Theory of Computing Systems

, Volume 61, Issue 4, pp 1376–1426 | Cite as

On the Uniform Computational Content of Computability Theory

  • Vasco BrattkaEmail author
  • Matthew Hendtlass
  • Alexander P. Kreuzer
Part of the following topical collections:
  1. Special Issue on Computability, Complexity and Randomness (CCR 2015)


We demonstrate that the Weihrauch lattice can be used to classify the uniform computational content of computability-theoretic properties as well as the computational content of theorems in one common setting. The properties that we study include diagonal non-computability, hyperimmunity, complete consistent extensions of Peano arithmetic, 1-genericity, Martin-Löf randomness, and cohesiveness. The theorems that we include in our case study are the low basis theorem of Jockusch and Soare, the Kleene-Post theorem, and Friedberg’s jump inversion theorem. It turns out that all the aforementioned properties and many theorems in computability theory, including all theorems that claim the existence of some Turing degree, have very little uniform computational content: they are located outside of the upper cone of binary choice (also known as LLPO); we call problems with this property indiscriminative. Since practically all theorems from classical analysis whose computational content has been classified are discriminative, our observation could yield an explanation for why theorems and results in computability theory typically have very few direct consequences in other disciplines such as analysis. A notable exception in our case study is the low basis theorem which is discriminative. This is perhaps why it is considered to be one of the most applicable theorems in computability theory. In some cases a bridge between the indiscriminative world and the discriminative world of classical mathematics can be established via a suitable residual operation and we demonstrate this in the case of the cohesiveness problem and the problem of consistent complete extensions of Peano arithmetic. Both turn out to be the quotient of two discriminative problems.


Computable analysis Weihrauch lattice Computability theory 



We would like to thank Arno Pauly for helpful comments on an earlier version of this article and the anonymous referee for her or his very careful proof reading that helped us to improve some results and the presentation of the article.


  1. 1.
    Ambos-Spies, K., Kjos-Hanssen, B., Lempp, S., Slaman, T.A.: Comparing DNR and WWKL. J. Symb. Log. 69(4), 1089–1104 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bienvenu, L., Patey, L.: Diagonally non-computable functions and fireworks. Inf. Comput. 253(part 1), 64–77 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bishop, E., Bridges, D.S.: Constructive Analysis volume 279 of Grundlehren der Mathematischen Wissenschaften. Springer, Berlin (1985)Google Scholar
  4. 4.
    Brattka, V.: Computable invariance. Theor. Comput. Sci. 210, 3–20 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brattka, V.: Effective Borel measurability and reducibility of functions. Math. Log. Q. 51(1), 19–44 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Brattka, V., de Brecht, M., Pauly, A.: Closed choice and a uniform low basis theorem. Ann. Pure Appl. Logic 163, 986–1008 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Brattka, V., Gherardi, G.: Borel complexity of topological operations on computable metric spaces. J. Log. Comput. 19(1), 45–76 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Brattka, V., Gherardi, G.: Effective choice and boundedness principles in computable analysis. Bull. Symb. Log. 17(1), 73–117 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Brattka, V., Gherardi, G.: Weihrauch degrees, omniscience principles and weak computability. J. Symb. Log. 76(1), 143–176 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Brattka, V., Gherardi, G., Hölzl, R.: Las Vegas computability and algorithmic randomness. In: Mayr, E.W., Ollinger, N. (eds.) 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015), volume 30 of Leibniz International Proceedings in Informatics (LIPIcs), pages 130–142, Dagstuhl, Germany, 2015. Schloss Dagstuhl–Leibniz-Zentrum für InformatikGoogle Scholar
  11. 11.
    Brattka, V., Gherardi, G., Hölzl, R.: Probabilistic computability and choice. Inf. Comput. 242, 249–286 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Brattka, V., Gherardi, G., Marcone, A.: The Bolzano-Weierstraß theorem is the jump of weak Kőnig’s lemma. Ann. Pure Appl. Logic 163, 623–655 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Brattka, V., Hendtlass, M., Kreuzer, P.: On the uniform computational content of the Baire category theorem. Notre Dame Journal of Formal Logic, (accepted for publication) (2016)Google Scholar
  14. 14.
    Brattka, V., Hertling, P.: Topological properties of real number representations. Theor. Comput. Sci. 284(2), 241–257 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Brattka, V., Roux, S.L., Pauly, A.: On the computational content of the Brouwer Fixed Point Theorem. In: Cooper, S.B., Dawar, A., Löwe, B. (eds.) How the World Computes, volume 7318 of Lecture Notes in Computer Science, pages 57–67, Berlin, 2012. Springer. Turing Centenary Conference and 8th Conference on Computability in Europe, CiE, 2012, Cambridge, UK, June (2012)Google Scholar
  16. 16.
    Brattka, V., Pauly, A.: Computation with advice. In: Zheng, X., Zhong, N. (eds.) CCA 2010, Proceedings of the Seventh International Conference on Computability and Complexity in Analysis, Electronic Proceedings in Theoretical Computer Science, pp. 41–55 (2010)Google Scholar
  17. 17.
    Brattka, V., Pauly, A.: On the algebraic structure of Weihrauch degrees. arXiv:1604.08348 (2016)
  18. 18.
    Brattka, V., Presser, G.: Computability on subsets of metric spaces. Theor. Comput. Sci. 305, 43–76 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Cholak, P.A., Jockusch, C.G., Slaman, T.A.: On the strength of Ramsey’s theorem for pairs. J. Symb. Log. 66(1), 1–55 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Cooper, S.B.: Minimal degrees and the jump operator. J. Symb. Log. 38, 249–271 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Cooper, B.S.: Computability theory. Chapman & Hall/CRC, Boca Raton (2004)zbMATHGoogle Scholar
  22. 22.
    Dorais, F.G., Dzhafarov, D.D., Hirst, J.L., Mileti, J.R., Shafer, P.: On uniform relationships between combinatorial problems. Trans. Am. Math. Soc. 368 (2), 1321–1359 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Dorais, F.G., Hirst, J.L., Shafer, P.: Comparing the strength of diagonally nonrecursive functions in the absence of \({{\Sigma }^{0}_{2}}\) induction. J. Symb. Log. 80(4), 1211–1235 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Downey, R.G., Greenberg, N., Jockusch Jr., C.G., Milans, K.G.: Binary subtrees with few labeled paths. Combinatorica 31(3), 285–303 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Downey, R.G., Hirschfeldt, D.R.: Algorithmic randomness and complexity. Theory and Applications of Computability. Springer, New York (2010)CrossRefzbMATHGoogle Scholar
  26. 26.
    Friedberg, R.: A criterion for completeness of degrees of unsolvability. J. Symb. Logic 22, 159–160 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Gherardi, G., Marcone, A.: How incomputable is the separable Hahn-Banach theorem Notre Dame J. Formal Logic 50(4), 393–425 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Hendtlass, M., Lubarsky, R.: Separating fragments of WLEM, LPO, and MP. J. Symb. Log. 81(4), 12 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Hertling, P.: Unstetigkeitsgrade von Funktionen in der effektiven Analysis. Informatik Berichte 208. FernUniversität Hagen, Hagen (1996). DissertationGoogle Scholar
  30. 30.
    Higuchi, K., Kihara, T.: Inside the Muchnik degrees II: The degree structures induced by the arithmetical hierarchy of countably continuous functions. Ann. Pure Appl. Logic 165(6), 1201–1241 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Higuchi, K., Pauly, A.: The degree structure of Weihrauch reducibility. Log. Methods Comput. Sci. 9(2), 2:02, 17 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Hirschfeldt, D.R.: Slicing the Truth, On the Computable and Reverse Mathematics of Combinatorial Principles, volume 28 of Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore. World Scientific, Singapore (2015)Google Scholar
  33. 33.
    Hoyrup, M., Rojas, C., Weihrauch, K.: Computability of the Radon-Nikodym derivative. Computability 1(1), 3–13 (2012)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Jockusch, C., Stephan, F.: A cohesive set which is not high. Math. Log. Q. 39(4), 515–530 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Jockusch, C., Stephan, F.: Correction to A cohesive set which is not high. Math. Log. Q. 43(4), 569 (1997)CrossRefGoogle Scholar
  36. 36.
    Jockusch Jr., C.G.: Upward closure and cohesive degrees. Israel J. Math. 15, 332–335 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Jockusch Jr., C.G.: Degrees of functions with no fixed points. In: Logic, methodology and philosophy of science, VIII (Moscow, 1987), volume 126 of Stud. Logic Found. Math., pages 191–201, Amsterdam, 1989. North-HollandGoogle Scholar
  38. 38.
    Jockusch Jr., C.G., Lewis, E.M.A: Diagonally non-computable functions and bi-immunity. J. Symb. Log. 78(3), 977–988 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Jockusch Jr., C.G., Soare, R.I.: \({\Pi }^{0}_{1}\) classes and degrees of theories. Trans. Am. Math. Soc. 173, 33–56 (1972)zbMATHGoogle Scholar
  40. 40.
    Kleene, S.C., Post, E.L.: The upper semi-lattice of degrees of recursive unsolvability. Ann. Math. 59(2), 379–407 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Kreuzer, A.P.: The cohesive principle and the Bolzano-Weierstraß principle. Math. Log. Q. 57(3), 292–298 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Kreuzer, A.P.: On the strength of weak compactness. Computability 1(2), 171–179 (2012)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Kučera, A.: Measure, \({{\Pi }^{0}_{1}}\)-classes and complete extensions of PA. In: Recursion theory week (Oberwolfach, 1984), volume 1141 of Lecture Notes in Math., pages 245–259, Berlin, 1985. SpringerGoogle Scholar
  44. 44.
    Kurtz, S.A.: Randomness and Genericity in the Degrees of Unsolvability. University of Illinois at Urbana-Champaign, PhD thesis (1981)Google Scholar
  45. 45.
    Miller, J.S.: Pi-0-1 Classes in Computable Analysis and Topology. PhD thesis, Cornell University, Ithaca, USA (2002)Google Scholar
  46. 46.
    Nies, A.: Computability and Randomness, volume 51 of Oxford Logic Guides. Oxford University Press, New York (2009)CrossRefGoogle Scholar
  47. 47.
    Odifreddi, P.: Classical Recursion Theory volume 125 of Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam (1989)Google Scholar
  48. 48.
    Pauly, A.: How incomputable is finding Nash equilibria J. Universal Comput. Sci. 16(18), 2686–2710 (2010)MathSciNetzbMATHGoogle Scholar
  49. 49.
    Pauly, A.: On the (semi)lattices induced by continuous reducibilities. Math. Log. Q. 56(5), 488–502 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Pauly, A.: Computable Metamathematics and its Application to Game Theory. PhD thesis, University of Cambridge, Computer Laboratory, Clare College, Cambridge (2011)Google Scholar
  51. 51.
    Simpson, S.G.: Degrees of unsolvability: A survey of results. In: Barwise, J. (ed.) Handbook of Mathematical Logic, volume 90 of Studies in Logic and the Foundations of Mathematics, pages 631–652. North-Holland, Amsterdam (1977)Google Scholar
  52. 52.
    Simpson, S.G.: Mass problems and measure-theoretic regularity. Bull. Symb. Log. 15(4), 385–409 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Simpson, S.G.: Degrees of unsolvability: a tutorial. In: Evolving computability, volume 9136 of Lecture Notes in Computer Science, pages 83–94. Springer, Cham (2015)Google Scholar
  54. 54.
    von Stein, T.: Vergleich nicht konstruktiv lösbarer Probleme in der Analysis. PhD thesis, Fachbereich Informatik, FernUniversität Hagen. Diplomarbeit (1989)Google Scholar
  55. 55.
    Tavana, N.R., Weihrauch, K.: Turing machines on represented sets, a model of computation for analysis. Log. Methods Comput. Sci. 7(2), 2:19:21 (2011)MathSciNetCrossRefGoogle Scholar
  56. 56.
    van Lambalgen, M.: The axiomatization of randomness. J. Symb. Log. 55(3), 1143–1167 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Weihrauch, K.: The degrees of discontinuity of some translators between representations of the real numbers. Technical Report TR-92-050. International Computer Science Institute, Berkeley (1992)Google Scholar
  58. 58.
    Weihrauch, K.: The TTE-interpretation of three hierarchies of omniscience principles. Informatik Berichte 130. FernUniversität Hagen, Hagen (1992)Google Scholar
  59. 59.
    Klaus Weihrauch: Computable Analysis. Springer, Berlin (2000)CrossRefzbMATHGoogle Scholar
  60. 60.
    Liang, Y.: Lowness for genericity. Arch. Math. Log. 45(2), 233–238 (2006)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  • Vasco Brattka
    • 1
    • 2
    Email author
  • Matthew Hendtlass
    • 3
  • Alexander P. Kreuzer
    • 4
  1. 1.Faculty of Computer ScienceUniversität der Bundeswehr MünchenNeubibergGermany
  2. 2.Department of Mathematics & Applied MathematicsUniversity of Cape TownRondeboschSouth Africa
  3. 3.School of Mathematics and StatisticsUniversity of CanterburyChristchurchNew Zealand
  4. 4.Department of MathematicsNational University of SingaporeSingaporeSingapore

Personalised recommendations