  In [3] a short proof is given that some strings have maximal plain Kolmogorov
complexity but not maximal prefix-free complexity. The proof uses Levin's
symmetry of information, Levin's formula relating plain and prefix complexity
and Gacs' theorem that complexity of complexity given the string can be high.
We argue that the proof technique and results mentioned above are useful to
simplify existing proofs and to solve open questions.
  We present a short proof of Solovay's result [21] relating plain and prefix
complexity: $K (x) = C (x) + CC (x) + O(CCC (x))$ and $C (x) = K (x) - KK (x) +
O(KKK (x))$, (here $CC(x)$ denotes $C(C(x))$, etc.).
  We show that there exist $\omega$ such that $\liminf C(\omega_1\dots
\omega_n) - C(n)$ is infinite and $\liminf K(\omega_1\dots \omega_n) - K(n)$ is
finite, i.e. the infinitely often C-trivial reals are not the same as the
infinitely often K-trivial reals (i.e. [1,Question 1]).
  Solovay showed that for infinitely many $x$ we have $|x| - C (x) \le O(1)$
and $|x| + K (|x|) - K (x) \ge \log^{(2)} |x| - O(\log^{(3)} |x|)$, (here $|x|$
denotes the length of $x$ and $\log^{(2)} = \log\log$, etc.). We show that this
result holds for prefixes of some 2-random sequences.
  Finally, we generalize our proof technique and show that no monotone relation
exists between expectation and probability bounded randomness deficiency (i.e.
[6, Question 1]).
