| Cellular Automation is closely related to the development of relevant theories and methods. On the one hand, the development of Cellular Automation is benefited from the research of relevant theories, such as the theory of logical mathematics, discrete mathematics, and computer automation, on the other hand, the development of cellular automation has also promoted the development of some relevant disciplines and theories (such as artificial intelligence, nonlinear science, and complexity science, it even directly leads to the emergence of artificial life sciences. In addition, in terms of performance, the Cellular Automation Model has great similarity or relativity with some theoretical methods. Next, we will briefly discuss some relevant theoretical methods of cellular automation and their relationships with the Cellular Automation Model. 1. Cellular Automation and Human Life Research
Artificial life is a new science that was just born in 1990s. It is one of the pillar disciplines of complex science research. Artificial life is a science that studies artificial systems that can demonstrate the behavioral characteristics of natural life systems, it tries to simulate, synthesize, and collaborate with biological organisms on artificial media such as computers and robots, such as self-replication, parasitic, competition, evolution, and collaboration, study and observe "possible life phenomena" (life-as-it-cocould-be ), in this way, people can better understand "life-as-we-know-it" (longton, C · g ·, 1987; Wu jianbing, 1998 ).
Cellular Automation is an important research tool and theoretical method branch of artificial life. Christopher Langton and others proposed and developed artificial life based on their in-depth research on Cellular Automation. At the same time, the development of artificial life has given new meanings to cellular automation. The Cellular Automation Model has been recognized and recognized by scientists, in 1990s, it became a cutting-edge topic in scientific research, and its theories and methods were further improved. In addition, Cellular Automation is very similar to other methods of artificial life research. Like other artificial life methods such as Neural Networks and genetic algorithms, Cellular Automation models are used to study the overall behavior of the system based on local interactions. In addition, cellular automata, neural networks, and l-systems can all be classified as network dynamics models in nonlinear dynamics. They are interrelated and closely related. Currently, a model called Cellular Neural Network (CNN) is the product of the combination of cellular automation and neural networks. 2. Cellular Automation and "chaotic edge" "The edge of chaos (Langton C. G ., 1992; M. waldrop, 1997) "is an important achievement and symbolic slogan of current research on complexity science. It is the flag of Santa Fe School. The so-called "Chaos" is not a "Chaos" in the scientific sense, but the original meaning of the chaos itself, that is, the concept of "Chaos" and "disorder" relative to the order. Therefore, "chaotic edge" should be understood as "chaotic edge ". Or "disordered edges", but not directly related to the "Chaos" of Chaos dynamics. In fact, the complete meaning of "chaotic edge" refers to the existence and production of complex phenomena such as life and complex systems ". The order is not complex, and the disorder is also not complicated, and the complexity exists on the edge of the disorder.
The concept of "chaotic edge" is proposed by Norman Packard and chhstopher Langton Based on the in-depth study of Cellular Automation. Here we will give a brief introduction.
Langton is. based on the analysis and research of wolfram's dynamic behavior classification, this article puts forward the concept of "chaotic edge, in particular, the fourth type of Cellular Automation is the most creative dynamic system-the complex state, which is precisely between order and chaos. In most non-linear systems, there is often a conversion parameter from order to chaos. For example, the dripping water on the tap in our daily life shows different and stable complex dynamic behaviors with the change of water flow speed, such as one-point, two-point, or multiple-point periods, and even chaos and extreme disorder, apparently, the flow speed here. Or water pressure is the state parameter of this nonlinear system. Langton defines a parameter for the conversion function, so as to compare the function space parameters of the cellular automation. When this parameter changes, the cellular automata can display different dynamic behaviors and obtain the parameter space similar to the graph in the continuous dynamics system. The Langton method is added (TAN Yuejin, 1996 ):
First, define the cellular static (quiescent state ). The static state of the metacell has such characteristics if all fields of the metacell are static. Then the metacell will remain in this static state at the next moment (similar to the fixed point in the ing ). One-dimensional cellular automation is considered. Each cellular Has k States (State set is Σ), and each cellular is connected to N adjacent cellular units. A total of Kn neighboring states exist. Select any kind of S in k States and call it static sq. Assume that for the conversion function, a total of NQ transformations map the neighborhood to the static state, the remaining kn-NQ statuses are randomly and evenly mapped to every State in Σ-{sq. It can be defined as follows: λ = (k ^ N-NQ)/K ^ n
In this way, any conversion function is used. Defines a corresponding parameter value. With the change of the parameter λ from 0 to 1, the behavior of the cellular automata can change from the point state disturbance to the periodic disturbance, and achieve the chaotic disturbance through the fourth complex pattern, the fourth type is the complex mode with a local structure. The relationship between order and chaos is called the edge of chaos in the preceding parameter space. The dynamic behavior of Cellular Automata (qualitative 1 has an evolutionary pattern that attracts point-10-period confluence-> "complex pattern"-> chaotic confluence. At the same time, it gives a new meaning to the classification of the dynamic behavior of Cellular Automata: that is, If λ is lower than a certain value (about 0.6 here), then the system will be too simple. In other words, too much order leads to a lack of creativity in the system; in another extreme case, Lambda approaches 1. The system becomes too disordered to identify structural features. Therefore, the system is extremely complicated only when λ is near a certain value, the so-called "chaotic edge. It is also possible that "Life symptom" exists only at this time. On this basis, landton proposed and developed artificial life sciences. In modern system science. The concept of dissipation structure indicates that "life" makes a living by negative entropy, while Langton creatively proposes that life exists on the "chaotic edge ". The complex phenomena of life are further explored from another perspective.
3. Cellular Automata and Differential Equations Differential equations have a history of more than three hundred years. A group of great scientists, such as Euler and caus. Langrange, Laplace, and Poisson have made outstanding contributions. Moreover, the development of Partial Differential universal procedures is of great significance to the development of modern physics, such as quantum mechanics. A large number of physical laws are exclusively expressed by partial differential equations, for example, 2D equation. Engels also pointed out that "the unity of nature is shown in the 'striking similarity 'of Differential Equations in various phenomena ". In short, differential equations are the language of Modern Science and one of the most important research tools in scientific research.
The main feature of differential equations is that time and space are continuous (if there are spatial factors in the equation), which is based on the Philosophical Understanding of Time and Space continuity. Cellular Automata, however, are completely spatial and temporal discretization. In this sense, the differential equation and a pair of relative computing methods (toffoli. T., 1987) are used ).
In the case of manual computing. The (partial) Differential Equations composed of symbols can be used to flexibly perform reduction and other symbol operations to obtain accurate quantitative solutions. This is an advantage. However, as modern computers develop increasingly and have become an important tool for scientific research, differential equations have encountered an embarrassing problem. That is to say, the computer is based on discretization, and the differential equations have to be discretization in their own time and space During computation to establish the difference equations. Or, they can be expanded into power series equations and part of them can be truncated; or use a discrete structure to represent continuous variables. This transformation process is complex and impossible to solve, but most importantly, in this process, the differential equation also loses its most important feature-accuracy and continuity.
For Cellular Automation, it is almost impossible to perform computation without the computer environment. However, it is natural and reasonable to use a computer for computation, and even the prototype of the Next Generation parallel computer. Therefore, in the computing environment of modern computers, discrete computing methods represented by Cellular Automation have greater advantages in solving problems, especially dynamic system simulation. Although Cellular Automation has the completeness of computing in theory, it does not yet have complete theoretical support to meet the specific purpose of modeling. Its construction is often an intuitive process. It is very difficult to obtain a quantitative result by using cellular automation. Even if possible, Cellular Automation will also fall into an embarrassment. The status and rules of Cellular Automation will inevitably become complicated, this will inevitably lead to the loss of its simple and vivid features.
However, the evidence, as said by the physicist Boer, is that "the opposite is not necessarily a conflict, but may be complementary and complete ". They both have advantages and disadvantages and complement each other. They both have reasons for their existence. However, in the modern computer environment, Cellular Automation is still in its infancy and requires more attention and support. In geography, the spatial dynamics models of Lowry, Wilson, and Zhang Xinsheng (Zhang Xinsheng, 1997) are all based on differential equations, because most of these models are complex nonlinear differential equations, the analytical solution cannot be obtained. One-step or multiple-step differential operation must be performed on the differential equations using the Euler's method or the Runge-Kutta method to complete the corresponding computer model or spatial analysis model supported by GIS. For these models, we can build the corresponding Cellular Automation Model. 4. Cellular Automata and fragment dimension Cellular Automata are closely related to the theory of fragment dimension. The self-replication and chaos features of Cellular Automation often lead to the ability of the Cellular Automation Model to exhibit Self-Similar fragment features in spatial configuration, that is to say, the simulation results of Cellular Automation can be quantitatively described using the fragtal theory. At the same time, in the classic example of fragment dimension, some models are, or very close to the cellular automata model. For example, the cohesive diffusion model we mentioned below. Therefore, some Cellular Automation models are fragtal dynamic models. However, Cellular Automation is significantly different from the theory of fragment.
Cellular Automata focuses on the simulation and analysis of the imagination mechanism; fragment Dimension focuses on the expression of phenomena. When modeling Cellular Automation, we start with the pattern of phenomena to build a Cellular Automation Model with specific meanings. However, the fragment dimension model is constructed from physical or mathematical laws and rules, then it is applied to a specific complex phenomenon. Most of its application methods are to describe the self-similarity and fragment dimension characteristics of the phenomenon. However, how much more valuable information can these dimensions provide to us? The problem of further application of the fractal theory is still solved (yi xianxiang, 1995 ).
In addition, both emphasize a process from the local to the whole, but in essence, there is a huge difference between the two. The essence of fragment theory is self-similarity. This self-similarity is not limited to ry, but has a broader and deeper meaning. It is a local (partial) it is statistically similar to the overall form, function, information and structural characteristics. Therefore, the methodology provided by the fractal theory to analyze the problem is to infer the overall feature from the local structure (declarative Peng, 1998 ). On the contrary, the essence of Cellular Automation lies in the overall "abrupt" complex behavior produced by a simple local structure under certain local rules; that is, the system (overall) at the macro level, some or part of the sum does not have a property (TAN Yuejin, etc., 1996 ). Therefore, the fractal theory emphasizes the similarity and correlation between the local and the whole. However, Cellular Automata focuses on the "abrupt" feature, that is, the uncertainty and nonlinear relationship between the local behavior structure and the overall behavior. 5. Cellular Automata and Markov (chain) Processes A Markov process is a typical random process. X (t) is a random process. When the Process status t0 is known at the moment, T (t> t0) the status is irrelevant to the status of the process before t0. This feature becomes ineffective. An ineffective random process is called a Markov process. The time and state in the Markov process can be continuous and discrete. We call the Markov process of time discretization and State discretization as a Markov chain. In a Markov chain, the state transition at each time point is controlled by a probability matrix of state transfer.
Markov chains and cellular automation are both dynamic models of time discretization and State discretization. They have certain conceptual similarities. In particular, for random cellular automata, the behavior of each cellular can be regarded as a Markov chain that not only has no time effect, but also has no space effect.
However, even random cellular automata are quite different from Markov chains. First, Markov chains do not have the concept of space, and there is only one state variable. The State quantity of Cellular Automation is closely related to the concept of space location. Second, the probability of state transfer in a Markov chain is usually preset, while the probability of state transfer in a random cellular machine is determined by the neighbor configuration of the current cellular. 6. Cellular Automation, Random Walk Model, and cohesive Diffusion Model
The Random Walk Model simulates a mathematical model that provides the "most likely State" in statistical mathematics. Its basic idea is: a particle in a given space: Its moving vector (including the direction and distance) in space is controlled by the random amount of transition probability, this allows us to simulate complex processes such as molecular Brown motion in nature and random motion of electrons in metals. Its theoretical research focuses on the motion of a single particle. However, there may be many particles in the random walking model, but they follow a unified random procedure, and the motion between them is independent of each other and does not affect each other. If the interaction between them is considered, other models based on random walk may be constructed, such as the cohesive diffusion model.
The diffusion-Limited Aggregation Model (DLA) can be regarded as a multi-particle random walking model, and its computing space is often a discrete grid. It was first proposed by A. Written and Sander in 1981. The basic idea is as follows: given an initial point as a consortium point and taking it as the center of the circle as a large circle, releasing a particle at a random point on the circumference, for the sake of simplicity, its motion is usually defined as a random walking process, until it moves adjacent to an existing consortium point, changes its state to a consortium point, and no longer moves; then it releases a particle randomly until the consortium. Repeat the above process to obtain a consortium connected set, which is similar to the ice on the glass in winter. The cohesive diffusion model can also have different forms. For example, the release point can be placed on the top of a quadrilateral, thus saving the growth of a thorny bush below. In 1984, the multi-particle diffusion aggregation model proposed by R · f · Voss was an improvement and development of the aggregation diffusion model. The basic idea is: in a given discrete space, free particles are randomly distributed according to a certain density. A consortium point can be set as a seed point in the center, and several consortium points can be randomly deployed as seeds, then the free particles walk randomly. Once adjacent to the consortium point, it becomes a new consortium point until all the free particles are "Consortium ". Cellular Automata, Random Walk models, and cohesive diffusion models are typical methods for generating fragment graphs. In many cases, they can generate similar complex patterns. However, there are still some differences between them.
The difference between the random walking model and cellular automatic machine is as follows: first, the random walking model generally only considers the motion of a single particle, however, the Cellular Automation Model usually has a large number of cellular units. Second, even if there are multiple particles in the model, the random walking model usually does not consider the interaction between particles, the motion of particles is independent of each other. Third, particles in random walking are the concept of motion, while the cellular mechanism of Cellular Automation is usually a process of state change; fourth, the moving space of particles in a random walk can be discrete or continuous. However, in Cellular Automation, cellular units are distributed on discrete spatial grids.
Cohesive diffusion model. In particular, the multi-particle cohesive diffusion model is very similar to the Cellular Automation Model: temporal space discretization; there are particle interactions in the model, and this effect has local characteristics, that is, when a free particle has a consortium point as a neighbor, the state changes to a consortium point. In particular, this transformation is only a one-way transformation, and the cohesive Diffusion Model eventually reaches a fixed-state scalar. The motion of particles follows the same law and can be calculated synchronously. Therefore. In a broad sense, the cohesive diffusion model can be classified as a special case of cellular automation. However, there are still several differences between them: 1. The Cellular Automation Model is oriented to the entire grid space, while the cohesive diffusion model is oriented to the movement of specific particles; 2. cellular machines usually only have state changes, and their spatial locations are fixed. particles in the cohesive diffusion model are not only stateful but also moving particles. 3. In aggregation and diffusion, multiple particles usually occupy a grid space point at the same time. In the cellular automata model, each grid point can have only one cellular cell. Therefore, in a sense, the cohesive diffusion model is more similar to the multi-subject model mentioned below, it can be seen as a "Brainless" subject model with no objective, competition, collaboration, and other intelligent features between particles. 7. Cellular Automation and multi-subject system Multi-Agent System (MAS) is a hot topic of Distributed Artificial Intelligence (Shi zhongzhi. 1998), mainly studying the interaction of intelligent behavior collaboration and competition among independent intelligent subjects for the purpose of mutual or different goals. Agent-based model (ABM), also known as entity-based model (EBM ), or an individual-based model (lndividual based model, abbreviated as IBM) is a subset of multi-subject systems, its main characteristic is that each subject represents an intelligent and autonomous entity or individual in the real world, such as an individual in a group, an individual in a plant, an individual in an ecosystem, an individual in an animal, and an automobile in a traffic flow, computers in the computing network and operators in the economic system. In the multi-subject system
In the system, the individual that makes up the system can be any part of the system. For example, an expert system is composed of comments. Some subjects in subject-based models have spatial concepts, automobiles in traffic flows, animals and plants in the ecosystem, and some do not have spatial concepts, such as computers in computing networks. For those entities with spatial concepts, the spatial representation can be continuous, such as a group of real-number coordinate pairs, or discrete, that is, the row and column values in the grid space. Cellular Automation is very similar to this entity model with the concept of discrete space. Both of them study the interaction between individuals in discrete space and form a complex behavior as a whole. However, there are still many differences;
(L) the subject in the subject model may be movable, such as an individual animal, but it may also not be movable; however, the meta-cellular individuals in the cellular automation model cannot be moved. The overall motion of the meta-Cellular Automation is achieved through the state changes of the meta-cellular individual.
(2) In the subject Model Based on Grid space, grid is only used as the Space Positioning of the subject. Multiple Subjects can occupy one grid site. In the Cellular Automation Model, each grid can have only one metacell in a specific State.
(3) In essence, it can be said that the subject model is oriented to (usually sparse, individual distributed on the grid space, while the cellular automation is oriented to the entire grid space. When the model runs, the main model only considers individual behaviors, while the Cellular Automation considers the status of each grid (cellular) in the whole cellular space. 8. Cellular Automation and system dynamic learning model Systemdynamics (SD) is a discipline that analyzes and studies the feedback system. It is also a comprehensive discipline that recognizes system problems and solves system problems. It was initially developed by Professor Jay W. forrestr of the Massachusetts Institute of Technology in 1956. It is characterized by introducing the concept of system analysis and emphasizing information feedback control, it is a comprehensive product of system theory, information theory, control theory, and decision theory. It is very suitable for studying the relationship between the structure, function, and dynamic behavior of complex systems. By analyzing the system structure, selecting appropriate factors, establishing the feedback relationship between them, and establishing a series of Differential Equations on this basis, the system dynamic learning equations are constructed, and the system is further investigated in different parameters and different
The system dynamically changes behaviors and trends when policy factors are input to provide decision-making support for decision makers. Because it can carry out dynamic simulation on the actual system, the system dynamics model can be used as the actual system, especially the "Laboratory" of the complex social, economic, and ecological systems. J · W ·, 1969; Yan xiangbin, 1999; Li yizizhi, 1987 ).
The system dynamic model has a wide range of practicability in the study of Earth science. Because it focuses on the overall best goal of the system, rather than simply pursuing the best goal of individual subsystems, it helps to achieve coordination between population, resources, environment and social and economic subsystems, comprehensive Research with no dimension is adopted. At the same time, this model still uses the first-order differential equations with delay functions and table functions, and can introduce the concept of input-output feedback loop, it can handle some complicated non-linear Problems intuitively and visually (declarative Peng, 1991 ). However, system dynamics also have inherent limitations, which limit its application in the earth sciences.
(1) first, SD's description of the system is subjective. The Modeler's understanding of the system structure mainly includes the selection of factors and the description of their correlation, which is directly reflected in the model. The uncertainty and non-linearity of a complex system determine that its system structure is chaotic. Different people may have different descriptions of it, system Dynamics will inevitably be subject to personal subjective interference in learning modeling, which affects the simulation results of the model.
(2) SD lacks a comprehensive indicator system. There are many qualitative factors in a complex system and a quantum process is required. Then, the classification and classification standards of multiple related factors need to be coordinated from the height of the system, which is often a challenge for dynamic system learning models.
(3) Finally, the lack of processing functions of spatial factors makes it difficult to describe the interactions and mutual feedback of elements in a spatial system (Zhang Xinsheng, 1997; Yan xiangbin, 1999 ). This is a fatal constraint on its application to research on spatial complex systems.
Both the system dynamic learning model and Cellular Automation adopt the "bottom-up" research idea, and use the feedback among system elements to simulate and predict the overall dynamic behavior of the system, they are both tools to study the dynamic changes of complex systems. However, the two are different: first, in terms of the model mechanism, the CA model is based on the spatial interaction between system elements, while SD considers the relationship between indicator attributes; second, in the form of model representation, CA is time, space, and State discretization, and the conversion rules are often displayed as reference tables, while SD is represented as a series of differential equations, the time, attribute, and feedback relationship between elements are continuous.
Now, the CA model is represented by the dynamic evolution of the spatial structure of the system, and the result of the SD Model is the dynamic change of a certain socio-economic index of the system. Finally, in the application, the CA model is mostly used to simulate the evolution of Time and Space in complex systems. The SD model lacks the concept of space and is more suitable for the simulation and Prediction of socio-economic systems.
From http://www.blog.edu.cn/user1/10235/archives/2005/203989.shtml |
