Game Characterizations and Lower Cones in the Weihrauch Degrees

  • Hugo NobregaEmail author
  • Arno Pauly
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10307)


We introduce generalized Wadge games and show that each lower cone in the Weihrauch degrees is characterized by such a game. These generalized Wadge games subsume (a variant of) the original Wadge game, the eraser and backtrack games as well as Semmes’s tree games. In particular, we propose that the lower cones in the Weihrauch degrees are the answer to Andretta’s question on which classes of functions admit game characterizations. We then discuss some applications of such generalized Wadge games.



We are grateful to Benedikt Löwe, Luca Motto Ros, Takayuki Kihara and Raphaël Carroy for helpful and inspiring discussions. We would also like to thank the anonymous referees for the many corrections which have significantly improved the paper.


  1. 1.
    Andretta, A.: The SLO principle and the Wadge hierarchy. In: Bold, S., Löwe, B., Räsch, T., van Benthem, J. (eds.) Foundations of the Formal Sciences V: Infinite Games, pp. 1–38. College Publications (2007)Google Scholar
  2. 2.
    Brattka, V., Gherardi, G.: Effective choice and boundedness principles in computable analysis. Bull. Symbolic Log. 1, 73–117 (2011). arXiv:0905.4685 MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Brattka, V., Gherardi, G.: Weihrauch degrees, omniscience principles and weak computability. J. Symbolic Log. 76, 143–176 (2011). arXiv:0905.4679 MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brattka, V., Gherardi, G., Hölzl, R.: Probabilistic computability and choice. Inf. Comput. 242, 249–286 (2015)., arXiv:1312.7305 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brattka, V., Gherardi, G., Marcone, A.: The Bolzano-Weierstrass theorem is the jump of weak König’s lemma. Ann. Pure Appl. Log. 163(6), 623–625 (2012)CrossRefzbMATHGoogle Scholar
  6. 6.
    Brattka, V., Pauly, A.: On the algebraic structure of Weihrauch degrees (2016). arXiv 1604.08348,
  7. 7.
    de Brecht, M.: Quasi-Polish spaces. Ann. Pure Appl. Log. 164(3), 354–381 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    de Brecht, M.: Levels of discontinuity, limit-computability, and jump operators. In: Brattka, V., Diener, H., Spreen, D. (eds.) Logic, Computation, Hierarchies, pp. 79–108 (2014). de Gruyter, arXiv:1312.0697
  9. 9.
    Carroy, R.: A quasi-order on continuous functions. J. Symbolic Log. 78(2), 633–648 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cenzer, D., Remmel, J.: Recursively presented games and strategies. Math. Soc. Sci. 24(2–3), 117–139 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Duparc, J.: Wadge hierarchy and Veblen hierarchy part I: Borel sets of finite rank. J. Symbolic Log. 66(1), 56–86 (2001). MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gale, D., Stewart, F.M.: Infinite games with perfect information. In: Kuhn, H.W., Tucker, A.W. (eds.) Contributions to the Theory of Games, vol. 2, pp. 245–266. Princeton University Press (1953)Google Scholar
  13. 13.
    Gherardi, G., Marcone, A.: How incomputable is the separable Hahn-Banach theorem? Notre Dame J. Formal Log. 50(4), 393–425 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hertling, P.: Unstetigkeitsgrade von Funktionen in der effektiven Analysis. Ph.D. thesis, Fernuniversität, Gesamthochschule in Hagen (Oktober 1996)Google Scholar
  15. 15.
    Higuchi, K., Pauly, A.: The degree-structure of Weihrauch-reducibility. Log. Methods Comput. Sci. 9(2), 1–17 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kanamori, A.: The Higher Infinite: Large Cardinals in Set Theory from their Beginnings. Springer Monographs in Mathematics, 2nd edn. Springer, Heidelberg (2005)zbMATHGoogle Scholar
  17. 17.
    Kechris, A.S.: Classical Descriptive Set Theory. Graduate Texts in Mathematics, vol. 156. Springer, Heidelberg (1995)zbMATHGoogle Scholar
  18. 18.
    Kihara, T., Pauly, A.: Point degree spectra of represented spaces (2014). arXiv:1405.6866
  19. 19.
    Le Roux, S., Pauly, A.: Weihrauch degrees of finding equilibria in sequential games. In: Beckmann, A., Mitrana, V., Soskova, M. (eds.) CiE 2015. LNCS, vol. 9136, pp. 246–257. Springer, Cham (2015). doi: 10.1007/978-3-319-20028-6_25 CrossRefGoogle Scholar
  20. 20.
    Martin, D.A.: Borel determinacy. Ann. Math. 102(2), 363–371 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Moschovakis, Y.N.: Classical descriptive set theory as a refinement of effective descriptive set theory. Ann. Pure Appl. Log. 162, 243–255 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Ros, L.M.: Borel-amenable reducibilities for sets of reals. J. Symbolic Log. 74(1), 27–49 (2009). MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Ros, L.M.: Game representations of classes of piecewise definable functions. Math. Log. Q. 57(1), 95–112 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Ros, L.M.: Bad Wadge-like reducibilities on the baire space. Fundam. Math. 224(1), 67–95 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Ros, L.M., Schlicht, P., Selivanov, V.: Wadge-like reducibilities on arbitrary quasi-Polish spaces. Mathematical Structures in Computer Science pp. 1–50 (2014)., arXiv:1204.5338
  26. 26.
    Neumann, E., Pauly, A.: A topological view on algebraic computation models (2016). arXiv:1602.08004
  27. 27.
    Nobrega, H., Pauly, A.: Game characterizations and lower cones in the Weihrauch degrees (2015). arXiv:1511.03693
  28. 28.
    Pauly, A.: Computable Metamathematics and its Application to Game Theory. Ph.D. thesis, University of Cambridge (2012)Google Scholar
  29. 29.
    Pauly, A.: The descriptive theory of represented spaces (2014). arXiv:1408.5329
  30. 30.
    Pauly, A.: Many-one reductions and the category of multivalued functions. Mathematical Structures in Computer Science (2015). arXiv:1102.3151
  31. 31.
    Pauly, A.: On the topological aspects of the theory of represented spaces. Computability 5(2), 159–180 (2016).
  32. 32.
    Pauly, A., de Brecht, M.: Towards synthetic descriptive set theory: An instantiation with represented spaces (2013).
  33. 33.
    Pauly, A., de Brecht, M.: Descriptive set theory in the category of represented spaces. In: 30th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), pp. 438–449 (2015)Google Scholar
  34. 34.
    Pauly, A., Ziegler, M.: Relative computability and uniform continuity of relations. J. Log. Anal. 5, 1–39 (2013)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Pequignot, Y.: A Wadge hierarchy for second countable spaces. Arch. Math. Log. 54(5), 1–25 (2015). MathSciNetzbMATHGoogle Scholar
  36. 36.
    Schröder, M.: Extended admissibility. Theoret. Comput. Sci. 284(2), 519–538 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Schröder, M., Selivanov, V.: Hyperprojective hierarchy of QCB\(_{0}\)-spaces (2014). arXiv:1404.0297,
  38. 38.
    Schröder, M., Selivanov, V.L.: Some hierarchies of QCB\(_{0}\)-spaces. Math. Struct. Comput. Sci. 25(8), 1–25 (2014). arXiv:1304.1647 MathSciNetGoogle Scholar
  39. 39.
    Semmes, B.: A game for the Borel functions. Ph.D. thesis, University of Amsterdam (2009)Google Scholar
  40. 40.
    Wadge, W.W.: Reducibility and determinateness on the Baire space. Ph.D. thesis, University of California, Berkeley (1983)Google Scholar
  41. 41.
    Weihrauch, K.: The degrees of discontinuity of some translators between representations of the real numbers. Informatik Berichte 129, FernUniversität Hagen, Hagen, July 1992Google Scholar
  42. 42.
    Weihrauch, K.: The TTE-interpretation of three hierarchies of omniscience principles. Informatik Berichte 130, FernUniversität Hagen, Hagen, September 1992Google Scholar
  43. 43.
    Weihrauch, K.: Computable Analysis. Springer, Heidelberg (2000)CrossRefzbMATHGoogle Scholar
  44. 44.
    Wesep, R.: Wadge degrees and descriptive set theory. In: Kechris, A.S., Moschovakis, Y.N. (eds.) Cabal Seminar 76–77. LNM, vol. 689, pp. 151–170. Springer, Heidelberg (1978). doi: 10.1007/BFb0069298 CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Institute for Logic, Language, and ComputationUniversity of AmsterdamAmsterdamThe Netherlands
  2. 2.Département D’informatiqueUniversité Libre de BruxellesBrusselsBelgium

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