Similarity as Thermodynamic Work: Between Depth and Diversity -- from Information Distance to Ugly Duckling

Abstract

Defining similarity is a fundamental challenge in information science. Watanabe's Ugly Duckling Theorem highlights diversity, while algorithmic information theory emphasizes depth through Information Distance. We propose a statistical-mechanical framework that treats program length as energy, with a temperature parameter unifying these two aspects: in the low-temperature limit, similarity approaches Information Distance; in the high-temperature limit, it recovers the indiscriminability of the Ugly Duckling theorem; and at the critical point, it coincides with the Solomonoff prior. We refine the statistical-mechanical framework by introducing regular universal machines and effective degeneracy ratios, allowing us to separate redundant from core diversity. This refinement yields new tools for analyzing similarity and opens perspectives for information distance, model selection, and non-equilibrium extensions.