An operational characterization of the notion of probability by algorithmic randomness and its applications (Proof Theory, Computation Theory and Related Topics)
孝太郎 只木 · Kyoto University Research Information Repository (Kyoto University) · 2015
The notion of probability plays an important role in almost all areas of science.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 algo- rithmic randomness we present an operational characterization of the notion of probability.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-L\"of randomness with respect to Bernoulli measure to present the operational charac- terization.As the first step of the research of this line, in this paper we only consider the case of finite probability space, i.e., the case where the sample space of the underlying probability space is finite, for simplicity.In the paper we make applications of our formalism to information theory, cryptography, and quan- tum mechanics.