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 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 R-t(CD), R-t(CND), and R-t(CS) as the set of strings x such that CDt (x), CNDt (x), and CSs (x) is greater than or equal to the length of x for s and t polynomials. We show the following: . MA subset of or equal to NPtRCD, where MA is the class of Merlin Arthur games defined by Babai. . AM subset of or equal to NPtRCND, where AM is the class of Arthur Merlin games. . PSPACE subset of or equal to NPscRCS. In the last item cR(sCS) is the set of pairs x, y 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.
randomness kolmogorov complexity cd complexity complexity classes cnd complexity interactive proofs merlin-arthur arthur-merlin relativization