Calculating Kolmogorov complexity from the output frequency distributions of small Turing machines

Abstract

Drawing on various notions from theoretical computer science, we present a novel numerical approach, motivated by the notion of algorithmic probability, to the problem of approximating the Kolmogorov-Chaitin complexity of short strings. The method is an alternative to the traditional lossless compression algorithms, which it may complement, the two being serviceable for different string lengths. We provide a thorough analysis for allP11 n~1 2n binary strings of length nv12 and for most strings of length 12ƒnƒ16 by running all *2:5|1013 Turing machines with 5 states and 2 symbols (8|229 with reduction techniques) using the most standard formalism of Turing machines, used in for example the Busy Beaver problem. We address the question of stability and error estimation, the sensitivity of the continued application of the method for wider coverage and better accuracy, and provide statistical evidence suggesting robustness. As with compression algorithms, this work promises to deliver a range of applications, and to provide insight into the question of complexity calculation of finite (and short) strings. Additional material can be found at the Algorithmic Nature Group website at http:// www.algorithmicnature.org. An Online Algorithmic Complexity Calculator implementing this technique and making the data available to the research community is accessible at http://www.complexitycalculator.com.