Suppose we have a binary string $s$. We wish to partition this string in a series of $0$s followed by $1$s (alternatively: we wish to sort), using only one operation: moving three consecutive elements to the end. E.g. if our string is $ABCDEFGH$ we can do our operation at index $1$ to get $DEFGH...

What is the mutual information between a random bitstring $R$ and any Kolmogorov minimal structure function $S$? Specifically, is it ever true that $I(R:S)>0$? I am interested in the case when $S$ is generated by processing $R$ somehow. I think the answer is either no, or rarely, since if $K(S...