Limits on the Computational Power of Random Strings

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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.