FRG: Collaborative Research: Algorithmic Randomness

FRG：协作研究：算法随机性

• 批准号: 0652582
• 负责人: Eric Allender
• 金额: $ 2.46万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0652582&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Algorithmic; Randomness

This Focused Research Group is a collaborative effort by researchers at many sites who bring ideas from recursion theory, complexity theory, and other specialties to bear on questions about algorithmic randomness. Important background notions include the ideas of Kolmogorov complexity and Martin-Lof randomness, which have separately and jointly received large amounts of attention, and which come together in many of the examples and problems described in this proposal. Issues to be studied during the project include relationships between Martin-Lof random sets and Hausdorff dimension or other measures of dimension, methods for extracting randomness from a semi-random source of data, dimensions and other properties of complexity classes of strings, distinctive properties of sets with low Kolmogorov complexity, and relationships between algorithmic randomness and reverse mathematics, which seeks to understand the axiomatic strength required by particular theories.The forms of randomness studied by this group of researchers are based on some appealing ideas regarding infinite strings, such as the record of an infinitely repeated series of coin tosses. Intuitively, the Kolmogorov complexity of a binary string like the record of heads and tails from coin tosses is the length of the shortest definitive description of the string. Digitization methods for voice and picture transmission take advantage of the regularity and repetition in typical voice signals or digitized images, using much less space or time to record the sound or image data than might seem necessary.From the point of view of Kolmogorov complexity, a genuinely random binary string is probably its own shortest description, or nearly so.Some of the problems studied by this research group seek to establish properties of subsets of strings that have the same complexity, such as their dimension. Activities of the group will include workshops, summer schools for graduate students, and travel for collaboration.

这个重点研究小组是来自多个地点的研究人员的共同努力，他们将递归理论、复杂性理论和其他专业的思想引入到算法随机性问题的研究中。 重要的背景概念包括柯尔莫哥洛夫复杂性和马丁-洛夫随机性的思想，它们分别和共同受到了大量的关注，并且在本提案中描述的许多示例和问题中得到了结合。 该项目期间要研究的问题包括 Martin-Lof 随机集与 Hausdorff 维数或其他维数度量之间的关系、从半随机数据源中提取随机性的方法、字符串复杂性类别的维数和其他属性、字符串的独特属性具有低柯尔莫哥洛夫复杂度的集合，以及算法随机性和逆向数学之间的关系，旨在理解特定理论所需的公理强度。这组研究人员研究的随机性形式基于一些关于无限的吸引人的想法字符串，例如无限重复的一系列抛硬币的记录。 直观地说，二进制字符串（如抛硬币的正面和反面记录）的柯尔莫哥洛夫复杂度是该字符串的最短确定描述的长度。 用于语音和图像传输的数字化方法利用了典型语音信号或数字化图像中的规律性和重复性，使用比看起来必要的少得多的空间或时间来记录声音或图像数据。从柯尔莫哥洛夫复杂度的角度来看，真正的随机二进制字符串可能是它自己最短的描述，或者几乎是这样。该研究小组研究的一些问题试图建立具有相同复杂性的字符串子集的属性，例如它们的维度。 该小组的活动将包括研讨会、研究生暑期学校以及合作旅行。