AI 中文总结
本文在对称群背景下建立经典编码定理类似物，通过定义对称先验，证明其为通用下半可计算半测度，建立伽罗瓦连接等，统一算法信息论与群论，为研究对称诱导复杂性度量提供框架。
AI 中文摘要
本文在对称群的背景下建立了经典编码定理的直接类似物。我们考虑二进制字符串集合上的可计算双射，称为对称性，并将字符串的对称先验定义为从给定群中随机选择的对称性以该字符串为其唯一不动点的概率。我们表明，对于任何固定可收缩对称群，即允许一个可计算截面为每个字符串选择一个隔离对称性的群，对称先验是一个通用的下半可计算半测度。在这种情况下，几何编码定理成立。我们还在G的子群和二进制字符串子集之间建立了伽罗瓦连接，刻画了闭点和最大闭子群，并探索了稠密子群的并半格。我们的结果将算法信息论与群论统一起来，并为研究对称诱导的复杂性度量提供了一个框架。本文是关于计算算法统计（CAS）系列的第一篇。
英文摘要
This paper establishes a direct analogue of the classical Coding Theorem in the setting of symmetry groups. We consider computable bijections on the set of binary strings and define the symmetry prior of a string x as the probability that a randomly chosen symmetry from a given group G has x as its unique fixed point. We show that for any fix-retractable symmetry group G, a group admitting a computable section that selects an isolating symmetry for every string, the symmetry prior is a universal lower semi-computable semi-measure. In this case, the Geometric Coding Theorem holds. This result is a coding-theoretic restatement of a theorem of Trejo, Kreinovich and Longpré, who showed that the complexity of describing a string by a symmetry with that string as its unique fixed point equals its Kolmogorov complexity. Our contribution is to recast it in terms of algorithmic probability and to treat the symmetry group as a parameter, identifying fix-retractability as exactly the condition under which symmetry complexity collapses onto Kolmogorov complexity.
