Two-dimensional Kolmogorov complexity and an empirical validation of the Coding theorem method by compressibility

Uncomputability and instability of K

A technical inconvenience of K as a function taking s to the length of the shortest program that produces s is its uncomputability (Chaitin, 1969). In other words, there is no program which takes a string s as input and produces the integer K(s) as output. This is usually considered a major problem, but one ought to expect a universal measure of complexity to have such a property. On the other hand, K is more precisely upper semi-computable, meaning that one can find upper bounds, as we will do by applying a technique based on another semi-computable measure to be presented in the ‘Solomonoff–Levin Algorithmic Probability’.

The invariance theorem guarantees that complexity values will only diverge by a constant c (e.g., the length of a compiler, a translation program between U₁ and U₂) and that they will converge at the limit.

Invariance Theorem (Calude, 2002; Li & Vitányi, 2009): If U₁ and U₂ are two universal Turing machines and K_U1(s) and K_U2(s) the algorithmic complexity of s for U₁ and U₂, there exists a constant c such that for all s: (2)${\displaystyle |K_{U_1}\left(s\right)-K_{U_2}\left(s\right)|<c.}$

Hence the longer the string, the less important c is (i.e., the choice of programming language or universal Turing machine). However, in practice c can be arbitrarily large because the invariance theorem tells nothing about the rate of convergence between K_U1 and K_U2 for a string s of increasing length, thus having an important impact on short strings.

Deterministic 2-dimensional Turing machines (Turmites)

Turmites or 2-dimensional (2D) Turing machines run not on a 1-dimensional tape but in a 2-dimensional unbounded grid or array. At each step they can move in four different directions (up, down, left, right) or stop. Transitions have the format {n₁, m₁} → {n₂, m₂, d}, meaning that when the machine is in state n₁ and reads symbols m₁, it writes m₂, changes to state n₂ and moves to a contiguous cell following direction d. If n₂ is the halting state then d is stop. In other cases, d can be any of the other four directions.

Let (n, m)_2D be the set of Turing machines with n states and m symbols. These machines have nm entries in the transition table, and for each entry {n₁, m₁} there are 4nm + m possible instructions, that is, m different halting instructions (writing one of the different symbols) and 4nm non-halting instructions (4 directions, n states and m different symbols). So the number of machines in (n, m)_2D is (4nm + m)^nm. It is possible to enumerate all these machines in the same way as 1D Turing machines (e.g., as has been done in Wolfram (2002) and Joosten (2012)). We can assign one number to each entry in the transition table. These numbers go from 0 to 4nm + m − 1 (given that there are 4nm + m different instructions). The numbers corresponding to all entries in the transition table (irrespective of the convention followed in sorting them) form a number with nm digits in base 4nm + m. Then, the translation of a transition table to a natural number and vice versa can be done through elementary arithmetical operations.

We take as output for a 2D Turing machine the minimal array that includes all cells visited by the machine. Note that this probably includes cells that have not been visited, but it is the more natural way of producing output with some regular format and at the same time reducing the set of different outputs.

Figure 1 shows an example of the transition table of a Turing machine in (3, 2)_2D and its execution over a ‘0’-filled grid. We show the portion of the grid that is returned as the output array. Two of the six cells have not been visited by the machine.

Setting the runtime

The Busy Beaver runtime value for (4, 2) is 107 steps upon halting. But no equivalent Busy Beavers are known for 2-dimensional Turing machines (although variations of Turmite’s Busy Beaver functions have been proposed (Pegg, 2013)). So to set the runtime in our experiment we generated a sample of 334 × 10⁸ random machines in the reduced enumeration. We used a runtime of 2,000 steps for the runtime sample, this is 10.6% of the machines in the reduced enumeration for (4, 2)_2D, but 1,500 steps for running all (4, 2)₂D. These machines were generated instruction by instruction. As we have explained above, it is possible to assign a natural number to every instruction. So to generate a random machine in the reduced enumeration for (n, m)_2D we produce a random number from 0 to m(n − 1) − 1 for the initial transition and from 0 to 4nm + m − 1 for the other nm − 1 transitions. We used the implementation of the Mersenne Twister in the Boost C++ library. The output of this sample was the distribution of the runtime of the halting machines.

Figure 1 shows the probability that a random halting machine will halt in at most the number of steps indicated on the horizontal axis. For 100 steps this probability is 0.9999995273. Note that the machines in the sample are in the reduced enumeration, a large number of very trivial machines halting in just one step having been removed. So in the complete enumeration the probability of halting in at most 100 steps is even greater.

But we found some high runtime values—precisely 23 machines required more than 1,000 steps. The highest value was a machine progressing through 1,483 steps upon halting. So we have enough evidence to believe that by setting the runtime at 2,000 steps we have obtained almost all (if not all) output arrays. We ran all 6 × 34⁷ Turing machines in the reduced enumeration for (4, 2)_2D. Then we applied the completions explained before.

Evaluating 2-dimensional Kolmogorov complexity

D(4, 2)_2D denotes the frequency distribution (a calculated Universal Distribution) from the output of deterministic 2-dimensional Turing machines, with associated complexity measure K_m,2D. D(4, 2)_2D distributes 1,068,618 arrays into 1,272 different complexity values, with a minimum complexity value of 2.22882 bits (an explanation of non-integer program-size complexity is given in Soler-Toscano et al. (2014) and Soler-Toscano et al. (2013)), a maximum value of 36.2561 bits and a mean of 35.1201. Considering the number of possible square binary arrays given by the formula 2^d×d (without considering any symmetries), D(4, 2)_2D can be said to produce all square binary arrays of length up to 3 × 3, that is $\sum _{d=1}^3{2}^{d\times d}=530$ square arrays, and 60,016 of the 2^(4×4) = 65,536 square arrays with side of length (or dimension) d = 4. It only produces 84,104 of the 33,554,432 possible square binary arrays of length d = 5 and only 11,328 of the possible 68,719,476,736 of dimension d = 6. The largest square array produced in D(4, 2)_2D is of side length d = 13 (Left of Fig. 3) out of a possible 748 × 10⁴⁸; it has a K_m,2D value equal to 34.2561.

What one would expect from a distribution where simple patterns are more frequent (and therefore have lower Kolmogorov complexity after application of the Coding theorem) would be to see patterns of the “checkerboard” type with high frequency and low random complexity (K), and this is exactly what we found (see Fig. 3), while random looking patterns were found at the bottom among the least frequent ones (Fig. 4).

We have coined the informal notion of a “climber” as an object in the frequency classification (from greatest to lowest frequency) that appears better classified among objects of smaller size rather than with the arrays of their size, this is in order to highlight possible candidates for low complexity, hence illustrating how the process make low complexity patterns to emerge. For example, “checkerboard” patterns (see Fig. 3) seem to be natural “climbers” because they come significantly early (more frequent) in the classification than most of the square arrays of the same size. In fact, the larger the checkerboard array, the more of a climber it seems to be. This is in agreement with what we have found in the case of strings (Zenil, 2011; Delahaye & Zenil, 2012; Soler-Toscano et al., 2014) where patterned objects emerge (e.g., (01)ⁿ, that is, the string 01 repeated n times), appearing relatively increasingly higher in the frequency classifications the larger n is, in agreement with the expectation that patterned objects should also have low Kolmogorov complexity.

An attempt of a definition of a climber is a pattern P of size a × b with small complexity among all a × b patterns, such that there exists smaller patterns Q (say c × d, with cd < ab) such that K_m(P) < K_m(Q) < median(K_m(all ab patterns)).

For example, Fig. 5 shows arrays that come together among groups of much shorter arrays, thereby demonstrating, as expected from a measure of randomness, that array—or string—size is not what determines complexity (as we have shown before in Zenil (2011), Delahaye & Zenil (2012) and Soler-Toscano et al. (2014) for binary strings). The fact that square arrays may have low Kolmogorov complexity can be understood in several ways, some of which strengthen the intuition that square arrays should be less Kolmogorov random, such as for example, the fact that for square arrays one only needs the information of one of its dimensions to determine the other, either height or width.

Figure 5 shows cases in which square arrays are significantly better classified towards the top than arrays of similar size. Indeed, 100% of the squares of size 2 × 2 are in the first fifth (F1), as are the 3 × 3 arrays. Square arrays of 4 × 4 are distributed as follows when dividing (4, 2)_2D in 5 equal parts: 72.66%, 15.07%, 6.17359%, 2.52%, 3.56%.

Comparison of K_m and approaches based on compression

It is also not uncommon to detect instabilities in the values retrieved by a compression algorithm for short strings, as explained in ‘Uncomputability and instability of K’, strings which the compression algorithm may or may not compress. This is not a malfunction of a particular lossless compression algorithm (e.g., Deflate, used in most popular computer formats such as ZIP and PNG) or its implementation, but a commonly encountered problem when lossless compression algorithms attempt to compress short strings.

When researchers have chosen to use compression algorithms for reasonably long strings, they have proven to be of great value, for example, for DNA false positive repeat sequence detection in genetic sequence analysis (Rivals et al., 1996), in distance measures and classification methods (Cilibrasi & Vitanyi, 2005), and in numerous other applications (Li & Vitányi, 2009). However, this effort has been hamstrung by the limitations of compression algorithms–currently the only method used to approximate the Kolmogorov complexity of a string–given that this measure is not computable.

In this section we study the relation between K_m and approaches to Kolmogorov complexity based on compression. We show that both approaches are consistent, that is, strings with higher K_m value are less compressible than strings with lower values. This is as much validation of K_m and our Coding theorem method as it is for the traditional lossless compression method as approximation techniques to Kolmogorov complexity. The Coding theorem method is, however, especially useful for short strings where losses compression algorithms fail, and the compression method is especially useful where the Coding theorem is too expensive to apply (long strings).

Compressing strings of length 10–15

For this experiment we have selected the strings in D(5) with lengths ranging from 10 to 15. D(5) is the frequency distribution of strings produced by all 1-dimensional deterministic Turing machines as described in Soler-Toscano et al. (2014). Table 1 shows the number of D(5) strings with these lengths. Up to length 13 we have almost all possible strings. For length 14 we have a considerable number and for length 15 there are less than 50% of the 2¹⁵ possible strings. The distribution of complexities is shown in Fig. 7.

Table 1:

Number of strings of length 10–15 found in D(5).

Length (l)	Strings
10	1,024
11	2,048
12	4,094
13	8,056
14	13,068
15	14,634

DOI: 10.7717/peerj-cs.23/table-1

As expected, the longer the strings, the greater their average complexity. The overlapping of strings with different lengths that have the same complexity correspond to climbers. The experiment consisted in creating files with strings of different K_m-complexity but equal length (files with more complex (random) strings are expected to be less compressible than files with less complex (random) strings). This was done in the following way. For each l (10 ≤ l ≤ 15), we let S(l) denote the list of strings of length l, sorted by increasing K_m complexity. For each S(l) we made a partition of 10 sets with the same number of consecutive strings. Let’s call these partitions P(l, p), 1 ≤ p ≤ 10.

Then for each P(l, p) we have created 100 files, each with 100 random strings in P(l, p) in random order. We called these files F(l, p, f), 1 ≤ f ≤ 100. Summarizing, we now have:

• 6 different string lengths l, from 10 to 15, and for each length

• 10 partitions (sorted by increasing complexity) of the strings with length l, and

• 100 files with 100 random strings in each partition.

This makes for a total of 6,000 different files. Each file contains 100 different binary strings, hence with length of 100 × l symbols.

A crucial step is to replace the binary encoding of the files by a larger alphabet, retaining the internal structure of each string. If we compressed the files F(l, p, f) by using binary encoding then the final size of the resulting compressed files would depend not only on the complexity of the separate strings but on the patterns that the compressor discovers along the whole file. To circumvent this we chose two different symbols to represent the ‘0’ and ‘1’ in each one of the 100 different strings in each file. The same set of 200 symbols was used for all files. We were interested in using the most standard symbols we possibly could, so we created all pairs of characters from ‘a’ to ‘p’ (256 different pairs) and from this set we selected 200 two-character symbols that were the same for all files. This way, though we do not completely avoid the possibility of the compressor finding patterns in whole files due to the repetition of the same single character in different strings, we considerably reduce the impact of this phenomenon.

The files were compressed using the Mathematica function Compress, which is an implementation of the Deflate algorithm (Lempel–Ziv plus Huffman coding). Figure 7 shows the distributions of lengths of the compressed files for the different string lengths. The horizontal axis shows the 10 groups of files in increasing K_m. As the complexity of the strings grows (right part of the diagrams), the compressed files are larger, so they are harder to compress. The relevant exception is length 15, but this is probably related to the low number of strings of that length that we have found, which are surely not the most complex strings of length 15.

We have used other compressors such as GZIP (which uses Lempel–Ziv algorithm LZ77) and BZIP2 (Burrows–Wheeler block sorting text compression algorithm and Huffman coding), with several compression levels. The results are similar to those shown in Fig. 7.

Comparing (4, 2)_2D and (4, 2)

We shall now look at how 1-dimensional arrays (hence strings) produced by 2D Turing machines correlate with strings that we have calculated before (Zenil, 2011; Delahaye & Zenil, 2012; Soler-Toscano et al., 2014) (denoted by D(5)). In a sense this is like changing the Turing machine formalism to see whether the new distribution resembles distributions following other Turing machine formalisms, and whether it is robust enough.

All Turing machines in (4, 2) are included in (4, 2)_2D because these are just the machines that do not move up or down. We first compared the values of the 1,832 output strings in (4, 2) to the 1-dimensional arrays found in (4, 2)_2D. We are also interested in the relation between the ranks of these 1,832 strings in both (4, 2) and (4, 2)_2D.

Figure 8 shows the link between K_m,2D with 2D Turing machines as a function of ordinary K_m,1D (that is, simply K_m as defined in Soler-Toscano et al. (2014)). It suggests a strong almost-linear overall association. The correlation coefficient r = 0.9982 confirms the linear association, and the Spearman correlation coefficient r_s = 0.9998 proves a tight and increasing functional relation.

The length l of strings is a possible confounding factor. However Fig. 9 suggests that the link between one and 2-dimensional complexities is not explainable by l. Indeed, the partial correlation r_Km,1DKm,2D.l = 0.9936 still denotes a tight association.

Figure 9 also suggests that complexities are more strongly linked with longer strings. This is in fact the case, as Table 2 shows: the strength of the link increases with the length of the resulting strings. One and 2-dimensional complexities are remarkably correlated and may be considered two measures of the same underlying feature of the strings. How these measures vary is another matter. The regression of K_m,2D on K_m,1D gives the following approximate relation: K_m,2D ≈ 2.64 + 1.11K_m,1D. Note that this subtle departure from identity may be a consequence of a slight non-linearity, a feature visible in Fig. 8.

Table 2:

Correlation coefficients between one and 2-dimensional complexities by length of strings.

Length (l)	Correlation
5	0.9724
6	0.9863
7	0.9845
8	0.9944
9	0.9977
10	0.9952
11	1
12	1

DOI: 10.7717/peerj-cs.23/table-2

Comparison of K_m and compression of cellular automata

A 1-dimensional CA can be represented by an array of cells x_i where i ∈ ℤ (integer set) and each x takes a value from a finite alphabet Σ. Thus, a sequence of cells {x_i} of finite length n describes a string or global configuration c on Σ. This way, the set of finite configurations will be expressed as Σⁿ. An evolution comprises a sequence of configurations {c_i} produced by the mapping Φ:Σⁿ → Σⁿ; thus the global relation is symbolized as: (8)${\displaystyle \Phi \left(c^t\right)\to c^{t+1}}$ where t represents time and every global state of c is defined by a sequence of cell states. The global relation is determined over the cell states in configuration c^t updated simultaneously at the next configuration c^t+1 by a local function φ as follows: (9)${\displaystyle \phi \left(x_{i-r}^t,\dots ,x_i^t,\dots ,x_{i+r}^t\right)\to x_i^{t+1}.}$ Wolfram (2002) represents 1-dimensional cellular automata (CA) with two parameters (k, r) where k = |Σ| is the number of states, and r is the neighborhood radius. Hence this type of CA is defined by the parameters (2, 1). There are Σⁿ different neighborhoods (where n = 2r + 1) and k^kn distinct evolution rules. The evolutions of these cellular automata usually have periodic boundary conditions. Wolfram calls this type of CA Elementary Cellular Automata (denoted simply by ECA) and there are exactly k^kn = 256 rules of this type. They are considered the most simple cellular automata (and among the simplest computing programs) capable of great behavioral richness.

1-dimensional ECA can be visualized in 2-dimensional space–time diagrams where every row is an evolution in time of the ECA rule. By their simplicity and because we have a good understanding about them (e.g., at least one ECA is known to be capable of Turing universality (Cook, 2004; Wolfram, 2002)) they are excellent candidates to test our measure K_m,2D, being just as effective as other methods that approach ECA using compression algorithms (Zenil, 2010) that have yielded the results that Wolfram obtained heuristically.

K_m,2D comparison with compressed ECA evolutions

We have seen that our Coding theorem method with associated measure K_m (or K_m,2D in this paper for 2D Kolmogorov complexity) is in agreement with bit string complexity as approached by compressibility, as we have reported in ‘Comparison of K_m and approaches based on compression’.

The Universal Distribution from Turing machines that we have calculated (D(4, 2)_2D) will help us to classify Elementary Cellular Automata. Classification of ECA by compressibility has been done before in Zenil (2010) with results that are in complete agreement with our intuition and knowledge of the complexity of certain ECA rules (and related to Wolfram’s (2002) classification). In Zenil (2010) both classifications by simplest initial condition and random initial condition were undertaken, leading to a stable compressibility classification of ECAs. Here we followed the same procedure for both simplest initial condition (single black cell) and random initial condition in order to compare the classification to the one that can be approximated by using D(4, 2)_2D, as follows.

We will say that the space–time diagram (or evolution) of an Elementary Cellular Automaton c after time t has complexity: (10)${\displaystyle K_{m,2D_{d\times d}}\left(c^t\right)=\sum _{q\in {\left\{\stackrel{t}{c}\right\}}_{d\times d}}{K}_{m,2D}\left(q\right).}$ That is, the complexity of a cellular automaton c is the sum of the complexities of the q arrays or image patches in the partition matrix {c^t}_d×d from breaking {c^t} into square arrays of length d produced by the ECA after t steps. An example of a partition matrix of an ECA evolution is shown in Fig. 13 for ECA Rule 30 and d = 3 where t = 6. Notice that the boundary conditions for a partition matrix may require the addition of at most d − 1 empty rows or d − 1 empty columns to the boundary as shown in Fig. 13 (or alternatively the dismissal of at most d − 1 rows or d − 1 columns) if the dimensions (height and width) are not multiples of d, in this case d = 3.

If the classification of all rules in ECA by K_m,2D yields the same classification obtained by compressibility, one would be persuaded that K_m,2D is a good alternative to compressibility as a method for approximating the Kolmogorov complexity of objects, with the signal advantage that K_m,2D can be applied to very short strings and very short arrays such as images. Because all possible 2⁹ arrays of size 3 × 3 are present in K_m,2D we can use this arrays set to try to classify all ECAs by Kolmogorov complexity using the Coding Theorem method. Figure 6 shows all relevant (non-symmetric) arrays. We denote by K_m,2D3×3 this subset from K_m,2D.

Figure 11 displays the scatterplot of compression complexity against K_m,2D3×3 calculated for every cellular automaton. It shows a positive link between the two measures. The Pearson correlation amounts to r = 0.8278, so the determination coefficient is r² = 0.6853. These values correspond to a strong correlation, although smaller than the correlation between 1- and 2-dimensional complexities calculated in ‘Comparison of K_m and approaches based on compression’.

Concerning orders arising from these measures of complexity, they too are strongly linked, with a Spearman correlation of r_s = 0.9200. The scatterplots (Fig. 11) show a strong agreement between the Coding theorem method and the traditional compression method when both are used to classify ECAs by their approximation to Kolmogorov complexity.

The anomalies found in the classification of Elementary Cellular Automata (e.g., Rule 77 being placed among ECA with high complexity according to K_m,2D3×3) is a limitation of K_m,2D3×3 itself and not of the Coding theorem method which for d = 3 is unable to “see” beyond 3-bit squares using, which is obviously very limited. And yet the degree of agreement with compressibility is surprising (as well as with intuition, as a glance at Fig. 10 shows, and as the distribution of ECAs starting from random initial conditions in Fig. 13 confirms). In fact an average ECA has a complexity of about 20K bits, which is quite a large program-size when compared to what we intuitively gauge to be the complexity of each ECA, which may suggest that they should have smaller programs. However, one can think of D(4, 2)_2D3×3 as attempting to reconstruct the evolution of each ECA for the given number of steps with square arrays only 3 bits in size, the complexity of the three square arrays adding up to approximate K_m,2D of the ECA rule. Hence it is the deployment of D(4, 2)_2D3×3 that takes between 500 to 50K bits to reconstruct every ECA space–time evolution depending on how random versus how simple it is.

Other ways to exploit the data from D(4, 2)_2D (e.g., non-square arrays) can be utilized to explore better classifications. We think that constructing a Universal Distribution from a larger set of Turing machines, e.g., D(5, 2)_2D4×4 will deliver more accurate results but here we will also introduce a tweak to the definition of the complexity of the evolution of a cellular automaton.

Splitting ECA rules in array squares of size 3 is like trying to look through little windows 9 pixels wide one at a time in order to recognize a face, or training a “microscope” on a planet in the sky. One can do better with the Coding theorem method by going further than we have in the calculation of a 2-dimensional Universal Distribution (e.g., calculating in full or a sample of D(5, 2)_2D4×4), but eventually how far this process can be taken is dictated by the computational resources at hand. Nevertheless, one should use a telescope where telescopes are needed and a microscope where microscopes are needed.

Block Decomposition Method

One can think of an improvement in resolution of K_m,2D(c) for growing space–time diagrams of cellular automaton by taking the log₂(n) of the sum of the arrays where n is the number of repeated arrays, instead of simply adding the complexity of the image patches or arrays. That is, one penalizes repetition to improve the resolution of K_m,2D for larger images as a sort of “optical lens”. This is possible because we know that the Kolmogorov complexity of repeated objects grows by log₂(n), just as we explained with an example in ‘Kolmogorov–Chaitin Complexity’. Adding the complexity approximation of each array in the partition matrix of a space–time diagram of an ECA provides an upper bound on the ECA Kolmogorov complexity, as it shows that there is a program that generates the ECA evolution picture with the length equal to the sum of the programs generating all the sub-arrays (plus a small value corresponding to the code length to join the sub-arrays). So if a sub-array occurs n times we do not need to consider its complexity n times but log₂(n). Taking into account this, Eq. (10) can be then rewritten as: (11)${\displaystyle K_{m,2D_{d\times d}}^{\prime }\left(c^t\right)=\sum _{\left(\underset{u}{r},n_u\right)\in {\left\{c^t\right\}}_{d\times d}}{K}_m\left(r_u\right)+{log}_2\left(n_u\right)}$ where r_u are the different square arrays in the partition {c^t}_d×d of the matrix c^t and n_u the multiplicity of r_u, that is the number of repetitions of d × d-length patches or square arrays found in c^t. From now on we will use K′ for squares of size greater than 3 and it may be denoted only by K or by BDM standing for Block decomposition method. BDM has now been applied successfully to measure, for example, the Kolmogorov complexity of graphs and complex networks (Zenil et al., 2014) by way of their adjacency matrices (a 2D grid) and was shown to be consistent with labelled and unlabelled (up to isomorphisms) graphs.

Now complexity values of $K_{m,2D_{d\times d}}^{\prime }$ range between 70 and 3K bits with a mean program-size value of about 1K bits. The classification of ECA, according to Eq. (11), is presented in Fig. 12. There is an almost perfect agreement with a classification by lossless compression length (see Fig. 13) which makes even one wonder whether the Coding theorem method is actually providing more accurate approximations to Kolmogorov complexity than lossless compressibility for this objects length. Notice that the same procedure can be extended for its use on arbitrary images. We denominate this technique Block Decomposition Method. We think it will prove to be useful in various areas, including machine learning as an of Kolmogorov complexity (other contributions to ML inspired in Kolmogorov complexity can be found in Hutter (2003)).

Also worth notice that the fact that ECA can be successfully classified by K_m,2D with an approximation of the Universal Distribution calculated from Turing machines (TM) suggests that output frequency distributions of ECA and TM cannot be but strongly correlated, something that we had found and reported before in Zenil & Delahaye (2010) and Delahaye & Zenil (2007b).

Another variation of the same K_m,2D measure is to divide the original image into all possible square arrays of a given length rather than taking a partition. This would, however, be exponentially more expensive than the partition process alone, and given the results in Fig. 12 further variations do not seem to be needed, at least not for this case.

Robustness of the approximations to m(s)

One important question that arises when positing the soundness of the Coding theorem method as an alternative to having to pick a universal Turing machine to evaluate the Kolmogorov complexity K of an object, is how many arbitrary choices are made in the process of following one or another method and how important they are. One of the motivations of the Coding theorem method is to deal with the constant involved in the Invariance theorem (Eq. (2)), which depends on the (prefix-free) universal Turing machine chosen to measure K and which has such an impact on real-world applications involving short strings. While the constant involved remains, given that after application of the Coding theorem (Eq. (3)) we reintroduce the constant in the calculation of K, a legitimate question to ask is what difference it makes to follow the Coding theorem method compared to simply picking the universal Turing machine.

On the one hand, one has to bear in mind that no other method existed for approximating the Kolmogorov complexity of short strings. On the other hand, we have tried to minimize any arbitrary choice, from the formalism of the computing model to the informed runtime, when no Busy Beaver values are known and therefore sampling the space using an educated runtime cut-off is called for. When no Busy Beaver values are known the chosen runtime is determined according to the number of machines that we are ready to miss (e.g., less than .01%) for our sample to be significative enough as described in ‘Setting the runtime’. We have also shown in Soler-Toscano et al. (2014) that approximations to the Universal Distribution from spaces for which Busy Beaver values are known are in agreement with larger spaces for which Busy Beaver values are not known.

Among the possible arbitrary choices it is the enumeration that may perhaps be questioned, that is, calculating D(n) for increasing n (number of Turing machine states), hence by increasing size of computer programs (Turing machines). On the one hand, one way to avoid having to make a decision on the machines to consider when calculating a Universal Distribution is to cover all of them for a given number of n states and m symbols, which is what we have done (hence the enumeration in a thoroughly (n, m) space becomes irrelevant). While it may be an arbitrary choice to fix n and m, the formalisms we have followed guarantee that n-state m-symbol Turing machines are in (n + i, m + j) with i, j ≥ 0 (that is, the space of all n + i-state m + j-symbol Turing machines). Hence the process is incremental, taking larger spaces and constructing an average Universal Distribution. In fact, we have demonstrated (Soler-Toscano et al., 2014) that D(5) (that is, the Universal Distribution produced by the Turing machines with 2 symbols and 5 states) is strongly correlated to D(4) and represents an improvement in accuracy of the string complexity values in D(4), which in turn is in agreement with and an improvement on D(3) and so on. We have also estimated the constant c involved in the invariance theorem (Eq. (2)) between these D(n) for n > 2, which turned out to be very small in comparison to all the other calculated Universal Distributions (Soler-Toscano et al., 2013).

Real-world evidence

We have provided here some theoretical and statistical arguments to show the reliability, validity and generality of our measure, more empirical evidence has also been produced, in particular in the field of cognition and psychology where researchers often have to deal with too short strings or too small patterns for compression methods to be used. For instance, it was found that the complexity of a (one-dimensional) string better predicts its recall from short-term memory that the length of the string (Chekaf et al., 2015). Incidentally, a study on the conspiracy theory believers mindset also revealed that human perception of randomness is highly linked to our one-dimensional measure of complexity (Dieguez, Wagner-Egger & Gauvrit, 2015). Concerning the two-dimensional version introduced in this paper, it has been fruitfully used to show how language iterative learning triggers the emergence of linguistic structures (Kempe, Gauvrit & Forsyth, 2015). A direct link between the perception of two-dimensional randomness, our complexity measure, and natural statistics was also established in two experiments (Gauvrit, Soler-Toscano & Zenil, 2014). These findings further support the complexity metrics presented herein. Furthermore, more theoretical arguments have been advanced in Soler-Toscano et al. (2013) and Soler-Toscano & Zenil (2015).