Derandomizing from random strings

In this paper we show that BPP is truth-table reducible to the set of Kolmogorov random strings R(K). It was previously known that PSPACE, and hence BPP is Turing-reducible to R(K). The earlier proof relied on the adaptivity of the Turing-reduction to find a Kolmogorov-random string of polynomial length using the set R(K) as oracle. Our new non-adaptive result relies on a new fundamental fact about the set R(K), namely each initial segment of the characteristic sequence of R(K) has high Kolmogorov complexity. As a partial converse to our claim we show that strings of very high Kolmogorov-complexity when used as advice are not much more useful than randomly chosen strings.