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Causal emergence refers to a special kind of [[emergence]] phenomenon in dynamical systems, that is, the system will exhibit stronger causal characteristics on a macroscopic scale. In particular, for a class of [[Markov dynamical systems]], after appropriate [[coarse-graining]] of its state space, the formed macroscopic dynamics will exhibit stronger causal characteristics than the microscopic ones. Then it is said that causal emergence occurs in this system <ref name=":0">Hoel E P, Albantakis L, Tononi G. Quantifying causal emergence shows that macro can beat micro[J]. Proceedings of the National Academy of Sciences, 2013, 110(49): 19790-19795.</ref><ref name=":1">Hoel E P. When the map is better than the territory[J]. Entropy, 2017, 19(5): 188.</ref>. At the same time, the causal emergence theory is also a theory that uses causal effect measures to quantify emergence phenomena in complex systems.

Causal emergence refers to a specific type of [[emergence]] phenomenon in dynamical systems, where stronger causal characteristcs emerge at the system's macroscopic scale. In particular, for a class of [[Markov dynamical systems]], after appropriate [[coarse-graining]] of the state space, the resulting macroscopic dynamics demonstrate stronger causal characteristics than their microscopic counterparts. This is referred to as causal emergence <ref name=":0">Hoel E P, Albantakis L, Tononi G. Quantifying causal emergence shows that macro can beat micro[J]. Proceedings of the National Academy of Sciences, 2013, 110(49): 19790-19795.</ref><ref name=":1">Hoel E P. When the map is better than the territory[J]. Entropy, 2017, 19(5): 188.</ref>. Furthermore, the theory of causal emergence uses causal effect measurements to quantify emergence phenomena in complex systems.

=== Development of related concepts ===

=== Development of related concepts ===

The causal emergence theory is a theory that attempts to answer the question of what emergence is from a phenomenological perspective using a causal-based quantitative research method. Therefore, the development of causal emergence is closely related to people's understanding and development of concepts such as emergence and causality.

The theory of causal emergence seeks to answer the question of what emergence is from a phenomenological perspective,  employing a causal-based quantitative research methodology. Consequently, the development of causal emergence is closely tied to our understanding of the improvement of concepts such as emergence and causality.

==== Emergence ====

==== Emergence ====

[[Emergence]] has always been an important characteristic in [[complex systems]] and a core concept in many discussions about system [[complexity]] and the relationship between the macroscopic and microscopic levels <ref>Meehl P E, Sellars W. The concept of emergence[J]. Minnesota studies in the philosophy of science, 1956, 1239-252.</ref><ref name=":7">Holland J H. Emergence: From chaos to order[M]. OUP Oxford, 2000.</ref>. Emergence can be simply understood as the whole being greater than the sum of its parts, that is, the whole exhibits new characteristics that the individuals constituting it do not possess <ref>Anderson P W. More is different: broken symmetry and the nature of the hierarchical structure of science[J]. Science, 1972, 177(4047): 393-396.</ref>. Although scholars have pointed out the existence of emergence phenomena in various fields <ref name=":7" /><ref>Holland, J.H. Hidden Order: How Adaptation Builds Complexity; Addison Wesley Longman Publishing Co., Inc.: Boston, MA, USA, 1996.</ref>, such as the [[collective behavior]] of birds <ref>Reynolds, C.W. Flocks, herds and schools: A distributed behavioral model. In Proceedings of the 14th Annual Conference on Computer Graphics and Interactive Techniques, Anaheim, CA, USA, 27–31 July 1987; pp. 25–34.</ref>, the formation of [[consciousness]] in the brain, and the [[emergent]] capabilities of large language models <ref>Wei, J.; Tay, Y.; Bommasani, R.; Raffel, C.; Zoph, B.; Borgeaud, S.; Yogatama, D.; Bosma, M.; Zhou, D.; Metzler, D.; et al. Emergent abilities of large language models. arXiv 2022, arXiv:2206.07682.</ref>, there is currently no universally accepted unified understanding of this phenomenon. Previous research on emergence mostly stayed at the qualitative stage. For example, Bedau et al. <ref name=":9">Bedau, M.A. Weak emergence. Philos. Perspect. 1997, 11, 375–399. [CrossRef] </ref><ref>Bedau, M. Downward causation and the autonomy of weak emergence. Principia Int. J. Epistemol. 2002, 6, 5–50. </ref> conducted classified research on emergence, dividing emergence into [[nominal emergence]] <ref name=":10">Harré, R. The Philosophies of Science; Oxford University Press: New York, NY, USA, 1985.</ref><ref name=":11">Baas, N.A. Emergence, hierarchies, and hyperstructures. In Artificial Life III, SFI Studies in the Science of Complexity, XVII; Routledge: Abingdon, UK, 1994; pp. 515–537.</ref>, [[weak emergence]] <ref name=":9" /><ref>Newman, D.V. Emergence and strange attractors. Philos. Sci. 1996, 63, 245–261. [CrossRef]</ref>, and [[strong emergence]] <ref name=":12">Kim, J. ‘Downward causation’ in emergentism and nonreductive physicalism. In Emergence or Reduction; Walter de Gruyter: Berlin, Germany, 1992; pp. 119–138. </ref><ref name=":13">O’Connor, T. Emergent properties. Am. Philos. Q. 1994, 31, 91–104</ref>.

[[Emergence]] has long been recognized as a key characteristic of [[complex systems]] and a central concept in discussions about system [[complexity]] and the relationship between macroscopic and microscopic levels <ref>Meehl P E, Sellars W. The concept of emergence[J]. Minnesota studies in the philosophy of science, 1956, 1239-252.</ref><ref name=":7">Holland J H. Emergence: From chaos to order[M]. OUP Oxford, 2000.</ref>. Emergence can be understood as the phenomenon where the whole is greater than the sum of its parts, meaning that the whole exhibits new characteristics that the individual components  do not possess <ref>Anderson P W. More is different: broken symmetry and the nature of the hierarchical structure of science[J]. Science, 1972, 177(4047): 393-396.</ref>. While scholars have identified emergent phenomena across various fields ([4])([6]), such as the [[collective flocking behavior]] of birds <ref>Reynolds, C.W. Flocks, herds and schools: A distributed behavioral model. In Proceedings of the 14th Annual Conference on Computer Graphics and Interactive Techniques, Anaheim, CA, USA, 27–31 July 1987; pp. 25–34.</ref>, the formation of [[consciousness]] in the brain, and the [[emergent]] capabilities of large language models <ref>Wei, J.; Tay, Y.; Bommasani, R.; Raffel, C.; Zoph, B.; Borgeaud, S.; Yogatama, D.; Bosma, M.; Zhou, D.; Metzler, D.; et al. Emergent abilities of large language models. arXiv 2022, arXiv:2206.07682.</ref>, there is no universally accepted or unified understanding of this phenomenon. Earlier research on emergence has largely remained qualitative. For instance, Bedau et al. <ref name=":9">Bedau, M.A. Weak emergence. Philos. Perspect. 1997, 11, 375–399. [CrossRef] </ref><ref>Bedau, M. Downward causation and the autonomy of weak emergence. Principia Int. J. Epistemol. 2002, 6, 5–50. </ref> conducted a classified research of emergence, dividing it into [[nominal emergence]] <ref name=":10">Harré, R. The Philosophies of Science; Oxford University Press: New York, NY, USA, 1985.</ref><ref name=":11">Baas, N.A. Emergence, hierarchies, and hyperstructures. In Artificial Life III, SFI Studies in the Science of Complexity, XVII; Routledge: Abingdon, UK, 1994; pp. 515–537.</ref>, [[weak emergence]] <ref name=":9" /><ref>Newman, D.V. Emergence and strange attractors. Philos. Sci. 1996, 63, 245–261. [CrossRef]</ref>, and [[strong emergence]] <ref name=":12">Kim, J. ‘Downward causation’ in emergentism and nonreductive physicalism. In Emergence or Reduction; Walter de Gruyter: Berlin, Germany, 1992; pp. 119–138. </ref><ref name=":13">O’Connor, T. Emergent properties. Am. Philos. Q. 1994, 31, 91–104</ref>.

* [[Nominal emergence]] can be understood as attributes and patterns that can be possessed by the macroscopic level but not by the microscopic level. For example, the shape of a circle composed of several pixels is a kind of nominal emergence <ref name=":10" /><ref name=":11" />.

* [[Nominal emergence]] refers to attributes and patterns that can exist at the macroscopic level but not at the microscopic level. For example, the shape of a circle formed by several pixels is an instance of nominal emergence <ref name=":10" /><ref name=":11" />.

* [[Weak emergence]] refers to the fact that macroscopic-level attributes or processes are generated by complex interactions between individual components. Or weak emergence can also be understood as a characteristic that can be simulated by a computer in principle. Due to the principle of [[computational irreducibility]], even if weak emergence characteristics can be simulated, they still cannot be easily reduced to microscopic-level attributes. For weak emergence, the causes of its pattern generation may come from both microscopic and macroscopic levels <ref name=":12" /><ref name=":13" />. Therefore, the causal relationship of emergence may coexist with microscopic causal relationships.

* [[Weak emergence]] refers to macroscopic attributes or processes that arise from complex interactions between individual components. It can also be understood as a characteristic that could, in principle, be simulated by a computer. Due to [[computational irreducibility]], even though weak emergence traits can be simulated, they cannot easily be reduced to microscopic-level attributes. In the case of weak emergence, the causes of pattern formation may stem from both microscopic and macroscopic levels <ref name=":12" /><ref name=":13" />. Therefore, the causal relationship underlying emergence may coexist with microscopic causal relationships.

* As for [[strong emergence]], there are many controversies. It refers to macroscopic-level attributes that cannot be reduced to microscopic-level attributes in principle, including the interactions between individuals. In addition, Jochen Fromm further interprets strong emergence as the [[causal effect]] of [[downward causation]] <ref>Fromm, J. Types and forms of emergence. arXiv 2005, arXiv:nlin/0506028</ref>. [[Downward causation]] refers to the [[causal force]] from the macroscopic level to the microscopic level. However, there are many controversies about the concept of [[downward causation]] itself <ref>Bedau, M.A.; Humphreys, P. Emergence: Contemporary Readings in Philosophy and Science; MIT Press: Cambridge, MA, USA, 2008. </ref><ref>Yurchenko, S.B. Can there be a synergistic core emerging in the brain hierarchy to control neural activity by downward causation? TechRxiv 2023. [CrossRef] </ref>.

* [[Strong emergence]] remains  highly controversial. It refers to macroscopic-level attributes that cannot, in principle, be reduced to microscopic-level attributes, including the interactions between individuals. Additionally, Jochen Fromm further interprets strong emergence as the [[causal effect]] of [[downward causation]] <ref>Fromm, J. Types and forms of emergence. arXiv 2005, arXiv:nlin/0506028</ref>. [[Downward causation]] refers to the [[causal force]] influence exerted from the macroscopic level to the microscopic level. However, the concept of [[downward causation]] itself is subject to significant debate <ref>Bedau, M.A.; Humphreys, P. Emergence: Contemporary Readings in Philosophy and Science; MIT Press: Cambridge, MA, USA, 2008. </ref><ref>Yurchenko, S.B. Can there be a synergistic core emerging in the brain hierarchy to control neural activity by downward causation? TechRxiv 2023. [CrossRef] </ref>.

From these early studies, it can be seen that emergence has a natural and profound connection with causality.

These early studies reveal a deep and inherent connection between emergence and causality.

==== Causality and its measurement ====

==== Causality and its measurement ====

The so-called causality refers to the mutual influence between events. Causality is not equal to correlation, which is manifested in that not only will B occur when A occurs, but also if A does not occur, then B will not occur. Only by intervening in event A and then examining the result of B can people detect whether there is a causal relationship between A and B.

Causality refers to the mutual influence between events. Causality differs from correlation because B occurs only when A occurs, and if A does not occur, B will not occur. Only by intervening in event A and then examining the result of B can one determine whether there is a causal relationship between A and B.

With the further development of [[causal science]] in recent years, people can use a mathematical framework to quantify causality. [[Causality]] describes the [[causal effect]] of a dynamical process <ref name=":14">Pearl J. Causality[M]. Cambridge university press, 2009.</ref><ref>Granger C W. Investigating causal relations by econometric models and cross-spectral methods[J]. Econometrica: journal of the Econometric Society, 1969, 424-438.</ref><ref name=":8">Pearl J. Models, reasoning and inference[J]. Cambridge, UK: CambridgeUniversityPress, 2000, 19(2).</ref>. Judea Pearl <ref name=":8" /> uses [[probabilistic graphical models]] to describe causal interactions. Pearl uses different models to distinguish and quantify three levels of causality. Here we are more concerned with the second level in the [[causal ladder]]: [[intervening]] in the input distribution. In addition, due to the uncertainty and ambiguity behind the discovered causal relationships, measuring the degree of causal effect between two variables is another important issue. Many independent historical studies have addressed the issue of measuring causal relationships. These measurement methods include [[Hume]]'s [[concept of constant connection]] <ref>Spirtes, P.; Glymour, C.; Scheines, R. Causation Prediction and Search, 2nd ed.; MIT Press: Cambridge, MA, USA, 2000.</ref> and value function-based methods <ref>Chickering, D.M. Learning equivalence classes of Bayesian-network structures. J. Mach. Learn. Res. 2002, 2, 445–498.</ref>, Eells and Suppes' probabilistic causal measures <ref>Eells, E. Probabilistic Causality; Cambridge University Press: Cambridge, UK, 1991; Volume 1</ref><ref>Suppes, P. A probabilistic theory of causality. Br. J. Philos. Sci. 1973, 24, 409–410.</ref>, and Judea Pearl's [[causal measure]] indicators, etc. <ref name=":14" />.

With the further development of [[causal science]] in recent years, it is now possible to use a mathematical framework to quantify causality.[[Causality]] describes the [[causal effect]] of a dynamical process <ref name=":14">Pearl J. Causality[M]. Cambridge university press, 2009.</ref><ref>Granger C W. Investigating causal relations by econometric models and cross-spectral methods[J]. Econometrica: journal of the Econometric Society, 1969, 424-438.</ref><ref name=":8">Pearl J. Models, reasoning and inference[J]. Cambridge, UK: CambridgeUniversityPress, 2000, 19(2).</ref>. Judea Pearl <ref name=":8" /> uses [[probabilistic graphical models]] to describe causal interactions. Pearl uses different models to distinguish and quantify three layers of causality. Here we are more concerned with the second layer in the [[causal ladder]]: [[intervening]] in the input distribution. Quantifying the causal effect between two variables is challenging due to inherent uncertainty and ambiguity. Many independent historical studies have addressed the issue of measuring causal relationships. These measurement methods include [[Hume]]'s [[concept of constant connection]] <ref>Spirtes, P.; Glymour, C.; Scheines, R. Causation Prediction and Search, 2nd ed.; MIT Press: Cambridge, MA, USA, 2000.</ref> and value function-based methods <ref>Chickering, D.M. Learning equivalence classes of Bayesian-network structures. J. Mach. Learn. Res. 2002, 2, 445–498.</ref>, Eells and Suppes' probabilistic causal measures <ref>Eells, E. Probabilistic Causality; Cambridge University Press: Cambridge, UK, 1991; Volume 1</ref><ref>Suppes, P. A probabilistic theory of causality. Br. J. Philos. Sci. 1973, 24, 409–410.</ref>, and Judea Pearl's [[causal measure]] indicators among others. <ref name=":14" />.

==== Causal emergence ====

==== Causal emergence ====

As mentioned earlier, emergence and causality are interconnected. Specifically, the connection exists in the following aspects: on the one hand, emergence is the causal effect of [[complex nonlinear interactions]] among the components of a [[Complex Systems|complex system]]; on the other hand, the emergent properties will also have a causal effect on individual elements in complex systems. In addition, in the past, people were accustomed to attributing macroscopic factors to the influence of microscopic factors. However, macroscopic emergent patterns often cannot find microscopic attributions, so corresponding causes cannot be found. Thus, there is a profound connection between emergence and causality. Moreover, although we have a qualitative classification of [[emergence]], we cannot quantitatively characterize the occurrence of emergence. Therefore, we can use causality to quantitatively characterize the occurrence of emergence.

As mentioned before, emergence and causality are interconnected. Specifically, the connection exists in the following aspects: Emergence arises as the causal effect of [[complex nonlinear interactions]] among the components of a [[Complex Systems|complex system]].Additionally, emergent properties exert causal effects on individual elements within complex systems. In addition, in the past, it was common to attribute macroscopic factors to the influence of microscopic factors. Macroscopic emergent patterns often lack identifiable microscopic causes. Thus, there is a profound connection between emergence and causality. Moreover, although we have a qualitative classification of [[emergence]], we cannot quantitatively characterize the occurrence of emergence. Causality provides a framework to quantitatively characterize emergence.

In 2013, [[Erik hoel|Erik Hoel]], an American theoretical neurobiologist, tried to introduce causality into the measurement of emergence, proposed the concept of causal emergence, and used [[effective information]] (EI for short) to quantify the strength of causality in system dynamics <ref name=":0" /><ref name=":1" />. '''Causal emergence can be described as: when a system has a stronger causal effect on a macroscopic scale compared to its microscopic scale, causal emergence occurs.''' Causal emergence well characterizes the differences and connections between the macroscopic and microscopic states of a system. At the same time, it combines the two core concepts - causality in [[artificial intelligence]] and emergence in complex systems - together. Causal emergence also provides scholars with a quantitative perspective to answer a series of philosophical questions. For example, the top-down causal characteristics in life systems or social systems can be discussed with the help of the causal emergence framework. The top-down causation here refers to [[downward causation]], indicating the existence of macroscopic-to-microscopic causal effects. For example, in the phenomenon of a gecko breaking its tail. When encountering danger, the gecko directly breaks off its tail regardless of its condition. Here, the whole is the cause and the tail is the effect. Then there is a [[causal force]] from the whole pointing to the part.

In 2013, [[Erik hoel|Erik Hoel]], an American theoretical neurobiologist, introduced causality into the measurement of emergence. He proposed the concept of causal emergence and used [[effective information]] (EI for short) to quantify the extent of causality in system dynamics <ref name=":0" /><ref name=":1" />. '''Causal emergence occurs when a system has a stronger causal effect on a macroscopic scale compared to its microscopic scale.''' Causal emergence effectively captures the differences and connections between the macroscopic and microscopic states of a system. At the same time, it combines the two core concepts - causality in [[artificial intelligence]] and emergence in complex systems. Causal emergence also provides scholars with a quantitative perspective to answer a series of philosophical questions. For example, the top-down causal characteristics in life systems or social systems are often analyzed using the causal emergence framework. The top-down causation here refers to [[downward causation]], which involves macroscopic-to-microscopic causal effects. For example, consider the phenomenon of a gecko breaking its tail, which illustrates how a macroscopic system influences its microscopic components. When encountering danger, the gecko directly breaks off its tail regardless of its condition. Here, the whole is the cause and the tail is the effect. Then there is a [[causal force]] from the whole pointing to the part.

=== Early work on quantifying emergence ===

=== Early work on quantifying emergence ===

There have been some related works in the early stage 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 discusses related concepts 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, that is, how to construct 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" />.

 

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 hopes to reconstruct and predict the observed random sequence in a certain degree of accuracy. However, the randomness of the sequence makes it impossible for us to obtain a 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>, which can divide <math>\overleftarrow{S}</math> into several mutually exclusive subsets (all mutually exclusive subsets form the complete set), and the formed set is denoted as <math>\mathcal{R}</math>.