CAS II: Symmetric Partitions as Kolmogorov Models

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Original abstract

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. Vereshchagin's strong models, those computable from the data by a total algorithm, are essentially the cells of simple partitions. We read a partition of binary strings as a hypothesis, with the cell containing x as its model, and 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 a lattice of symmetric partitions, with canonical certificates, canonical costs, and an algebra of hypotheses. The resulting structure function and symmetric sophistication measure which part of the regularity of x is symmetric. For the full symmetric group every partition is symmetric: cells recover all Kolmogorov models, cells of cheap partitions recover exactly the strong models, and normal and strange strings are characterized by symmetry. For GL(n,2) the cells are exactly the linearly homogeneous sets, so linear symmetry is a restricted model class. For nonzero x, the linear-symmetry structure function lies in a band between the sufficiency line and the trivial bound, and both edges are attained: there are stochastic normal strings whose simple structure is invisible to linear symmetry. We also give coordinates on the space of permutation groups: each group is an element of a Burnside ring (its type) together with a permutation (its placement), and restriction moves refine partitions via the Mackey formula. In these coordinates the collapse for the symmetric group is a statement about placement, a linear hypothesis is determined by its type up to n^2 bits, and the maximal gap theorem shows that any space of symmetry hypotheses small enough to search is small enough to miss simple structure.

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