Bounds on the Kolmogorov complexity function for infinite words

The Kolmogorov complexity function of an infinite word ξ maps a natural number to the complexity K(ξ n) of the n-length prefix of ξ. We investigate the maximally achievable complexity function if ξ is taken from a constructively describable set of infinite words. Here we are interested in linear upper bounds where the slope is the Hausdorff dimension of the set. As sets we consider Π1-definable sets obtained by dilution and sets obtained from constructively describable infinite iterated function systems. In these cases, for a priori and monotone complexity, the upper bound coincides (up to an additive constant) with the lower bound, thus verifying the existence of oscillation-free maximally complex infinite words. ∗email: [email protected] ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Report No. 13 (2016)