Theory of Computing Systems

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

On the Uniform Computational Content of Computability Theory

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

Abstract

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.

Keywords

Computable analysis Weihrauch lattice Computability theory 

Notes

Acknowledgments

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.

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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  • Vasco Brattka
    • 1
    • 2
  • 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

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