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=== Early work on quantifying emergence ===

=== Early work on quantifying emergence ===

There have been some related works in the early stages that attempted to quantitatively analyze emergence. The [[computational mechanics]] theory proposed by Crutchfield et al. <ref name=":3">J. P. Crutchfield, K. Young, Inferring statistical complexity, Physical review letters 63 (2) (1989) 105.</ref> considers [[causal states]]. This method explores similar ideas based on the division of state space and is very similar to Erik Hoel's causal emergence theory. On the other hand, Seth et al. proposed the G-emergence theory <ref name=":4">A. K. Seth, Measuring emergence via nonlinear granger causality., in: alife, Vol. 2008, 2008, pp. 545–552.</ref> to quantify emergence by using [[Granger causality]].

There have been some related works in the early stages that attempted to quantitatively analyze emergence. The [[computational mechanics]] theory proposed by Crutchfield et al. <ref name=":3">J. P. Crutchfield, K. Young, Inferring statistical complexity, Physical review letters 63 (2) (1989) 105.</ref> considers [[causal states]]. This theory explores similar ideas based on the division of state space and is very similar to Erik Hoel's causal emergence theory. On the other hand, Seth et al. proposed the [[G-emergence]] theory <ref name=":4">A. K. Seth, Measuring emergence via nonlinear granger causality., in: alife, Vol. 2008, 2008, pp. 545–552.</ref> to quantify emergence by using [[Granger causality]].

==== 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 laws of causality in emergence with 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>.

The goal of computational mechanics is to establish a model that reconstructs and predicts the observed random sequence with high accuracy. However, the randomness of the sequence prevents perfect reconstruction. Therefore, we need a coarse-grained mapping to capture the ordered structure in the random sequence. This coarse-grained mapping can be characterized by a partitioning function <math>\eta: \overleftarrow{S}→\mathcal{R}</math>. It can divide <math>\overleftarrow{S}</math> into several mutually exclusive subsets, where all such subsets collectively form the complete set denoted as <math>\mathcal{R}</math>.

The goal of computational mechanics is to establish a model that reconstructs and predicts the observed time series with high accuracy. However, the randomness of the sequence prevents perfect reconstruction. Therefore, we need a coarse-grained mapping to capture the ordered structure in the random sequence. This coarse-grained mapping can be characterized by a partitioning function <math>\eta: \overleftarrow{S}→\mathcal{R}</math>. It can divide <math>\overleftarrow{S}</math> into several mutually exclusive subsets, where all such subsets collectively form the complete set denoted as <math>\mathcal{R}</math>.

Computational mechanics regards any subset <math>R \in \mathcal{R}</math> as a macroscopic state. For a set of macroscopic states <math>\mathcal{R}</math>, computational mechanics uses Shannon entropy to define an index <math>C_\mu</math> to measure the statistical complexity of these states:

Computational mechanics regards any subset <math>R \in \mathcal{R}</math> as a macroscopic state. For a set of macroscopic states <math>\mathcal{R}</math>, computational mechanics uses Shannon entropy to define an index <math>C_\mu</math> to measure the statistical complexity of these states:

It can be proved that when a set of states is used to build a predictive model, their statistical complexity is approximately equivalent to the size of the prediction model.

It can be proved that when a set of states is used to build a predictive model, their statistical complexity is approximately equivalent to the size of the prediction model.

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>.

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 map <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>, 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.

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 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>.

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 authors introduced 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 authors demonstrated 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 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.

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 a process <math>{\overleftarrow{s}}'</math> has higher prediction efficiency than process <math>\overleftarrow{s}</math>, then emergence occurs in the 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>), where <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 authors explained 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 EI 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 state transitions <math>T</math> corresponds to effective macroscopic dynamics. The characteristic of minimum randomness characterizes the determinism of macroscopic dynamics and can be measured by EI in causal emergence.

==== G-emergence ====

==== G-emergence ====