CAS II: Orbits as Models: Kolmogorov's Structure Function under Symmetry

Romie Banerjee

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…