Limits on the computational power of random strings

  • Authors:

  • Eric Allender;Luke Friedman;William Gasarch

  • Affiliations:

  • Dept. of Computer Science, Rutgers University, New Brunswick, NJ 08855, USA;Dept. of Computer Science, Rutgers University, New Brunswick, NJ 08855, USA;Dept. of Computer Science, University of Maryland, College Park, MD 20742, USA

  • Venue:
  • Information and Computation
  • Year:
  • 2013

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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 P^R and NP^R. The two most widely-studied notions of Kolmogorov complexity are the ''plain'' complexity C(x) and ''prefix'' complexity K(x); this gives rise to two common ways to define the set of random strings ''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) has shown:*BPP@?{A:A=