Thermodynamic depth of causal states: Objective complexity via minimal representations

Thermodynamic depth is an appealing but flawed structural complexity measur e. 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 consid ered how to select these states. Depth, therefore, is at root arbitrary. Co mputational mechanics, an alternative approach to structural complexity, pr ovides a definition for a system's minimal, necessary causal states and a p rocedure for finding them. We show that the rate of increase in thermodynam ic depth, or dive, is the system's reverse-time Shannon entropy rate, and s o depth only measures degrees of macroscopic randomness, not structure. To hx this, we redefine the depth in terms of the causal state representation- epsilon-machines-and show that this representation gives the minimum dive c onsistent with accurate prediction. Thus, E-machines are optimally shallow. [S1063-651X(99)12401-2].