  Symmetry of information establishes a relation between the information that x
has about y (denoted I(x : y)) and the information that y has about x (denoted
I(y : x)). In classical information theory, the two are exactly equal, but in
algorithmical information theory, there is a small excess quantity of
information that differentiates the two terms, caused by the necessity of
packaging information in a way that makes it accessible to algorithms. It was
shown in [Zim11] that in the case of strings with simple complexity (that is
the Kolmogorov complexity of their Kolmogorov complexity is small), the
relevant information can be packed in a very economical way, which leads to a
tighter relation between I(x : y) and I(y : x) than the one provided in the
classical symmetry-of-information theorem of Kolmogorov and Levin. We give here
a simpler proof of this result, using a suggestion of Alexander Shen. This
result implies a van Lambalgen- type theorem for finite strings and plain
complexity: If x is c-random and y is c-random relative to x, then xy is
O(c)-random. We show that a similar result holds for prefix-free complexity and
weak-K-randomness.
