Access the full text.
Sign up today, get DeepDyve free for 14 days.
Loading next page...
/lp/springer-journals/pointwise-complexity-of-the-derivative-of-a-computable-function-BQO00pg0ho
References (21)
-
M. Hoyrup, Cristobal Rojas, K. Weihrauch (2011)
Computability of the Radon-Nikodym Derivative -
V. Brattka, Joseph Miller, A. Nies (2011)
Randomness and DifferentiabilityArXiv, abs/1104.4465
-
Rod Downey, D. Hirschfeldt (2010)
Algorithmic Randomness and Complexity -
M. Hoyrup, Cristobal Rojas (2009)
An Application of Martin-Löf Randomness to Effective Probability Theory -
A. Pauly, G. Davie, W. Fouché (2015)
Weihrauch-completeness for layerwise computabilityLog. Methods Comput. Sci., 14
-
(1975)
The differentiability of constructive functions of weakly bounded variation on pseudo numbers -
MB Pour-El, JI Richards (1989)
10.1007/978-3-662-21717-7Computability in Analysis and Physics
-
Kenshi Miyabe (2012)
Characterization of Kurtz Randomness by a Differentiation TheoremTheory of Computing Systems, 52
-
A. Nies (2009)
Computability and randomness -
M Hoyrup, C Rojas, K Weihrauch, B Löwe, D Normann, I Soskov, A Soskova (2011)
Computability of the Radon–Nikodym derivativeModels Comput. Context
-
M. Pour-El, J. Richards (1989)
Computability in analysis and physics -
Noopur Pathak, Cristobal Rojas, S. Simpson (2013)
Schnorr randomness and the Lebesgue differentiation theorem, 142
-
M. Hoyrup, Cristobal Rojas (2009)
Applications of Effective Probability Theory to Martin-Löf Randomness -
Jason Rute (2013)
Topics in algorithmic randomness and computable analysis -
N. Ackerman, Cameron Freer, Daniel Roy (2010)
On the Computability of Conditional ProbabilityJournal of the ACM (JACM), 66
-
Jason Rute (2018)
On the close interaction between algorithmic randomness and constructive/computable measure theoryArXiv, abs/1812.03375
-
O Demuth (1975)
583Comment. Math. Univ. Carolin., 16
-
K. Weihrauch (2014)
Computable Analysis: An IntroductionComputable Analysis
-
(2012)
Algorithmic randomness and complexity. Theory and Applications of Computability -
J. Clarkson (1947)
A property of derivativesBulletin of the American Mathematical Society, 53
-
Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations
- Publisher
- Springer Journals
- Copyright
- Copyright © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2021
- ISSN
- 0933-5846
- eISSN
- 1432-0665
- DOI
- 10.1007/s00153-021-00769-4
- Publisher site
- See Article on Publisher Site
Abstract
We explore the relationship between analytic behavior of a computable real valued function and the computability-theoretic complexity of the individual values of its derivative (the function’s slopes) almost-everywhere. Given a computable function f, the values of its derivative f′(x)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$f'(x)$$\end{document}, where they are defined, are uniformly computable from x′\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$x'$$\end{document}, the Turing jump of the input. It is known that when f is C2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {C}}^2$$\end{document}, the values of f′(x)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$f'(x)$$\end{document} are actually computable from x. We construct a C1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {C}}^1$$\end{document} function f so that, almost everywhere, f′(x)≥Tx′\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$f'(x)\ge _T x'$$\end{document}. Although the values f′(x)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$f'(x)$$\end{document} at each point x cannot uniformly compute the corresponding jumps x′\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$x'$$\end{document} of the inputs x almost everywhere for any C1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {C}}^1$$\end{document} function f, we produce an example of a C1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {C}}^1$$\end{document} function f such that f(x)≥T∅′\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$f(x)\ge _T \emptyset '$$\end{document} uniformly on subsets of arbitrarily large measure, effectively (using the notion of a Schnorr test). We also explore analogous questions for weaker smoothness conditions, such as for f differentiable everywhere, and f differentiable almost everywhere.
Journal
Archive for Mathematical Logic – Springer Journals
Published: Nov 1, 2021
Keywords: Computable analysis; Computability theory; Differentiation; Halting set; Algorithmic randomness; Layerwise computability; 03D30