Thermodynamic depth of causal states: Objective complexity via minimal representations

Thermodynamic depth is an appealing but flawed structural 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 arbitrary. Computational mechanics, an alternative approach to structural complexity, 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. To fix this, we redefine the depth in terms of the causal state representation---{ε}-machines---and show that this representation gives the minimum dive consistent with accurate prediction. Thus, {ε}-machines are optimally shallow.