Archive for Mathematical Logic

, Volume 45, Issue 2, pp 179–190 | Cite as

The Medvedev lattice of computably closed sets

Article

Abstract

Simpson introduced the lattice Open image in new window of Π01 classes under Medvedev reducibility. Questions regarding completeness in Open image in new window 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 Open image in new window does not allow an implication operator (i.e. that Open image in new window is not a Heyting algebra). We also discuss properties of the class of PA-complete sets that are relevant in this context.

Keywords or phrases

Π01 classes Medvedev reducibility intuitionistic propositional logic 

Mathematics Subject Classification (2000)

03D30 03B55 03G10 

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References

  1. 1.
    Balbes, R., Dwinger, P.: Distributive lattices. University of Missouri Press, 1974Google Scholar
  2. 2.
    Binns, S.: A splitting theorem for the Medvedev and Muchnik lattices. Mathematical Logic Quarterly 49, 327–335 (2003)Google Scholar
  3. 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. 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. 37–85Google Scholar
  5. 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. 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.)Google Scholar
  7. 7.
    Jockusch, C. G. Jr, McLaughlin, T. G.: Countable retracing functions and Π02 predicates. Pacific Journal of Mathematics 30, 67–93 (1969)Google Scholar
  8. 8.
    Carl J. G., Jr., Robert Soare, I.: Degrees of members of Π01 classes. Pacific Journal of Mathematics 40 (3), 605–616 (1972)Google Scholar
  9. 9.
    Jockusch, C. G. Jr, Soare, R. I.: Π01 classes and degrees of theories, Transactions of the American Mathematical Society 173, 35–56 (1972)Google Scholar
  10. 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. 245–259Google Scholar
  11. 11.
    Medvedev, Yu. T., Degrees of difficulty of the mass problems. Dokl. Akad. Nauk. SSSR 104 (4), 501–504 (1955)Google Scholar
  12. 12.
    Medvedev, Yu. T.: Finite problems. Dokl. Akad. Nauk. SSSR (NS) 142 (5), 1015–1018 (1962)Google Scholar
  13. 13.
    Muchnik, A. A.: On strong and weak reducibility of algorithmic problems. Sibirsk. Math. Zh 4, 1328–1341 (1963)Google Scholar
  14. 14.
    Odifreddi, P. G.: Classical recursion theory Vol. I. Studies in logic and the foundations of mathematics 125, North-Holland, 1989Google Scholar
  15. 15.
    Odifreddi, P. G.: Classical Recursion Theory Vol. II. Studies in logic and the foundations of mathematics 143, North-Holland, 1999Google Scholar
  16. 16.
    Rogers, H, Jr.: Theory of recursive functions and effective computability, McGraw-Hill, 1967Google Scholar
  17. 17.
    Simpson, S. G.: slides for a talk posted on http://www.math.psu.edu/simpson/talks/umn0105, 2003
  18. 18.
    Simpson, S. G.: Π01 sets and models of WKL0 to appear in: Reverse Mathematics 2001. Lecture Notes in Logic, ASLGoogle Scholar
  19. 19.
    Simpson, S. G., Slaman, T. A.: Medvedev degrees of Π01 subsets of 2ω. unpublished manuscriptGoogle Scholar
  20. 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. 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. 22.
    Sorbi, A.: Embedding Brouwer algebras in the Medvedev lattice. Notre Dame Journal of Formal Logic 32 (2), 266–275 (1991)Google Scholar
  23. 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. 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. 289–312Google Scholar
  25. 25.
    Sorbi, A.: personal communication, Siena, July 2004Google Scholar
  26. 26.
    Terwijn, S. A.: Complexity and Randomness. Rendiconti del Seminario Matematico di Torino 62 (1), 1–38 (2004)Google Scholar
  27. 27.
    Terwijn, S. A.: Constructive logic and the Medvedev lattice, Journal of Formal Logic (in press) Notre DameGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Institute of Discrete Mathematics and GeometryTechnical University of ViennaViennaAustria

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