Randomness is hard

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 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 R/sub t//sup CD/, R/sub t//sup CND/, and R/sub s//sup CS/ as the set of strings x such that CD/sup t/(x), CND/sup t/(x), and CS/sup s/(x) is greater than or equal to the length of x, for s and t polynomials. We show the following: MA/spl sube/NP(R/sub t//sup CD/), where MA is the class of Merlin-Arthur games defined by Babai. AM/spl sube/NP(R/sub t//sup CND/), where AM is the class of Arthur-Merlin games. PSPACE/spl sube/NP(/sub s//sup CS/). 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.