Lower Bound on the difference between uniform and prefix-free complexity
Lance Forthnow's example works to separate decision complexity and prefix-complexity: $Ku(0^n) \le O(1)$ for all $n$, and at the same time $K(0^n) &gt; \log n$ for infinitely many $n$.

Note that $Ku(x) \le C(x) + O(1)$ where $C(x)$ denotes the plain complexity of $x$.

Decision complexity can also be lower than the logarithm of a priori probability. The argument is similar as the standard argument to separate plain and prefix complexity: we take advantage of the length of a program to encode an initial segment of a string. More formally, consider a long anough string $x$ such that for each $i &lt; |x|$ we have $M(x_1...x_i) \le M(x_1...x_{i-1})/2$. Clearly we have $KM(x_1...x_i) \ge i$. Let $n \le |x|$ be a number whose binary representation is an initial segment of $x$. A uniform program of length $n - \log n + O(\log \log n)$ is obtained as follows: if $i \le \log n$ return the $i$-th bit of $n$ in binary, if $i \ge \log n$, return the corresponding bit in a literal representation of $x_{1 + \log n}...x_n$. For this program, we need to encrypt $\log n$ and $x_{1 + \log n}...x_n$. [The value $n$ can be recovered from $\log n$ and the length of the program.]