汉明球中的最大Kolmogorov复杂度

中文摘要 AI 辅助

给定字符串x的汉明距离r内的字符串的最小Kolmogorov复杂度是x的算法率失真函数，Vereshchagin和Vitanyi完全刻画了其可能的形状。本文关注相反的极端情况。对于长度为n的二进制字符串x，令g_x(r)表示x的汉明距离r内的字符串的最大Kolmogorov复杂度；我们研究该量能取哪些值，更一般地，作为r的函数能取哪些形式。首先，我们在加性误差O(log n)内刻画三元组(C(x),r,g_x(r))的可能取值：记r_k为基数约为2^k的汉明球的半径，三元组(k,r,l)可实现当且仅当log V(r_k + r) < l < min{n, k+log V(r)}，其中V(a)是半径为a的球的基数。特别地，当r_k+r > n/2时，两个界都坍缩为n，仅l = n可实现。该区间的两个端点对应于在立方体中放置复杂度为k的集合的两种极端方式：单个汉明球，其中下界来自Harper的等周不等式；以及纠错码，对于中间参数我们将其放宽为具有有界覆盖重数的中心族，类似于列表解码。然后我们转向整个函数r -> g_x(r)：我们建立它始终满足的四个性质，并表明与这些性质一致的最小和最大函数在每个复杂度水平k下都能达到。哪些中间轮廓可实现仍然开放。

英文摘要

The minimal Kolmogorov complexity of a string within Hamming distance r of a given string x is the algorithmic rate-distortion function of x, and Vereshchagin and Vitanyi characterized completely which shapes it can have. This paper is about the opposite extreme. For a binary string x of length n let g_x(r) denote the maximal Kolmogorov complexity of a string within Hamming distance r of x; we study which values, and more generally which functions of r, this quantity can attain. First we characterize, up to an additive error O(log n), the possible values of the triple (C(x),r,g_x(r)): writing r_k for the radius of a Hamming ball of cardinality about 2^k, a triple (k,r,l) is realizable if and only if log V(r_k + r) < l < min{n, k+log V(r)}, where V(a) is the cardinality of a ball of radius a. In particular, for r_k+r > n/2 both bounds collapse to n and only l = n is realizable. The two ends of this interval correspond to the two extreme ways of placing a set of complexity k in the cube: a single Hamming ball, where the lower bound comes from Harper's isoperimetric inequality, and an error-correcting code, which for the intermediate parameters we relax to a family of centers with bounded covering multiplicity, in the spirit of list decoding. Then we turn to the function r -> g_x(r) as a whole: we establish four properties that it always has, and show that the minimal and the maximal functions consistent with these properties are both attained, for every complexity level k. Which intermediate profiles are attainable remains open.

发表机构

• Centro Nacional de Inteligencia Artificial (CENIA)（智利国家人工智能中心）
• Moscow State University（莫斯科国立大学）
• HSE University（高等经济大学）
• Yandex

机构由 AI 辅助整理，请以论文原文为准。