How to find the minimal description for an array?
The phenomenon you're describing is Kolmogorov complexity.  Essentially, if you fix a programming language (or, more formally, a coding of Turing machines) then the Kolmogorov complexity&nbsp;$K(s)$ of a string&nbsp;$s$ is the length of the shortest program that outputs&nbsp;$s$ when started with no input. It turns out that, within reason, it doesn't matter what programming language you use, since using a different language makes at most a constant additive difference to&nbsp;$K(s)$. If you want to define Kolmogorov complexity with respect to some whacko language, I can just use that language to write an interpreter for a sensible language, so the Kolmogorov complexity of a string with respect to your language cannot be worse than the complexity in my language plus the fixed, constant overhead of the interpreter.

Like all interesting properties of Turing machines, the Kolmogorov complexity of a string is undecidable. However, this doesn't stop it being a useful concept and there's been a large amount of research on the subject.