Archive for Mathematical Logic

, Volume 50, Issue 3–4, pp 319–340 | Cite as

Topological aspects of the Medvedev lattice

  • Andrew E. M. Lewis
  • Richard A. Shore
  • Andrea Sorbi
Original Article

Abstract

We study the Medvedev degrees of mass problems with distinguished topological properties, such as denseness, closedness, or discreteness. We investigate the sublattices generated by these degrees; the prime ideal generated by the dense degrees and its complement, a prime filter; the filter generated by the nonzero closed degrees and the filter generated by the nonzero discrete degrees. We give a complete picture of the relationships of inclusion holding between these sublattices, these filters, and this ideal. We show that the sublattice of the closed Medvedev degrees is not a Brouwer algebra. We investigate the dense degrees of mass problems that are closed under Turing equivalence, and we prove that the dense degrees form an automorphism base for the Medvedev lattice. The results hold for both the Medvedev lattice on the Baire space and the Medvedev lattice on the Cantor space.

Keywords

Medvedev reducibility Baire space Cantor space 

Mathematics Subject Classification (2000)

03D30 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Balbes R., Dwinger P.: Distributive Lattices. University of Missouri Press, Columbia (1974)MATHGoogle Scholar
  2. 2.
    Bianchini C., Sorbi A.: A note on closed degrees of difficulty of the Medvedev lattice. Math. Log. Q. 42, 127–133 (1996)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Cooper S.B.: Computability Theory. Chapman & Hall/CRC Mathematics, Boca Raton, London, New York, Washington, DC (2003)Google Scholar
  4. 4.
    Dyment E.Z.: Certain properties of the Medvedev lattice. Mathematics of the USSR Sbornik, 30, 321–340 (1976) English TranslationCrossRefGoogle Scholar
  5. 5.
    Ershov Y.L.: Note on a problem of Rogers. Algebra Log. 8, 285 (1969)CrossRefGoogle Scholar
  6. 6.
    Kechris A.S.: Classical Descriptive Set Theory, volume 156 of Graduate Texts in Mathematics. Springer–Verlag, Heidelberg (1995)Google Scholar
  7. 7.
    Lerman M.: Degrees of Unsolvability. Perspectives in Mathematical Logic. Springer–Verlag, Heidelberg (1983)Google Scholar
  8. 8.
    Medevdev Y.T.: Degrees of difficulty of the mass problems. Dokl. Nauk. SSSR 104(4), 501–504 (1955)Google Scholar
  9. 9.
    Muchnik A.A.: On strong and weak reducibility of algorithmic problems. Sibirskii Matematicheskii Zhurnal 4, 1328–1341 (1963) RussianMATHGoogle Scholar
  10. 10.
    Platek R.A.: A note on the cardinality of the Medvedev lattice. Proc. Am. Math. Soc. 25, 917 (1970)MATHMathSciNetGoogle Scholar
  11. 11.
    Rogers H. Jr: Theory of Recursive Functions and Effective Computability. McGraw-Hill, New York (1967)MATHGoogle Scholar
  12. 12.
    Rozinas, M.: The semilattice of e-degrees. In: Recursive Functions, pp. 71–84. Ivanov. Gos. Univ., Ivanovo (1978). (Russian) MR 82i:03057Google Scholar
  13. 13.
    Rozinas M.G.: Partial degrees of immune and hyperimmune sets. Siberian Math. J. 19, 613–616 (1978)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Shafer, P.: Characterizing the join-irreducible Medvedev degrees. To appear in Notre Dame J. Formal Log.Google Scholar
  15. 15.
    Simpson S.G.: Mass problems and intuitionism. Notre Dame J. Formal Log. 49(2), 127–136 (2008)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Skvortsova E.Z.: Faithful interpretation of the intitionistic prpositional calculus by an initial segment of the Medvedev lattice. Sibirsk. Mat. Zh. 29(1), 171–178 (1988) RussianMATHMathSciNetGoogle Scholar
  17. 17.
    Soare R.I.: Recursively Enumerable Sets and Degrees. Perspectives in Mathematical Logic, Omega Series. Springer–Verlag, Heidelberg (1987)Google Scholar
  18. 18.
    Sorbi A.: On some filters and ideals of the Medvedev lattice. Arch. Math. Log. 30, 29–48 (1990)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Sorbi A.: Embedding Brouwer algebras in the Medvedev lattice. Notre Dame J. Formal Log. 32(2), 266–275 (1991)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Sorbi A.: Some quotient lattices of the Medvedev lattice. Z. Math. Logik Grundlag. Math. 37, 167–182 (1991)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Sorbi A., Terwijn S.: Intermediate logics and factors of the Medvedev lattice. Ann. Pure Appl. Log. 155(2), 69–86 (2008)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Terwijn S.A.: The Medvedev lattice of computably closed sets. Arch. Math. Log. 45, 179–190 (2006)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Terwijn S.A.: On the structure of the Medvedev lattice. J. Symbolic Log. 73(2), 543–558 (2008)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Andrew E. M. Lewis
    • 1
  • Richard A. Shore
    • 2
  • Andrea Sorbi
    • 3
  1. 1.Department of Pure MathematicsUniversity of LeedsLeedsUK
  2. 2.Department of MathematicsCornell UniversityIthacaUSA
  3. 3.Department of Mathematics and Computer Science “R. Magari”University of SienaSienaItaly

Personalised recommendations