Introduction
“The edge of chaos” is a dynamic mechanism that lies between order and disorder (chaos), and it is a core concept in complex systems science. Is there a universal “edge of chaos” in living systems? If so, would this be a fundamental principle that self-organization, evolution, and complex natural and artificial systems must follow? This article will review some literature, mainly focusing on the basic principles of computation in natural and artificial systems at the “edge of chaos”. In the 1980s, Norman Packard coined the term “edge of chaos”. Since then, the concept of “adapting to the edge of chaos” has been validated and studied in many fields, where both simple and complex systems receive some form of feedback. In addition to reviewing ancient and modern literature, this article will also address criticisms of this concept.
Keywords: Edge of Chaos, Chaos
Christof Teuscher | Author
朱欣怡 | Translator
刘培源 | Reviewer
邓一雪 | Editor
Paper Title:Revisiting the edge of chaos: Again?Paper Address:https://www.sciencedirect.com/science/article/abs/pii/S0303264722000806
Table of Contents
1. Back To The Theoretical Starting Point Of The Edge Of Chaos: From Mechanical Computers To Cellular Automata
2. Early Edge Of Chaos: Cellular Automata And Random Boolean Networks
3. Voices Of Criticism
4. How To Accurately Define The Edge Of Chaos?
5. Are Living Systems At A Critical State?
1. Back To The Theoretical Starting Point Of The Edge Of Chaos:
From Mechanical Computers To Cellular Automata
In 1665, Robert Hooke published Micrographia, in which he made the famous description of plant cells. The history of computation was just beginning to receive some attention: Pascal designed the first mechanical calculator, the Pascaline, in 1642, which could perform addition and subtraction. Later, Leibniz improved upon Pascal’s invention, but it wasn’t until the 1820s that Charles Babbage designed the difference engine, a much more complex mechanical calculator that could tabulate polynomial functions. More than a decade later, in 1936, Alan M. Turing laid the key foundations of “modern computation” in his famous paper, namely the concept of the Turing machine (TM) and the universal Turing machine (UTM) (Turing, 1936-1937). At this point, significant advancements were also made in the field of cell biology, as less than a year later, Hans Adolf Krebs discovered the citric acid cycle, also known as the tricarboxylic acid cycle or Krebs cycle (Krebs and Johnson, 1937), a series of chemical reactions that serve as a major energy source for cells. The fusion of cell biology and computation naturally created the fields of biological computing, molecular computing, and cellular computing. In Turing’s pioneering work on intelligent machinery (Turing, 1969) and morphogenesis (Turing, 1952), he was the first to introduce the aforementioned concepts. Soon after, Liberman became a pioneer in the development of paradigms for biological computing and the concept of “cells as molecular computers”. His earliest work on this subject can be traced back to 1972, where he introduced cells as molecular calculators capable of controlling themselves using molecular language (Liberman, 1972). Later, in his 1979 paper (Liberman, 1979), Liberman described what he called the molecular cellular computer (MCC) and listed more details and its basic characteristics. He also proposed that “living cells follow a reality or information physics”. At the end of this article, Michael Conrad commented: “Dr. Liberman presents an unavoidable and controversial point of view that what life systems require is a kind of information physics, which may be modified from ordinary physics to some extent” (Liberman, 1979). Liberman’s views from 1972 raised interesting questions about the intersection of physics, biology, and information theory, some of which remain unanswered to this day (in part or in whole). Can the laws of physics and biology be derived from information theory? Can physics and biology develop new ways to process information? What is biological information? Does biological information processing occur at some “sweet spot” or “edge of chaos”—a dynamic mechanism that lies between order and chaos? If so, is this “sweet spot” the basis for self-organization, evolution, and the adaptation of complex physical, biological, and artificial systems to their environments? What are the trade-off principles that drive systems to reach certain “sweet spots”? Over 20 years later, Liberman has found some answers. For example, in his 1996 paper co-authored with Minina (Liberman and Minina, 1996), he proposed four unified principles describing biology: “It seems beneficial to construct a ‘new science’ that describes biology and the material world from a unified perspective through the establishment of fundamental principles. We believe these principles are: 1. The principle of minimal cost for computation and measurement; 2. The principle of optimality; 3. The principle of minimal irreversibility; 4. The principle of causality (new phrasing).” Although Liberman never used the term “edge of chaos”, both he and Michael Conrad’s pioneering work hinted at an optimal state or “sweet spot” where information processing occurs in living systems. This viewpoint is clearly reflected in Conrad’s 1989 paper “The Brain-Machine Disanalogy” (Conrad, 1989), which outlines the trade-off principles between evolution (or adaptability), efficiency, and (structural) programmability. Physical constraints lead to such trade-offs. “Unlike the programmable domains familiar to current machines, the trade-off principles imply high efficiency and adaptability in information processing” (Conrad, 1989). In this article, we will review the literature on the basic computational principles of the “edge of chaos” in natural and artificial systems. This term was coined by Norman Packard in the late 1980s (Packard, 1988). Since then, the concept of “adapting to the edge of chaos” has been validated and studied in many fields, where both simple and complex systems receive some form of feedback. In addition to reviewing ancient and modern literature, we will also review criticisms of this concept. Please note that this article is not about chaos, self-organized criticality, or many other subfields related to the theory of nonlinear dynamical systems. We will only touch the edges of these topics while primarily following the storyline of the “edge of chaos”.
2. Early Edge Of Chaos:
Cellular Automata And Random Boolean Networks
The early research on the edge of chaos primarily focused on cellular automata (CA) (Toffoli and Margolus, 1987) and other discrete dynamical systems, such as random Boolean networks (RBN) (Kauffman, 1969). In this section, we will highlight some early works that directed research on the edge of chaos in different directions. In 1988, in a book published by Norman Packard, the term “edge of chaos” was born in a chapter titled “Adapting to the Edge of Chaos” (Packard, 1988). Packard proposed a simple adaptive model that uses genetic algorithms (GA) to evolve cellular automata rules (Holland, 1975). His simulations indicated that “populations of rules always move toward a region in the rule space that marks the boundary between chaotic and non-chaotic rules”. This behavior can be interpreted as, firstly, it is observed that communication is necessary for significant computation in cellular automata. Packard used a density classification task, where the cellular automata must decide whether the initial density of “1” in the initial state is greater than 50%. If it is greater than 50%, the density classification task requires the cellular automata to evolve towards a state where all positions are “1”; if the density is less than 50%, it requires the cellular automata to evolve towards a state where all positions are “0”. Looking at each experiment individually is meaningless, as it can always compute the proportion of “1”. However, the cells of the cellular automata can only see nearby cells (for example, with a radius of 3), which makes the task valuable, as it requires cells to achieve a globally consistent state through local communication. Secondly, it can be intuitively understood that cellular automata rules that do not communicate with neighbors (or communicate very little) cannot solve the density task. Packard termed these rules as “ineffective”. We could say that the system is too “rigid” or too “ordered” to do anything useful. On the other hand, if the prescribed communication is too much, any local disturbance will significantly affect the state of the cellular automata, which also leads to an inability to solve the task at hand. Packard referred to these rules as “very active”, and we could say they belong to the “chaotic” state in the rule space. His conclusion was: “Therefore, it is reasonable to expect that any computation requiring such communication can only be accomplished through rules that are close to the edge of chaos” (Packard, 1988). Interestingly, Wolfram published a highly influential paper in 1984 (Wolfram, 1984), in which he introduced four different universal classes for one-dimensional cellular automata and proposed that four classes of cellular automata could perform universal (or “complex”) computation. However, Packard did not cite it. Although neither Wolfram nor Langton used the term “edge of chaos” in their 1986 paper titled “Studying Artificial Life with Cellular Automata” (Langton, 1986), they laid much of the groundwork for its study. For example, Langton proposed an artificial biochemical sample composed of virtual state machines (VSM) as a “molecular operator”. He may have implied the edge of chaos by stating, “One example of a system seems to balance well between stasis and chaos… In order to dynamically maintain this balance, there must be some ordered self-regulating mechanism.” But perhaps more importantly, Langton established the connection between simulating biochemistry and artificial molecules through cellular automata. “Biochemistry studies how life emerges from the interactions of lifeless molecules… Cellular automata provide us with a great artificial universe in which we can embed artificial molecules in the form of virtual automata” (Langton, 1986). A few years later, Langton fully caught up with the edge of chaos trend and published a now widely cited paper (Langton, 1990). This paper demonstrated that his views aligned with Packard’s: Intuitively, rules close to the edge of chaos possess the best information communication capabilities and are therefore most suitable for computation. Langton again employed the framework of simple, unified, and synchronous cellular automata: “This paper presents research on cellular automata, showing that near phase transitions are the best conditions for achieving information transmission, storage, and modification.” At the end of the article, Langton noted: “Systems exhibit a range of behaviors near phase transitions that very well reflect computational phenomenology; we believe we can define the computation at the edge of chaos in the dynamics behavior spectrum” (Langton, 1990). Langton used the parameter λ (initially introduced in Langton (1986)) to explore and characterize the rule space of cellular automata from the most homogeneous to the most heterogeneous rules. He then indicated that the parameter λ could also support Wolfram’s work and connected Wolfram’s four classes of cellular automata with his own space. Langton raised two points in the discussion section that are very relevant to this article: 1. He wondered what the relationship between his work and Bak’s self-organized criticality was (Bak et al., 1988). 2. He pointed out that phase transition phenomena are ubiquitous in living cells. Secondly, let us return to Stuart Kauffman’s 1969 paper (Kauffman, 1969), in which he introduced formal genetic networks—also known as random Boolean networks (RBN), NK networks, or NK models—as models of genetic regulatory networks composed of binary “on-off” genes, to study cellular control processes. In its simplest and most primitive form, a random Boolean network is a discrete dynamical system composed of N automata (or nodes), where each node receives input from K randomly selected other nodes. The result is that each node has an average of K outputs to other nodes. Each node is a Boolean variable with only two possible states: {0, 1}. The dynamics satisfy F = (f₁, …, fᵢ, …, f_N), where each fᵢ is represented by a lookup table of Kᵢ inputs randomly selected from the set of N nodes. Initially, the Kᵢ neighbors and lookup tables are randomly assigned to each node: Each node’s state is updated using its corresponding Boolean function: 
Each node synchronously updates with its corresponding Boolean function: The number of states in classic synchronous RBN is finite, and after a certain number of time steps, T will fall into fixed or cyclic attractors. Kauffman studied the cycles and perturbations of three major classes of genetic networks: (1) fully connected K=N, (2) K=1, (3) K=2. He concluded that fully connected networks (K=N) are biologically impossible, K=1 networks require cycles that no organism on Earth possesses, and K=2 networks are biologically reasonable. Kauffman believed that the parameters of gene networks in organisms are adjusted through evolution, so they will operate at critical stages near the edge of chaos. More broadly, his findings indicate that biological networks at the edge of chaos exhibit stability, evolvability, and information processing efficiency, which is consistent with the trade-off principles proposed by Conrad (Conrad, 1989). Derrida and Pomeau spent nearly 20 years, along with many others toward the end of that century, beginning to analyze radial basis neural networks from a more mathematical perspective (Derrida and Pomeau, 1986). Kauffman concluded his article with the statement: “Networks composed of large, randomly combined binary elements exhibit simple, stable, and orderly characteristics. Nature has utilized this possible and reliable system to create evolution and protect its offspring” (Kauffman, 1969). Let us recap: Early works on the edge of chaos primarily focused on discrete dynamical systems and provided evidence that complex dynamics in such dynamical systems occur near certain phase transitions, “dependent on the fundamental ability to process information” (Langton, 1990). Before continuing the story, let us first hear the voices of criticism.
3. Voices Of Criticism
The article by Mitchell et al. is often cited as a criticism of the edge of chaos (Mitchell et al., 1993). However, this is not the case. These articles are more like specific critiques of Packard’s findings (Packard, 1988), which involve the evolution, dynamics, and computational capabilities of unified, one-dimensional, synchronous cellular automata. Mitchell et al. attempted to replicate Packard’s results by conducting a set of similar experiments. Unfortunately, Packard’s paper omitted quite a bit of technical detail, forcing Mitchell et al. to make “reasonable” assumptions about several parameters. But unfortunately, Mitchell et al. were unable to reproduce Packard’s results at all: “Our experiments produced completely different results, and we believe the interpretation of the original results is incorrect” (Mitchell et al., 1993). While they observed that more complex rules evolved, these rules seemed not to exist at the edge of chaos. They further pointed out that “… we do not know what caused the discrepancy between our results and the original experimental results.” Mitchell et al. explicitly stated in their general discussion that “the results we present do not contradict the hypothesis: computational capabilities are related to phase transitions in the rule space of cellular automata.” This hypothesis has since been confirmed by some authors. In 1995, Gutowitz and Langton re-examined whether there exists an edge of chaos and whether evolution can guide systems to the edge of chaos (Gutowitz and Langton, 1995). They argued that a new measure of complexity is needed to determine these issues and suggested that convergence time is an appropriate metric. By using mean field methods, they demonstrated that as evolution progresses, cellular automata rules do indeed become increasingly complex, and that the average convergence speed of critical rules is slower than that of non-critical rules. Kaneko and Suzuki (1994) and Suzuki and Kaneko (1994) used logistic graphs to identify the edge of chaos and obtained similar results. They also found that under appropriate conditions, evolutionary processes do indeed drive a system toward the edge of chaos. In 1997, Siper reported in his book that he obtained results supporting Mitchell et al. (Siper, 1997). He conducted experiments using his own “cellular programming algorithm”, which relies on local adaptability measures (as opposed to global measures in evolutionary algorithms). Siper found that the rules concentrated around λ=0.5, most between 0.45-0.55, which aligns with the findings of Mitchell et al. The value of the density λ task may be close to 0.5. In his experiments, more complex rules again evolved, but there was disagreement regarding Packard’s results about the edge of chaos. In concluding this section, we emphasize once again that the article by Mitchell et al. is a critique of Packard’s experiments and a failed attempt. Their arguments and findings entirely relied on unified cellular automata, whose rules were evolved using genetic algorithms. This article does not question the widespread existence of the edge of chaos, nor does it question that evolution or other adaptive feedback mechanisms can drive systems toward the edge of chaos. Therefore, citing this article as a criticism of the edge of chaos, as it often is, is misleading. For example, the Wikipedia article on the edge of chaos indeed states, “However, Melanie Mitchell and others have questioned the universality and significance of this idea” (Wikipedia, 2022). This is an inaccurate statement and does not reflect the conclusions of Mitchell et al. In fact, in a subsequent paper, Mitchell et al. (Mitchell et al., 1994) included some additional arguments and viewpoints, reiterating that “… our research does not exclude the possibility that someone in the future can provide a rigorous and useful definition of the term ‘edge of chaos’ in the context of cellular automata. They also do not exclude the possibility of discovering that the edge of chaos is related to the enhanced computational capabilities of cellular automata. Finally, they do not exclude the possibility that adaptive systems evolve into such dynamic regions to exploit the inherent computational capabilities there.” Earlier, Crutchfield and Young provided substantial evidence (Crutchfield and Young, 1990).
4. How To Accurately Define The Edge Of Chaos?
So far, we have not formally defined the edge of chaos. Now let us discuss this issue. We will explore two main approaches to delineate what is generally considered the edge of chaos: (1) Derrida’s simulated annealing and (2) Lyapunov exponent measurement. The propagation of (typically small) local disturbances (also known as “damage” or noise”) in dynamical systems is an established way to determine what state the system is in. To do this, create two copies of the system and introduce some disturbance in one of them. Then compare the distance between the two state trajectories over time: (1) when the system is in a (deterministic) chaotic state, disturbances spread; (2) if the system is in an ordered (or “frozen”) region, disturbances will disappear; and (3) if the system is in a critical/complex state at the edge of chaos, disturbances will remain at a constant size. Therefore, the main question in determining the edge of chaos is: do disturbances disappear, remain, or grow? Or in other words, how sensitive is the dynamical system to initial conditions? In the critical phase, i.e., the edge of chaos, disturbances will propagate over long time and space scales without decaying or amplifying. Kauffman studied disturbances in his 1969 paper and used them to classify random Boolean networks (Kauffman, 1969). In 1986, Derrida and Pomeau proposed the simulated annealing method (AA), which is an average field method (Derrida and Pomeau, 1986), which has since been expanded by others, e.g., (Solé and Luque, 1995; Luque and Solé, 1997). Derrida and Pomeau’s analysis showed that in the thermodynamic limit, i.e., as the network size tends to infinity, the critical point can be precisely determined. For Kauffman’s random networks, it was found that they exhibit dynamic order-disorder transitions at sparse critical connectivity. However, in practical applications of problems, the thermodynamic limit is often not very important. Recent work has focused more on finite size scaling, both for random Boolean networks (RBN) and random threshold networks (RTN), see (Rohlf and Bornholdt, 2002; Rämö et al., 2006; Samuelsson and Socolar, 2006; Lu and Teuscher, 2014; Ishii et al., 2019; Rohlf et al., 2007; Goudarzi et al., 2012). We will later see how random Boolean networks achieve critical connectivity through self-organization. The Lyapunov exponent λ measures the rate of separation between two trajectories of a dynamical system (in phase space). Thus, this exponent can measure the degree of instability and determine the edge of chaos. The Lyapunov exponent describes the short-term behavior of the system, while the maximum Lyapunov exponent (MLE) describes the long-term dynamics. A positive (maximum) Lyapunov exponent indicates that the system is sensitive to initial conditions and thus in a chaotic state. On the other hand, if λ < 0, the system is stable or ordered. If the maximum Lyapunov exponent of a dynamical system is 0, it is considered to be in a critical state at the edge of chaos. The Lyapunov exponent can also be derived from discrete systems such as CAs and RBNs, see (Bagnoli et al., 1992; Luque and Solé, 2000). It is worth noting that much of the work in chaos theory is based on the assumption of continuous and infinite space, which we will not discuss here. Last but not least, defining what complexity is and what it is not remains an open or at least controversial question. Solé et al. pointed out: “Several definitions of (complexity) have been proposed, all sharing an intuitive concept that complexity is neither completely ordered nor completely disordered” (Solé et al., 1996).
5. Are Living Systems At A Critical State?
As mentioned above, Kauffman was the first to claim and demonstrate that living systems, particularly genetic regulatory networks, operate in a critical dynamical mechanism between order and chaos (Kauffman, 1969). One of Kauffman’s hypotheses is that cell types correspond to the state attractors of Kauffman’s random Boolean networks. Kauffman went further than Packard, who believed that his cellular automata rules were the result of natural selection. Kauffman’s findings indicate that the nature of genetic regulatory networks results from intrinsic dynamic processes. Since then, there has been ample evidence that quasi-static non-equilibrium living matter operates at some critical state near or at the edge of chaos, and that critical states emerge after continuous balancing. Intuitively, this makes sense, as both living and non-living systems exhibit different phase transitions in response to environmental changes (Griffiths and Wheeler, 1970; Jacobs and Frenkel, 2017). Later, we will mention some specific publications to emphasize the concept of criticality in living systems. Shortly after Packard’s paper on the edge of chaos was published, Ito and Gunji provided further evidence that self-organization as an evolutionary process drives living systems toward the edge of chaos. They also hypothesized that their self-organized criticality is a generalization of Bak’s self-organized criticality (Ito and Gunji, 1994). In 2005, Shmulevich et al. (2005) were the first to demonstrate that “… the fundamental genetic network of HeLa cells seems to operate at an ordered state or at the boundary between order and chaos, but not in chaos.” Kesseli and Yli-Harja proposed a new method to study the criticality of gene regulatory networks: “This method is one of the first attempts to quantify whether cells are dynamically ordered, critical, or chaotic using biological measurement data” (Rämö et al., 2006). Nykter et al. (2008) used gene expression data from macrophages and found that their dynamics were critical. Hanel et al. (2010) showed that their minimally nonlinear (MNL) system used as a model for biological response networks was also critical. Torres-Sosa et al. (2012) discussed how gene networks truly achieve critical behavior? This remains a somewhat elusive open question. They proposed an “… evolutionary model in which criticality naturally emerges from the balance between the two fundamental components of evolvability: phenotypic conservatism under mutation and phenotypic innovation”. “Our results indicate that this interaction between expanding new phenotypes (innovation) and retaining existing phenotypes (conservation) is sufficient to drive all networks in the population toward a critical state.” Although Mora and Bialek did not mention the edge of chaos, their article posed the question: “Are living systems poised at criticality?” (Mora and Bialek, 2011). They discussed empirical evidence of criticality in a wide range of living systems, including protein networks, neural networks, and flocks of birds. “The enticing possibility is that many systems are not deep in one phase or another, but rather balance near the critical surface of natural parameter space.” Today, there remains a half-open question: how do living systems reach a critical state? Because critical states are often a very small region in the entire parameter space (if not a point). Most evidence suggests that it is a combination of feedback mechanisms and balanced evolution (Ulanowicz, 2002; Conrad, 1989).
6. The Golden Age Of Edge Of Chaos Research
Kauffman began researching random Boolean networks in the late 1960s (Kauffman, 1969), while the golden age of theoretical analysis did not begin until the early 21st century. Random Boolean networks are simple enough to formalize and analyze, yet complex enough to be interesting, making them a perfect playground for many researchers. The modern tools of statistical physics combined with the newly emerging infinite computational resources have allowed the field to flourish for many years. Physicists have consistently been interested in small systems, such as logistic mappings, or self-organizing evolution to the edge of chaos (Pierre and Hübler, 1994; Baym and Hübler, 2006; Melby et al., 2000). In addition, over the years, random Boolean networks have also derived many new properties. For instance, at K_c, the phase space structure of attractor cycles (Albert and Barabási, 2000), the number of different attractors (Samuelsson and Troein, 2003), and the distribution of chaotic attractors are complex (Bastolla and Parisi, 1998), displaying many properties of biological networks (Kauffman, 1993). In 2000, Bornholdt and Rohlf (2000) demonstrated that a simple local reconnection rule (static nodes gain edges, dynamic nodes lose edges) can drive random threshold networks (RTNs)—a subset of random Boolean networks—to achieve critical average connectivity K_c=2 in the limit of large systems. This result was later generalized to random Boolean networks (Rohlf et al., 2007). This conclusion first proved that the local rules of a simple self-organizing network can lead them into critical states. Liu and Bassler (2006) used the same reconnection rule but in a co-evolutionary process. They found that “during evolution, the emergence of critical states arises from the interaction of topology and dynamics.” Goudarzi et al. (2012) went further, solving a long-standing question: they computed that for large-scale (N) systems, adaptive information processing drives random Boolean networks’ critical connectivity K_c=2. For finite-sized networks, the connectivity approaches the critical value and has a power-law relationship with the system size N. Moreover, they also showed that if the task complexity and the amount of information provided exceed a certain complexity threshold, network learning and generalization are optimized when close to critical states. Around 2001, a new research direction emerged—Reservoir Computing (RC) (Jaeger, 2001; Bertschinger and Natschläger, 2004; Maass et al., 2002; Legenstein and Maass, 2007). The main idea behind reservoir computing is: 1) a recurrent neural network acts as a fixed, typically nonlinear “reservoir” that projects inputs into a higher-dimensional space; 2) training the linear output layer to map the high-dimensional space to the desired output greatly reduces the computational cost of training. Reservoir computing has evolved from a fringe concept into a fairly mainstream computational framework (Büsing et al., 2010; Goudarzi and Teuscher, 2016). Reservoir computing excels in the field of temporal signal processing; however, the fact that the reservoir may be unstructured, imperfect, and unreliable has drawn attention from the hardware community, as it may allow for constructing reservoirs to be easier and cheaper than traditional hardware. This has led to another new field: immaterial computing (Lilak et al., 2021; Dale et al., 2017; Hochstetter et al., 2021). The neuromorphic computing community is particularly interested in this field, as it directly utilizes physics to realize the potential of neural networks, rather than building traditional von Neumann architectures and then “simulating” neural networks (Nakajima et al., 2021; Hochstetter et al., 2021; Cao et al., 2022; Tran and Teuscher, 2021; Slavova and Ignatov, 2022). Reservoir computing has become one of the latest playgrounds for researchers studying the edge of chaos. Boedecker et al. (2012) used an information-theoretic framework to show that when the reservoir is between stable and unstable states, i.e., at the edge of chaos, the computational capacity of echo state networks (a variant of reservoir computing) is maximized. Snyder et al. (2013) pointed out that the intrinsic computational capabilities of random Boolean networks in reservoir computing peak when the system is at the edge of chaos. Carroll’s recent research (2020) seems to contradict many earlier studies related to reservoir computing at the edge of chaos: “The optimal computational capability of dynamical systems occurs at the edge of stability, a concept that is typically incorrect. This paper shows that optimal performance does not occur at the edge of stability in reservoir computing.” Carroll redefined the edge of chaos (EoC) as the edge of stability (EoS), the point where the maximum Lyapunov exponent becomes positive. It is noteworthy that his results are based only on a single task and a single reservoir size. Whether this result applies to more general situations remains to be seen.
7. Conclusion And Outlook
In many scientific fields, the study of the balance between structure and randomness, order and chaos is common. Of course, the field of computation is no exception. The concept of computation at the edge of chaos has attracted considerable attention due to its rather abstract origins—stemming from algorithms. Over the years, the concept of the edge of chaos has also been applied well beyond information processing, such as Dambisa Moyo’s book on socioeconomic issues (Moyo, 2018), Oswald et al.’s book on project management at the edge of chaos (Oswald et al., 2018), and Pamela McCorduck’s novel (McCorduck, 2007) which also illustrates this. Does the edge of chaos follow Gartner’s hype cycle (Blosch and Fenn, 2018)? This cycle includes five stages: (1) technology trigger, (2) peak of inflated expectations, (3) trough of disillusionment, (4) slope of enlightenment, (5) plateau of productivity. We believe the answer is no. While there may have been some inflated expectations peaking initially, the edge of chaos does not seem to have experienced a trough of disillusionment. Instead, following the technological trigger in the late 1980s, it has gradually reached the plateau of productivity over the years. Mitchell et al.’s paper (Mitchell et al., 1993) may have caused a slight bump on the road of edge of chaos development, but it does not seem to have altered the overall progress of the research. Many complex systems, especially living cells, exhibit extraordinary stability in maintaining their spatiotemporal organization, yet they also adapt to various changes in their environment. The survival instinct of a living system requires it to integrate and process limited resources and respond to environmental changes. Therefore, systems must manage feedback and balances, such as limited energy, resources, space, and time, to survive. They roughly align with the four unified biological principles proposed by Liberman and Minina (1996). It has also been hypothesized that the basis of natural selection in the evolutionary process of living systems is the ability to perform complex information processing near the edge of chaos. Several mechanisms have been proposed and studied that drive living systems as well as abstract non-living systems (such as RBN) into critical states. How much of this self-organization is goal-directed, and how much is not? remains an open question. A concept pioneered and developed by Liberman is that cells are molecular computers—often viewed as models of how living systems organize and operate (Liberman, 1972). Under this view, computers are seen as physical systems in critical states. They must support arbitrarily long time correlations within certain macroscopic “computational” degrees of freedom. This is achieved by decoupling these degrees of freedom from the thermal bath degrees of freedom that produce errors. Computers are a type of physical system that exists in a continuous phase transition state in an entropy-disordered environment. In such an environment, computers gain the driving force to push computation forward. But at the same time, they must protect computation from fluctuations caused by the environment (Crutchfield and Young, 1990). In summary: According to Betteridge’s law and Hinchcliffe’s rule (Cook and Plourde, 2016), when the title of an article poses a yes or no question, the answer is usually no. Currently, it does not seem necessary to re-seek the edge of chaos. We can confidently say that this concept has been established and validated in various fields through various methods.