Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-25T06:19:06.027Z Has data issue: false hasContentIssue false
  • English
  • Français

The approximation structure of a computably approximable real

Published online by Cambridge University Press:  12 March 2014

George Barmpalias
George Barmpalias*
Affiliation:
School of Mathematics, University of Leeds, Leeds LS2 9JT, U.K., E-mail: georgeb@maths.leeds.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

A new approach for a uniform classification of the computably approximable real numbers is introduced. This is an important class of reals, consisting of the limits of computable sequences of rationals, and it coincides with the 0′-computable reals. Unlike some of the existing approaches, this applies uniformly to all reals in this class: to each computably approximable real x we assign a degree structure, the structure of all possible ways available to approximate x. So the main criterion for such classification is the variety of the effective ways we have to approximate a real number. We exhibit extreme cases of such approximation structures and prove a number of related results.

Keywords

Computably approximable realsapproximation structureimmunity properties
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 68 , Issue 3 , September 2003 , pp. 885 - 922
Copyright
Copyright © Association for Symbolic Logic 2003

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] Barmpalias, G., On 0′-computable reals, Electronic Notes in Theoretical Computer Science, vol. 66 issue 1 (2002), This is the issue containing the Proceedings of the Fifth Workshop on Computability and Complexity in Analysis, edited by Brattka, V., Schröder, M. and Weihrauch, K., and is available electronically at http://www.elsevier.com/gej-ng/31/29/23/120/49/show/Products/notes/ Google Scholar
[2] Downey, R. G., Some computability-theoretical aspects of reals and randomness, Notes based upon a short course of lectures given in the fall of 2000 at the University of Notre Dame, 09 10, 2001 (see http://www.mcs.vuw.ac.nz/research/maths-pubs.shtml).Google Scholar
[3] Downey, R. G., Hirschfeldt, D. R., and LaForte, G., Randomness and reducibility, Mathematical foundations of computer science, August 27–31, 2001, Marianske Lázně, Czech Republic (Sgall, J., Pultr, A., and Kolman, P., editors), Lecture Notes in Computer Science, vol. 2136, Springer, Berlin, 2001, pp. 316–327.CrossRefGoogle Scholar
[4] Dunlop, A. J. and Pour-El, M. B., The degree of unsolvability of a real number, Computability and complexity in analysis (Swansea, 2000) (Blanck, J., Brattka, V., and Hertling, P., editors), Lecture Notes in Computer Science, vol. 2064, Springer, Berlin, 2001, pp. 16–29.CrossRefGoogle Scholar
[5] Ho, C.-K., Relatively recursive reals and real functions, Theoretical Computer Science, vol. 210 (1999), pp. 99–120.Google Scholar
[6] Odifreddi, P., Classical recursion theory, North-Holland, Amsterdam, Oxford, 1989.Google Scholar
[7] Odifreddi, P., Classical recursion theory. Vol. II, North-Holland, Amsterdam, Oxford, 1999.Google Scholar
[8] Rettinger, R. and Zheng, X., Hierarchy of the monotonically computable real numbers, Mathematical foundations of computer science, August 27–31, 2001, Marianske Lázně, Czech Republic (Sgall, J., Pultr, A., and Kolman, P., editors), Lecture Notes in Computer Science, vol. 2136, Springer, Berlin, 2001, pp. 633–644.Google Scholar
[9] Rettinger, R., Monotonically computable real numbers, Mathematical Logic Quarterly, vol. 48 (2002), no. 3, pp. 459–479.Google Scholar
[10] Soare, R. I., Computational complexity, speedable and levelable sets, this Journal, vol. 41 (1977), pp. 545–563.Google Scholar
[11] Zheng, X., Recursive approximability of real numbers, Mathematical Logic Quarterly, vol. 48 (2002), no. S1, pp. 131–156.Google Scholar