We present a novel formal interpretation of dynamical hierarchies based on information theory, in which each level is a near-state-determined system, and levels are related to one another in a partial ordering. This reformulation moves away from previous definitions, which have considered unique hierarchies of structures or objects arranged in aggregates. Instead, we consider hierarchies of dynamical systems: these are more suited to describing living systems, which are not mere aggregates, but organizations. Transformations from lower to higher levels in a hierarchy are redescriptions that lose information. There are two criteria for partial ordering. One is a state-dependence criterion enforcing predictability within a level. The second is a distinctness criterion enforcing the idea that the higher-level description must do more than just throw information away. We hope this will be a useful tool for empirical studies of both computational and physical dynamical hierarchies.

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