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==== 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 [3][4]. 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 [5]. Although scholars have pointed out the existence of emergence phenomena in various fields [4][6], such as the collective behavior of birds [7], the formation of consciousness in the brain, and the emergent capabilities of large language models [8], 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. [9][10] conducted classified research on emergence, dividing emergence into nominal emergence [11][12], weak emergence [9][13], and strong emergence [14][15].

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>.

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 [11][12].

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" />.

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 [14][15]. Therefore, the causal relationship of emergence may coexist with microscopic causal relationships.

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.

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 [16]. 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 [17][18].

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>.

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

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

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.

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.

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 [19][20][21]. Judea Pearl [21] 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 [22] and value function-based methods [23], Eells and Suppes' probabilistic causal measures [24][25], and Judea Pearl's causal measure indicators, etc. [19].

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" />.

==== Causal emergence ====

==== Causal emergence ====

Emergence and causality are also interconnected: on the one hand, emergence is the causal effect of complex nonlinear interactions among the components of a complex system; on the other hand, emergent characteristics will also have causal relationships with individuals 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. On the other hand, 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 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 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.

In 2013, 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 [1][2]. 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 of 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 can be discussed with the help of the causal emergence framework. The top-down causation here refers to downward causation [26], indicating the existence of macroscopic-to-microscopic causal effects. For example, in the phenomenon of a gecko breaking its tail. When in danger, the gecko breaks its tail directly without asking for the tail's advice. 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, 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 [26], 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.

=== Early work on quantifying emergence ===

=== Early work on quantifying emergence ===