On the Kolmogorov complexity of continuous real functions

Farjudian, Amin (2013) On the Kolmogorov complexity of continuous real functions. Annals of Pure and Applied Logic, 164 (5). pp. 566-576. ISSN 0168-0072

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Abstract

Kolmogorov complexity was originally defined for finitely-representable objects. Later, the definition was extended to real numbers based on the asymptotic behaviour of the sequence of the Kolmogorov complexities of the finitely-representable objects-such as rational numbers-used to approximate them.This idea will be taken further here by extending the definition to continuous functions over real numbers, based on the fact that every continuous real function can be represented as the limit of a sequence of finitely-representable enclosures, such as polynomials with rational coefficients.Based on this definition, we will prove that for any growth rate imaginable, there are real functions whose Kolmogorov complexities have higher growth rates. In fact, using the concept of prevalence, we will prove that 'almost every' continuous real function has such a high-growth Kolmogorov complexity. An asymptotic bound on the Kolmogorov complexities of total single-valued computable real functions will be presented as well.

Item Type: Article
Keywords: Kolmogorov complexity; Algorithmic randomness; Computable analysis; Domain theory; Measure theory; Prevalence
Schools/Departments: University of Nottingham Ningbo China > Faculty of Science and Engineering > School of Computer Science
Identification Number: https://doi.org/10.1016/j.apal.2012.11.003
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Depositing User: Yu, Tiffany
Date Deposited: 01 Apr 2019 13:20
Last Modified: 01 Apr 2019 13:20
URI: http://eprints.nottingham.ac.uk/id/eprint/56380

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