Skip to main content
×
Home
    • Aa
    • Aa
  • Get access
    Check if you have access via personal or institutional login
  • Cited by 26
  • Cited by
    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Beros, Achilles A. 2015. A DNC function that computes no effectively bi-immune set. Archive for Mathematical Logic, Vol. 54, Issue. 5-6, p. 521.


    KIHARA, TAKAYUKI 2015. Comparing the Medvedev and Turing degrees of Π0 1 classes. Mathematical Structures in Computer Science, Vol. 25, Issue. 08, p. 1649.


    Simpson, Stephen G. and Stephan, Frank 2015. Cone avoidance and randomness preservation. Annals of Pure and Applied Logic, Vol. 166, Issue. 6, p. 713.


    Higuchi, K. and Kihara, T. 2014. Inside the Muchnik degrees I: Discontinuity, learnability and constructivism. Annals of Pure and Applied Logic, Vol. 165, Issue. 5, p. 1058.


    Higuchi, Kojiro Hudelson, W.M. Phillip Simpson, Stephen G. and Yokoyama, Keita 2014. Propagation of partial randomness. Annals of Pure and Applied Logic, Vol. 165, Issue. 2, p. 742.


    Higuchi, Kojiro and Kihara, Takayuki 2014. On effectively closed sets of effective strong measure zero. Annals of Pure and Applied Logic, Vol. 165, Issue. 9, p. 1445.


    Higuchi, K. and Kihara, T. 2014. Inside the Muchnik degrees II: The degree structures induced by the arithmetical hierarchy of countably continuous functions. Annals of Pure and Applied Logic, Vol. 165, Issue. 6, p. 1201.


    HUDELSON, W. M. PHILLIP 2014. MASS PROBLEMS AND INITIAL SEGMENT COMPLEXITY. The Journal of Symbolic Logic, Vol. 79, Issue. 01, p. 20.


    SIMPSON, STEPHEN G. 2014. Medvedev degrees of two-dimensional subshifts of finite type. Ergodic Theory and Dynamical Systems, Vol. 34, Issue. 02, p. 665.


    Cenzer, Douglas Dashti, Ali Toska, Ferit and Wyman, Sebastian 2012. Computability of Countable Subshifts in One Dimension. Theory of Computing Systems, Vol. 51, Issue. 3, p. 352.


    Higuchi, Kojiro 2012. Effectively closed mass problems and intuitionism. Annals of Pure and Applied Logic, Vol. 163, Issue. 6, p. 693.


    Downey, Rodney G. Greenberg, Noam Jockusch, Carl G. and Milans, Kevin G. 2011. Binary subtrees with few labeled paths. Combinatorica, Vol. 31, Issue. 3, p. 285.


    Cole, Joshua A. and Kihara, Takayuki 2010. The ∀∃-theory of the effectively closed Medvedev degrees is decidable. Archive for Mathematical Logic, Vol. 49, Issue. 1, p. 1.


    Barmpalias, G. Cenzer, D. Remmel, J. B. and Weber, R. 2009. K-Triviality of Closed Sets and Continuous Functions. Journal of Logic and Computation, Vol. 19, Issue. 1, p. 3.


    Cenzer, D. Laforte, G. and Wu, G. 2009. Pseudojumps and Formula Classes. Journal of Logic and Computation, Vol. 19, Issue. 1, p. 77.


    Binns, Stephen 2008. Π1 0 classes with complex elements. The Journal of Symbolic Logic, Vol. 73, Issue. 04, p. 1341.


    Brodhead, Paul and Cenzer, Douglas 2008. Effectively closed sets and enumerations. Archive for Mathematical Logic, Vol. 46, Issue. 7-8, p. 565.


    Cenzer, Douglas and Hinman, Peter G. 2008. Degrees of difficulty of generalized r.e. separating classes. Archive for Mathematical Logic, Vol. 46, Issue. 7-8, p. 629.


    Cole, Joshua A. 2008. Embedding FD(ω) into $${\mathcal{P}_s}$$ densely. Archive for Mathematical Logic, Vol. 46, Issue. 7-8, p. 649.


    Downey, Rod Greenberg, Noam and Miller, Joseph S. 2008. The upward closure of a perfect thin class. Annals of Pure and Applied Logic, Vol. 156, Issue. 1, p. 51.


    ×

Mass Problems and Randomness

  • Stephen G. Simpson (a1)
  • DOI: http://dx.doi.org/10.2178/bsl/1107959497
  • Published online: 15 January 2014
Abstract
Abstract

A mass problem is a set of Turing oracles. If P and Q are mass problems, we say that P is weakly reducible to Q if every member of Q Turing computes a member of P. We say that P is strongly reducible to Q if every member of Q Turing computes a member of P via a fixed Turing functional. The weak degrees and strong degrees are the equivalence classes of mass problems under weak and strong reducibility, respectively. We focus on the countable distributive lattices ω and s of weak and strong degrees of mass problems given by nonempty subsets of 2ω. Using an abstract Gödel/Rosser incompleteness property, we characterize the subsets of 2ω whose associated mass problems are of top degree in ω and s, respectively Let R be the set of Turing oracles which are random in the sense of Martin-Löf, and let r be the weak degree of R. We show that r is a natural intermediate degree within ω. Namely, we characterize r as the unique largest weak degree of a subset of 2ω of positive measure. Within ω we show that r is meet irreducible, does not join to 1, and is incomparable with all weak degrees of nonempty thin perfect subsets of 2ω. In addition, we present other natural examples of intermediate degrees in ω. We relate these examples to reverse mathematics, computational complexity, and Gentzen-style proof theory.

Copyright
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[1]K. Ambos-Spies , G. H. Müller , and G. E. Sacks (editors), Recursion theory week, Lecture Notes in Mathematics, no. 1432, Springer-Verlag, 1990.

[4]Stephen Binns , A splitting theorem for the Medvedev and Muchnik lattices, Mathematical Logic Quarterly, vol. 49 (2003), pp. 327335.

[6]Stephen Binns and Stephen G. Simpson , Embeddings into the Medvedev and Muchnik lattices of classes, Archive for Mathematical Logic, vol. 43 (2004), pp. 399414.

[7]Douglas K. Brown , Mariagnese Giusto , and Stephen G. Simpson , Vitali's theorem and WWKL, Archive for Mathematical Logic, vol. 41 (2002), pp. 191206.

[9]Douglas Cenzer and Peter G. Hinman , Density of the Medvedev lattice of classes, Archive for Mathematical Logic, vol. 42 (2003), pp. 583600.

[11]Peter Cholak , Richard Coles , Rod Downey , and Eberhard Herrmann , Automorphisms of the lattice of classes; perfect thin classes and ANC degrees, Transactions of the American Mathematical Society, vol. 353 (2001), pp. 48994924.

[13]S. B. Cooper , T. A. Slaman , and S. S. Wainer (editors), Computability, enumerability, unsolvability: Directions in recursion theory, London Mathematical Society Lecture Notes, no. 224, Cambridge University Press, 1996.

[19]H.-D. Ebbinghaus , G. H. Müller , and G. E. Sacks (editors), Recursion theory week, Lecture Notes in Mathematics, no. 1141, Springer-Verlag, 1985.

[26]Carl G. Jockusch Jr. and Robert I. Soare , classes and degrees of theories, Transactions of the American Mathematical Society, vol. 173 (1972), pp. 3556.

[34]Antonín Kučera , On relative randomness, Annals of Pure and Applied Logic, vol. 63 (1993), pp. 6167.

[36]Manuel Lerman , Degrees of unsolvability, Perspectives in Mathematical Logic, Springer-Verlag, 1983.

[37]Ming Li and Paul Vitányi , An introduction to Kolmogorov complexity and its applications, 2nd ed., Graduate Texts in Computer Science, Springer-Verlag, 1997.

[39]Per Martin-Löf , The definition of random sequences, Information and Control, vol. 9 (1966), pp. 602619.

[44]Emil L. Post , Recursively enumerable sets of positive integers and their decision problems, Bulletin of the American Mathematical Society, vol. 50 (1944), pp. 284316.

[49]Stephen G. Simpson , Subsystems of second order arithmetic, Perspectives in Mathematical Logic, Springer-Verlag, 1999.

[56]Robert I. Soare , Recursively enumerable sets and degrees, Perspectives in Mathematical Logic, Springer-Verlag, 1987.

[60]Stanley S. Wainer , A classification of the ordinal recursive functions, Archiv für Mathematische Logik und Grundlagenforschung, vol. 13 (1970), pp. 136153.

[61]Xiaokang Yu and Stephen G. Simpson , Measure theory and weak Königs lemma, Archive for Mathematical Logic, vol. 30 (1990), pp. 171180.

Recommend this journal

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

Bulletin of Symbolic Logic
  • ISSN: 1079-8986
  • EISSN: 1943-5894
  • URL: /core/journals/bulletin-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×