Limits on the Computational Power of Random Strings

Abstract

How powerful is the set of random strings? What can one say about a set A that is efficiently reducible to R, the set of Kolmogorov-random strings? We present the first upper bound on the class of computable sets in $\mbox{\rm P}^R$ and $\mbox{\rm NP}^R$.

The two most widely-studied notions of Kolmogorov complexity are the “plain” complexity C(x) and “prefix” complexity K(x); this gives two ways to define the set “R”: R _C and R _K . (Of course, each different choice of universal Turing machine U in the definition of C and K yields another variant ${R_{{C}_U}}$ or ${R_{{K}_U}}$.) Previous work on the power of “R” (for any of these variants [1,2,9]) has shown

• $\mbox{\rm BPP} \subseteq \{A : A \mbox{$\leq^{\rm p}_{\it tt}$} R\}$.

• $\mbox{\rm PSPACE} \subseteq \mbox{\rm P}^R$.

• $\mbox{\rm NEXP} \subseteq \mbox{\rm NP}^R$.

Since these inclusions hold irrespective of low-level details of how “R” is defined, we have e.g.: $\mbox{\rm NEXP} \subseteq \mbox{$\Delta^0_1$} \cap \bigcap_U \mbox{\rm NP}^{{R_{{K}_U}}}$. ($\mbox{$\Delta^0_1$}$ is the class of computable sets.)

Our main contribution is to present the first upper bounds on the complexity of sets that are efficiently reducible to ${R_{{K}_U}}$. We show:

• $\mbox{\rm BPP} \subseteq \mbox{$\Delta^0_1$} \cap \bigcap_U \{A : A \mbox{$\leq^{\rm p}_{\it tt}$} {R_{{K}_U}}\}\subseteq \mbox{\rm PSPACE}$.

• $\mbox{\rm NEXP} \subseteq \mbox{$\Delta^0_1$} \cap \bigcap_U \mbox{\rm NP}^{{R_{{K}_U}}}\subseteq \mbox{\rm EXPSPACE}$.

Hence, in particular, $\mbox{\rm PSPACE}$ is sandwiched between the class of sets Turing- and truth-table-reducible to R.

As a side-product, we obtain new insight into the limits of techniques for derandomization from uniform hardness assumptions.