The $5000 Compression Challenge (2001)

There's another interesting loophole to these general compression challenges. A universal Turing machine will necessarily compress some number of strings (despite the fact that almost all strings are incompressible). The set of compressible strings varies depending on the choice of UTM, and if your UTM if fixed, you're out of luck for random data. But if the UTM is unspecified, then there exist an infinite number of UTMs that will compress any specific string.

To some extent, this resembles the approach of "hiding the data in the decompressor". But the key difference here is that you can make it less obvious by selecting a particular programming language capable of universal computation, and it is the choice of language that encodes the missing data. For example, suppose we have ~17k programming languages to choose from—the language selection itself encodes about 14 bits (log2(17000)) of information.

If there are m bits of truly random data to compress and n choices of programming languages capable of universal computation, then as n/m approaches infinity, the probability of at least one language being capable of compressing an arbitrary string approaches 1. This ratio is likely unrealistically large for any amount of random data more than a few bits in length.

There's also the additional caveat that we're assuming the set of compressible strings is algorithmically independent for each choice of UTM. This certainly isn't the case. The invariance theorem states that ∀x|K_U(x) - K_V(x)| < c for UTMs U and V, where K is Kolmogorov complexity and c is a constant that depends only on U and V. So in our case, c is effectively the size of the largest minimal transpiler between any two programming languages that we have to choose from, and I'd imagine c is quite small.

Oh, and this all requires computing the Kolmogorov complexity of a string of random data. Which we can't, because that's uncomputable.

Nevertheless it's an interesting thought experiment. I'm curious what the smallest value of m is such that we could realistically compress a random string of length 2^m given the available programming languages out there. Unfortunately, I imagine it's probably like 6 bits, and no one is going to give you an award for compressing 6 bits of random data.

The issue with the invariance theorem you point out always bugged me.

Let s be an algorithmically random string relative to UTM A. Is it the case that there exists some pathological UTM S, such that K(s|S) (the Kolmogorov complexity of s relative to S) is arbitrarily small? I.e. the blank print statement of S produces s. And there always exists such an S for any s?

Is there some way of defining a meta-complexity measure, the complexity of some UTM without a reference UTM? It seems intuitive that although some pathological UTM might exist that can "compress" whichever string you have, its construction appears very unnatural. Is there some way of formalizing this "naturalness"?

You are correct to be bugged IMO, I agree with you. My thoughts: https://forwardscattering.org/page/Kolmogorov%20complexity

Kolmogorov complexity is useless as an objective measure of complexity.

Nice blog post. I wasn’t aware of those comments by Yann LeCunn and Murray Gell-Mann, but it’s reassuring to know there are some experts who have been wondering about this “flaw” in Kolmogorov complexity as well.

I wouldn’t go so far as to say Kolmogorov complexity is useless as an objective complexity measure however. The invariance theorem does provide a truly universal and absolute measure of algorithmic complexity — but it’s the complexity between two things rather than of one thing. You can think of U and V as “representatives” of any two partial recursive functions u(x) and v(x) capable of universal computation. The constant c(u, v) is interesting then because it is a natural number that depends only on the two abstract functions themselves and not the specific Turing machines that compute the functions.

What does that mean philosophically? I’m not sure. It might mean that the notion of absolute complexity for a finite string isn’t a coherent concept, i.e., complexity is fundamentally a property of the relationship between things rather than of a thing.