Random strings make hard instances

H. Buhrman, P. Orponen
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引用次数: 17

Abstract

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)=/spl omega/(n log n), ic/sup t/(x:A)/spl ges/K/sup t/(x)-c holds for some constant c and all x/spl isin/C, where ic/sup t/ and K/sup t/ are the t-bounded instance complexity and Kolmogorov complexity measures, respectively. For r.e. complete sets A we obtain an infinite set C/spl sube/A~ such that ic/sup /spl infin(x:A)/spl ges/K/sup /spl infin(x)-c holds for some constant c and all x/spl isin/C. The proofs are based on the observation that Kolmogorov random strings are individually hard to recognize by bounded computations.<>
随机字符串产生硬实例
在dext完备集w.r.t.多项式时间计算和r.e.完备集w.r.t.递归计算的情况下,证明了“实例复杂度猜想”的成立。具体地说,我们得到了每个dext完备集A的指数密集子集C,使得对于每个非递减多项式t(n)=/spl ω /(n log n), ic/sup t/(x:A)/spl ges/K/sup t/(x)-c对于某些常数C和所有x/spl isin/C成立,其中ic/sup t/和K/sup t/分别是t有界实例复杂度和Kolmogorov复杂度度量。对于r.e.完全集合A,我们得到了一个无限集C/spl sub /A~,使得ic/sup /spl infin(x:A)/spl ges/K/sup /spl infin(x)-c对某常数C和所有x/spl isin/C成立。这些证明是基于观察到Kolmogorov随机字符串很难通过有界计算来识别。
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