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==== Computational mechanics ====

==== Computational mechanics ====

The [[computational mechanics]] theory attempts to express the causal laws of emergence in a quantitative framework, specifically by constructing a coarse-grained [[causal model]] from a random process so that this model can generate the time series of the observed random process <ref name=":3" />.

The [[computational mechanics]] theory attempts to express the causal laws of emergence in a quantitative framework, specifically by constructing a coarse-grained [[causal model]] from a random process so that this model can generate the time series of the observed random process <ref name=":3" />.

Here, the random process can be represented by <math>\overleftrightarrow{s}</math>. Based on time <math>t</math>, the [[random process]] can be divided into two parts: the process before time <math>t</math>, <math>\overleftarrow{s_t}</math>, and the process after time <math>t</math>, <math>\overrightarrow{s_t}</math>. The computational mechanics framework denotes the set of all possible historical processes <math>\overleftarrow{s_t}</math> as <math>\overleftarrow{S}</math>, and the set of all future processes <math>\overrightarrow{s_t}</math> as <math>\overrightarrow{S}</math>.

Here, the random process can be represented by <math>\overleftrightarrow{s}</math>. Based on time <math>t</math>, the [[random process]] can be divided into two parts: the process before time <math>t</math>, <math>\overleftarrow{s_t}</math>, and the process after time <math>t</math>, <math>\overrightarrow{s_t}</math>. The computational mechanics framework denotes the set of all possible historical processes <math>\overleftarrow{s_t}</math> as <math>\overleftarrow{S}</math>, and the set of all future processes <math>\overrightarrow{s_t}</math> as <math>\overrightarrow{S}</math>.

Furthermore, in order to achieve the best balance between predictability and simplicity for the set of macroscopic states, computational mechanics defines the concept of [[causal equivalence]]. If <math>P\left ( \overrightarrow{s}|\overleftarrow{s}\right )=P\left ( \overrightarrow{s}|{\overleftarrow{s}}'\right )</math>, then <math>\overleftarrow{s}</math> and <math>{\overleftarrow{s}}'</math> are causally equivalent. This equivalence relation can divide all historical processes into equivalence classes and define them as [[causal states]]. All causal states of the historical process <math>\overleftarrow{s}</math> can be characterized by a mapping <math>\epsilon \left ( \overleftarrow{s} \right )</math>. Here, <math>\epsilon: \overleftarrow{\mathcal{S}}\rightarrow 2^{\overleftarrow{\mathcal{S}}}</math> is a function that maps the historical process <math>\overleftarrow{s}</math> to the causal state <math>\epsilon(\overleftarrow{s})\in 2^{\overleftarrow{\mathcal{S}}}</math>.

Furthermore, to balance predictability and simplicity, computational mechanics defines [[causal equivalence]]. If <math>P\left ( \overrightarrow{s}|\overleftarrow{s}\right )=P\left ( \overrightarrow{s}|{\overleftarrow{s}}'\right )</math>, then <math>\overleftarrow{s}</math> and <math>{\overleftarrow{s}}'</math> are causally equivalent. This equivalence relation can divide all historical processes into equivalence classes and define them as [[causal states]]. All causal states of the historical process <math>\overleftarrow{s}</math> can be characterized by a mapping <math>\epsilon \left ( \overleftarrow{s} \right )</math>. Here, <math>\epsilon: \overleftarrow{\mathcal{S}}\rightarrow 2^{\overleftarrow{\mathcal{S}}}</math> is a function that maps the historical process <math>\overleftarrow{s}</math> to the causal state <math>\epsilon(\overleftarrow{s})\in 2^{\overleftarrow{\mathcal{S}}}</math>.

Further, we can denote the causal transition probability between two [[causal states]] <math>S_i</math> and <math>S_j</math> as <math>T_{ij}^{\left ( s \right )}</math>, which is similar to a coarsened macroscopic dynamics. The <math>\epsilon</math>-machine of a random process is defined as an ordered pair <math>\left \{ \epsilon,T \right \}</math>. This is a pattern discovery machine that can achieve prediction by learning the <math>\epsilon</math> and <math>T</math> functions. This is equivalent to defining the so-called identification problem of emergent causality. Here, the <math>\epsilon</math>-machine is a machine that attempts to discover emergent causality in data.

Further, we can denote the causal transition probability between two [[causal states]] <math>S_i</math> and <math>S_j</math> as <math>T_{ij}^{\left ( s \right )}</math>, which is similar to a coarsened macroscopic dynamics. The <math>\epsilon</math>-machine of a random process is defined as an ordered pair <math>\left \{ \epsilon,T \right \}</math>, which is a pattern discovery machine that can achieve prediction by learning the <math>\epsilon</math> and <math>T</math> functions. This is equivalent to defining the identification problem of emergent causality. Here, the <math>\epsilon</math>-machine identifies emergent causality in data.

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 <ref name="The_calculi_of_emergence">{{cite journal|author1=Crutchfield, J.P|title=The calculi of emergence: computation, dynamics and induction|journal=Physica D: Nonlinear Phenomena|year=1994|volume=75|issue=1-3|page=11-54|url=https://www.sciencedirect.com/science/article/abs/pii/0167278994902739}}</ref>.

Computational mechanics demonstrates that the causal states derived from the ϵ-machine possess three key characteristics: '''maximum predictability ''',  '''minimum statistical complexity ''', and  '''minimum randomness ''' , which are considered optimal. In addition, the author introduces a hierarchical machine reconstruction algorithm that can calculate causal states and ϵ-machines from observational data. Although this algorithm may not be applicable to all scenarios, the author demonstrates its application using examples such as chaotic dynamics, hidden Markov models, and cellular automata, providing numerical results and reconstruction paths <ref name="The_calculi_of_emergence">{{cite journal|author1=Crutchfield, J.P|title=The calculi of emergence: computation, dynamics and induction|journal=Physica D: Nonlinear Phenomena|year=1994|volume=75|issue=1-3|page=11-54|url=https://www.sciencedirect.com/science/article/abs/pii/0167278994902739}}</ref>.

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.

Although the original computational mechanics does not give a clear definition and quantitative theory of emergence, some researchers 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 information about past states 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.

==== G-emergence ====

==== G-emergence ====