Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-22T21:35:17.398Z Has data issue: false hasContentIssue false
  • English
  • Français

Relative randomness and real closed fields

Published online by Cambridge University Press:  12 March 2014

Alexander Raichev
Alexander Raichev*
Affiliation:
Department of Mathematics, University of Wisconsin-Madison, 480 Lincoln DR, Madison, Wisconsin 53706, USA, E-mail: raichev@math.wisc.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that for any real number, the class of real numbers less random than it, in the sense of rK-reducibility, forms a countable real closed subfield of the real ordered field. This generalizes the well-known fact that the computable reals form a real closed field.

With the same technique we show that the class of differences of computably enumerable reals (d.c.e. reals) and the class of computably approximable reals (c.a. reals) form real closed fields. The d.c.e. result was also proved nearly simultaneously and independently by Ng (Keng Meng Ng, Master's Thesis, National University of Singapore, in preparation).

Lastly, we show that the class of d.c.e. reals is properly contained in the class or reals less random than Ω (the halting probability), which in turn is properly contained in the class of c.a. reals, and that neither the first nor last class is a randomness class (as captured by rK-reducibility).

Keywords

relative randomnessreal closed fieldrK-reducibilityd.c.e. realc.a. real
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 70 , Issue 1 , March 2005 , pp. 319 - 330
Copyright
Copyright © Association for Symbolic Logic 2005

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, Klaus, Weihrauch, Klaus, and Zheng, Xizhong, Weakly computable real numbers. Journal of Complexity, vol. 16 (2000), no. 4, pp. 676–690.CrossRefGoogle Scholar
[2]Downey, Rod G., Hirschfeldt, Denis R., and Laforte, Geoff, Randomness and reducibility. Journal of Computer and System Sciences, vol. 68 (2004), no. 1, pp. 96–114.CrossRefGoogle Scholar
[3]HO, Chun-Kuen, Relatively recursive reals and real functions. Theoretical Computer Science, vol. 210 (1999), no. 1, pp. 99–120.CrossRefGoogle Scholar
[4]Lempp, Steffen, Priority arguments in computability theory, model theory, and complexity theory, manuscript, available at http://www.math.wisc.edu/~lempp/papers/prio.pdf.Google Scholar
[5]Li, Ming and Vitányi, Paul, An introduction to Kolmogorov complexity and its applications, second ed., Graduate Texts in Computer Science, Springer-Verlag, New York, 1997.CrossRefGoogle Scholar
[6]Marker, David, Model theory. Graduate Texts in Mathematics, vol. 217, Springer-Verlag, New York, 2002.Google Scholar
[7]Ng, Keng Meng, Master's thesis, National University of Singapore, In preparation.Google Scholar
[8]Pour-El, Marian B. and Richards, J. Ian, Computability in analysis and physics. Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1989.CrossRefGoogle Scholar
[9]Rudin, Walter, Principles of mathematical analysis, third ed., McGraw-Hill Book Co., New York, 1976, International Series in Pure and Applied Mathematics.Google Scholar
[10]Zheng, Xizhong and Weihrauch, Klaus, The arithmetical hierarchy of real numbers. Mathematical Logic Quarterly, vol. 47 (2001), no. 1, pp. 51–65.3.0.CO;2-W>CrossRefGoogle Scholar