Cellular automaton

1118: 997: 826:, both of which involve modifying the definition of a cellular automaton in some way. Although such automata do not strictly satisfy the definition given above, it can be shown that they can be emulated by conventional cellular automata with sufficiently large neighborhoods and numbers of states, and can therefore be considered a subset of conventional cellular automata. Conversely, it has been shown that every reversible cellular automaton can be emulated by a block cellular automaton. 449:. Von Neumann's initial design was founded upon the notion of one robot building another robot. This design is known as the kinematic model. As he developed this design, von Neumann came to realize the great difficulty of building a self-replicating robot, and of the great cost in providing the robot with a "sea of parts" from which to build its replicant. Neumann wrote a paper entitled "The general and logical theory of automata" for the 469: 256: 1243:. This unit hypercube is the cellular automaton rule space. For next-nearest-neighbor cellular automata, a rule is specified by 2 = 32 bits, and the cellular automaton rule space is a 32-dimensional unit hypercube. A distance between two rules can be defined by the number of steps required to move from one vertex, which represents the first rule, and another vertex, representing another rule, along the 1284: 240: 1033: 1457: 154:(generally, a mathematical function) that determines the new state of each cell in terms of the current state of the cell and the states of the cells in its neighborhood. Typically, the rule for updating the state of cells is the same for each cell and does not change over time, and is applied to the whole grid simultaneously, though exceptions are known, such as the 337: 1255:

number of bit-1 in the 8-bit string for elementary rules (or 32-bit string for the next-nearest-neighbor rules). Drawing the rules in different Wolfram classes in these slices of the rule space show that class 1 rules tend to have lower number of bit-1s, thus located in one region of the space, whereas class 3 rules tend to have higher proportion (50%) of bit-1s.

4456:– Home to free MCell and MJCell cellular automata explorer software and rule libraries. The software supports a large number of 1D and 2D rules. The site provides both an extensive rules lexicon and many image galleries loaded with examples of rules. MCell is a Windows application, while MJCell is a Java applet. Source code is available. 349:

cells in the grid. One possible method is to allow the values in those cells to remain constant. Another method is to define neighborhoods differently for these cells. One could say that they have fewer neighbors, but then one would also have to define new rules for the cells located on the edges. These cells are usually handled with

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consequence of computation universality in a 1-dimensional CA. Intended as the introduction to the German edition of von Neumann's book on CA, he wrote a survey of the field with dozens of references to papers, by many authors in many countries over a decade or so of work, often overlooked by modern CA researchers.

1213:. This result is interesting because rule 110 is an extremely simple one-dimensional system, and difficult to engineer to perform specific behavior. This result therefore provides significant support for Wolfram's view that class 4 systems are inherently likely to be universal. Cook presented his proof at a 1254:

Cellular automaton rule space allows us to ask the question concerning whether rules with similar dynamical behavior are "close" to each other. Graphically drawing a high dimensional hypercube on the 2-dimensional plane remains a difficult task, and one crude locator of a rule in the hypercube is the

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There have been several attempts to classify cellular automata in formally rigorous classes, inspired by Wolfram's classification. For instance, Culik and Yu proposed three well-defined classes (and a fourth one for the automata not matching any of these), which are sometimes called Culik–Yu classes;

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adjacent cells. The latter includes the von Neumann neighborhood as well as the four diagonally adjacent cells. For such a cell and its Moore neighborhood, there are 512 (= 2) possible patterns. For each of the 512 possible patterns, the rule table would state whether the center cell will be black or

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in particular) in June 1983. The unexpected complexity of the behavior of these simple rules led Wolfram to suspect that complexity in nature may be due to similar mechanisms. His investigations, however, led him to realize that cellular automata were poor at modelling neural networks. Additionally,

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completed a Stanford PhD dissertation on Cellular Automata Theory, the first mathematical treatment of CA as a general class of computers. Many papers came from this dissertation: He showed the equivalence of neighborhoods of various shapes, how to reduce a Moore to a von Neumann neighborhood or how

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Ulam and von Neumann created a method for calculating liquid motion in the late 1950s. The driving concept of the method was to consider a liquid as a group of discrete units and calculate the motion of each based on its neighbors' behaviors. Thus was born the first system of cellular automata. Like

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Cellular automata are often simulated on a finite grid rather than an infinite one. In two dimensions, the universe would be a rectangle instead of an infinite plane. The obvious problem with finite grids is how to handle the cells on the edges. How they are handled will affect the values of all the

208:, automata in which patterns evolve into mostly stable or oscillating structures, automata in which patterns evolve in a seemingly chaotic fashion, and automata in which patterns become extremely complex and may last for a long time, with stable local structures. This last class is thought to be 1640:

For a random starting pattern, these maze-generating cellular automata will evolve into complex mazes with well-defined walls outlining corridors. Mazecetric, which has the rule B3/S1234 has a tendency to generate longer and straighter corridors compared with Maze, with the rule B3/S12345. Since

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Class 4: Nearly all initial patterns evolve into structures that interact in complex and interesting ways, with the formation of local structures that are able to survive for long periods of time. Class 2 type stable or oscillating structures may be the eventual outcome, but the number of steps

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These definitions are qualitative in nature and there is some room for interpretation. According to Wolfram, "...with almost any general classification scheme there are inevitably cases which get assigned to one class by one definition and another class by another definition. And so it is with

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The simplest nontrivial cellular automaton would be one-dimensional, with two possible states per cell, and a cell's neighbors defined as the adjacent cells on either side of it. A cell and its two neighbors form a neighborhood of 3 cells, so there are 2 = 8 possible patterns for a

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who identified these 4 classes of thermodynamical systems: (1) systems in thermodynamic equilibrium, (2) spatially/temporally uniform systems, (3) chaotic systems, and (4) complex far-from-equilibrium systems with dissipative structures (see figure 1 in the 1974 paper of Nicolis, Prigogine's

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and several papers dating from the mid-1980s, defined four classes into which cellular automata and several other simple computational models can be divided depending on their behavior. While earlier studies in cellular automata tended to try to identify types of patterns for specific rules,

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that two-dimensional CA are computation universal, introduced 1-dimensional CA, and showed that they too are computation universal, even with simple neighborhoods. He showed how to subsume the complex von Neumann proof of construction universality (and hence self-reproducing machines) into a

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developed a model of excitable media with some of the characteristics of a cellular automaton. Their specific motivation was the mathematical description of impulse conduction in cardiac systems. However their model is not a cellular automaton because the medium in which signals propagate is

1519:, of two or three dimensions; other tilings are possible, but not yet used. Cell states are determined only by interactions with adjacent neighbor cells. No means exists to communicate directly with cells farther away. One such cellular automaton processor array configuration is the 1024:

cellular automata are particularly interesting. The images below show the history of rules 30 and 110 when the starting configuration consists of a 1 (at the top of each image) surrounded by 0s. Each row of pixels represents a generation in the history of the automaton, with

901:, those hexagons could be used as cells. In many cases the resulting cellular automata are equivalent to those with rectangular grids with specially designed neighborhoods and rules. Another variation would be to make the grid itself irregular, such as with 1647:

Like some of the graph-theory based methods described above, these cellular automata typically generate mazes from a single starting pattern; hence it will usually be relatively easy to find the way to the starting cell, but harder to find the way anywhere

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is the evolution of a finite CA whose inverse is believed to be hard to find. Given the rule, anyone can easily calculate future states, but it appears to be very difficult to calculate previous states. Cellular automata have also been applied to design

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Visualization of a lattice gas automaton. The shades of grey of the individual pixels are proportional to the gas particle density (between 0 and 4) at that pixel. The gas is surrounded by a shell of yellow cells that act as reflectors to create a closed

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pigments according to the activating and inhibiting activity of its neighbor pigment cells, obeying a natural version of a mathematical rule. The cell band leaves the colored pattern on the shell as it grows slowly. For example, the widespread species

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can be used to generate mazes. Two well-known such cellular automata, Maze and Mazectric, have rulestrings B3/S12345 and B3/S1234. In the former, this means that cells survive from one generation to the next if they have at least one and at most five

572:, proposing that the physical laws of the universe are discrete by nature, and that the entire universe is the output of a deterministic computation on a single cellular automaton; "Zuse's Theory" became the foundation of the field of study called 1437:, acidified bromate, and a ceric salt were mixed together and left undisturbed, fascinating geometric patterns such as concentric circles and spirals propagate across the medium. In the "Computer Recreations" section of the August 1988 issue of 637:, arrangements of cells that essentially move themselves across the grid. It is possible to arrange the automaton so that the gliders interact to perform computations, and after much effort it has been shown that the Game of Life can emulate a 1485:

become demagnetized when heated. Moreover, results from studying the demagnetization phase transition can be transferred to other phase transitions, like the evaporation of a liquid into a gas; this convenient cross-applicability is known as

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cellular automata: there are occasionally rules...that show some features of one class and some of another." Wolfram's classification has been empirically matched to a clustering of the compressed lengths of the outputs of cellular automata.

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is the number of neighboring cells (including the cell to be calculated itself) used to determine the cell's next state. Thus, in the two-dimensional system with a Moore neighborhood, the total number of automata possible would be 2, or

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arrangement: when one goes off the top, one comes in at the corresponding position on the bottom, and when one goes off the left, one comes in on the right. (This essentially simulates an infinite periodic tiling, and in the field of

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Cellular automaton processors are physical implementations of CA concepts, which can process information computationally. Processing elements are arranged in a regular grid of identical cells. The grid is usually a square tiling, or

1008:, a standard naming convention invented by Wolfram that gives each rule a number from 0 to 255. A number of papers have analyzed and compared the distinct cases among the 256 cellular automata (many are trivially isomorphic). The 333:. More generally, it is sometimes assumed that the universe starts out covered with a periodic pattern, and only a finite number of cells violate that pattern. The latter assumption is common in one-dimensional cellular automata. 927:

In cellular automata, the new state of a cell is not affected by the new state of other cells. This could be changed so that, for instance, a 2 by 2 block of cells can be determined by itself and the cells adjacent to itself.

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supports von Neumann, Nobili, GOL, and a great many other systems of cellular automata. Developed by Tomas Rokicki and Andrew Trevorrow. This is the only simulator currently available that can demonstrate von Neumann type

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The neighborhood or rules could change over time or space. For example, initially the new state of a cell could be determined by the horizontally adjacent cells, but for the next generation the vertical cells would be used.

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Class 2: Nearly all initial patterns evolve quickly into stable or oscillating structures. Some of the randomness in the initial pattern may filter out, but some remains. Local changes to the initial pattern tend to remain

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discussed a cellular automaton developed by Martin Gerhardt and Heike Schuster of the University of Bielefeld (Germany). This automaton produces wave patterns that resemble those in the Belousov-Zhabotinsky reaction.

226:, in which the future value of individual cells only depends on the total value of a group of neighboring cells. Cellular automata can simulate a variety of real-world systems, including biological and chemical ones. 718:

Class 3: Nearly all initial patterns evolve in a pseudo-random or chaotic manner. Any stable structures that appear are quickly destroyed by the surrounding noise. Local changes to the initial pattern tend to spread

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Ilina, Olga; Gritsenko, Pavlo G.; Syga, Simon; Lippoldt, Jürgen; La Porta, Caterina A. M.; Chepizhko, Oleksandr; Grosser, Steffen; Vullings, Manon; Bakker, Gert-Jan; Starruß, Jörn; Bult, Peter (September 2020).

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in that patterns that do not have a living cell adjacent to 1, 4, or 5 other living cells in any generation will behave identically to it. However, for large patterns, it behaves very differently from Life.

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compiled many results following this point of view in what is still considered as a seminal paper for the mathematical study of cellular automata. The most fundamental result is the characterization in the

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at quantum scales), or any other physically useful means. This can be done in several ways so that no wires are needed between any elements. This is very unlike processors used in most computers today (

920: + 1. Sometimes a simpler rule is used; for example: "The rule is the Game of Life, but on each time step there is a 0.001% probability that each cell will transition to the opposite color." 950:

have a continuum of locations. The state of a location is a finite number of real numbers. Time is also continuous, and the state evolves according to differential equations. One important example is

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For one-dimensional cellular automata there are known algorithms for deciding whether a rule is reversible or irreversible. However, for cellular automata of two or more dimensions reversibility is

641:. It was viewed as a largely recreational topic, and little follow-up work was done outside of investigating the particularities of the Game of Life and a few related rules in the early 1970s. 1217:

conference on Cellular Automata in 1998, but Wolfram blocked the proof from being included in the conference proceedings, as Wolfram did not want the proof announced before the publication of

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cellular automata. The state of each cell in a totalistic cellular automaton is represented by a number (usually an integer value drawn from a finite set), and the value of a cell at time

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Wiener, N.; Rosenblueth, A. (1946). "The mathematical formulation of the problem of conduction of impulses in a network of connected excitable elements, specifically in cardiac muscle".

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behavior, which is neither completely random nor completely repetitive. Localized structures appear and interact in various complicated-looking ways. In the course of the development of

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that a particular pattern would make endless copies of itself within the given cellular universe by designing a 200,000 cell configuration that could do so. This design is known as the

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Crutchfeld, James P.; Mitchell, Melanie; Das, Rajarshi (2002). "The Evolutionary Design of Collective Computation in Cellular Automata". In Crutchfield, J. P.; Schuster, P. K. (eds.).

33: 3150:, who used it in a broader sense to refer to outer totalistic automata, not necessarily of two dimensions. The more specific meaning given here was used e.g. in several chapters of 886: 329:

It is usually assumed that every cell in the universe starts in the same state, except for a finite number of cells in other states; the assignment of state values is called a

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For larger cellular automaton rule space, it is shown that class 4 rules are located between the class 1 and class 3 rules. This observation is the foundation for the phrase

782:, a topological characterization of cellular automata. For cellular automata in which not every configuration has a preimage, the configurations without preimages are called 723:

required to reach this state may be very large, even when the initial pattern is relatively simple. Local changes to the initial pattern may spread indefinitely. Wolfram has

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boundary conditions.) This can be visualized as taping the left and right edges of the rectangle to form a tube, then taping the top and bottom edges of the tube to form a

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The primary classifications of cellular automata, as outlined by Wolfram, are numbered one to four. They are, in order, automata in which patterns generally stabilize into

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neighborhood. A rule consists of deciding, for each pattern, whether the cell will be a 1 or a 0 in the next generation. There are then 2 = 256 possible rules.

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continuous, and wave fronts are curves. A true cellular automaton model of excitable media was developed and studied by J. M. Greenberg and S. P. Hastings in 1978; see

795:; that is, there is no algorithm that takes as input an automaton rule and is guaranteed to determine correctly whether the automaton is reversible. The proof by 2264:

Davidenko, J. M.; Pertsov, A. V.; Salomonsz, R.; Baxter, W.; Jalife, J. (1992). "Stationary and drifting spiral waves of excitation in isolated cardiac muscle".

1645:, each maze generated is uniquely determined by its random starting pattern. This is a significant drawback since the mazes tend to be relatively predictable. 305:, which includes the two closest cells in each orthogonal direction, for a total of eight. The general equation for the total number of automata possible is 4411:. Proceedings of the First International Conference on Evolutionary Computation and Its Applications (EvCA'96). Moscow, Russia: Russian Academy of Sciences. 3079: 1632:. In the latter, this means that cells survive if they have one to four neighbours. If a cell has exactly three neighbours, it is born. It is similar to 778:. If a cellular automaton is reversible, its time-reversed behavior can also be described as a cellular automaton; this fact is a consequence of the 1886: 492:

working within a cellular automaton with a small neighborhood (only those cells that touch are neighbors; for von Neumann's cellular automata, only

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independently began working on cellular automata in mid-1981 after considering how complex patterns seemed formed in nature in violation of the

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Chowdhury, D. Roy; Basu, S.; Gupta, I. Sen; Chaudhuri, P. Pal (June 1994). "Design of CAECC - cellular automata based error correcting code".

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along with a set of rules for the cells to follow. Each square is called a "cell" and each cell has two possible states, black and white. The

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is a prototypical example, in which each cell can be in either of two states called "up" and "down", making an idealized representation of a

1227:(Vol. 15, No. 1), over ten years after Cook came up with it. Rule 110 has been the basis for some of the smallest universal Turing machines. 944:). The state of a location is a finite number of real numbers. Certain cellular automata can yield diffusion in liquid patterns in this way. 818:

and others. Several techniques can be used to explicitly construct reversible cellular automata with known inverses. Two common ones are the

774:). If one thinks of a cellular automaton as a function mapping configurations to configurations, reversibility implies that this function is 4376:"Building Efficient Computational Cellular Automata Models of Complex Systems: Background, Applications, Results, Software, and Pathologies" 1481:. By adjusting the parameters of the model, the proportion of cells being in the same state can be varied, in ways that help explicate how 3970:

Tomassini, M.; Sipper, M.; Perrenoud, M. (2000). "On the generation of high-quality random numbers by two-dimensional cellular automata".

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An elementary cellular automaton rule is specified by 8 bits, and all elementary cellular automaton rules can be considered to sit on the

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Gerhardt, M.; Schuster, H. (1989). "A cellular automaton describing the formation of spatially ordered structures in chemical systems".

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There are known examples of continuous spatial automata, which exhibit propagating phenomena analogous to gliders in the Game of Life.

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Reversible cellular automata are often used to simulate such physical phenomena as gas and fluid dynamics, since they obey the laws of

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Class 1: Nearly all initial patterns evolve quickly into a stable, homogeneous state. Any randomness in the initial pattern disappears.

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are handled similarly. This solves boundary problems with neighborhoods, but another advantage is that it is easily programmable using

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Tomita, Kohji; Kurokawa, Haruhisa; Murata, Satoshi (2009). "Graph-Rewriting Automata as a Natural Extension of Cellular Automata".

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Barral, Bernard; Chaté, Hugues; Manneville, Paul (1992). "Collective behaviors in a family of high-dimensional cellular automata".

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is an example of an outer totalistic cellular automaton with cell values 0 and 1; outer totalistic cellular automata with the same

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can be simulated with a two-state, two-dimensional cellular automata, each state corresponding to either an expanded or retracted

1117: 524:. The original work of Wiener and Rosenblueth contains many insights and continues to be cited in modern research publications on 4520: 4034: 4327: 4273: 4254: 4231: 4199: 4176: 4153: 4081: 3866: 3824: 3277: 3131: 2923: 2739: 2706: 2679: 1859: 1577: 1379:

Additionally, biological phenomena which require explicit modeling of the agents' velocities (for example, those involved in

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Wolfram's classification was the first attempt to classify the rules themselves. In order of complexity the classes are:

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and spots on leopards. When these are approximated by cellular automata, they often yield similar patterns. MacLennan

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a table, using states {0,1,2}), continuous functions are used, and the states become continuous (usually values in

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functions. For example, in a 1-dimensional cellular automaton like the examples below, the neighborhood of a cell

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Kroc, Jiří; Jiménez-Morales, Francisco; Guisado, José Luis; Lemos, María Carmen; Tkáč, Jakub (December 2019).

2087:, ed., Cerebral Mechanisms in Behavior – The Hixon Symposium, John Wiley & Sons, New York, 1951, pp. 1–31. 810:. Such cellular automata have rules specially constructed to be reversible. Such systems have been studied by 629:

Despite its simplicity, the system achieves an impressive diversity of behavior, fluctuating between apparent

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behavior, meaning even simple input patterns such as that shown lead to chaotic, seemingly random histories.

987: 761: 663: 485: 218: 186: 181:, a two-dimensional cellular automaton, that interest in the subject expanded beyond academia. In the 1980s, 155: 3661:"Cell adhesion heterogeneity reinforces tumour cell dissemination: novel insights from a mathematical model" 3385:

Wentian Li; Norman Packard; Chris G Langton (1990). "Transition phenomena in cellular automata rule space".

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if, for every current configuration of the cellular automaton, there is exactly one past configuration (

4515: 4505: 4191: 3858: 1850: 1808: 1802: 1717: 1531:) which are divided into sections with elements that can communicate with distant elements over wires. 747:

The idea that there are 4 classes of dynamical system came originally from Nobel-prize winning chemist

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Letichevskii, A. A.; Reshodko, L. V. (1974). "N. Wiener's theory of the activity of excitable media".

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of a cell is the nearby, usually adjacent, cells. The two most common types of neighborhoods are the

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The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics

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to understand in depth. Other cellular automata that have been of significance in physics include

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Some examples of biological phenomena modeled by cellular automata with a simple state space are:

937:. These are like totalistic cellular automata, but instead of the rule and states being discrete ( 4510: 4168: 4035:

Cellular automata in generative electronic music and sonic art: a historical and technical review

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of the values of the cells in its neighborhood (possibly including the cell itself) at time

446: 438: 4061: 134:). The grid can be in any finite number of dimensions. For each cell, a set of cells called its 4367:

Evolutionary Dynamics: Exploring the Interplay of Selection, Neutrality, Accident, and Function

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and order. One of the most apparent features of the Game of Life is the frequent occurrence of

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Also, rules can be probabilistic rather than deterministic. Such cellular automata are called

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Cellular Automaton Processor Based Systems for Genetic Sequence Comparison/Database Searching

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bear similarities to cellular automata, as each fibroblast only interacts with its neighbors.

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Proceedings of 6th International Conference in Software Engineering for Defence Applications

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A. K. Dewdney, The hodgepodge machine makes waves, Scientific American, p. 104, August 1988.

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Any dead cell with exactly three live neighbours becomes a live cell, as if by reproduction.

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are two-dimensional, with his self-replicator implemented algorithmically. The result was a

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An animation of the way the rules of a 1D cellular automaton determine the next generation

8: 4050:." Proceedings of 2001 Workshop on Artificial Life Models for Musical Applications. 2001. 1629: 1466: 1439: 1422: 1383:) may be modeled by cellular automata with a more complex state space and rules, such as 916:, the probabilities that the central cell will transition to each possible state at time 792: 741: 728: 608: 549: 292:. The former, named after the founding cellular automaton theorist, consists of the four 131: 102: 4429: 4418:

Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences

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Plants regulate their intake and loss of gases via a cellular automaton mechanism. Each

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Any live cell with fewer than two live neighbours dies, as if caused by underpopulation.

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One way to simulate a two-dimensional cellular automaton is with an infinite sheet of

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engaged in a systematic study of one-dimensional cellular automata, or what he calls

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for the blue cell. The range-2 "cross neighborhood" includes the pink cells as well.

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Reher, David; Klink, Barbara; Deutsch, Andreas; Voss-Böhme, Anja (11 August 2017).

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Pivato, M: "RealLife: The continuum limit of Larger than Life cellular automata",

3173: 3080:"Representing reversible cellular automata with reversible block cellular automata" 3060: 3030: 2964: 2892: 2829: 2819: 2644: 2640: 2547: 2533: 2450: 2327: 2301: 2281: 2238: 2114: 1906: 1772: 1687: 1662: 1595: 1557: 1263: 1248: 963: 898: 598:

became widely known, particularly among the early computing community. Invented by

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Hatzikirou, H.; Basanta, D.; Simon, M.; Schaller, K.; Deutsch, A. (1 March 2012).

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Any live cell with more than three live neighbours dies, as if by overpopulation.

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Models of massive parallelism: analysis of cellular automata and neural networks

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Any live cell with two or three live neighbours lives on to the next generation.

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Wainwright, Robert. "Conway's game of life: early personal recollections". In

3677: 3619: 1910: 1029:=0 being the top row. Each pixel is colored white for 0 and black for 1. 4499: 4319: 3876: 3848: 3834: 3743: 3686: 3627: 3051:(1999). "On the circuit depth of structurally reversible cellular automata". 1805: – Method in computational solid mechanics based on the discrete concept 1642: 1444: 1359: 1336: 1302:

Several biological processes occur—or can be simulated—by cellular automata.

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A cellular automaton based on hexagonal cells instead of squares (rule 34/2)

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is a popular version of this model. Another common neighborhood type is the

222:, where only a single configuration leads directly to a subsequent one, and 4492:

contains an extensive list of academic and professional reference material.

4437: 4409:

Evolving Cellular Automata with Genetic Algorithms: A Review of Recent Work

3900: 3751: 3720:"'Go or Grow': the key to the emergence of invasion in tumour progression?" 3704: 3645: 3586:

Yves Bouligand (1986). "Fibroblasts, Morphogenesis and Cellular Automata".

3514: 3330: 3064: 2843: 2824: 2215: 1542: 1538: 1516: 1434: 1206: 1005: 882:

There are many possible generalizations of the cellular automaton concept.

689: 501: 190: 2293: 1369:, and complex behaviors such as recognition and learning can be simulated. 854:

depends on both its own state and the total of its neighbors at time

3955:

Muhtaroglu, Ali (August 1996). "4.1 Cellular Automaton Processor (CAP)".

3048: 3010: 2318:(1969). "Endomorphisms and automorphisms of the shift dynamical system". 1482: 1474: 955: 796: 563: 556: 273: 205: 36: 2421:"Introduction to and Survey of Cellular Automata or Polyautomata Theory" 4209:

Eppstein, David. "Growth and decay in life-like cellular autometa". In

3384: 2758: 2331: 2242: 1562: 1392: 1372: 1355: 1315: 724: 673: 630: 493: 371: 293: 97:. Cellular automata have found application in various areas, including 4486:(includes discussion on triangular grids, and larger neighborhood CAs) 4453: 4020: 3983: 1769: – Discrete (i.e., incremental) version of infinitesimal calculus 142: = 0) is selected by assigning a state for each cell. A new 4223: 4048:

Evolving cellular automata music: From sound synthesis to composition

2285: 1732: 1523:. Cell interaction can be via electric charge, magnetism, vibration ( 1425:

that can be simulated by means of a cellular automaton. In the 1950s

1330: 1240: 800: 775: 531:

In the 1960s, cellular automata were studied as a particular type of

32: 3321: 3294: 468: 2083:

John von Neumann, "The general and logical theory of automata," in

1712: 1545:. Two-dimensional cellular automata can be used for constructing a 1384: 1321: 1310: 1221:. In 2004, Cook's proof was finally published in Wolfram's journal 1126: 1021: 1017: 771: 681: 594:

In the 1970s a two-state, two-dimensional cellular automaton named

194: 3914: 2635: 1568:

Other problems that can be solved with cellular automata include:

1473:

to study phenomena like fluid dynamics and phase transitions. The

138:

is defined relative to the specified cell. An initial state (time

4469:– An atlas of various types of one-dimensional cellular automata. 3146:

The phrase "life-like cellular automaton" dates back at least to

1722: 1707: 1534: 1524: 1348: 1341: 1326: 1209:

proved that some of these structures were rich enough to support

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considers continuous spatial automata as a model of computation.

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Statistical Mechanics: Entropy, Order Parameters, and Complexity

3013:(1990). "Reversibility of 2D cellular automata is undecidable". 2665: 1433:) discovered that when a thin, homogenous layer of a mixture of 1387:. These include phenomena of great medical importance, such as: 1247:

of the hypercube. This rule-to-rule distance is also called the

727:

that many class 4 cellular automata, if not all, are capable of

672:

during this period Wolfram formulated the concepts of intrinsic

445:, Ulam's colleague at Los Alamos, was working on the problem of 4407:

Mitchell, Melanie; Crutchfeld, James P.; Das, Rajarshi (1996).

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cells reside in a narrow band along the shell's lip. Each cell

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These 256 cellular automata are generally referred to by their

893:

One way is by using something other than a rectangular (cubic,

461:

performed many of the earliest explorations of these models of

3717: 2763:"The structure of the elementary cellular automata rule space" 1509: 958:

to explain how chemical reactions could create the stripes on

858: − 1 then the cellular automaton is properly called 731:. This has been proven for Rule 110 and Conway's Game of Life. 548:

of the set of global rules of cellular automata as the set of

457:

system for creating a reductionist model of self-replication.

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Turing, A. M. (1952). "The Chemical Basis of Morphogenesis".

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Rule 110, like the Game of Life, exhibits what Wolfram calls

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to reduce any neighborhood to a von Neumann neighborhood. He

437:

in the 1940s, studied the growth of crystals, using a simple

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Proposes reaction-diffusion, a type of continuous automaton.

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Kier, Lemont B.; Seybold, Paul G.; Cheng, Chao-Kun (2005).

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Computational analysis of one-dimensional cellular automata

2536:(4 October 2002). "Is the Universe a Universal Computer?". 1672: 239: 3600: 2757: 1939:

Cellular Automata Machines: A New Environment for Modeling

4163:

Bialynicki-Birula, Iwo; Bialynicka-Birula, Iwona (2004).

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Cellular Automata in Hyperbolic Spaces – Tome I, Volume 1

1617:

Maze generation algorithm § Cellular automaton algorithms

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cells), and with 29 states per cell. Von Neumann gave an

3969: 336: 3205:"First gliders navigate ever-changing Penrose universe" 1992: 1990: 1988: 1986: 912:. A probabilistic rule gives, for each pattern at time 165:

The concept was originally discovered in the 1940s by

4062:"Evolving Diverse Cellular Automata Based Level Maps" 3155: 3147: 3084:

Discrete Mathematics and Theoretical Computer Science

2197: 2195: 2035: 688:—a fact proved later by Wolfram's research assistant 4477: 3259: 2228: 1983: 1798:

Pages displaying wikidata descriptions as a fallback

1789:

Pages displaying wikidata descriptions as a fallback

3264:. Understanding Complex Systems. pp. 291–309. 2097:Kemeny, John G. (1955). "Man viewed as a machine". 1292:

exhibits a cellular automaton pattern on its shell.

112:A cellular automaton consists of a regular grid of 4284: 4283:von Neumann, John (1966). Burks, Arthur W. (ed.). 4114:Nathaniel Johnston; et al. (21 August 2010). 4113: 3896:"A-D-E Classification of Conformal Field Theories" 2694: 2659: 2192: 1365:Threshold automata have been invented to simulate 1325:, are generated by natural cellular automata. The 535:and the connection with the mathematical field of 4266:Modeling Chemical Systems using Cellular Automata 2532: 2201: 850: − 1. If the state of the cell at time 655:found in brains. He published his first paper in 313:is the number of possible states for a cell, and 4497: 3773: 3351: 2573: 2571: 2569: 1881: 1498:has been of particular interest, as it requires 453:in 1948. Ulam was the one who suggested using a 4059: 3954: 3475:Proceedings of the National Academy of Sciences 2986:"De Bruijn Graphs and Linear Cellular Automata" 2142: 1966:Cellular Automata: A Discrete View of the World 1935: 1758: – Study of abstract machines and automata 981: 4188:Cellular Automata Modeling of Physical Systems 3588:Disordered Systems and Biological Organization 3585: 3567: 3184: 2504: 2502: 2500: 2498: 2496: 2469:http://www.igblan.free-online.co.uk/igblan/ca/ 2385:"Simple Computation-Universal Cellular Spaces" 2171: 2169: 1817: – Computerised aid to land use decisions 1205:, as a research assistant to Wolfram in 1994, 3889: 3468: 2753: 2751: 2666:G. Cattaneo; E. Formenti; L. Margara (1998). 2583: 2566: 2481: 2403:"Simple Nontrivial Self-Reproducing Machines" 2007: 2005: 1811: – Abstract model of quantum computation 1781: – Tool for simulating cellular automata 974:are extensions of cellular automata based on 954:textures, differential equations proposed by 539:was established for the first time. In 1969, 422:is the index (horizontal) in one generation. 4480:from the newsgroup comp.theory.cell-automata 2688: 2044:. Sterling Publishing Company, Inc. p. 1887:"Statistical Mechanics of Cellular Automata" 1465:Probabilistic cellular automata are used in 1402:in the development of aggressive carcinomas. 4282: 4165:Modeling Reality: How Computers Mirror Life 4060:Ashlock, Daniel; Kreitzer, Matthew (2020). 4037:." Digital Creativity 16.3 (2005): 165-185. 3077: 2950: 2911: 2862: 2860: 2595: 2514: 2493: 2187: 2166: 2154: 1936:Toffoli, Tommaso; Margolus, Norman (1987). 1823: – Computing by new or unusual methods 1583: 1510:Computer science, coding, and communication 4490:Cosma Shalizi's Cellular Automata Notebook 4299: 3438:The Geometry and Pigmentation of Seashells 3292: 3119: 3098: 2932: 2881:"Tessellations with local transformations" 2878: 2748: 2721: 2715: 2487: 2346: 2067: 2065: 2002: 1657:Specific cellular automata rules include: 27:Discrete model studied in computer science 4301: 4210: 4140: 3931: 3913: 3694: 3676: 3635: 3504: 3494: 3430: 3428: 3426: 3406: 3320: 3151: 2968: 2951:Amoroso, Serafino; Patt, Yale N. (1972). 2896: 2833: 2823: 2634: 2367:"Cellular Automata Complexity Trade-Offs" 2161:Bialynicki-Birula, Bialynicka-Birula 2004 2012:Bialynicki-Birula, Bialynicka-Birula 2004 1877: 1875: 1552:Cellular automata have been proposed for 834:A special class of cellular automata are 216:. Special types of cellular automata are 4220:Cellular Automata: Theory and Experiment 4208: 3190: 2918:. Archives contemporaines. p. 134. 2857: 2222: 2130: 2017: 1455: 1385:biological lattice-gas cellular automata 1282: 1116: 1031: 995: 884: 467: 335: 31: 4263: 4186:Chopard, Bastien; Droz, Michel (2005). 3115: 3113: 2794: 2788: 2467:Paul Chapman. Life universal computer. 2436: 2314: 2062: 2029: 1996: 1537:was originally suggested as a possible 1391:Characterization of different modes of 870:structure as Life are sometimes called 14: 4498: 4243:Cellular Automata: A Discrete Universe 4033:Burraston, Dave, and Ernest Edmonds. " 3847: 3808: 3559:: CS1 maint: archived copy as title ( 3434: 3423: 3221: 3202: 3123:Cellular automata: a discrete universe 2983: 2096: 1872: 4346:Berto, Francesco; Tagliabue, Jacopo. 3959:. Cornell University. pp. 62–74. 3286: 3148:Barral, Chaté & Manneville (1992) 2613: 1969:. Wiley & Sons, Inc. p. 40. 1340:bears a pattern resembling Wolfram's 522:Greenberg-Hastings cellular automaton 370:(doughnut shape). Universes of other 4369:. New York: Oxford University Press. 3469:Peak, West; Messinger, Mott (2004). 3435:Coombs, Stephen (15 February 2009), 3352:Weinberg, Steven (24 October 2002). 3226:(3rd ed.). New York: Springer. 3126:. World Scientific. pp. 44–45. 3110: 3047: 3009: 2670:. In M. Delorme; J. Mazoyer (eds.). 2181: 1752: – Type of computational models 1594:Cellular automata have been used in 1573:Firing squad synchronization problem 1354:Moving wave patterns on the skin of 799:is related to the tiling problem by 116:, each in one of a finite number of 73:. Cellular automata are also called 4357:Stanford Encyclopedia of Philosophy 4287:Theory of Self-Reproducing Automata 2672:Cellular automata: a parallel model 1641:these cellular automaton rules are 897:) grid. For example, if a plane is 877: 612:article, its rules are as follows: 24: 4338: 2455:10.1038/scientificamerican1070-120 1775: – Nonlinear dynamical system 1609: 173:while they were contemporaries at 25: 4532: 4447: 4185: 3724:Mathematical Medicine and Biology 2418: 2400: 2382: 2364: 2119:10.1038/scientificamerican0455-58 1652: 695: 562:In 1969, German computer pioneer 506:von Neumann universal constructor 418:is the time step (vertical), and 303:extended von Neumann neighborhood 297:white on the next time interval. 3250:, 372 (1), March 2007, pp. 46–68 1615:This section is an excerpt from 1131:(binary 01101110 = decimal 110) 581:Also in 1969 computer scientist 490:universal copier and constructor 441:as his model. At the same time, 254: 238: 4240: 4053: 4040: 4027: 3998: 3963: 3948: 3883: 3841: 3802: 3767: 3758: 3711: 3652: 3594: 3579: 3573: 3521: 3462: 3378: 3345: 3295:"What Kind of Science is This?" 3253: 3240: 3215: 3196: 3140: 3071: 3041: 3003: 2977: 2944: 2905: 2872: 2701:. World Scientific. p. 8. 2607: 2589: 2577: 2526: 2461: 2430: 2412: 2394: 2376: 2358: 2308: 2257: 2148: 2090: 2077: 1815:Spatial decision support system 1273: 1046:(binary 00011110 = decimal 30) 910:probabilistic cellular automata 820:second-order cellular automaton 486:von Neumann's cellular automata 160:asynchronous cellular automaton 150:by 1), according to some fixed 4218:Gutowitz, Howard, ed. (1991). 4162: 4146:Game of Life Cellular Automata 4133: 4008:IEEE Transactions on Computers 3972:IEEE Transactions on Computers 3444:, pp. 3–4, archived from 2645:10.25088/ComplexSystems.19.1.1 2160: 2036:Pickover, Clifford A. (2009). 2011: 1956: 1929: 1839: 1728:Von Neumann cellular automaton 1506:, which simulate fluid flows. 1490:. The phase transition in the 1313:, like the ones in the genera 435:Los Alamos National Laboratory 360:partial differential equations 175:Los Alamos National Laboratory 13: 1: 4521:Computational fields of study 4217: 3854:Statistical Physics of Fields 3354:"Is the Universe a Computer?" 2970:10.1016/s0022-0000(72)80013-8 2912:Margenstern, Maurice (2007). 2898:10.1016/S0022-0000(72)80009-6 2866: 1828: 1547:pseudorandom number generator 1419:Belousov–Zhabotinsky reaction 1230: 988:Elementary cellular automaton 829: 780:Curtis–Hedlund–Lyndon theorem 762:Reversible cellular automaton 755: 546:Curtis–Hedlund–Lyndon theorem 212:, or capable of simulating a 156:stochastic cellular automaton 4308: 4293:University of Illinois Press 3796:10.1016/0167-2789(89)90081-x 3417:10.1016/0167-2789(90)90175-O 3359:The New York Review of Books 3270:10.1007/978-3-642-01284-6_14 3248:Theoretical Computer Science 3178:10.1016/0375-9601(92)91013-H 3104: 3078:Durand-Lose, Jérôme (2001). 3035:10.1016/0167-2789(90)90195-U 2601: 2520: 2508: 2175: 1833: 1412: 1262:, and is reminiscent of the 982:Elementary cellular automata 678:computational irreducibility 664:elementary cellular automata 649:Second Law of Thermodynamics 362:is sometimes referred to as 351:periodic boundary conditions 187:elementary cellular automata 46:" in the cellular automaton 7: 4380:Advances in Complex Systems 4241:Ilachinski, Andrew (2001). 4074:10.1007/978-3-030-14687-0_2 3120:Ilachinski, Andrew (2001). 2938: 2695:Burton H. 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M. Zhabotinsky 1309:Patterns of some 1193: 1192: 1110:Rule 30 exhibits 1108: 1107: 569:Calculating Space 541:Gustav A. Hedlund 537:symbolic dynamics 59:cellular automata 18:Cellular automata 16:(Redirected from 4528: 4441: 4412: 4403: 4370: 4361: 4352:Zalta, Edward N. 4333: 4310:Wolfram, Stephen 4305: 4302:Adamatzky (2010) 4296: 4290: 4279: 4260: 4247:World Scientific 4237: 4214: 4211:Adamatzky (2010) 4205: 4182: 4159: 4128: 4127: 4125: 4123: 4111: 4096: 4095: 4057: 4051: 4044: 4038: 4031: 4025: 4024: 4002: 3996: 3995: 3967: 3961: 3960: 3952: 3946: 3945: 3935: 3917: 3887: 3881: 3880: 3845: 3839: 3838: 3806: 3800: 3799: 3771: 3765: 3762: 3756: 3755: 3715: 3709: 3708: 3698: 3680: 3656: 3650: 3649: 3639: 3614:(9): 1103–1115. 3598: 3592: 3591: 3583: 3577: 3571: 3565: 3564: 3558: 3550: 3548: 3546: 3540: 3534:. Archived from 3533: 3525: 3519: 3518: 3508: 3498: 3466: 3460: 3459: 3458: 3456: 3450: 3443: 3432: 3421: 3420: 3410: 3382: 3376: 3375: 3373: 3371: 3349: 3343: 3342: 3324: 3290: 3284: 3283: 3257: 3251: 3244: 3238: 3237: 3219: 3213: 3212: 3200: 3194: 3193:, pp. 72–73 3188: 3182: 3181: 3152:Adamatzky (2010) 3144: 3138: 3137: 3117: 3108: 3102: 3096: 3095: 3075: 3069: 3068: 3045: 3039: 3038: 3021:(1–3): 379–385. 3007: 3001: 3000: 2990: 2981: 2975: 2974: 2972: 2948: 2942: 2936: 2930: 2929: 2909: 2903: 2902: 2900: 2876: 2870: 2864: 2855: 2854: 2852: 2850: 2837: 2827: 2810:(7): 2748–2751. 2801: 2795:Nicolis (1974). 2792: 2786: 2785: 2783: 2781: 2767: 2755: 2746: 2745: 2729: 2719: 2713: 2712: 2692: 2686: 2685: 2663: 2657: 2656: 2638: 2620: 2611: 2605: 2599: 2593: 2587: 2581: 2575: 2564: 2563: 2530: 2524: 2518: 2512: 2506: 2491: 2485: 2479: 2465: 2459: 2458: 2434: 2428: 2427: 2425: 2416: 2410: 2409: 2407: 2398: 2392: 2391: 2389: 2380: 2374: 2373: 2371: 2362: 2356: 2350: 2344: 2343: 2312: 2306: 2305: 2286:10.1038/355349a0 2261: 2255: 2254: 2226: 2220: 2219: 2199: 2190: 2188:von Neumann 1966 2185: 2179: 2173: 2164: 2158: 2152: 2146: 2140: 2134: 2128: 2122: 2094: 2088: 2081: 2075: 2069: 2060: 2059: 2043: 2033: 2027: 2021: 2015: 2009: 2000: 1994: 1981: 1980: 1960: 1954: 1953: 1933: 1927: 1926: 1924: 1922: 1913:. Archived from 1883:Wolfram, Stephen 1879: 1870: 1843: 1799: 1790: 1773:Excitable medium 1602:composition and 1596:generative music 1578:Majority problem 1558:one-way function 1264:phase transition 1249:Hamming distance 1136:current pattern 1133: 1124: 1051:current pattern 1048: 1039: 878:Related automata 860:outer totalistic 533:dynamical system 473:John von Neumann 443:John von Neumann 325: 323: 258: 242: 171:John von Neumann 95:iterative arrays 65:) is a discrete 21: 4536: 4535: 4531: 4530: 4529: 4527: 4526: 4525: 4496: 4495: 4450: 4445: 4415: 4406: 4373: 4364: 4345: 4341: 4339:Further reading 4336: 4330: 4276: 4257: 4234: 4202: 4179: 4156: 4136: 4131: 4121: 4119: 4112: 4099: 4084: 4058: 4054: 4045: 4041: 4032: 4028: 4003: 3999: 3968: 3964: 3953: 3949: 3888: 3884: 3869: 3846: 3842: 3827: 3807: 3803: 3772: 3768: 3763: 3759: 3716: 3712: 3657: 3653: 3599: 3595: 3584: 3580: 3574:Ilachinsky 2001 3572: 3568: 3552: 3551: 3544: 3542: 3541:on 25 July 2010 3538: 3531: 3529:"Archived copy" 3527: 3526: 3522: 3467: 3463: 3454: 3452: 3448: 3441: 3433: 3424: 3383: 3379: 3369: 3367: 3350: 3346: 3322:10.1038/417216a 3291: 3287: 3280: 3258: 3254: 3245: 3241: 3234: 3220: 3216: 3201: 3197: 3189: 3185: 3145: 3141: 3134: 3118: 3111: 3103: 3099: 3094:on 15 May 2011. 3076: 3072: 3046: 3042: 3008: 3004: 2993:Complex Systems 2988: 2982: 2978: 2949: 2945: 2937: 2933: 2926: 2910: 2906: 2877: 2873: 2865: 2858: 2848: 2846: 2799: 2793: 2789: 2779: 2777: 2770:Complex Systems 2765: 2756: 2749: 2742: 2720: 2716: 2709: 2693: 2689: 2682: 2664: 2660: 2623:Complex Systems 2618: 2612: 2608: 2600: 2596: 2590:Ilachinsky 2001 2588: 2584: 2578:Ilachinsky 2001 2576: 2567: 2546:(5591): 65–68. 2531: 2527: 2519: 2515: 2507: 2494: 2488:Wainwright 2010 2486: 2482: 2476:Wayback Machine 2466: 2462: 2435: 2431: 2423: 2417: 2413: 2405: 2399: 2395: 2387: 2381: 2377: 2369: 2363: 2359: 2351: 2347: 2326:(4): 320–3751. 2313: 2309: 2262: 2258: 2227: 2223: 2200: 2193: 2186: 2182: 2174: 2167: 2159: 2155: 2149:Ilachinski 2001 2147: 2143: 2135: 2131: 2095: 2091: 2082: 2078: 2070: 2063: 2056: 2034: 2030: 2022: 2018: 2010: 2003: 1995: 1984: 1977: 1961: 1957: 1950: 1934: 1930: 1920: 1918: 1880: 1873: 1844: 1840: 1836: 1831: 1826: 1797: 1788: 1756:Automata theory 1745: 1739: 1737: 1693:Langton's loops 1655: 1650: 1649: 1620: 1612: 1610:Maze generation 1592: 1586: 1512: 1454: 1415: 1300: 1281: 1276: 1233: 1224:Complex Systems 1130: 1045: 990: 984: 880: 832: 816:Norman Margolus 812:Tommaso Toffoli 764: 758: 698: 653:neural networks 645:Stephen Wolfram 575:digital physics 498:existence proof 463:artificial life 451:Hixon Symposium 439:lattice network 428: 413: 403: 394: 383: 353:resulting in a 321: 319: 270: 269: 268: 267: 266: 259: 251: 250: 243: 232: 199:Turing-complete 183:Stephen Wolfram 75:cellular spaces 71:automata theory 28: 23: 22: 15: 12: 11: 5: 4534: 4524: 4523: 4518: 4513: 4511:Systems theory 4508: 4494: 4493: 4487: 4481: 4475: 4470: 4464: 4457: 4449: 4448:External links 4446: 4444: 4443: 4424:(641): 37–72. 4413: 4404: 4371: 4362: 4342: 4340: 4337: 4335: 4334: 4328: 4306: 4297: 4280: 4274: 4261: 4255: 4238: 4232: 4215: 4206: 4200: 4183: 4177: 4160: 4154: 4144:, ed. (2010). 4137: 4135: 4132: 4130: 4129: 4097: 4082: 4052: 4039: 4026: 4015:(6): 759–764. 3997: 3962: 3947: 3882: 3867: 3849:Kardar, Mehran 3840: 3825: 3801: 3782:(3): 209–221. 3766: 3757: 3710: 3665:Biology Direct 3651: 3593: 3578: 3566: 3520: 3481:(4): 918–922. 3461: 3422: 3408:10.1.1.15.2786 3393:(1–3): 77–94. 3377: 3344: 3285: 3278: 3252: 3239: 3232: 3214: 3195: 3183: 3164:(4): 279–285. 3139: 3132: 3109: 3097: 3070: 3040: 3002: 2976: 2963:(5): 448–464. 2943: 2931: 2924: 2904: 2891:(5): 373–388. 2871: 2856: 2787: 2747: 2740: 2714: 2707: 2687: 2680: 2658: 2606: 2594: 2582: 2565: 2525: 2513: 2492: 2480: 2460: 2449:(4): 120–123. 2429: 2411: 2393: 2375: 2357: 2345: 2316:Hedlund, G. A. 2307: 2256: 2237:(5): 856–864. 2221: 2191: 2180: 2165: 2153: 2151:, p. xxix 2141: 2129: 2089: 2076: 2061: 2055:978-1402757969 2054: 2028: 2016: 2001: 1982: 1975: 1955: 1948: 1928: 1897:(3): 601–644. 1871: 1846:Daniel Dennett 1837: 1835: 1832: 1830: 1827: 1825: 1824: 1818: 1812: 1806: 1800: 1791: 1782: 1776: 1770: 1764: 1759: 1753: 1746: 1744: 1741: 1736: 1735: 1730: 1725: 1720: 1715: 1710: 1705: 1700: 1695: 1690: 1685: 1680: 1675: 1670: 1665: 1659: 1654: 1653:Specific rules 1651: 1621: 1613: 1611: 1608: 1590:Generative art 1585: 1582: 1581: 1580: 1575: 1521:systolic array 1511: 1508: 1453: 1450: 1431:B. P. Belousov 1414: 1411: 1410: 1409: 1403: 1396: 1377: 1376: 1370: 1363: 1352: 1345: 1280: 1277: 1275: 1272: 1268:thermodynamics 1232: 1229: 1191: 1190: 1187: 1184: 1181: 1178: 1175: 1172: 1169: 1166: 1162: 1161: 1158: 1155: 1152: 1149: 1146: 1143: 1140: 1137: 1106: 1105: 1102: 1099: 1096: 1093: 1090: 1087: 1084: 1081: 1077: 1076: 1073: 1070: 1067: 1064: 1061: 1058: 1055: 1052: 986:Main article: 983: 980: 879: 876: 831: 828: 808:thermodynamics 785:Garden of Eden 760:Main article: 757: 754: 749:Ilya Prigogine 733: 732: 720: 716: 712: 697: 696:Classification 694: 692:in the 1990s. 661:investigating 627: 626: 623: 620: 617: 604:Martin Gardner 583:Alvy Ray Smith 513:Norbert Wiener 431:Stanislaw Ulam 427: 424: 408: 399: 389: 381: 260: 253: 252: 244: 237: 236: 235: 234: 233: 231: 228: 214:Turing machine 167:Stanislaw Ulam 107:microstructure 26: 9: 6: 4: 3: 2: 4533: 4522: 4519: 4517: 4514: 4512: 4509: 4507: 4504: 4503: 4501: 4491: 4488: 4485: 4482: 4479: 4476: 4474: 4471: 4468: 4467:Wolfram Atlas 4465: 4461: 4458: 4455: 4452: 4451: 4439: 4435: 4431: 4427: 4423: 4419: 4414: 4410: 4405: 4401: 4397: 4393: 4389: 4385: 4381: 4377: 4372: 4368: 4363: 4359: 4358: 4353: 4349: 4344: 4343: 4331: 4325: 4321: 4320:Wolfram Media 4317: 4316: 4311: 4307: 4303: 4298: 4294: 4289: 4288: 4281: 4277: 4271: 4267: 4262: 4258: 4252: 4248: 4244: 4239: 4235: 4229: 4225: 4221: 4216: 4212: 4207: 4203: 4197: 4193: 4189: 4184: 4180: 4174: 4170: 4166: 4161: 4157: 4151: 4147: 4143: 4139: 4138: 4117: 4110: 4108: 4106: 4104: 4102: 4093: 4089: 4085: 4079: 4075: 4071: 4067: 4063: 4056: 4049: 4043: 4036: 4030: 4022: 4018: 4014: 4010: 4009: 4001: 3993: 3989: 3985: 3981: 3977: 3973: 3966: 3958: 3951: 3943: 3939: 3934: 3929: 3925: 3921: 3916: 3911: 3907: 3903: 3902: 3897: 3893: 3886: 3878: 3874: 3870: 3864: 3860: 3856: 3855: 3850: 3844: 3836: 3832: 3828: 3822: 3818: 3814: 3813: 3805: 3797: 3793: 3789: 3785: 3781: 3777: 3770: 3761: 3753: 3749: 3745: 3741: 3737: 3733: 3729: 3725: 3721: 3714: 3706: 3702: 3697: 3692: 3688: 3684: 3679: 3674: 3670: 3666: 3662: 3655: 3647: 3643: 3638: 3633: 3629: 3625: 3621: 3617: 3613: 3609: 3605: 3597: 3589: 3582: 3576:, p. 275 3575: 3570: 3562: 3556: 3537: 3530: 3524: 3516: 3512: 3507: 3502: 3497: 3492: 3488: 3484: 3480: 3476: 3472: 3465: 3447: 3440: 3439: 3431: 3429: 3427: 3418: 3414: 3409: 3404: 3400: 3396: 3392: 3388: 3381: 3365: 3361: 3360: 3355: 3348: 3340: 3336: 3332: 3328: 3323: 3318: 3314: 3310: 3306: 3302: 3301: 3296: 3289: 3281: 3275: 3271: 3267: 3263: 3256: 3249: 3243: 3235: 3233:0-387-95228-4 3229: 3225: 3218: 3210: 3209:New Scientist 3206: 3199: 3192: 3191:Eppstein 2010 3187: 3179: 3175: 3171: 3167: 3163: 3159: 3153: 3149: 3143: 3135: 3129: 3125: 3124: 3116: 3114: 3106: 3101: 3093: 3089: 3085: 3081: 3074: 3066: 3062: 3058: 3054: 3050: 3044: 3036: 3032: 3028: 3024: 3020: 3016: 3012: 3006: 2998: 2994: 2987: 2980: 2971: 2966: 2962: 2958: 2954: 2947: 2941:, p. 103 2940: 2935: 2927: 2921: 2917: 2916: 2908: 2899: 2894: 2890: 2886: 2882: 2875: 2869:, p. 379 2868: 2863: 2861: 2845: 2841: 2836: 2831: 2826: 2821: 2817: 2813: 2809: 2805: 2798: 2791: 2775: 2771: 2764: 2760: 2754: 2752: 2743: 2737: 2733: 2728: 2727: 2718: 2710: 2704: 2700: 2699: 2691: 2683: 2677: 2673: 2669: 2662: 2654: 2650: 2646: 2642: 2637: 2632: 2628: 2624: 2617: 2610: 2604:, p. 231 2603: 2598: 2591: 2586: 2579: 2574: 2572: 2570: 2561: 2557: 2553: 2549: 2545: 2541: 2540: 2535: 2529: 2523:, p. 881 2522: 2517: 2511:, p. 880 2510: 2505: 2503: 2501: 2499: 2497: 2489: 2484: 2478:November 2002 2477: 2473: 2470: 2464: 2456: 2452: 2448: 2444: 2440: 2433: 2422: 2415: 2404: 2397: 2386: 2379: 2368: 2361: 2355:, p. 182 2354: 2349: 2341: 2337: 2333: 2329: 2325: 2321: 2317: 2311: 2303: 2299: 2295: 2291: 2287: 2283: 2279: 2275: 2271: 2267: 2260: 2252: 2248: 2244: 2240: 2236: 2232: 2225: 2217: 2213: 2210:(3): 205–65. 2209: 2205: 2198: 2196: 2189: 2184: 2178:, p. 876 2177: 2172: 2170: 2162: 2157: 2150: 2145: 2138: 2133: 2126: 2120: 2116: 2112: 2108: 2104: 2100: 2093: 2086: 2085:L.A. Jeffress 2080: 2073: 2068: 2066: 2057: 2051: 2047: 2042: 2041: 2032: 2025: 2020: 2013: 2008: 2006: 1998: 1993: 1991: 1989: 1987: 1978: 1976:9781118030639 1972: 1968: 1967: 1959: 1951: 1949:9780262200608 1945: 1941: 1940: 1932: 1916: 1912: 1908: 1904: 1900: 1896: 1892: 1888: 1884: 1878: 1876: 1869: 1868:0-14-016734-X 1865: 1861: 1857: 1853: 1852: 1847: 1842: 1838: 1822: 1819: 1816: 1813: 1810: 1807: 1804: 1801: 1795: 1794:Lattice model 1792: 1786: 1783: 1780: 1777: 1774: 1771: 1768: 1765: 1763: 1760: 1757: 1754: 1751: 1748: 1747: 1740: 1734: 1731: 1729: 1726: 1724: 1721: 1719: 1716: 1714: 1711: 1709: 1706: 1704: 1701: 1699: 1696: 1694: 1691: 1689: 1688:Langton's ant 1686: 1684: 1683:Day and Night 1681: 1679: 1676: 1674: 1671: 1669: 1666: 1664: 1663:Brian's Brain 1661: 1660: 1658: 1646: 1644: 1643:deterministic 1638: 1635: 1631: 1626: 1618: 1607: 1605: 1601: 1597: 1591: 1579: 1576: 1574: 1571: 1570: 1569: 1566: 1564: 1559: 1555: 1550: 1548: 1544: 1540: 1536: 1532: 1530: 1526: 1522: 1518: 1507: 1505: 1501: 1497: 1493: 1489: 1484: 1480: 1476: 1472: 1468: 1458: 1449: 1446: 1445:A. K. Dewdney 1442: 1441: 1436: 1432: 1428: 1424: 1420: 1407: 1404: 1401: 1400:heterogeneity 1397: 1394: 1390: 1389: 1388: 1386: 1382: 1374: 1371: 1368: 1364: 1361: 1360:chromatophore 1357: 1353: 1350: 1346: 1343: 1339: 1338: 1337:Conus textile 1332: 1328: 1324: 1323: 1318: 1317: 1312: 1308: 1307: 1306: 1303: 1299: 1291: 1290: 1289:Conus textile 1285: 1271: 1269: 1265: 1261: 1260:edge of chaos 1256: 1252: 1250: 1246: 1242: 1238: 1228: 1226: 1225: 1220: 1216: 1212: 1208: 1204: 1203: 1198: 1188: 1185: 1182: 1179: 1176: 1173: 1170: 1167: 1164: 1163: 1159: 1156: 1153: 1150: 1147: 1144: 1141: 1138: 1135: 1134: 1128: 1119: 1115: 1113: 1103: 1100: 1097: 1094: 1091: 1088: 1085: 1082: 1079: 1078: 1074: 1071: 1068: 1065: 1062: 1059: 1056: 1053: 1050: 1049: 1043: 1034: 1030: 1028: 1023: 1019: 1015: 1011: 1007: 998: 994: 989: 979: 977: 973: 969: 966: 964: 961: 957: 953: 949: 945: 943: 940: 936: 935: 929: 925: 921: 919: 915: 911: 906: 904: 903:Penrose tiles 900: 896: 887: 883: 875: 873: 869: 865: 861: 857: 853: 849: 845: 841: 837: 827: 825: 821: 817: 813: 809: 804: 802: 798: 794: 789: 787: 786: 781: 777: 773: 769: 763: 753: 750: 745: 743: 737: 730: 726: 721: 719:indefinitely. 717: 713: 710: 709: 708: 705: 704: 693: 691: 687: 683: 679: 675: 670: 666: 665: 660: 659: 654: 650: 646: 642: 640: 636: 632: 624: 621: 618: 615: 614: 613: 611: 610: 605: 601: 597: 592: 589: 584: 579: 577: 576: 571: 570: 565: 560: 558: 554: 553:endomorphisms 551: 547: 542: 538: 534: 529: 527: 523: 518: 514: 509: 507: 503: 499: 495: 491: 487: 478: 474: 470: 466: 464: 460: 456: 452: 448: 444: 440: 436: 432: 423: 421: 417: 411: 407: 402: 398: 392: 388: 384: 377: 373: 369: 365: 361: 356: 352: 343: 338: 334: 332: 331:configuration 327: 316: 312: 308: 304: 300: 295: 291: 290: 285: 284: 279: 275: 264: 257: 248: 241: 227: 225: 221: 220: 215: 211: 207: 202: 200: 196: 192: 188: 184: 180: 176: 172: 168: 163: 161: 157: 153: 149: 145: 141: 137: 133: 129: 125: 121: 120: 115: 110: 108: 104: 100: 96: 92: 88: 84: 80: 76: 72: 68: 64: 60: 56: 49: 45: 41: 38: 34: 30: 19: 4421: 4417: 4408: 4383: 4379: 4366: 4355: 4314: 4286: 4268:. Springer. 4265: 4242: 4219: 4187: 4164: 4148:. Springer. 4145: 4120:. Retrieved 4065: 4055: 4042: 4029: 4012: 4006: 4000: 3975: 3971: 3965: 3956: 3950: 3908:(4): 10314. 3905: 3901:Scholarpedia 3899: 3885: 3853: 3843: 3811: 3804: 3779: 3775: 3769: 3760: 3730:(1): 49–65. 3727: 3723: 3713: 3668: 3664: 3654: 3611: 3607: 3596: 3587: 3581: 3569: 3545:14 September 3543:. Retrieved 3536:the original 3523: 3478: 3474: 3464: 3453:, retrieved 3446:the original 3437: 3390: 3386: 3380: 3368:. Retrieved 3363: 3357: 3347: 3304: 3298: 3288: 3261: 3255: 3247: 3242: 3223: 3217: 3208: 3203:Jacob Aron. 3198: 3186: 3161: 3157: 3142: 3122: 3107:, p. 60 3105:Wolfram 2002 3100: 3092:the original 3087: 3083: 3073: 3056: 3052: 3049:Kari, Jarkko 3043: 3018: 3014: 3011:Kari, Jarkko 3005: 2996: 2992: 2979: 2960: 2956: 2946: 2934: 2914: 2907: 2888: 2884: 2874: 2847:. Retrieved 2807: 2803: 2790: 2778:. Retrieved 2773: 2769: 2725: 2717: 2697: 2690: 2671: 2661: 2626: 2622: 2609: 2602:Wolfram 2002 2597: 2592:, p. 13 2585: 2580:, p. 12 2543: 2537: 2528: 2521:Wolfram 2002 2516: 2509:Wolfram 2002 2490:, p. 16 2483: 2463: 2446: 2442: 2432: 2414: 2396: 2378: 2360: 2348: 2323: 2319: 2310: 2269: 2265: 2259: 2234: 2230: 2224: 2207: 2203: 2183: 2176:Wolfram 2002 2156: 2144: 2132: 2124: 2105:(4): 58–67. 2102: 2098: 2092: 2079: 2039: 2031: 2026:, p. 41 2019: 1999:, p. 15 1965: 1958: 1938: 1931: 1919:. Retrieved 1915:the original 1894: 1890: 1849: 1841: 1738: 1656: 1639: 1622: 1593: 1567: 1551: 1543:cryptography 1539:block cipher 1533: 1517:tessellation 1513: 1488:universality 1483:ferromagnets 1464: 1438: 1435:malonic acid 1416: 1398:The role of 1378: 1335: 1320: 1314: 1304: 1301: 1287: 1274:Applications 1257: 1253: 1234: 1222: 1218: 1211:universality 1207:Matthew Cook 1200: 1196: 1194: 1111: 1109: 1026: 1006:Wolfram code 1003: 991: 971: 970: 967: 946: 938: 932: 930: 926: 922: 917: 913: 907: 894: 892: 881: 859: 855: 851: 847: 843: 839: 835: 833: 805: 790: 783: 767: 765: 746: 738: 734: 701: 700:Wolfram, in 699: 690:Matthew Cook 662: 656: 643: 634: 628: 607: 596:Game of Life 593: 580: 573: 567: 561: 557:shift spaces 530: 510: 502:tessellation 482: 454: 429: 419: 415: 409: 405: 400: 396: 390: 386: 379: 363: 354: 347: 330: 328: 314: 310: 306: 302: 294:orthogonally 287: 281: 278:neighborhood 277: 271: 223: 217: 203: 193:showed that 191:Matthew Cook 164: 151: 147: 143: 139: 136:neighborhood 135: 127: 123: 117: 113: 111: 94: 90: 86: 82: 78: 74: 62: 58: 54: 52: 29: 4473:Conway Life 4134:Works cited 3455:2 September 2939:Schiff 2011 2759:Li, Wentian 2629:(1): 1–28. 2353:Schiff 2011 2231:Cybernetics 2163:, p. 8 2139:, p. 3 2137:Schiff 2011 2074:, p. 1 2072:Schiff 2011 2024:Schiff 2011 2014:, p. 9 1921:28 February 1541:for use in 1475:Ising model 1467:statistical 1373:Fibroblasts 1356:cephalopods 956:Alan Turing 797:Jarkko Kari 793:undecidable 742:undecidable 725:conjectured 600:John Conway 564:Konrad Zuse 274:graph paper 206:homogeneity 69:studied in 4500:Categories 4291:. Urbana: 4118:. 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