A flow chart illustrating the <i>Coding Theorem Method</i>, a never-ending algorithm for evaluating the (Kolmogorov) complexity of a (short) string making use of several concepts and results from theoretical computer science, in particular the halting probability, the Busy Beaver problem, Levin's semi-measure and the Coding theorem.

<p>The Busy Beaver values can be used up to 4 states for which they are known, for more than 4 states an informed maximum runtime is used as described in this paper, informed by theoretical <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0096223#pone.0096223-Calude2" target="_blank">[3]</a> and experimental (Busy Beaver values) results. Notice that are the probability values calculated dynamically by running an increasing number of Turing machines. is intended to be an approximation to out of which we build after application of the Coding theorem.</p

All the 2<i>ⁿ</i> strings for <i>n</i> = 7 from <i>D</i>(5) sorted from highest frequency (hence lowest complexity) to lowest frequency (hence highest (random) complexity).

<p>Strings in each row have the same frequency (hence the same Kolmogorov complexity). There are 31 different groups representing the different complexities of the 2⁷ = 128 strings.</p

Cumulative probability of all <i>n</i>-long strings against <i>n</i>.

<p>Cumulative probability of all <i>n</i>-long strings against <i>n</i>.</p

The 20 strings for which .

<p>The 20 strings for which .</p

Proportion of all <i>n</i>-long strings appearing in <i>D</i>(5) against <i>n</i>.

<p>Proportion of all <i>n</i>-long strings appearing in <i>D</i>(5) against <i>n</i>.</p

Distributions of the number of zeros in <i>n</i>-long binary sequences according to a truly random drawing (red, dotted), or a <i>D</i>(5) drawing (black, solid) for length 4 to 12.

<p>Distributions of the number of zeros in <i>n</i>-long binary sequences according to a truly random drawing (red, dotted), or a <i>D</i>(5) drawing (black, solid) for length 4 to 12.</p

Distribution of runtimes from <i>D</i>(2) to <i>D</i>(5).

<p>On the <i>y</i>-axes are the number of Turing machines and on the <i>x</i>-axes the number of steps upon halting. For 5-state Turing machines no Busy Beaver values are known, hence <i>D</i>(5) (Fig. d) was produced by Turing machines with 5 states that ran for at most steps. These plots show, however, that the runtime cutoff for the production of <i>D</i>(5) covers most of the halting Turing machines when taking a sample of machines letting them run for up to steps, hence the missed machines in <i>D</i>(5) must be a negligible number for .</p

The 147 most frequent strings from <i>D</i>(5) (by row).

<p>The first column is a counter to help locate the rank of each string.</p

20 random strings (sorted from lowest to highest complexity values) from the first half of <i>D</i>(5) to which the coding theorem has been applied (extreme right column) to approximate <i>K</i>(<i>s</i>).

<p>20 random strings (sorted from lowest to highest complexity values) from the first half of <i>D</i>(5) to which the coding theorem has been applied (extreme right column) to approximate <i>K</i>(<i>s</i>).</p

<i>R</i>₅ (rank according to <i>D</i>(5)) against <i>R</i>₄.

<p>The grayscale indicates the length of the strings: the darker the point, the shorter the string.</p