Random Strings Make Hard Instances

We establish the truth of the \instance complexity conjecture" in the case of DEXT-complete sets w.r.t. polynomial time computations, and r.e. complete sets w.r.t. recursive computations. Speciically, we obtain for every DEXT-complete set A an exponentially dense subset C such that for every nondecreasing polynomial t(n) = !(n log n), ic t (x : A) K t (x) ? c holds for some constant c and all x 2 C, where ic t and K t are the t-bounded instance complexity and Kol-mogorov complexity measures, respectively. For r.e. complete sets A we obtain an innnite set C A such that ic 1 (x : A) K 1 (x) ? c holds for some constant c and all x 2 C. The proofs are based on the observation that Kolmogorov random strings are individually hard to recognize by bounded computations.