On the Kolmogorov complexity of continuous real functionsTools Farjudian, Amin (2013) On the Kolmogorov complexity of continuous real functions. Annals of Pure and Applied Logic, 164 (5). pp. 566576. ISSN 01680072
Official URL: http://dx.doi.org/10.1016/j.apal.2012.11.003
AbstractKolmogorov complexity was originally defined for finitelyrepresentable objects. Later, the definition was extended to real numbers based on the asymptotic behaviour of the sequence of the Kolmogorov complexities of the finitelyrepresentable objectssuch as rational numbersused 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 finitelyrepresentable 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 highgrowth Kolmogorov complexity. An asymptotic bound on the Kolmogorov complexities of total singlevalued computable real functions will be presented as well.
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