> You cannot specify the “size” of an arbitrary Turing machine

Related to compression this is, as I said, irrelevant. Kolmogorov complexity, in the first order naive sense, picks a machine then talks about length. In the complete sense, due to Kolmorogov invariance which extends the naive version to any possible encoding, it shows that there is a fundamental minimal length over any machine Then one proves that the vast majority of strings are incompressible. Withtout this machine invariance there could be no notion of inherent Kolmorogov complexity of a string.

This has all been known since the 1960s.

So no, it matters not which UTM, FSM, TM, Java, Hand coded wizard machine, or whatever you choose. This line of reasoning has been investigated decades ago and thrown out since it does not work.

Your claim

>then there exist an infinite number of UTMs that will compress any specific string

combined with your claim

> For example, suppose we have ~17k programming languages to choose from—the language selection itself encodes about 14 bits (log2(17000)) of information.

Does not let you use that information to compress the string you wish to compress, unless you're extremely lucky that the string matches the particular information in the language choice matches, say, the string prefix so you know how to use the language encoded information to apply to the string. Otherwise you need to also encode the information of how the language choice maps to the prefix (or any other part), and that information is going to be as large or larger than what you want. Your program, to reconstruct the data you think language choice hides, will have to restore that data - how does your program choice exactly do this? Does it list all 17k languages and look up the proper information? Whoops, once again your program is too big.

So no, you cannot get a free ride with this choice.

By your argument, Kolmogorov complexity (the invariance version over any possible machine) allows compression of any string, which has been proven impossible for decades.

Interesting. I guess then we would only be interested in the normalized complexity of infinite strings, e.g. lim n-> \infty K(X|n)/n where X is an infinite set of numbers (e.g. the decimal expansion of some real number), and K(X|n) is the complexity of the first n of them. This quantity should still be unique w/o reference to the choice of UTM, no?

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.

Hmm. At least it's still fine to define limits of the complexity for infinite strings. That should be unique, e.g.:

lim n->\infty K(X|n)/n

Possible solutions that come tom mind:

1) UTMs are actually too powerful, and we should use a finitary abstraction to have a more sensible measure of complexity for finite strings.

2) We might need to define a kind of "relativity of complexity". This is my preferred approach and something I've thought about to some degree. That is, that we want a way of describing the complexity of something relative to our computational resources.