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Computational mechanics can prove that the causal states obtained through the <math>\epsilon</math>-machine have three important characteristics: "maximum predictability", "minimum statistical complexity", and "minimum randomness", and it is verified that it is optimal in a certain sense. In addition, the author introduces a hierarchical machine reconstruction algorithm that can calculate causal states and <math>\epsilon</math>-machines from observational data. Although this algorithm may not be applicable to all scenarios, the author takes chaotic dynamics, hidden Markov models, and cellular automata as examples and gives numerical calculation results and corresponding machine reconstruction paths.

Computational mechanics can prove that the causal states obtained through the <math>\epsilon</math>-machine have three important characteristics: "maximum predictability", "minimum statistical complexity", and "minimum randomness", and it is verified that it is optimal in a certain sense. In addition, the author introduces a hierarchical machine reconstruction algorithm that can calculate causal states and <math>\epsilon</math>-machines from observational data. Although this algorithm may not be applicable to all scenarios, the author takes chaotic dynamics, hidden Markov models, and cellular automata as examples and gives numerical calculation results and corresponding machine reconstruction paths.

Although the original computational mechanics does not give a clear definition and quantitative theory of emergence, some researchers later further advanced the development of this theory. Shalizi et al.<ref name="The_calculi_of_emergence"></ref> discussed the relationship between computational mechanics and emergence in their work. If process <math>{\overleftarrow{s}}'</math> has higher prediction efficiency than process <math>\overleftarrow{s}</math>, then emergence occurs in process <math>{\overleftarrow{s}}'</math>. The prediction efficiency <math>e</math> of a process is defined as the ratio of its excess entropy to its statistical complexity (<math>e=\frac{E}{C_{\mu}}</math>). <math>e</math> is a real number between 0 and 1. We can regard it as a part of the historical memory stored in the process. In two cases, <math>C_{\mu}=0</math>. One is that this process is completely uniform and deterministic; the other is that it is independently and identically distributed. In both cases, there cannot be any interesting predictions, so we set <math>e=0</math>. At the same time, the author explains that emergence can be understood as a dynamical process in which a pattern gains the ability to adapt to different environments.

Although the original computational mechanics does not give a clear definition and quantitative theory of emergence, some researchers later further advanced the development of this theory. Shalizi et al. discussed the relationship between computational mechanics and emergence in their work. If process <math>{\overleftarrow{s}}'</math> has higher prediction efficiency than process <math>\overleftarrow{s}</math>, then emergence occurs in process <math>{\overleftarrow{s}}'</math>. The prediction efficiency <math>e</math> of a process is defined as the ratio of its excess entropy to its statistical complexity (<math>e=\frac{E}{C_{\mu}}</math>). <math>e</math> is a real number between 0 and 1. We can regard it as a part of the historical memory stored in the process. In two cases, <math>C_{\mu}=0</math>. One is that this process is completely uniform and deterministic; the other is that it is independently and identically distributed. In both cases, there cannot be any interesting predictions, so we set <math>e=0</math>. At the same time, the author explains that emergence can be understood as a dynamical process in which a pattern gains the ability to adapt to different environments.

The causal emergence framework has many similarities with computational mechanics. All historical processes <math>\overleftarrow{s}</math> can be regarded as microscopic states. All <math>R \in \mathcal{R}</math> correspond to macroscopic states. The function <math>\eta</math> can be understood as a possible coarse-graining function. The causal state <math>\epsilon \left ( \overleftarrow{s} \right )</math> is a special state that can at least have the same predictive power as the microscopic state <math>\overleftarrow{s}</math>. Therefore, <math>\epsilon</math> can be understood as an effective coarse-graining strategy. Causal transfer <math>T</math> corresponds to effective macroscopic dynamics. The characteristic of minimum randomness characterizes the determinism of macroscopic dynamics and can be measured by effective information in causal emergence.

The causal emergence framework has many similarities with computational mechanics. All historical processes <math>\overleftarrow{s}</math> can be regarded as microscopic states. All <math>R \in \mathcal{R}</math> correspond to macroscopic states. The function <math>\eta</math> can be understood as a possible coarse-graining function. The causal state <math>\epsilon \left ( \overleftarrow{s} \right )</math> is a special state that can at least have the same predictive power as the microscopic state <math>\overleftarrow{s}</math>. Therefore, <math>\epsilon</math> can be understood as an effective coarse-graining strategy. Causal transfer <math>T</math> corresponds to effective macroscopic dynamics. The characteristic of minimum randomness characterizes the determinism of macroscopic dynamics and can be measured by effective information in causal emergence.