Hostname: page-component-788cddb947-wgjn4 Total loading time: 0 Render date: 2024-10-08T19:03:17.770Z Has data issue: false hasContentIssue false
  • English
  • Français

A MINIMAL PAIR IN THE GENERIC DEGREES

Published online by Cambridge University Press:  12 November 2019

DENIS R. HIRSCHFELDT
DENIS R. HIRSCHFELDT*
Affiliation:
DEPARTMENT OF MATHEMATICS THE UNIVERSITY OF CHICAGO CHICAGO, IL, USA E-mail: drh@math.uchicago.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that there is a minimal pair in the nonuniform generic degrees, and hence also in the uniform generic degrees. This fact contrasts with Igusa’s result that there are no minimal pairs for relative generic computability and answers a basic structural question mentioned in several papers in the area.

Keywords

03D30generic computabilitygeneric reducibilityminimal pairs
Type
Articles
Information
The Journal of Symbolic Logic , Volume 85 , Issue 1 , March 2020 , pp. 531 - 537
Copyright
Copyright © The Association for Symbolic Logic 2019 

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

Astor, Eric. P., Hirschfeldt, Denis. R.,, and Jockusch, Carl. G. Jr., Dense computability, upper cones, and minimal pairs,. Computability, vol. 8 (2019), no. 2, pp. 155–177.CrossRefGoogle Scholar
Downey, Rodney. G. and Hirschfeldt, Denis. R., Algorithmic Randomness and Complexity, Theory and Applications of Computability, Springer, New York, 2010.CrossRefGoogle Scholar
Downey, Rodney. G., Jockusch, Carl. G. Jr., and Schupp, Paul E. Jr. and Schupp, P. E., Asymptotic density and computably enumerable sets,. Journal of Mathematical Logic, vol. 13 (2013), no. 2, pp. 1350005, 43.CrossRefGoogle Scholar
Dzhafarov, Damir. D. and Igusa, Gregory., Notions of robust information coding,. Computability, vol. 6 (2017), no. 2, pp. 105–124.CrossRefGoogle Scholar
Hirschfeldt, Denis. R., Some questions in computable mathematics, Computability and Complexity (Day, A., Fellows, M., Greenberg, N., Khoussainov, B., Melnikov, A., and Rosamond, F., editors), Sci.Lecture Notes in Computer Science, vol. 10010, Springer, Cham, 2017, pp. 22–55.Google Scholar
Hirschfeldt, Denis. R., Jockusch, Carl. G. Jr., Kuyper, R., and Schupp, P.E., Coarse reducibility and algorithmic randomness, this Journal, vol. 81 (2016), no. 3, pp. 1028–1046.Google Scholar
Igusa, Gregory., Nonexistence of minimal pairs for generic computability, this Journal, vol. 78 (2013), no. 2, pp. 511–522.Google Scholar
Igusa, Gregory., The generic degrees of density-1 sets, and a characterization of the hyperarithmetic reals, this Journal, vol. 80 (2015), no. 4, pp. 1290–1314.Google Scholar
Jockusch, C. G. Jr. and Schupp, P. E., Generic computability, Turing degrees, and asymptotic density,. Journal of the London Mathematical Society. Second Series, vol. 85 (2012), no. 2, pp. 472–490.Google Scholar
Jockusch, C. G. Jr. and Schupp, P. E., Asymptotic density and the theory of computability: A partial survey, Computability and Complexity (Day, A., Fellows, M., Greenberg, N., Khoussainov, B., Melnikov, A., and Rosamond, F., editors), Lecture Notes in Computer Science, vol. 10010, Springer, Cham, 2017, pp. 501–520.CrossRefGoogle Scholar
Kapovich, Ilya., Myasnikov, Alexei., Schupp, Paul., and Shpilrain, Vladimir., Generic-case complexity, decision problems in group theory, and random walks,. Journal of Algebra, vol. 264 (2003), no. 2, pp. 665–694.CrossRefGoogle Scholar
Lynch, Nancy., Approximations to the halting problem,. Journal of Computer and System Sciences, vol. 9 (1974), pp. 143–150.CrossRefGoogle Scholar
Meyer, Albert. R., An open problem on creative sets,. Recursive Function Theory Newsletter, vol. 4 (1973), pp. 15–16.Google Scholar
Terwijn, Sebastiaan. A., Computability and measure, Ph.D. dissertation, University of Amsterdam, 1998.Google Scholar