  20. 20.
    Le Roux, S., Pauly, A.: Finite choice, convex choice and finding roots. Log. Methods Comput. Sci. (2015).
  21. 21.
    Neumann, E.: Computational problems in metric fixed point theory and their Weihrauch degrees. Log. Methods Comput. Sci. 11(4) (2015).
  22. 22.
    Neumann, E., Pauly, A.: A topological view on algebraic computations models. J. Complex. 44 (2018). Scholar
  23. 23.
    Pauly, A.: How incomputable is finding Nash equilibria? J. Univers. Comput. Sci. 16(18), 2686–2710 (2010). Scholar
  24. 24.
    Pauly, A.: On the (semi)lattices induced by continuous reducibilities. Math. Log. Q. 56(5), 488–502 (2010). Scholar
  25. 25.
    Pauly, A.: On the topological aspects of the theory of represented spaces. Computability 5(2), 159–180 (2016). Scholar
  26. 26.
    Pauly, A., Tsuiki, H.: \(T^\omega \)-representations of compact sets. arXiv:1604.00258 (2016)
  27. 27.
    Selivanov, V.L.: Total representations. Log. Methods Comput. Sci. 9(2) (2013)Google Scholar
  28. 28.
    Weihrauch, K.: The degrees of discontinuity of some translators between representations of the real numbers. Informatik Berichte 129, FernUniversität Hagen, Hagen, July 1992Google Scholar
  29. 29.
    Weihrauch, K.: Computable Analysis. Springer, Heidelberg (2000). Scholar
  30. 30.
    Ziegler, M.: Computable operators on regular sets. Math. Log. Q. 50, 392–404 (2004)MathSciNetCrossRefGoogle Scholar

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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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