On the Kolmogorov complexity of continuous real functions

Research output: Journal PublicationArticlepeer-review

1 Citation (Scopus)
3 Downloads (Pure)

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.

Original languageEnglish
Pages (from-to)566-576
Number of pages11
JournalAnnals of Pure and Applied Logic
Volume164
Issue number5
DOIs
Publication statusPublished - May 2013

Keywords

  • Algorithmic randomness
  • Computable analysis
  • Domain theory
  • Kolmogorov complexity
  • Measure theory
  • Prevalence

ASJC Scopus subject areas

  • Logic

Fingerprint

Dive into the research topics of 'On the Kolmogorov complexity of continuous real functions'. Together they form a unique fingerprint.

Cite this