# Thermodynamic Depth of Causal States: When Paddling Around in Occam's Pool Shallowness Is a Virtue

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## Abstract

Thermodynamic depth is an appealing but flawed complexity measure. It depends on a set of macroscopic states for a system, but neither its original introduction by Lloyd and Pagels nor any follow-up work has considered how to select these states. Depth, therefore, is at root subjective. Computational mechanics provides a definition for a system'[s minimal, necessary causal states and a procedure for finding them. We show that the rate of increase in thermodynamic depth, or "dive," is the system's reverse-time Shannon entropy rate, and so depth only measures degrees of macroscopic randomness, not structure. We redefine the depth in terms of the causal state representation--- epsilon-machines---and show that this representation gives the minimum dive consistent with accurate predition. Thus, epsilon-machines are optimally shallow.

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