The evaluation of the complexity of finite sequences is key in many areas of science. For example, the notions of structure, simplicity and randomness are common currency in biological systems epitomized by a sequence of fundamental nature and utmost importance: the DNA. Nevertheless, researchers have for a long time avoided any practical use of the current accepted mathematical theory of randomness, mainly because it has been considered to be useless in practice . Despite this belief, related notions such as lossless uncompressibility tests have proven relative success, in areas such as sequence pattern detection  and have motivated distance measures and classification methods  in several areas (see  for a survey), to mention but two examples among many others of even more practical use. The method presented in this paper aims to provide sound directions to explore the feasibility and stability of the evaluation of the complexity of strings by means different to that of lossless compressibility, particularly useful for short strings. The authors known of only two similar attempts to compute the uncomputable, one related to the estimation of a Chaitin Omega number , and of another seminal related measure of complexity, Bennett's Logical Depth , . This paper provides an approximation to the output frequency distribution of all Turing machines with 5 states and 2 symbols which in turn allow us to apply a central theorem in the theory of algorithmic complexity based in the notion of algorithmic probability (also known as Solomonoff's theory of inductive inference) that relates frequency of production of a string and its Kolmogorov complexity hence providing, upon application of the theorem, numerical estimations of Kolmogorov complexity by a method different to lossless compression algorithms.
Soler-Toscano F, Zenil H, Delahaye J-P, Gauvrit N (2014) Calculating Kolmogorov Complexity from the Output Frequency Distributions of Small Turing Machines. PLoS ONE 9(5): e96223. http://dx.doi.org/10.1371/journal.pone.0096223