| James P. Crutchfield Physics Department University of California Berkeley, California 94720-7300, USA and Santa Fe Institute 1399 Hyde Park Rd. Santa Fe, NM 87501, USA |
Cosma Rohilla Shalizi Physics Department University of Wisconsin Madison, WI 53706 USA and Santa Fe Institute 1399 Hyde Park Rd. Santa Fe, NM 87501, USA |
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 prediction. Thus, epsilon-machines are optimally shallow.