Bounds on the Kolmogorov Complexity Function for Infinite Words

Abstract

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 Π 1 -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. *

The Maximal Complexity of Quasiperiodic Infinite Words

“…Since the ω-languages P ω q are regular ones, the results of [22] show that there are ω-words ξ ∈ P ω q whose Kolmogorov complexity achieves their subword complexity. Moreover, as P ω q = q • R ω q where R ω q is a finite prefix code, the results of [22,26,27] give more detailed bounds for most complex quasiperiodic ω-words w.r.t. several notions of Kolmogorov complexity [28].…”