概率概念的运算随机性刻画及其应用
An operational characterization of the notion of probability by algorithmic randomness and its applications
摘要：
概率的概念在几乎所有的科学和技术领域都起着重要的作用。然而，在现代数学中，概率论只意味着测度论，概率概念的运算特征尚未建立。在本文中，基于工具包的算法随机性，我们提出了一个操作表征的概念的概率，称为合奏。数学随机性，也被称为算法信息论，是一个数学领域，它使我们能够考虑一个单独的无穷序列的随机性。我们使用的概念，马丁-勒夫随机性的伯努利措施，目前的业务特征。作为这条线研究的第一步，本文考虑有限概率空间的情况，即，为简单起见，基本概率空间的样本空间是有限的情况。给出了条件概率概念在系综下的一个自然操作刻画，并在此基础上给出了两个事件之间独立性概念的等价刻画，进而给出了任意数目的事件/随机变量之间独立性概念在系综下的等价刻画.此外，我们表明，事件/随机变量的独立性是等价的范Lambalgen定理的意义上的独立性，在潜在的有限概率空间是可计算的情况下。在本文中，我们使我们的框架的应用程序，信息论，密码学，并模拟一个有偏见的硬币使用公平的硬币，以证明我们的框架的广泛适用性，一般领域的科学和技术。
英文摘要：
The notion of probability plays an important role in almost all areas of science and technology. In modern mathematics, however, probability theory means nothing other than measure theory, and the operational characterization of the notion of probability is not established yet. In this paper, based on the toolkit of algorithmic randomness we present an operational characterization of the notion of probability, called an ensemble. Algorithmic randomness, also known as algorithmic information theory, is a field of mathematics which enables us to consider the randomness of an individual infinite sequence. We use the notion of Martin-Loef randomness with respect to Bernoulli measure to present the operational characterization. As the first step of the research of this line, in this paper we consider the case of finite probability space, i.e., the case where the sample space of the underlying probability space is finite, for simplicity. We give a natural operational characterization of the notion of conditional probability in terms of ensemble, and give equivalent characterizations of the notion of independence between two events based on it. Furthermore, we give equivalent characterizations of the notion of independence of an arbitrary number of events/random variables in terms of ensembles. Moreover, we show that the independence of events/random variables is equivalent to the independence in the sense of van Lambalgen's Theorem, in the case where the underlying finite probability space is computable. In the paper we make applications of our framework to information theory, cryptography, and the simulation of a biased coin using fair coins, in order to demonstrate the wide applicability of our framework to the general areas of science and technology.
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