Are there strings with known Kolmogorov complexity?

Question

I just looked into Kolmogorov complexity today and it appears to me that for a binary string of length $1$ (ex. '$0$') the Kolmogorov complexity must be $0$.

It follows that Kolmogorov complexity may not be computable in general but there must be some strings for which Kolmogorov complexity is trivially computed.

Is this correct?

Answer 1

One could enumerate all the algorithms (the set they form is countable) in a given language in some order of increasing length, and the strings that are outputted by these algorithms then have known Kolmogorov complexity. In general, there is no way to compute the Kolmogorov complexity of a given string. Of course, one runs into the problem of non-terminating algorithms, which is itself an undecidable property, so this is not sensible in practise.