# Thermodynamic depth of causal states: Objective complexity via minimal representations

@article{Crutchfield1999ThermodynamicDO, title={Thermodynamic depth of causal states: Objective complexity via minimal representations}, author={James P. Crutchfield and Cosma Rohilla Shalizi}, journal={Physical Review E}, year={1999}, volume={59}, pages={275-283} }

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{close_quote}s minimal, necessary causal states and a procedure for finding them. We show…

## 124 Citations

### What Is a Macrostate? Subjective Observations and Objective Dynamics

- Mathematics
- 2003

We consider the question of whether thermodynamic macrostates are objective consequences of dynamics, or subjective reflections of our ignorance of a physical system. We argue that they are both;…

### Structure or Noise?

- Computer ScienceArXiv
- 2007

It is shown how rate-distortion theory provides a mechanism for automated theory building by naturally distinguishing between regularity and randomness by constructing an objective function for model making whose extrema embody the trade-off between a model's structural complexity and its predictive power.

### The Thermodynamics of Network Coding, and an Algorithmic Refinement of the Principle of Maximum Entropy †

- Computer ScienceEntropy
- 2019

The analysis provides insight in that the reprogrammability asymmetry appears to originate from a non-monotonic relationship to algorithmic probability, which motivates further analysis of the origin and consequences of the aforementioned asymmetries, reprogmability, and computation.

### Optimally Predictive Causal Inference

- Computer Science
- 2009

It is shown that, in the limit in which a model complexity constraint is relaxed, the filtering method finds the causal architecture of a stochastic dynamical system, known as the causal state partition, and reconstructs exactly the system’s hidden, causal states.

### Minimal history, a theory of plausible explanation

- Computer ScienceComplex.
- 2007

In computational theory, time is defined in terms of steps, and steps are defined by the computational process, which allows time to be measured in bits, which in turn allows the definition of various computable complexity measures.

### Reductions of Hidden Information Sources

- Computer Science
- 2004

It is shown that generators of hidden stochastic processes can be reduced to a minimal form and compared to that provided by computational mechanics – the ε-machine and a new two-step reduction process is introduced.

### A generalized complexity measure based on Rényi entropy

- Computer Science
- 2014

A generalized LMC-Rényi complexity is proposed which overcomes the problem of physical aspects of the internal disorder in atomic and molecular systems which are not grasped by their mother LMC quantity.

### Optimal causal inference: estimating stored information and approximating causal architecture.

- Computer ScienceChaos
- 2010

We introduce an approach to inferring the causal architecture of stochastic dynamical systems that extends rate-distortion theory to use causal shielding--a natural principle of learning. We study…

### Information Symmetries in Irreversible Processes

- Computer ScienceChaos
- 2011

This work examines stationary processes represented or generated by edge-emitting, finite-state hidden Markov models, and shows how to directly calculate a process's fundamental information properties, many of which are otherwise only poorly approximated via process samples.

### High-Probability Trajectories in the Phase Space and the System Complexity

- Computer ScienceComplex Syst.
- 2013

To understand and facilitate the rapid adoption of increasingly more complex computing and communication systems, the metrics that allow us to assess the complexity of a system are investigated and the concept of entropy is discussed.

## References

SHOWING 1-10 OF 19 REFERENCES

### An Introduction to Kolmogorov Complexity and Its Applications

- Computer ScienceTexts and Monographs in Computer Science
- 1993

The book presents a thorough treatment of the central ideas and their applications of Kolmogorov complexity with a wide range of illustrative applications, and will be ideal for advanced undergraduate students, graduate students, and researchers in computer science, mathematics, cognitive sciences, philosophy, artificial intelligence, statistics, and physics.

### Complexity: Hierarchical Structures and Scaling in Physics

- Computer Science
- 1997

Part I. Phenomenology and Models: Examples of complex behaviour and Mathematical models and Thermodynamic formalism.

### Philosophical writings : a selection

- Philosophy
- 1964

Editorial note Editor's introduction Bibliographical note The Notion of Knowledge or Science Epistemological problems Logical problems The theory of 'Supposito' Truth Inferential Operations Being,…

### A First Course In Chaotic Dynamical Systems: Theory And Experiment

- Physics
- 1993

A Mathematical and Historical Tour of Dynamical Systems and Graphical Analysis of the Quadratic Family, and the Role of the Critical Orbit.

### J.E.Hopcroft, J.D. Ullman 著, "Introduction to Automata Theory, Languages, and Computation", Addison-Wesley, A5変形版, X+418, \6,670, 1979

- Computer Science
- 1980

### Phys. Lett. A

- Phys. Lett. A

### J. Stat. Phys

- J. Stat. Phys

### Ann. Phys

- Ann. Phys

### Int. J. Theor. Phys

- Int. J. Theor. Phys