On the Kolmogorov-Chaitin Complexity for short sequences
Résumé
A drawback to Kolmogorov-Chaitin complexity (K) is that it is uncomputable in general, and that limits its range of applicability. Moreover when strings are short, the dependence of K on a particular universal Turing machine U can be arbitrary. In practice one can approximate it by computable compression methods. However, such compression methods do not provide a good approximation for short sequences -- shorter for example than typical compiler lengths. In this paper we will suggest an empirical approach to overcome this difficulty and to obtain a stable definition of the Kolmogorov-Chaitin complexity for short sequences. Additionally, a correlation in terms of distribution frequencies was found across the output of several systems including abstract devices as cellular automata and Turing machines, as well as real-world data sources such as images and human DNA fragments. This could suggest that all them stand a single distribution in accordance with algorithmic probability.