What can be efficiently reduced to the Kolmogorov-random strings?

Abstract

We investigate the question of whether one can characterize complexity classes (such as PSPACE or NEXP) in terms of efficient reducibility to the set of Kolmogorov-random strings $R_C$. This question arises because $\mathrm{PSPACE}\subseteq P^{R_C}$ and $\mathrm{NEXP}\subseteq {\mathrm{NP}}^{R_C}$, and no larger complexity classes are known to be reducible to $R_C$ in this way. We show that this question cannot be posed without explicitly dealing with issues raised by the choice of universal machine in the definition of Kolmogorov complexity. What follows is a list of some of our main results.

• Although Kummer showed that, for every universal machine $U$ there is a time bound $t$ such that the halting problem is disjunctive truth-table reducible to $R_{C_U}$ in time $t$, there is no such time bound $t$ that suffices for every universal machine $U$. We also show that, for some machines $U$, the disjunctive reduction can be computed in as little as doubly-exponential time.

• Although for every universal machine $U$, there are very complex sets that are ${\le }_{\mathrm{dtt}}^P$-reducible to $R_{C_U}$, it is nonetheless true that $P=\mathrm{REC}\cap {\bigcap }_U\{A:A{\le }_{\mathrm{dtt}}^P{R}_{C_U}\}$. (A similar statement holds for parity-truth-table reductions.)

• Any decidable set that is polynomial-time monotone-truth-table reducible to $R_C$ is in $P/\mathrm{poly}$.

• Any decidable set that is polynomial-time truth-table reducible to $R_C$ via a reduction that asks at most $f\left(n\right)$ queries on inputs of size $n$ lies in $P/\left(f\left(n\right)2^{f\left(n\right)3logf\left(n\right)}\right)$.