Skip to main content
×
Home

Comparing the Medvedev and Turing degrees of Π0 1 classes

  • TAKAYUKI KIHARA (a1)
Abstract

Every co-c.e. closed set (Π0 1 class) in Cantor space is represented by a co-c.e. tree. Our aim is to clarify the interaction between the Medvedev and Muchnik degrees of co-c.e. closed subsets of Cantor space and the Turing degrees of their co-c.e. representations. Among other results, we present the following theorems: if v and w are different c.e. degrees, then the collection of the Medvedev (Muchnik) degrees of all Π0 1 classes represented by v and the collection represented by w are also different; the ideals generated from such collections are also different; the collections of the Medvedev and Muchnik degrees of all Π0 1 classes represented by incomplete co-c.e. sets are upward dense; the collection of all Π0 1 classes represented by K-trivial sets is Medvedev-bounded by a single Π0 1 class represented by an incomplete co-c.e. set; and the Π0 1 classes have neither nontrivial infinite suprema nor infima in the Medvedev lattice.

Copyright
References
Hide All
Alfeld C. P. (2007) Non-branching degrees in the Medvedev lattice of Π0 1 classes. Journal of Symbolic Logic 72 (1) 8197.
Barmpalias G., Douglas A. C., Jeffrey B. R. and Weber R. (2009) K-triviality of closed sets and continuous functions. Journal of Logic and Computation 19 (1) 316.
Barmpalias G. and Nies A. (2011) Upper bounds on ideals in the computably enumerable Turing degrees. Annals of Pure and Applied Logic 162 (6) 465473.
Binns S. (2003) A splitting theorem for the Medvedev and Muchnik lattices. Mathematical Logic Quarterly 49 (4) 327335.
Binns S. (2007) Hyperimmunity in inline-graphic $2^\mathbb{N}$ . Notre Dame Journal of Formal Logic 48 (2) 293316.
Cenzer D. A. and Hinman P. G. (2003) Density of the Medvedev lattice of Π0 1 classes. Archive for Mathematical Logic 42 (6) 583600.
Cholak P., Coles R., Downey R. and Herrman E. (2001) Automorphism of the lattice of Π0 1 classes: Perfect thin classes and anc degrees Transactions of the American Mathematical Society 353 48994924.
Jockusch C. G. and Soare R. I. (1972) Π0 1 classes and degrees of theories. Transactions of the American Mathematical Society 173 3356.
Medvedev Y. T. (1955) Degrees of difficulty of the mass problems. Doklady Akademii Nauk SSSR 104 501504 (in Russian).
Melnikov A. G. and Nies A. (2013) K-triviality in computable metric spaces. Proceedings of the American Mathematical Society 141 (8) 28852899.
Muchnik A. A. (1963) On strong and weak reducibility of algorithmic problems. Sibirskii Matematicheskii Zhurnal 4 13281341 (in Russian).
Nies A. (2009) Computability and Randomness, Oxford Logic Guides. Oxford University Press 433.
Simpson S. G. (2005) Mass problems and randomness. Bulletin of Symbolic Logic 11 (1) 127.
Simpson S. G. (2011) Mass problems associated with effectively closed sets. Tohoku Mathematical Journal 63 (4) 489517.
Soare R. I. (1987) Recursively Enumerable Sets and Degrees. Perspectives in Mathematical Logic. Springer, Heidelberg XVIII+437 pp.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematical Structures in Computer Science
  • ISSN: 0960-1295
  • EISSN: 1469-8072
  • URL: /core/journals/mathematical-structures-in-computer-science
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 9 *
Loading metrics...

Abstract views

Total abstract views: 77 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 23rd November 2017. This data will be updated every 24 hours.