ICALP 2011: Automata, Languages and Programming pp 293-304

# Limits on the Computational Power of Random Strings

• Eric Allender
• Luke Friedman
• William Gasarch
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6755)

## 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.

## Keywords

Boolean Function Turing Machine Complexity Class Winning Strategy Kolmogorov Complexity
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Authors and Affiliations

• Eric Allender
• 1
• Luke Friedman
• 1
• William Gasarch
• 2
1. 1.Department of Computer ScienceRutgers UniversityPiscatawayUSA
2. 2.Dept. of Computer ScienceUniversity of MarylandCollege ParkUSA