有意义的信息
Meaningful Information
摘要：
单个有限对象（如二进制字符串）中的信息通常由其柯尔莫哥洛夫复杂度来衡量。我们可以把这些信息分为两部分：说明对象中存在的有用规律性的信息和说明剩余的偶然信息的信息。可以有几种方式（模型类）来表达规则性。Kolmogorov提出了有限集合的模型类，后来推广到可计算的概率质量函数。由此产生的理论，被称为算法统计，分析算法充分统计时，统计数据被限制到给定的模型类。然而，最通用的方法可能是将有用的信息表示为递归函数。由此产生的测量被称为对象的“放大”。我们发展了递归函数统计、最大值和最小值、绝对非随机对象的存在性理论（具有最大复杂性-其中的所有信息都是有意义的，并且没有剩余随机性），确定其与有限集的更受限制的模型类的关系，以及可计算的概率分布，特别是关于算法（Kolmogorov）最小充分统计量，与停机问题的关系以及进一步的算法性质。
英文摘要：
The information in an individual finite object (like a binary string) is commonly measured by its Kolmogorov complexity. One can divide that information into two parts: the information accounting for the useful regularity present in the object and the information accounting for the remaining accidental information. There can be several ways (model classes) in which the regularity is expressed. Kolmogorov has proposed the model class of finite sets, generalized later to computable probability mass functions. The resulting theory, known as Algorithmic Statistics, analyzes the algorithmic sufficient statistic when the statistic is restricted to the given model class. However, the most general way to proceed is perhaps to express the useful information as a recursive function. The resulting measure has been called the ``sophistication'' of the object. We develop the theory of recursive functions statistic, the maximum and minimum value, the existence of absolutely nonstochastic objects (that have maximal sophistication--all the information in them is meaningful and there is no residual randomness), determine its relation with the more restricted model classes of finite sets, and computable probability distributions, in particular with respect to the algorithmic (Kolmogorov) minimal sufficient statistic, the relation to the halting problem and further algorithmic properties.
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