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The settling-time reducibility ordering

  • Barbara F. Csima (a1) and Richard A. Shore (a2)

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))).

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[1]Csima, B. F., Applications of computability theory to prime models and differential geometry, Ph.D. thesis, The University of Chicago, 2003.
[2]Csima, B. F. and Soare, R. I., Computability results used in differential geometry, this Journal, vol. 71 (2006), pp. 13941410.
[3]Nabutovsky, A. and Weinberger, S., The fractal nature of Riem/Diff I, Geometrica Dedicata, vol. 101 (2003), pp. 154.
[4]Rogers, H. Jr., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.
[5]Soare, R. I., Recursively enumerable sets and degrees: A study of computable functions and computably generated sets, Springer-Verlag, Heidelberg, 1987.
[6]Soare, R. I., Computability theory and differential geometry, The Bulletin of Symbolic Logic, vol. 10 (2004), pp. 457486.
[7]Soare, R. I., Computability theory and applications, Springer-Verlag, Heidelberg, to appear.
[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.

The settling-time reducibility ordering

  • Barbara F. Csima (a1) and Richard A. Shore (a2)

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