โ Scribed by Harry Buhrman; Pekka Orponen
No coin nor oath required. For personal study only.
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)=0(n k ), 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 any r.e. complete set 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, 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.
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