Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-18T07:55:07.688Z Has data issue: false hasContentIssue false
  • English
  • Français

Promptness does not imply superlow cuppability

Published online by Cambridge University Press:  12 March 2014

David Diamondstone
David Diamondstone*
Affiliation:
University of Chicago, Department of Mathematics, 5734 S. University Avenue Chicago, Illinois 60637, USA, E-mail: ded@math.uchicago.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

A classical theorem in computability is that every promptly simple set can be cupped in the Turing degrees to some complete set by a low c.e. set. A related question due to A. Nies is whether every promptly simple set can be cupped by a superlow c.e. set, i.e. one whose Turing jump is truth-table reducible to the halting problem ∅′. A negative answer to this question is provided by giving an explicit construction of a promptly simple set that is not superlow cuppable. This problem relates to effective randomness and various lowness notions.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 74 , Issue 4 , December 2009 , pp. 1264 - 1272
Copyright
Copyright © Association for Symbolic Logic 2009

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.)

References

REFERENCES

[1]Ambos-Spies, K., Jockusch, C. G. Jr., Shore, R. A., and Soare, R. 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), pp. 109–128.CrossRefGoogle Scholar
[2]Downey, R., Greenberg, N., Miller, J., and Weber, R., Prompt simplicity, array computability and cupping, Computational prospects of infinity (Chong, , Feng, , Slaman, , Woodin, , and Yang, , editors), Lecture Notes Series, vol. 15, World Scientific, Institute for Mathematical Sciences. National University of Singapore, 2008, pp. 59–78.CrossRefGoogle Scholar
[3]Downey, R., Greenberg, N., and Weber, R.. Totally ω-computably enumerable degrees I: bounding critical triples, Journal of Mathematical Logic, vol. 7 (2007), pp. 145–171.CrossRefGoogle Scholar
[4]Downey, R. and Hirschfeldt, D. R., Algorithmic randomness and complexity, Springer-Verlag, Berlin, to appear.Google Scholar
[5]Downey, R., Hirschfeldt, D. R., Nies, A., and Terwijn, S. A., Calibrating randomness. The Bulletin of Symbolic Logic, vol. 12 (2006), pp. 411–491.CrossRefGoogle Scholar
[6]Jockusch, C. Jr. and Soare, R. I., Π10 classes and degrees of theories. Transactions of the American Mathematical Society, vol. 173 (1972), pp. 33–56.Google Scholar
[7]Kučera, A. and Slaman, T. A., Low upper bounds of ideals, to appear.Google Scholar
[8]Mohrherr, Jeanleah, A refinement of low n and high n for the r.e. degrees, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 32 (1986), pp. 5–12.CrossRefGoogle Scholar
[9]Nies, André, Reals which compute little, Proceedings of logic colloquium 2002, Lecture Notes in Logic, vol. 27, 2002, pp. 261–275.Google Scholar
[10]Nies, André, Lowness properties and randomness, Advances in Mathenatics, vol. 197 (2005), pp. 274–305.CrossRefGoogle Scholar
[11]Nies, André, Computability and randomness, Clarendon Press, Oxford, to appear.CrossRefGoogle Scholar
[12]Soare, Robert I., Recursively enumerable sets and degrees, Springer-Verlag, Heidelberg, 1987.CrossRefGoogle Scholar
[13]Soare, Robert I., Computability theory and applications, Springer-Verlag, Heidelberg, to appear.Google Scholar