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 and . 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 or .) Previous work on the power of "R" (for any of these variants [1,2,9]) has shown . . . Since these inclusions hold irrespective of low-level details of how "R" is defined, we have e.g.: . ( 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 . We show: . . Hence, in particular, 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. © 2011 Springer-Verlag.