Skip to main content
Springer Nature Link
Log in

The Medvedev lattice of computably closed sets

  • Published:
Archive for Mathematical Logic Aims and scope Submit manuscript

Abstract

Simpson introduced the lattice of Π01 classes under Medvedev reducibility. Questions regarding completeness in are related to questions about measure and randomness. We present a solution to a question of Simpson about Medvedev degrees of Π01 classes of positive measure that was independently solved by Simpson and Slaman. We then proceed to discuss connections to constructive logic. In particular we show that the dual of does not allow an implication operator (i.e. that is not a Heyting algebra). We also discuss properties of the class of PA-complete sets that are relevant in this context.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Balbes, R., Dwinger, P.: Distributive lattices. University of Missouri Press, 1974

  2. Binns, S.: A splitting theorem for the Medvedev and Muchnik lattices. Mathematical Logic Quarterly 49, 327–335 (2003)

    Google Scholar 

  3. Binns, S, Simpson, S. G.: Embeddings into the Medvedev and Muchnik lattices of Π01 classes. Archive for Mathematical Logic 43, 399–414 (2004)

    Google Scholar 

  4. Cenzer, D.: Π01 classes in computability theory. Handbook of computability theory, Studies in Logic and the Foundations of Mathematics 140, North-Holland, Amsterdam, 1999 pp. 37–85

  5. Cenzer, D., Hinman, P.: Density of the Medvedev lattice of Π01 classes, Archive for Mathematical Logic 42 (6), 583–600 (2003)

    Google Scholar 

  6. Jankov, A.V.: Calculus of the weak law of the excluded middle. Izv. Akad. Nauk SSSR Ser. Mat. 32, 1044–1051 (1968) (In Russian.)

  7. Jockusch, C. G. Jr, McLaughlin, T. G.: Countable retracing functions and Π02 predicates. Pacific Journal of Mathematics 30, 67–93 (1969)

  8. Carl J. G., Jr., Robert Soare, I.: Degrees of members of Π01 classes. Pacific Journal of Mathematics 40 (3), 605–616 (1972)

  9. Jockusch, C. G. Jr, Soare, R. I.: Π01 classes and degrees of theories, Transactions of the American Mathematical Society 173, 35–56 (1972)

  10. Kučera, A.: Measure, Π10-classes and complete extensions of PA. In: H.-D. Ebbinghaus, G. H. Müller, and G. E. Sacks (eds), Recursion theory week, Lect. Notes in Math. 1141, Springer-Verlag, 1985, pp. 245–259

  11. Medvedev, Yu. T., Degrees of difficulty of the mass problems. Dokl. Akad. Nauk. SSSR 104 (4), 501–504 (1955)

    Google Scholar 

  12. Medvedev, Yu. T.: Finite problems. Dokl. Akad. Nauk. SSSR (NS) 142 (5), 1015–1018 (1962)

  13. Muchnik, A. A.: On strong and weak reducibility of algorithmic problems. Sibirsk. Math. Zh 4, 1328–1341 (1963)

    Google Scholar 

  14. Odifreddi, P. G.: Classical recursion theory Vol. I. Studies in logic and the foundations of mathematics 125, North-Holland, 1989

  15. Odifreddi, P. G.: Classical Recursion Theory Vol. II. Studies in logic and the foundations of mathematics 143, North-Holland, 1999

  16. Rogers, H, Jr.: Theory of recursive functions and effective computability, McGraw-Hill, 1967

  17. Simpson, S. G.: slides for a talk posted on http://www.math.psu.edu/simpson/talks/umn0105, 2003

  18. Simpson, S. G.: Π01 sets and models of WKL0 to appear in: Reverse Mathematics 2001. Lecture Notes in Logic, ASL

  19. Simpson, S. G., Slaman, T. A.: Medvedev degrees of Π01 subsets of 2ω. unpublished manuscript

  20. Skvortsova, E. Z.: A faithful interpretation of the intuitionistic propositional calculus by means of an initial segment of the Medvedev lattice, Sibirsk. Math. Zh. 29 (1), 171–178 (1988) (In Russian.)

    Google Scholar 

  21. Sorbi, A.: Some remarks on the algebraic structure of the Medvedev lattice. Journal of Symbolic Logic 55 (2), 831–853 (1990)

    Google Scholar 

  22. Sorbi, A.: Embedding Brouwer algebras in the Medvedev lattice. Notre Dame Journal of Formal Logic 32 (2), 266–275 (1991)

    Google Scholar 

  23. Sorbi, A.: Some quotient lattices of the Medvedev lattice. Zeitschrift für mathematische Logik und Grundlagen der Mathematik 37, 167–182 (1991)

    Google Scholar 

  24. Sorbi, A.: The Medvedev lattice of degrees of difficulty. In: S. B. Cooper, T. A. Slaman, and S.S. Wainer (eds.), Computability, Enumerability, Unsolvability: Directions in Recursion Theory, London Mathematical Society Lecture Notes 224, Cambridge University Press, 1996, pp. 289–312

  25. Sorbi, A.: personal communication, Siena, July 2004

  26. Terwijn, S. A.: Complexity and Randomness. Rendiconti del Seminario Matematico di Torino 62 (1), 1–38 (2004)

  27. Terwijn, S. A.: Constructive logic and the Medvedev lattice, Journal of Formal Logic (in press) Notre Dame

Download references

Author information

Authors and Affiliations

  1. Institute of Discrete Mathematics and Geometry, Technical University of Vienna, Wiedner Hauptstrasse 8–10/E104, 1040, Vienna, Austria

    Sebastiaan A. Terwijn

Authors
  1. Sebastiaan A. Terwijn
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sebastiaan A. Terwijn.

Additional information

Supported by the Austrian Research Fund (Lise Meitner grant M699-N05).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Terwijn, S. The Medvedev lattice of computably closed sets. Arch. Math. Logic 45, 179–190 (2006). https://doi.org/10.1007/s00153-005-0278-y

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00153-005-0278-y

Keywords or phrases

Mathematics Subject Classification (2000)