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The settling-time reducibility ordering

Published online by Cambridge University Press:  12 March 2014

Barbara F. Csima  and
Richard A. Shore
Barbara F. Csima
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1, Canada, URL: www.math.uwaterloo.ca/~csima, E-mail: csima@math.uwaterloo.ca
Richard A. Shore
Affiliation:
Department of Mathematics, Cornell University, Ithaca NY 14853. USA, URL: http://www.math.cornell.edu/~shore/, E-mail: shore@math.cornell.edu
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Abstract

To each computable enumerable (c.e.) set A with a particular enumeration {As}s∈ω there is associated a settling function mA(x), where mA(x) is the last stage when a number less than or equal to x was enumerated into A. One c.e. set A is settling time dominated by another set B (B >stA) if for every computable function f, for all but finitely many x, mB(x) > f(mA(x)). This settling-time ordering, which is a natural extension to an ordering of the idea of domination, was first introduced by Nabutovsky and Weinberger in [3] and Soare [6]. They desired a sequence of sets descending in this relationship to give results in differential geometry. In this paper we examine properties of the <st ordering. We show that it is not invariant under computable isomorphism, that any countable partial ordering embeds into it. that there are maximal and minimal sets, and that two c.e. sets need not have an inf or sup in the ordering. We also examine a related ordering, the strong settling-time ordering where we require for all computable f and g, for almost all x, mB(x) > f(mA(g(x))).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 72 , Issue 3 , September 2007 , pp. 1055 - 1071
Copyright
Copyright © Association for Symbolic Logic 2007

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References

REFERENCES

[1]Csima, B. F., Applications of computability theory to prime models and differential geometry, Ph.D. thesis, The University of Chicago, 2003.Google Scholar
[2]Csima, B. F. and Soare, R. I., Computability results used in differential geometry, this Journal, vol. 71 (2006), pp. 1394–1410.Google Scholar
[3]Nabutovsky, A. and Weinberger, S., The fractal nature of Riem/Diff I, Geometrica Dedicata, vol. 101 (2003), pp. 1–54.CrossRefGoogle Scholar
[4]Rogers, H. Jr., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.Google Scholar
[5]Soare, R. I., Recursively enumerable sets and degrees: A study of computable functions and computably generated sets, Springer-Verlag, Heidelberg, 1987.CrossRefGoogle Scholar
[6]Soare, R. I., Computability theory and differential geometry, The Bulletin of Symbolic Logic, vol. 10 (2004), pp. 457–486.CrossRefGoogle Scholar
[7]Soare, R. I., Computability theory and applications, Springer-Verlag, Heidelberg, to appear.Google Scholar
[8]Weinberger, S., Computers, rigidity and moduli. The large scale fractal geometry of Reimannian moduli space, M. B. Porter Lectures, Princeton University Press, Princeton, NJ, 2005.CrossRefGoogle Scholar