Limits on the Computational Power of Random Strings

  • Eric Allender
  • Luke Friedman
  • William Gasarch
Conference paper

DOI: 10.1007/978-3-642-22006-7_25

Volume 6755 of the book series Lecture Notes in Computer Science (LNCS)
Cite this paper as:
Allender E., Friedman L., Gasarch W. (2011) Limits on the Computational Power of Random Strings. In: Aceto L., Henzinger M., Sgall J. (eds) Automata, Languages and Programming. ICALP 2011. Lecture Notes in Computer Science, vol 6755. Springer, Berlin, Heidelberg

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”: RC and RK. (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.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

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