AI 中文总结
本文在对称划分上建立算法统计学，通过群作用刻画字符串规律性的对称部分，并证明线性对称模型类存在局限性，同时给出置换群空间的坐标表示。
AI 中文摘要
在算法统计学中，一个字符串 x 由包含它的有限集合来解释，而柯尔莫哥洛夫的结构函数记录了在每个复杂度水平上最小的此类模型。Vereshchagin 的强模型，即通过全算法从数据中可计算的模型，本质上是简单划分的单元。我们将二进制字符串的划分视为一个假设，其中包含 x 的单元作为其模型，并发展对称划分上的算法统计学：即群作用于字符串的轨道划分。子群与划分之间的伽罗瓦连接为每个环境群提供了一个对称划分的格，具有规范证书、规范代价和假设代数。由此产生的结构函数和对称复杂度度量刻画了 x 的规律性中哪些部分是对称的。对于完全对称群，每个划分都是对称的：单元恢复所有柯尔莫哥洛夫模型，廉价划分的单元恰好恢复强模型，并且正常字符串和奇异字符串通过对称性来刻画。对于 GL(n,2)，单元恰好是线性齐次集合，因此线性对称是一个受限的模型类。对于非零 x，线性对称结构函数位于充分性线和平凡界之间的带状区域，并且两个边界均可达到：存在随机正常字符串，其简单结构对线性对称不可见。我们还给出了置换群空间上的坐标：每个群是 Burnside 环的一个元素（其类型）连同置换（其位置），而限制移动通过 Mackey 公式细化划分。在这些坐标中，对称群的坍缩是关于位置的陈述，线性假设由其类型决定，误差不超过 n^2 位，而最大间隙定理表明，任何足够小以可搜索的对称假设空间，也足够小以至于会遗漏简单结构。
英文摘要
In algorithmic statistics a string x is explained by a finite set containing it, and Kolmogorov's structure function records the smallest such model at each level of complexity. Strong models, those computable from the data by a total algorithm, are essentially the cells of simple partitions, so a partition of {0,1}^n can be read as a hypothesis and the cell containing x as the model it assigns. We develop algorithmic statistics over symmetric partitions, the orbit partitions of groups acting on strings. The Galois connection between subgroups and partitions gives each ambient group G a lattice of symmetric partitions with canonical certificates, joins and meets. A set is a cell of a symmetric partition exactly when its setwise stabiliser in G acts transitively on it, so the resulting structure function is Kolmogorov's restricted to these G-homogeneous sets. With a symmetric sophistication, it measures which part of the regularity of x is symmetric. Under the full symmetric group, cells recover all models and cells of cheap partitions recover exactly the strong models, so normal and strange strings are characterised by symmetry. For GL(n,2) the homogeneous sets are the linearly homogeneous ones, whose XOR dependencies look the same from every point. The GL structure function of a nonzero x lies in a band between C(x) - alpha and n - alpha, and both edges are attained: some stochastic normal strings have simple structure invisible to linear symmetry, with sophistication near 0 but GL-sophistication near C(x). We coordinatise permutation groups by a Burnside ring element (type) and a permutation (placement). In these coordinates a linear hypothesis is determined by its type up to n^2 bits, and any space of symmetry hypotheses small enough to search is small enough to miss simple structure. This sets up learned search over orbit models, the subject of later papers in the series.
