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→Early work on quantifying emergence
=== 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 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]].
==== 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>.
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 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>.
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:
<math>
<math>
C_\mu(\mathcal{R})\triangleq -\sum_{\rho\in \mathcal{R}} P(\mathcal{R}=\rho)\log_2 P(\mathcal{R}=\rho)
C_\mu(\mathcal{R})\triangleq -\sum_{\rho\in \mathcal{R}} P(\mathcal{R}=\rho)\log_2 P(\mathcal{R}=\rho)
</math>
</math>
It can be proved that when a set of states is used to build a prediction 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 prediction 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 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>, 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 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 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 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 ====
The G-emergence theory was proposed by Seth in 2008 and is one of the earliest studies to quantify [[emergence]] from a causal perspective <ref name=":4" />. The basic idea is to use nonlinear [[Granger causality|Granger causality]] to quantify [[weak emergence]] in complex systems.
The G-emergence theory was proposed by Seth in 2008 and is one of the earliest studies to quantify [[emergence]] from a causal perspective <ref name=":4" />. The basic idea is to use nonlinear [[Granger causality|Granger causality]] to quantify [[weak emergence]] in complex systems.
Specifically, if we use a binary autoregressive model for prediction, it involves two variables, A and B. The [[autoregressive model]] comprises two equations, each corresponding to one variable. The current value of each variable is determined by its own past values and the past values of the other variable within a certain time lag range. In addition, the model also calculates residuals. Residuals, or prediction errors, measure the degree of Granger causal effect (G-causality) for each equation. The extent of the Granger causality of B on A is determined by the logarithm of the ratio of residual variances: one from A's model without B and the other from the full model including A and B. The concept of “G-autonomous” measures how well past values of a time series predict its future values. The strength of this autonomous predictive causal effect can be characterized in a similar way to G-causality.
Specifically, if we use a binary autoregressive model for prediction, it involves two variables, A and B. The [[autoregressive model]] comprises two equations, each corresponding to one variable. The current value of each variable is determined by its own past values and the past values of the other variable within a certain time lag range. In addition, the model also calculates residuals. Residuals, or prediction errors, measure the degree of Granger causal effect (G-causality) for each equation. The extent of the Granger causality of B on A is determined by the logarithm of the ratio of residual variances: one from A's model without B and the other from the full model including A and B. The concept of “G-autonomous” measures how well past values of a time series predict its future values. The strength of this autonomous predictive causal effect can be characterized in a similar way to G-causality.
[[文件:G Emergence Theory.png|G-emergence理论图|alt=G-emergence理论图|居左|400x300像素]]
[[文件:G Emergence Theory.png|G-emergence理论图|alt=G-emergence理论图|居左|400x300像素]]
As shown in the figure, the occurrence of emergence can be judged based on two key concepts in the G-causality. This measure of emergence, derived from [[Granger causality|Granger causality]], is denoted as G-emergence. A can be understood as a macroscopic variable, while B represents a microscopic variable. The conditions for emergence to occur are as follows: 1) A is G-autonomous with respect to B; 2) B is a G-cause of A. The degree of G-emergence is calculated by multiplying the degree of A's G-autonomous by the degree of B's average G-cause.
As shown in the figure, the occurrence of emergence can be judged based on two key concepts in the G-causality. This measure of emergence, derived from [[Granger causality|Granger causality]], is denoted as G-emergence. A can be understood as a macroscopic variable, while B represents a microscopic variable. The conditions for emergence to occur are as follows: 1) A is G-autonomous with respect to B; 2) B is a G-cause of A. The degree of G-emergence is calculated by multiplying the degree of A's G-autonomous by the degree of B's average G-cause.
gence theory proposed by Seth is the first attempt to use causal measures to quantify emergence phenomena. However, the causal relationship used by the author is Granger causality, which is not a strict causal relationship. At the same time, the measurement results also depend on the regression method used. In addition, the measurement index of this method is defined according to variables rather than dynamics, which means that the results will depend on the choice of variables. These limitations highlight the drawbacks of the G-emergence theory.
gence theory proposed by Seth is the first attempt to use causal measures to quantify emergence phenomena. However, the causal relationship used by the author is Granger causality, which is not a strict causal relationship. At the same time, the measurement results also depend on the regression method used. In addition, the measurement index of this method is defined according to variables rather than dynamics, which means that the results will depend on the choice of variables. These limitations highlight the drawbacks of the G-emergence theory.
The causal emergence framework also has similarities with the aforementioned G-emergence. The macroscopic states of both methods need to be manually selected. In addition, it should be noted that some of the above methods for quantifying emergence often do not consider true interventionist causality.
The causal emergence framework also has similarities with the aforementioned G-emergence. The macroscopic states of both methods need to be manually selected. In addition, it should be noted that some of the above methods for quantifying emergence often do not consider true interventionist causality.
==== Other theories for quantitatively characterizing emergence ====
==== Other theories for quantitatively characterizing emergence ====
Several other quantitative theories of emergence have been proposed. There are mainly two methods that are widely discussed. One approach views [[emergence]] as the transition from disorder to order. Moez Mnif and Christian Müller-Schloer <ref>Mnif, M.; Müller-Schloer, C. Quantitative emergence. In Organic Computing—A Paradigm Shift for Complex Systems; Springer: Basel, Switzerland, 2011; pp. 39–52. </ref> use [[Shannon entropy]] to measure order and disorder. In the [[self-organization]] process, emergence occurs when order increases. The increase in order is calculated by measuring the difference in Shannon entropy between the initial state and the final state. However, this method depends on the observation level and the system's initial conditions, which limits its applicability. To overcome these two difficulties, the authors propose a measurement method compared with the maximum entropy distribution. Inspired by the work of Moez mif and Christian Müller-Schloer, reference <ref>Fisch, D.; Jänicke, M.; Sick, B.; Müller-Schloer, C. Quantitative emergence–A refined approach based on divergence measures. In Proceedings of the 2010 Fourth IEEE International Conference on Self-Adaptive and Self-Organizing Systems, Budapest, Hungary, 27 September–1 October 2010; IEEE Computer Society: Washington, DC, USA, 2010; pp. 94–103. </ref> suggests using the divergence between two probability distributions to quantify emergence. They understand emergence as an unexpected or unpredictable distribution change based on the observed samples. However, this method suffers from high computational complexity and low accuracy. To solve these problems, reference <ref>Fisch, D.; Fisch, D.; Jänicke, M.; Kalkowski, E.; Sick, B. Techniques for knowledge acquisition in dynamically changing environments. ACM Trans. Auton. Adapt. Syst. (TAAS) 2012, 7, 1–25. [CrossRef] </ref> proposes an approximate method for estimating density using [[Gaussian mixture models]]. Additionally, it introduces [[Mahalanobis distance]] to characterize the difference between data and Gaussian components, resulting in improved outcomes. In addition, Holzer, de Meer et al. <ref>Holzer, R.; De Meer, H.; Bettstetter, C. On autonomy and emergence in self-organizing systems. In International Workshop on Self-Organizing Systems, Proceedings of the Third International Workshop, IWSOS 2008, Vienna, Austria, 10–12 December 2008; Springer: Berlin/Heidelberg, Germany, 2008; pp. 157–169.</ref><ref>Holzer, R.; de Meer, H. Methods for approximations of quantitative measures in self-organizing systems. In Proceedings of the Self-Organizing Systems: 5th International Workshop, IWSOS 2011, Karlsruhe, Germany, 23–24 February 2011; Proceedings 5; Springer: Berlin/Heidelberg, Germany, 2011; pp. 1–15.</ref> proposed another emergence measurement method based on Shannon entropy. They believe that a complex system is a self-organizing process in which different individuals interact through communication. Emergence can be measured as the ratio of the Shannon entropy of all inter-agent communications to the sum of individual Shannon entropies.
Several other quantitative theories of emergence have been proposed. There are mainly two methods that are widely discussed. One approach views [[emergence]] as the transition from disorder to order. Moez Mnif and Christian Müller-Schloer <ref>Mnif, M.; Müller-Schloer, C. Quantitative emergence. In Organic Computing—A Paradigm Shift for Complex Systems; Springer: Basel, Switzerland, 2011; pp. 39–52. </ref> use [[Shannon entropy]] to measure order and disorder. In the [[self-organization]] process, emergence occurs when order increases. The increase in order is calculated by measuring the difference in Shannon entropy between the initial state and the final state. However, this method depends on the observation level and the system's initial conditions, which limits its applicability. To overcome these two difficulties, the authors propose a measurement method compared with the maximum entropy distribution. Inspired by the work of Moez mif and Christian Müller-Schloer, reference <ref>Fisch, D.; Jänicke, M.; Sick, B.; Müller-Schloer, C. Quantitative emergence–A refined approach based on divergence measures. In Proceedings of the 2010 Fourth IEEE International Conference on Self-Adaptive and Self-Organizing Systems, Budapest, Hungary, 27 September–1 October 2010; IEEE Computer Society: Washington, DC, USA, 2010; pp. 94–103. </ref> suggests using the divergence between two probability distributions to quantify emergence. They understand emergence as an unexpected or unpredictable distribution change based on the observed samples. However, this method suffers from high computational complexity and low accuracy. To solve these problems, reference <ref>Fisch, D.; Fisch, D.; Jänicke, M.; Kalkowski, E.; Sick, B. Techniques for knowledge acquisition in dynamically changing environments. ACM Trans. Auton. Adapt. Syst. (TAAS) 2012, 7, 1–25. [CrossRef] </ref> proposes an approximate method for estimating density using [[Gaussian mixture models]]. Additionally, it introduces [[Mahalanobis distance]] to characterize the difference between data and Gaussian components, resulting in improved outcomes. In addition, Holzer, de Meer et al. <ref>Holzer, R.; De Meer, H.; Bettstetter, C. On autonomy and emergence in self-organizing systems. In International Workshop on Self-Organizing Systems, Proceedings of the Third International Workshop, IWSOS 2008, Vienna, Austria, 10–12 December 2008; Springer: Berlin/Heidelberg, Germany, 2008; pp. 157–169.</ref><ref>Holzer, R.; de Meer, H. Methods for approximations of quantitative measures in self-organizing systems. In Proceedings of the Self-Organizing Systems: 5th International Workshop, IWSOS 2011, Karlsruhe, Germany, 23–24 February 2011; Proceedings 5; Springer: Berlin/Heidelberg, Germany, 2011; pp. 1–15.</ref> proposed another emergence measurement method based on Shannon entropy. They believe that a complex system is a self-organizing process in which different individuals interact through communication. Emergence can be measured as the ratio of the Shannon entropy of all inter-agent communications to the sum of individual Shannon entropies.
Another approach defines emergence as "the whole being greater than the sum of its parts" <ref>Teo, Y.M.; Luong, B.L.; Szabo, C. Formalization of emergence in multi-agent systems. In Proceedings of the 1st ACM SIGSIM Conference on Principles of Advanced Discrete Simulation, Montreal, QC, Canada, 19–22 May 2013; pp. 231–240. </ref><ref>Szabo, C.; Teo, Y.M. Formalization of weak emergence in multiagent systems. ACM Trans. Model. Comput. Simul. (TOMACS) 2015, 26, 1–25. [CrossRef] </ref>. This approach focuses on interaction rules and the agent states rather than statistical measures of the system as a whole. Specifically, this measure consists of two subtracting terms. The first term represents the system's collective state, while the second term is the sum of its individual components. This measure emphasizes that emergence arises from the interactions and collective behavior of the system.
Another approach defines emergence as "the whole being greater than the sum of its parts" <ref>Teo, Y.M.; Luong, B.L.; Szabo, C. Formalization of emergence in multi-agent systems. In Proceedings of the 1st ACM SIGSIM Conference on Principles of Advanced Discrete Simulation, Montreal, QC, Canada, 19–22 May 2013; pp. 231–240. </ref><ref>Szabo, C.; Teo, Y.M. Formalization of weak emergence in multiagent systems. ACM Trans. Model. Comput. Simul. (TOMACS) 2015, 26, 1–25. [CrossRef] </ref>. This approach focuses on interaction rules and the agent states rather than statistical measures of the system as a whole. Specifically, this measure consists of two subtracting terms. The first term represents the system's collective state, while the second term is the sum of its individual components. This measure emphasizes that emergence arises from the interactions and collective behavior of the system.
=== Causal emergence theory based on effective information ===
=== Causal emergence theory based on effective information ===