The red cells are the Moore neighborhood for the blue cell. The red cells are the von Neumann neighborhood for the blue cell. The extended neighborhood includes the pink cells as well. One way to simulate a two-dimensional cellular automaton is with an infinite sheet of graph paper 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.
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The red cells are the Moore neighborhood for the blue cell. The red cells are the von Neumann neighborhood for the blue cell. The extended neighborhood includes the pink cells as well. One way to simulate a two-dimensional cellular automaton is with an infinite sheet of graph paper 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 neighborhood of a cell is the nearby, usually adjacent, cells. The two most common types of neighborhoods are the von Neumann neighborhood and the Moore neighborhood. For each of the possible patterns, the rule table would state whether the center cell will be black or white on the next time interval.
Another common neighborhood type is the extended von Neumann neighborhood, which includes the two closest cells in each orthogonal direction, for a total of eight.
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 configuration. The latter assumption is common in one-dimensional cellular automata. A torus , a toroidal shape 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 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 a toroidal 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 partial differential equations is sometimes referred to as periodic 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 torus doughnut shape.
Universes of other dimensions are handled similarly. This solves boundary problems with neighborhoods, but another advantage is that it is easily programmable using modular arithmetic functions. History Edit Stanislaw Ulam , while working at the Los Alamos National Laboratory in the s, studied the growth of crystals, using a simple lattice network as his model.
This design is known as the kinematic model. Neumann wrote a paper entitled "The general and logical theory of automata" for the Hixon Symposium in However their model is not a cellular automaton because the medium in which signals propagate is continuous, and wave fronts are curves.
Greenberg and S. Hastings in ; see Greenberg-Hastings cellular automaton. The original work of Wiener and Rosenblueth contains many insights and continues to be cited in modern research publications on cardiac arrhythmia and excitable systems.
In , Gustav A. Hedlund 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 Curtis—Hedlund—Lyndon theorem of the set of global rules of cellular automata as the set of continuous endomorphisms of shift spaces.
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 to reduce any neighborhood to a von Neumann neighborhood. Invented by John Conway and popularized by Martin Gardner in a Scientific American article,  its rules are as follows: Any live cell with fewer than two live neighbours dies, as if caused by underpopulation.
Any live cell with two or three live neighbours lives on to the next generation. Any live cell with more than three live neighbours dies, as if by overpopulation.
Any dead cell with exactly three live neighbours becomes a live cell, as if by reproduction. Despite its simplicity, the system achieves an impressive diversity of behavior, fluctuating between apparent randomness and order. One of the most apparent features of the Game of Life is the frequent occurrence of gliders, 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 universal Turing machine.
In order of complexity the classes are: Class 1: Nearly all initial patterns evolve quickly into a stable, homogeneous state. Any randomness in the initial pattern disappears. Some of the randomness in the initial pattern may filter out, but some remains. Local changes to the initial pattern tend to remain local. Any stable structures that appear are quickly destroyed by the surrounding noise.
Local changes to the initial pattern tend to spread indefinitely. Local changes to the initial pattern may spread indefinitely. Wolfram has conjectured that many class 4 cellular automata, if not all, are capable of universal computation. These definitions are qualitative in nature and there is some room for interpretation.
According to Wolfram, " And so it is with cellular automata: there are occasionally rules 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; membership in these proved undecidable. The proof by Jarkko Kari is related to the tiling problem by Wang tiles. Such cellular automata have rules specially constructed to be reversible. Such systems have been studied by Tommaso Toffoli , Norman Margolus and others.
Several techniques can be used to explicitly construct reversible cellular automata with known inverses. Two common ones are the second order cellular automaton and the block cellular automaton , 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. For example, if a plane is tiled with regular hexagons , 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 Penrose tiles. Such cellular automata are called probabilistic cellular automata.
Sometimes a simpler rule is used; for example: "The rule is the Game of Life, but on each time step there is a 0. 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. 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.
There are continuous automata. These are like totalistic cellular automata, but instead of the rule and states being discrete e.
The state of a location is a finite number of real numbers. Certain cellular automata can yield diffusion in liquid patterns in this way.
Continuous spatial automata have a continuum of locations. Time is also continuous, and the state evolves according to differential equations. One important example is reaction—diffusion textures, differential equations proposed by Alan Turing to explain how chemical reactions could create the stripes on zebras and spots on leopards. MacLennan  considers continuous spatial automata as a model of computation. There are known examples of continuous spatial automata, which exhibit propagating phenomena analogous to gliders in the Game of Life.
If so, return X of the rulestring for example: "". These CA work with brickwall neighborhoods. These CA types also act like Logic gate s.
A rule consists of deciding, for each pattern, whether the cell will be a 1 or a 0 in the next generation. These cellular automata are generally referred to by their Wolfram code , a standard naming convention invented by Wolfram that gives each rule a number from 0 to A number of papers have analyzed and compared these cellular automata. The rule 30 and rule cellular automata are particularly interesting.
The images below show the history of each when the starting configuration consists of a 1 at the top of each image surrounded by 0s.
Cellular Automata: A Discrete Universe
Zulucage In many cases the resulting cellular automata are equivalent to those with rectangular grids with specially designed neighborhoods and rules. To see what your friends thought of this book, please sign up. The images below show the history of each when the starting configuration consists of a 1 at the top of each image surrounded by 0s. Wikibooks has a book on the topic of: One important example is reaction-diffusion textures, differential equations proposed by Alan Turing to explain how chemical reactions could create the stripes on zebras and ilzchinski on leopards.
I have added a few links. Some people study them for their own sake; some use them to model real phenomena; and some speculate that they underlie fundamental physics. The present volume is the most comprehensive single-author book on CAs to date, and provides a useful unified reference to many ideas scattered through the literature. While aimed at an audience of physicists, it should be useful and comprehensible to mathematicians and computer scientists. While no one book could exhaust such a wide subject, there are several places where this one falls short, and others where it is too generous to ideas that, while popular ten years ago in the complex systems community, have not borne fruit. After an introduction and a lengthy chapter on formalism mostly discrete mathematics , the author begins with a phenomenological exploration of basic CA rules.
Kesida Many subtle issues are involved: For larger cellular automaton rule space, it is shown that class 4 rules are located between the class 1 and class 3 rules. For instance, Culik and Ilachibski proposed three well-defined classes and a fourth one for the automata not matching any of thesewhich are sometimes called Culik-Yu classes; membership in these proved undecidable. In two dimensions, the universe would be a rectangle instead of an infinite plane. Rule has been the basis for some of the smallest universal Turing machines.