# On the Kolmogorov-Chaitin Complexity for short sequences

## TL;DR

This paper proposes an empirical method to approximate Kolmogorov-Chaitin complexity for short sequences, addressing noncomputability and variability issues, and finds correlations between different computational models' outputs.

## Contribution

It introduces a stable, empirical approach for estimating complexity of short strings and compares output distributions of cellular automata and Turing machines.

## Key findings

- Empirical complexity estimates are more stable for short sequences.
- Distribution frequencies correlate across different computational models.

## Abstract

A drawback of Kolmogorov-Chaitin complexity (K) as a function from s to the shortest program producing s is its noncomputability which 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 always provide meaningful approximations--for strings shorter, for example, than typical compiler lengths. In this paper we 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 two models of abstract machines, namely unidimensional cellular automata and deterministic Turing machine.