[SideQuark]

The problem with the UTM argument designed for a specific string is that the UTM size grows without bound. You also don’t need a UTM, which will be much larger than a simple TM or finite automaton. And now we’re back to the original problem: designing a machine capable of compressing the specific string and with smaller description.

UTM provides no gain.

[Xcelerate]

You cannot specify the “size” of an arbitrary Turing machine without knowing the index of that machine in some encoding scheme. If we are to account for the possibility of any string being selected as the one to compress, and if we wish to consider all possible algorithmic ways to do this, then the encoding scheme necessarily corresponds to that of a universal Turing machine.

It’s a moot point anyway as most programming languages are capable of universal computation.

Again, I’m not claiming we can compress random data. I’m claiming we can (sometimes) make it look like we can compress random data by selecting a specific programming language from one of many possibilities. This selection itself constitutes the “missing” information.

[SideQuark]

> 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.