Randomness and Computability

随机性和可计算性

• 批准号: 0945187
• 负责人: Joseph Miller
• 金额: $ 3.04万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-01-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0945187&HistoricalAwards=false
• 关键词: Randomness; Computability

The two most fundamental notions in effective randomness are Kolmogorov complexity and Martin-Lof randomness. The first measures the information content of finite binary strings, while the second captures the intuition that a random infinite binary sequence has no "distinguishing features". Both notions are rooted in computability theory, so it is not surprising that the study of randomness draws methods and ideas from computability theory, as well as core mathematical subjects like measure theory and information theory.However, the theory of effective randomness has proved to be a rich field of study in its own right. Miller proposes to study the strength of non-monotonic betting strategies, the computable power of sequences with positive effective Hausdorff dimension, degrees of randomness, and K-triviality.A sequence of zeros and ones is considered random if the shortest(binary) computer program that generates the sequence has essentially the same length as the sequence itself. Intuitively, a random sequence has no patterns that can be exploited to give a compressed description. For example, the sequence consisting of a million zeros can be generated easily by a short program, so it is not random. On the other hand, there is a high probability that a sequence generated by flipping a coin a million times (tails is zero, heads is one) will be random. The length of the shortest program for a sequence is called the Kolmogorov complexity of the sequence; this notion was introduced in the 1960s. An infinite sequence is considered random if all of its finite initial segments are sufficiently random. This is equivalent to saying that no (semi-)computable betting strategy can win money trying to predict the digits of the sequence. The proposed research will address fundamental questions about randomness, including: (1) How powerful are computable betting strategies that are allowed to bet on the digits of a sequence in any order; (2) Is it possible to distill information out of a semi-random source to produce a random sequence or, at least, a sequence that has higher information density; (3) What does it mean to say that one infinite sequence is more random that another (4) How computationally weak are the infinite?

有效随机性的两个最基本的概念是柯尔莫哥洛夫复杂性和马丁-洛夫随机性。第一个度量有限二进制字符串的信息内容，而第二个捕获随机无限二进制序列没有“区别特征”的直觉。这两个概念都植根于可计算性理论，所以随机性的研究从可计算性理论以及测度论和信息论等核心数学学科中汲取方法和思想也就不足为奇了。然而，有效随机性理论本身已经被证明是一个丰富的研究领域。米勒建议研究非单调投注策略的强度，具有正有效Hausdorff维数、随机度和K-平凡性的序列的可计算能力。如果生成序列的最短（二进制）计算机程序具有与序列本身基本相同的长度，则0和1的序列被认为是随机的。直觉上，随机序列没有可以用来给出压缩描述的模式。例如，由一百万个零组成的序列可以很容易地由一个简短的程序生成，因此它不是随机的。另一方面，抛硬币一百万次（反面是0，正面是1）产生的序列很有可能是随机的。一个序列的最短程序的长度被称为序列的柯尔莫哥洛夫复杂度;这个概念是在20世纪60年代引入的。一个无限序列被认为是随机的，如果它的所有有限初始段是充分随机的。这相当于说，没有（半）可计算的投注策略可以赢得金钱试图预测的数字序列。拟议的研究将解决有关随机性的基本问题，包括：（1）允许以任何顺序对序列的数字进行投注的可计算投注策略有多强大;（2）是否有可能从半随机源中提取信息以产生随机序列，或者至少是具有更高信息密度的序列;（3）说一个无限序列比另一个更随机是什么意思？（4）无限序列在计算上有多弱？