Space-bounded online Kolmogorov complexity is additive

Abstract

The even online Kolmogorov complexity of a string $x = x_1 x_2 \cdots x_{n}$ is the minimal length of a program that for all $i\le n/2$, on input $x_1x_3 \cdots x_{2i-1}$ outputs $x_{2i}$. The odd complexity is defined similarly. The sum of the odd and even complexities is called the dialogue complexity. In [Bauwens, 2014] it is proven that for all $n$, there exist $n$-bit $x$ for which the dialogue complexity exceeds the Kolmogorov complexity by $n\log \frac 4 3 + O(\log n)$. Let $\mathrm C^s(x)$ denote the Kolmogorov complexity with space bound~$s$. Here, we prove that the space-bounded dialogue complexity with bound $s + 6n + O(1)$ is at most $\mathrm C^{s}(x) + O(\log (sn))$, where $n=|x|$.