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

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

and

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.

Citation

J. P. Crutchfield and C. R. Shalizi Thermodynamic Depth of Causal States: When Paddling around in Occam's Pool Shallowness Is a Virtue, Santa Fe Insitute Working Paper 98-06-047.

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