🤖 AI Summary
This work develops a novel algorithmic information theory within the framework of symmetric groups to characterize string complexity induced by symmetries. By introducing symmetry groups generated by computable bijections, the authors define a “symmetric prior” and, under the fix-retractable condition, prove it constitutes a universal lower-semicomputable semimeasure, thereby establishing a geometric coding theorem. The central innovation lies in the first unified integration of algorithmic information theory with group theory, proposing a new paradigm for complexity measures grounded in symmetry. Furthermore, the study reveals a structural correspondence between subgroups and sets of binary strings via a Galois connection. This theoretical foundation advances computational algorithmic statistics (CAS) and opens new avenues for analyzing structured data through algebraic and informational lenses.
📝 Abstract
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, called symmetries and define the symmetry prior of a string as the probability that a randomly chosen symmetry from a given group has the string as its unique fixed point. We show that for any fix-retractable symmetry group, 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.
We also develop a Galois connection between subgroups of G and subsets of binary strings, characterizing closed points and maximal closed subgroups, and explore the join-semilattice of dense subgroups. Our results unify algorithmic information theory with group theory and provide a framework for studying symmetry-induced complexity measures.
This paper is the first in a series on Computational Algorithmic Statistics (CAS).