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 \(f'(x)\), where they are defined, are uniformly computable from \(x'\), the Turing jump of the input. It is known that when f is \({\mathcal {C}}^2\), the values of \(f'(x)\) are actually computable from x. We construct a \({\mathcal {C}}^1\) function f so that, almost everywhere, \(f'(x)\ge _T x'\). Although the values \(f'(x)\) at each point x cannot uniformly compute the corresponding jumps \(x'\) of the inputs x almost everywhere for any \({\mathcal {C}}^1\) function f, we produce an example of a \({\mathcal {C}}^1\) function f such that \(f(x)\ge _T \emptyset '\) 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.
Similar content being viewed by others
Notes
Depending on what convention the reader prefers for assigning values at the discontinuities of step functions, this may be taken to hold for all x.
References
Ackerman, N., Freer, C., Roy, D.: On the computability of conditional probability. J. ACM 66, 1 (2010). https://doi.org/10.1145/3321699
Brattka, V., Miller, J.S., Nies, A.: Randomness and differentiability. Trans. Am. Math. Soc. 368(1), 581–605 (2016). https://doi.org/10.1090/tran/6484
Clarkson, J.A.: A property of derivatives. Bull. Am. Math. Soc. 53(2), 124–125 (1947)
Demuth, O.: The differentiability of constructive functions of weakly bounded variation on pseudo numbers. Comment. Math. Univ. Carolin. 16(3), 583–599 (1975). (Russian)
Downey, R.G., Hirschfeldt, D.R.: Algorithmic Randomness and Complexity. Theory and Applications of Computability. Springer (2010). https://doi.org/10.1007/978-0-387-68441-3
Hoyrup, M., Rojas, C.: An application of Martin-Löf randomness to effective probability theory. In: Mathematical Theory and Computational Practice. CiE 2009, Lecture Notes in Computer Science, vol. 5635, pp. 260–269. CiE. Springer, Berlin (2009)
Hoyrup, M., Rojas, C.: Applications of effective probability theory to Martin-Löf randomness. In: 36th International Colloquium on Automata, Languages and Programming, Lecture Notes in Computer Science, vol. 5555, pp. 549–561. ICALP, Springer, Berlin (2009)
Hoyrup, M., Rojas, C., Weihrauch, K.: Computability of the Radon–Nikodym derivative. In: Löwe, B., Normann, D., Soskov, I., Soskova, A. (eds.) Models Comput. Context, pp. 132–141. Springer, Berlin (2011)
Miyabe, K.: Characterization of Kurtz randomness by a differentiation theorem. Theory Comput. Syst. 52(1), 113–132 (2013). https://doi.org/10.1007/s00224-012-9422-3
Nies, A.: Computabil. Randomness. Oxford University Press Inc, New York, NY, USA (2009)
Pathak, N., Rojas, C., Simpson, S.: Schnorr randomness and the Lebesgue differentiation theorem. Proc. Am. Math. Soc. 142(1), 335–349 (2014). https://doi.org/10.1090/S0002-9939-2013-11710-7
Pauly, A., Fouché, W., Davie, G.: Weihrauch-completeness for layerwise computability. Logical Methods Comput. Sci. 14(2), 1 (2018). https://doi.org/10.23638/LMCS-14(2:11)2018
Pour-El, M.B., Richards, J.I.: Computability in Analysis and Physics. Springer, Berlin (1989)
Rute, J.: Topics in Algorithmic Randomness and Computable Analysis. Ph.D. thesis, Carnegie Mellon University (2013)
Rute, J.: On the close interaction between algorithmic randomness and constructive/computable measure theory. arXiv e-prints arXiv:1812.03375 (2018)
Weihrauch, K.: Computable Analysis: An Introduction. Springer, New York (2000)
Acknowledgements
The author wishes to thank the anonymous referee for his/her valuable input.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Data sharing statement
Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
Conflict of interest
The author declares that he has no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
McCarthy, E. Pointwise complexity of the derivative of a computable function . Arch. Math. Logic 60, 981–994 (2021). https://doi.org/10.1007/s00153-021-00769-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00153-021-00769-4
Keywords
- Computable analysis
- Computability theory
- Differentiation
- Halting set
- Algorithmic randomness
- Layerwise computability