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A LIGHTFACE ANALYSIS OF THE DIFFERENTIABILITY RANK

  • LINDA BROWN WESTRICK (a1)
Abstract

We examine the computable part of the differentiability hierarchy defined by Kechris and Woodin. In that hierarchy, the rank of a differentiable function is an ordinal less than ${\omega _1}$ which measures how complex it is to verify differentiability for that function. We show that for each recursive ordinal $\alpha > 0$ , the set of Turing indices of $C[0,1]$ functions that are differentiable with rank at most α is ${{\rm{\Pi }}_{2\alpha + 1}}$ -complete. This result is expressed in the notation of Ash and Knight.

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[1] Ash, C. J. and Knight, J., Computable structures and the hyperarithmetical hierarchy, Studies in Logic and the Foundations of Mathematics, vol. 144, North-Holland Publishing Co., Amsterdam, 2000. MR 1767842 (2001k:03090)
[2] Cenzer, Douglas and Remmel, Jeffrey B., Index sets for computable differential equations. Mathematical Logic Quarterly, vol. 50 (2004), no. 4–5, pp. 329344. MR 2090381 (2005h:03083)
[3] Greenberg, Noam, Montalbán, Antonio, and Slaman, Theodore A., The Slaman-Wehner theorem in higher recursion theory. Proceedings of the American Mathematical Society, vol. 139 (2011), no. 5, pp. 18651869. MR 2763773 (2012c:03119)
[4] Grzegorczyk, A., On the definitions of computable real continuous functions. Fundamenta Mathematicae, vol. 44 (1957), pp. 6171. MR 0089809 (19,723c)
[5] Ki, Haseo, On the Denjoy rank, the Kechris-Woodin rank and the Zalcwasser rank. Transactions of the American Mathematical Society, vol. 349 (1997), no. 7, pp. 28452870. MR 1390042 (97i:04001)
[6] Kechris, Alexander S. and Hugh Woodin, W., Ranks of differentiable functions. Mathematika, vol. 33 (1986), no. 2, pp. 252278. MR 882498 (88d:03097)
[7] Lacombe, Daniel, Extension de la notion de fonction récursive aux fonctions d’une ou plusieurs variables réelles. II, III. Comptes Rendus de l’Acad[start]é[end]mie des Sciences, vol. 241 (1955), pp. 1314, 151–153. MR 0072080 (17,225e)
[8] Lempp, Steffen, Hyperarithmetical index sets in recursion theory. Transactions of the American Mathematical Society, vol. 303 (1987), no. 2, pp. 559583.
[9] Mazurkiewicz, S., Über die Menge der differenzierbaren Funktionen.Fundamenta Mathematicae, vol. 27 (1936), pp. 244249.
[10] Myhill, J., A recursive function, defined on a compact interval and having a continuous derivative that is not recursive. Michigan Mathematical Journal, vol. 18 (1971), pp. 9798. MR 0280373 (43 #6093)
[11] Pour-El, Marian B. and Ian Richards, J., Computability in analysis and physics , Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1989. MR 1005942 (90k:03062)
[12] Sacks, Gerald E., Higher recursion theory , Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1990. MR 1080970 (92a:03062)
[13] Simpson, Stephen G., Subsystems of second order arithmetic, second ed., Perspectives in Logic, Cambridge University Press, Cambridge, 2009. MR 2517689 (2010e:03073)
[14] Soare, Robert I., Recursively enumerable sets and degrees , Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1987, A study of computable functions and computably generated sets. MR 882921 (88m:03003)
[15] Spector, Clifford, Recursive well-orderings, this Journal, vol. 20 (1955), pp. 151–163. MR 0074347 (17,570b)
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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