When Worlds Collide: Derandomization, Lower Bounds, and Kolmogorov Complexity
FST TCS '01 Proceedings of the 21st Conference on Foundations of Software Technology and Theoretical Computer Science
Nearly bounded error probabilistic sets
CIAC'03 Proceedings of the 5th Italian conference on Algorithms and complexity
The dot-depth and the polynomial hierarchy correspond on the delta levels
DLT'04 Proceedings of the 8th international conference on Developments in Language Theory
| Hi-index | 0.00 |
We study the set of incompressible strings for various resource bounded versions of Kolmogorov complexity. The resource bounded versions of Kolmogorov complexity we study are polynomial time CD complexity defined by Sipser, the nondeterministic variant CND due to Buhrman and Fortnow, and the polynomial space bounded Kolmogorov complexity CS introduced by Hartmanis. For all of these measures we define the set of random strings $\mathrm{R}^{\mathit{CD}}_t$, $\mathrm{R}^{\mathit{CND}}_t$, and $\mathrm{R}^{\mathit{CS}}_t$ as the set of strings $x$ such that $\mathit{CD}^t(x)$, $\mathit{CND}^t(x)$, and $\mathit{CS}^s(x)$ is greater than or equal to the length of $x$ for $s$ and $t$ polynomials. We show the following: $\mathrm{MA} \subseteq \mathrm{NP}^{\mathrm{R}^{\mathit{CD}}_t}$, where $\mathrm{MA}$ is the class of Merlin--Arthur games defined by Babai. $\mathrm{AM} \subseteq \mathrm{NP}^{\mathrm{R}^{\mathit{CND}}_t}$, where $\mathrm{AM}$ is the class of Arthur--Merlin games. $\mathrm{PSPACE} \subseteq \mathrm{NP}^{\mathrm{cR}^{\mathit{CS}}_s}$. In the last item $\mathrm{cR}^{\mathit{CS}}_s$ is the set of pairs $\langle x,y \rangle$ so that x is random given y. These results show that the set of random strings for various resource bounds is hard for complexity classes under nondeterministic reductions.This paper contrasts the earlier work of Buhrman and Mayordomo where they show that for polynomial time deterministic reductions the set of exponential time Kolmogorov random strings is not complete for EXP.