On the Kolmogorov Complexity of Continuous Real Functions

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 functions over real numbers, based on the fact that any 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’ 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. Furthermore, we will present a hierarchical structure within the space of real functions based on their Kolmogorov complexities, and discuss where the set of complex oscillations fits within this hierarchy.