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. Specifically, we obtain for every DEXT-complete set A an exponentially dense subset C such that for every nondecreasing polynomial t(n)=ω(n log n), ic^t(x:A)⩾K^t(x)-c holds for some constant c and all x∈C, where ic^t and K^t are the t-bounded instance complexity and Kolmogorov complexity measures, respectively. For r.e. complete sets A we obtain an infinite set C⊆A¯ such that ic^∞(x:A)⩾K^∞ (x)-c holds for some constant c and all x∈C. The proofs are based on the observation that Kolmogorov random strings are individually hard to recognize by bounded computations