The method introduced in my doctoral dissertation was featured in the French version of Scientific American Pour La Science in its July 2011 issue No. 405 under the title Le défi des faibles complexités.
Jean-Paul Delahaye points out that:
Comme les très petites durées ou longueurs, les faibles complexités sont délicates à évaluer. Paradoxalement, les méthodes d’évaluation demandent des calculs colossaux.
(Like long durations or very short lengths, weak complexities are tricky to evaluate. Paradoxically, the evaluation methods require colossal calculations.)
and he continues:
Pour les petites séquences, cette mesure est stable et conforme à notre idée de la complexité; pour les grandes, elle est, d’après le théorème mentionné [the invariance theorem, my comment], conforme à la meilleure mesure de complexité unanimement admise, la complexité de Kolmogorov. Que demander de plus?
(For short strings, this measure is stable and conforms to our idea of complexity; for long strings, according to the aforementioned theorem [the invariance theorem-- my comment], it conforms to the best and universally accepted measure of complexity, Kolmogorov complexity. What more can one ask for?)
Imagine you are given 2 short strings and are asked to tell which one looks more random and which more structured. The strings are 0101 and 1011. Applying the idea of algorithmic complexity, the shorter the description the less random-looking. It would seem that the first string has a repeating pattern that can be taken advantage of in describing it. In plain English one may therefore say that the first string can be described as “zero and one twice” while the second one would require a longer description. In fact, there seem to be fewer descriptions for the first string than for the second (and also notice some ambiguity in plain English). The first may also allow the description “A zero followed by a one followed by a zero followed by a one” or “zero and one followed by the same” and perhaps other variations. Descriptions of the second one can include “one and zero followed by two ones” or “one zero one one,” just as the first one could simply be “zero one zero one,” which doesn’t look like a compressed version of the string but rather a straight translation into an expanded form of plain English.
All this by way of asking whether any of the two strings is without a doubt simpler than the other, or whether the apparent repetition in the first string makes us think that the string has a pattern despite the pattern being repeated only twice. Perhaps when one looks at such a string one gets the impression that it may belong to a larger string comprising alternations of 01 and one concludes that it is simpler than the second one. To leave the subjective realm, one would need to evaluate the algorithmic complexity of the strings and compare their respective values. The algorithmic complexity of a string is the shortest program (measured in bits) producing th string in question running on a universal Turing machine. It is inconvenient that there is no algorithm that, given a string, gives you the length of the shortest program that produces it. This is by reduction to the halting problem. Which means one cannot really measure with absolute certainty the algorithmic complexity of a string because it is uncomputable. It doesn’t mean, however, that one cannot approach it; one can often do so in an effective and useful way.
Traditionally, the way to approach the algorithmic complexity of a string was by using lossless compression algorithms. Lossless means that one can recover the original string from the compressed version. The use of lossless algorithms is preferable because there may be inconsistent ways of compressing strings that give the impression of compressing a string without corresponding to a measure of complexity (e.g., noise deletion is an effective way, but noise may be algorithmically random and the aim of algorithmic complexity is to precisely measure how random something is and not merely to gauge whether it looks random). The result of a compression algorithm is an upper bound of its algorithmic complexity. While one cannot ever tell when a string is not compressible, if one succeeds at somehow shortening a string one can tell that its algorithmic complexity cannot be larger than the compressed length.
One does not want to cheat and claim that one can compress any string into a bit if the decompression algorithm interprets that bit into the desired string. A fair compression algorithm can be defined as one that transforms a string into two pieces: one is the compressed version and the other the instructions to decompress the string, together accounting for the final length of the compressed version. In other words, it would seem as if you were adding the decompression algorithm to the compressed string so that the compressed string comes with its own decompression instructions. In the long run, there is a theorem (invariance) that guarantees that complexity values converge.
But for short strings (which are often the ones useful for practical applications), adding the decompression instructions to the compressed version makes the compressed string often, if not always, longer than the string itself. If the string is, for example, shorter than the size of the decompression algorithm, there will be no way to compress the string into something shorter still, simply because the decompression instructions are at least of the length of the original string. Moreover, the result is so dependent on the size of the decompression algorithm (because it is the greatest contributor to the overall length) that the final value of the compressed length is too unstable under different lossless compression/decompression algorithms.
For example, if one tries to compress a short string using Mathematica, one gets the following results:
StringLength@Compress["0101"] = 30
Looking at the beginning of the compression line when plotting the lengths of strings (x axis) against their compressed lengths (y axis) one observes that it does not start at y=0 even when x=0.
This means that compressing the string 0101 with 4 bits took 46 characters (even more in bits). In Mathematica, strings begin to be compressed this way at about length 30. This is not a malfunction of Mathematica; it is the result of what I have explained. The Mathematica Compress function is actually based on the Deflate lossless compression algorithm, which is a combination of the LZ77 algorithm and Huffman coding, among the most popular lossless compression algorithms available, used in formats like ZIP, GZIP, GIF and PNG (these last two are therefore lossless compression image formats).
Zooming into the axis origin of the plot one can see that the beginning looks unstable. The string in question is a repetition of n 1s with n lying on the x axis (which represents different compression lengths rather than individual ones to avoid repetition and a step-like curve)
If one tries to compress 1011 one gets none other than the same value:
StringLength@Compress["1011"] = 30
The instructions obviously take up some space in the final compressed length and they cannot be compressed themselves (if they were, they would in any case be the same for all strings, taking us back to the same situation). There is a limit for compression algorithms to compress short strings. So if one wished to tell which of these two strings were objectively more or less randomly complex by approximating their algorithmic complexity using a compression algorithm, it turns out that there is no way to do so.
On the other hand, given that the definition of algorithmic complexity based on compressibility says that the less compressible the more randomly complex a string, one could immediately say that a single bit, 0 or 1, is random for certain, i.e. has maximal algorithmic complexity, given that there is no way to further compress a single bit. In other words, there is no program shorter than 1 bit that can produce 0 or 1. The shortest descriptions of 0 and 1 are therefore 0 and 1 themselves. Hence 0 and 1, according to the compressibility approach, are random strings. It may seem to make no sense to characterize a single bit as random.
On the one hand, a single bit does not carry any information and on these grounds one may think of it as somehow random. If one thinks about whether one would have been able to predict 0 or 1 as the outcome of a process, given that there is no context because they occur alone, one may also conclude that their occurrence is somehow random. In other words, if you see a string like 010101 you may bet that the next bit is 0, but if you are provided with nothing there is no way you can favor any position on whether the bit to come is 0 or 1. So much for justifying that a single bit is random.
It is hard, however, to justify how 0 could look more random than, say, any other possible string. If 0 is random how is it relatively more complex than 00? Or 01? Intuition tells us that short strings shouldn’t be that random (more random than, for example, longer random-looking strings), so if a single bit is the most random among all finite strings, how could it be that there is such a phase transition from maximal random complexity to very low complexity of, say, strings of length 2, 3 or 5 bits long?
Since intuition tells us that something random should also be uncommon and rare, what if one asks how common 0 or 1 are as results of a program? There is a measure that gives the probability of a program’s producing a given string running on a universal Turing machine. This is a measure we used to present a new method to evaluate the complexity of strings, as an alternative to the traditional use of compression algorithms. The new method aims particularly to solve the problem of the evaluation of the complexity of short strings, as we’ve discussed. It is based on Levin-Solomonoff’s algorithmic probability and is connected back to algorithmic (Kolmogorov-Chaitin) complexity by way of Chaitin-Levin’s coding theorem.
Algorithmic probability says that it is not the case that a single bit is the most complex random string, but actually the most structured possible one and, more importantly , that the complexity transition is smooth, more in accordance with intuition.
It may be that it makes sense that a single bit can be regarded as both the most simple and the most complex of strings from different perspectives, and the advantage of the algorithmic probability approach is that it provides not only a different notion of the complexity of a single bit, one that is in keeping with intuition, but also that it generates a different outcome to the compressibility approach, even when the two measures are intimately related and asymptomatically produce the same results in the long term (for longer strings). I think the two views reflect different aspects of what a single bit represents.
The paper presenting the novel method for evaluating the algorithmic complexity of short strings was first proposed and sketched in Greg Chaitin’s 60th anniversary festschrift edited by Cris Calude (J-P. Delahaye & H. Zenil, “On the Kolmogorov-Chaitin complexity for short sequences,” Randomness and Complexity: From Leibniz to Chaitin, edited by C.S. Calude, World Scientific, 2007). The method uses an exhaustive and systematic search of Turing machines inspired by Wolfram’s NKS dictum, from which a frequency distribution of the halting machines is built and the Levin-Chaitin coding theorem applied to evaluate the algorithmic complexity of binary strings.
Chaitin pointed out (regarding our method) that:
…the dreaded theoretical hole in the foundations of algorithmic complexity turns out, in practice, not to be as serious as was previously assumed.
The full technical article is available in ArXiv Numerical Evaluation of Algorithmic Complexity for Short Strings: A Glance into the Innermost Structure of Randomness.
You can also look at the slides of the presentation I delivered at the Alan Turing amphitheater at the Computer Science Department of the University of Lille 1: