On characterizations of randomized computation using plain Kolmogorov complexity

Allender, Friedman, and Gasarch recently proved an upper bound of PSPACE for the class DTTR K 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 DTTR K 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 DTTR C defined by plain Kolmogorov randomness. In this paper, we provide further evidence for these conjectures. First, we show that the time-bounded analogue of DTTR C sits between BPP and PSPACE ∩ P/poly. Next, we show that the class DTTR_C,α obtained from DTTR_C by imposing a super-constant minimum query length restriction on the reduction lies between BPP and PSPACE. Finally, we show that the class P/R_Ct ^=log obtained by further restricting the reduction to ask queries of logarithmic length lies between BPP and ∑₂ ^p ∩ P/poly.