Hostname: page-component-7dd5485656-glrdx Total loading time: 0 Render date: 2025-10-31T17:49:46.787Z Has data issue: false hasContentIssue false
  • English
  • Français

Benign cost functions and lowness properties

Published online by Cambridge University Press:  12 March 2014

Noam Greenberg  and
André Nies
Noam Greenberg
Affiliation:
School of Mathematics, Statistics and Computer Science, Victoria University of Wellington, Wellington, New Zealand, E-mail: greenberg@msor.vuw.ac.nz
André Nies
Affiliation:
Department of Computer Science, University of Auckland, Auckland, New Zealand, E-mail: andre@cs.auckland.ac.nz
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that the class of strongly jump-traceable c.e. sets can be characterised as those which have sufficiently slow enumerations so they obey a class of well-behaved cost functions, called benign. This characterisation implies the containment of the class of strongly jump-traceable c.e. Turing degrees in a number of lowness classes, in particular the classes of the degrees which lie below incomplete random degrees, indeed all LR-hard random degrees, and all ω-c.e. random degrees. The last result implies recent results of Diamondstone's and Ng's regarding cupping with superlow c.e. degrees and thus gives a use of algorithmic randomness in the study of the c.e. Turing degrees.

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 76 , Issue 1 , March 2011 , pp. 289 - 312
Copyright
Copyright © Association for Symbolic Logic 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

[1]Ambos-Spies, Klaus, Jockusch, Carl G. Jr., Shore, Richard A., and Soare, Robert I., An algebraic decomposition of the recursively enumerable degrees and the coincidence of several degree classes with the promptly simple degrees, Transactions of the American Mathematical Society, vol. 281 (1984), no. 1, pp. 109–128.CrossRefGoogle Scholar
[2]Beigel, Richard, Buhrman, Harry, Fejer, Peter, Fortnow, Lance, Grabowski, Piotr, Longpre, Luc, Muchnik, Andrej, Stephan, Frank, and Torenvliet, Leen, Enumerations of the Kolmogorov function, this Journal, vol. 71 (2006), no. 2, pp. 501–528.Google Scholar
[3]Cholak, Peter, Downey, Rod, and Greenberg, Noam, Strong jump-traceabilty. I. The computably enumerable case, Advances in Mathematics, vol. 217 (2008), no. 5, pp. 2045–2074.CrossRefGoogle Scholar
[4]Cholak, Peter, Groszek, Marcia, and Slaman, Theodore, An almost deep degree, this Journal, vol. 66 (2001), no. 2, pp. 881–901.Google Scholar
[5]Diamondstone, David, Promptness does not imply superlow cuppability, this Journal, vol. 74 (2009), no. 4, pp. 1264–1272.Google Scholar
[6]Downey, Rod G., Hirschfeldt, Denis R., Nies, André, and Stephan, Frank, Trivial Reals, Proceedings of the 7th and 8th Asian Logic Conferences, Singapore University Press, Singapore, 2003, pp. 103–131.CrossRefGoogle Scholar
[7]Figueira, Santiago, Nies, André, and Stephan, Frank, Lowness properties and approximations of the jump, Annals of Pure and Applied Logic, vol. 152 (2008), no. 1–3, pp. 51–66.CrossRefGoogle Scholar
[8]Franklin, Johanna N.Y., Greenberg, Noam, Stephan, Frank, and Wu, Guohua, Reducibilities with tiny use, In preparation.Google Scholar
[9]Greenberg, Noam, Hirschfeldt, Denis R., and Nies, André, Characterizing the strongly jump traceable sets via randomness, To appear.Google Scholar
[10]Hirschfeldt, Denis R., Nies, André, and Stephan, Frank, Using random sets as oracles, Journal of the London Mathematical Society. Second Series, vol. 75 (2007), no. 3, pp. 610–622.CrossRefGoogle Scholar
[11]Ishmukhametov, Shamil, Weak recursive degrees and a problem ofSpector, Recursion theory and complexity (Kazan, 1997), de Gruyter Series in Logic and its Applications, vol. 2, de Gruyter, Berlin, 1999, pp. 81–87.CrossRefGoogle Scholar
[12]Kučera, Antonín and Terwijn, Sebastiaan A., Lowness for the class of random sets, this Journal, vol. 64 (1999), no. 4, pp. 1396–1402.Google Scholar
[13]Miller, Joseph S. and Nies, André, Randomness and computability: open questions, The Bulletin of Symbolic Logic, vol. 12 (2006), no. 3, pp. 390–410.CrossRefGoogle Scholar
[14]Ng, Keng Meng, On strongly jump traceable reals, Annals of Pure and Applied Logic, vol. 154 (2008), no. 1, pp. 51–69.CrossRefGoogle Scholar
[15]Ng, Keng Meng, Beyond strong jump-traceability, Proceedings of the London Mathematical Society, To appear.Google Scholar
[16]Ng, Keng Meng, Almost superdeep degrees, In preparation.Google Scholar
[17]Nies, André, Lowness properties and randomness, Advances in Mathematics, vol. 197 (2005), no. 1, pp. 274–305.CrossRefGoogle Scholar
[18]Nies, André, Reals which compute little, Logic Colloquium '02, Lecture Notes in Logic, vol. 27, Association of Symbolic Logic, La Jolla, CA, 2006, pp. 261–275.Google Scholar
[19]Nies, André, Computability and randomness, Oxford University Press, 2009.CrossRefGoogle Scholar
[20]Nies, André, Being below Demuth random sets, In preparation.Google Scholar
[21]Nies, André, Calculus of cost functions, In preparation.Google Scholar
[22]Simpson, Stephen G., Almost everywhere domination and superhighness, Mathematical Logic Quarterly, vol. 53 (2007), no. 4–5, pp. 462–482.CrossRefGoogle Scholar
[23]Nies, André, Mass problems andalmost everywhere domination, Mathematical Logic Quarterly, vol. 53 (2007), no. 4–5, pp. 483–492.Google Scholar
[24]Terwijn, Sebastiaan A. and Zambella, Domenico, Computational randomness and lowness, this Journal, vol. 66 (2001), no. 3, pp. 1199–1205.Google Scholar