Note:  A short preliminary version of this work was announced at the 39th International Symposium on Mathematical Foundations of Computer Science (MFCS 2014).
Abstract: Allender, Friedman, and Gasarch recently proved an upper bound of PSPACE for the class DTTRK of decidable languages that are polynomial-time truth-table reducible to the set of prefix-free Kolmogorov-random strings regardless of the universal machine used in the definition of Kolmogorov complexity. It is conjectured that DTTRK in fact lies closer to BPP, a lower bound established earlier by Buhrman, Fortnow, Koucký, and Loff. It is also conjectured that we have similar bounds for the analogous class DTTRC defined by plain Kolmogorov randomness. In this paper, we provide further evidence for these conjectures. First, we show that the time-bounded analogue of DTTRC sits between BPP and PSPACE∩P/poly. Next, we show that the class DTTRC,α obtained from DTTRC by imposing a super-constant minimum query length restriction on the reduction lies between BPP and PSPACE. Finally, we show that the class P/RCt=log obtained by further restricting the reduction to ask queries of logarithmic length lies between BPP and Σ2p∩P/poly.