Random Strings Make Hard Instances

We establish the truth of the "instance complexity conjecture" in the case of DEXT-complete sets over polynomial time computations, and r.e. complete sets over recursive computations. Specifically, we obtain for every DEXT-complete set A an exponentially dense subset C and a constant k such that for every nondecreasing polynomial t(n) = \Omega\Gamma n k ), ic t (x : A) K t (x) \Gamma c holds for some constant c and all x 2 C, where ic t and K t are the t-bounded instance complexity and Kolmogorov complexity measures, respectively. For any r.e. complete set A we obtain an infinite set C ` ¯ A such that ic(x : A) K(x) \Gamma c holds for some constant c and all x 2 C, where ic and K denote the time-unbounded versions of instance and Kolmogorov complexities, respectively. The proofs are based on the observation that Kolmogorov random strings are individually hard to recognize by bounded computations. 1 Introduction The notion of "instance complexity" was introduced in [6] to ..