FRG: Collaborative Research: Algorithmic Randomness

这个重点研究小组是由许多网站的研究人员共同努力的结果，他们从递归理论、复杂性理论和其他专业中提出了关于算法随机性的问题。重要的背景概念包括Kolmogorov复杂性和Martin-Lof随机性，它们分别或共同受到了大量关注，并在本提案中描述的许多示例和问题中结合在一起。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、字符串复杂度类的维数和其他性质、低Kolmogorov复杂度集的独特性质、算法随机性与逆向数学之间的关系。它试图理解特定理论所需要的公理力量。这组研究人员所研究的随机性的形式是基于一些关于无限字符串的吸引人的想法，比如对一系列无限重复的抛硬币的记录。直观地说，二元字符串的Kolmogorov复杂度，比如抛硬币时正面和反面的记录，是对字符串的最短确定描述的长度。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的规律性和重复性，使用比可能需要的少得多的空间或时间来记录声音或图像数据。从Kolmogorov复杂度的角度来看，一个真正随机的二进制字符串可能是它自己最短的描述，或者几乎是这样。这个研究小组研究的一些问题寻求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、研究生暑期学校和合作旅行。

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.