随机性：它是什么以及为何重要？
Randomness: what is it and why does it matter?
AI总结：
本文针对随机性缺乏严格定义的问题，提出一种基于信息论、聚焦随机数生成物理过程的新定义，并引入“随机性偏差”度量生成质量，推导了马尔可夫过程下的表达式，为随机性的理解、证明与实际应用提供完整工具。
摘要：
随机性是众多重要应用的关键资源，例如蒙特卡洛模拟与计算、生成式人工智能以及密码学。但随机性究竟是什么？迄今为止，一个被广泛接受的定义仍未被研究者找到，然而若没有这样的定义，任何依赖随机性概念的工作都缺乏科学严谨性，其结果也值得怀疑。在此，我提出一种基于信息论的随机性定义，与以往尝试不同，该定义不列举所生成数字序列的期望性质，而是聚焦于随机数生成这一物理过程本身。这一方法所得出的定义符合我们对随机性的直觉认知。文中证明该定义非空且可验证。此外，引入一个名为“随机性偏差”的新量，可用于实际度量随机数生成过程或设备的质量。针对实践中常见的马尔可夫过程，推导出了该量的表达式。最后，一个面向过程的随机数序列定义补全了理解、证明和实际使用随机性所需的工具箱。
英文摘要：
Randomness is a crucial resource for a broad range of important applications, such as Monte Carlo simulation and computation, generative artificial intelligence and cryptography. But what is randomness? A widely accepted definition has eluded researchers thus far, yet without one any work that relies on notion of randomness lacks scientific rigor and its results are questionable. Here, I propose an information-theory-based definition of randomness which, unlike previous attempts, does not list desired properties of the generated number sequence, but rather focuses on the physical process of random number generation itself. This approach results in a definition which complies with our intuitive perception of randomness. It is demonstrated to be non-empty and verifiable. Moreover, a new quantity named "randomness deviation" allows for a practical measure of quality of a random number generating process or a device. An expression for it is derived for a Markovian process, which is frequently encountered in practice. Finally, a process-oriented definition of a random number sequence completes the toolbox needed for understanding, proving, and practical use of randomness.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
