Thermodynamic Depth of Causal States: When Paddling around in Occam's Pool Shallowness Is a Virtue
Abstract
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 {\it 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---$\epsilon$-machines---and show that this representation gives the minimum dive consistent with accurate prediction. Thus, $\epsilon$-machines are optimally shallow.
- Publication:
-
arXiv e-prints
- Pub Date:
- August 1998
- DOI:
- 10.48550/arXiv.cond-mat/9808147
- arXiv:
- arXiv:cond-mat/9808147
- Bibcode:
- 1998cond.mat..8147C
- Keywords:
-
- Condensed Matter - Statistical Mechanics;
- Adaptation;
- Noise;
- and Self-Organizing Systems;
- Chaotic Dynamics;
- Nonlinear Sciences - Adaptation and Self-Organizing Systems;
- Nonlinear Sciences - Chaotic Dynamics
- E-Print:
- 11 pages, 9 figures, RevTeX