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| [[Image:Gospers glider gun.gif|frame|right|[[Bill Gosper|Gosper's]] [[Gun (cellular automaton)|Glider Gun]] creating "[[Glider (Conway's Life)|gliders]]" in the cellular automaton [[Conway's Game of Life]]<ref>[[Daniel Dennett]] (1995), ''[[Darwin's Dangerous Idea]]'', Penguin Books, London, ISBN 978-0-14-016734-4, ISBN 0-14-016734-X</ref>]]
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| A '''cellular automaton''' (pl. '''cellular automata''', abbrev. '''CA''') is a [[discrete mathematics|discrete]] model studied in [[Computability theory (computer science)|computability theory]], [[mathematics]], [[physics]], [[Complexity Science|complexity science]], [[theoretical biology]] and [[microstructure]] modeling. Cellular automata are also called '''cellular spaces''', [[tessellation]] '''automata''', '''homogeneous structures''', '''cellular structures''', '''tessellation structures''', and '''iterative arrays'''.<ref name=reviews>{{Cite journal | url=http://www.stephenwolfram.com/publications/articles/ca/83-statistical/ | author=Wolfram, Stephen | authorlink=Stephen Wolfram | title=Statistical Mechanics of Cellular Automata | journal=Reviews of Modern Physics | issue=3 | volume=55 | pages=601–644 | year=1983 | doi=10.1103/RevModPhys.55.601 | bibcode=1983RvMP...55..601W | ref=harv}}</ref>
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| A cellular automaton consists of a regular grid of ''cells'', each in one of a finite number of ''[[State (computer science)|states]]'', such as ''on'' and ''off'' (in contrast to a [[coupled map lattice]]). The grid can be in any finite number of dimensions. For each cell, a set of cells called its ''neighborhood'' is defined relative to the specified cell. An initial state (time ''t''=0) is selected by assigning a state for each cell. A new ''generation'' is created (advancing ''t'' by 1), according to some fixed ''rule'' (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 [[stochastic cellular automaton]] and [[asynchronous cellular automaton]].
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| The concept was originally discovered in the 1940s by [[Stanislaw Ulam]] and [[John von Neumann]] while they were contemporaries at [[Los Alamos National Laboratory]]. While studied by some throughout the 1950s and 1960s, it was not until the 1970s and [[Conway's Game of Life]], a two-dimensional cellular automaton, that interest in the subject expanded beyond academia. In the 1980s, [[Stephen Wolfram]] engaged in a systematic study of one-dimensional cellular automata, or what he calls [[elementary cellular automaton|elementary cellular automata]]; his research assistant [[Matthew Cook]] showed that one of these rules is [[Turing-complete]]. Wolfram published ''[[A New Kind of Science]]'' in 2002, claiming that cellular automata have applications in many fields of science. These include computer processors and [[cryptography]].
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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 [[homogenity]], 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 are thought to be [[computationally universal]], or capable of simulating a [[Turing machine]]. Special types of cellular automata are those which are ''reversible'', in which only a single configuration leads directly to a subsequent one, and ''totalistic'', in which the future value of individual cells depend 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.
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| ==Overview==
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| [[File:CA-Moore.png|thumb|The red cells are the [[Moore neighborhood]] for the blue cell.]]
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| [[File:CA-von-Neumann.png|thumb|The red cells are the [[von Neumann neighborhood]] for the blue cell, while the extended neighborhood includes the pink cells as well.]]
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| 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]]''.<ref name=kier-seybold-cheng-15>{{Harvnb|Kier, Seybold, Cheng|2005|p=15|ref=kier-seybold-cheng}}</ref> The former, named after the founding cellular automaton theorist, consists of the four [[orthogonal]]ly adjacent cells.<ref name=kier-seybold-cheng-15/> The latter includes the von Neumann neighborhood as well as the four remaining cells surrounding the cell whose state is to be calculated.<ref name=kier-seybold-cheng-15/> For such a cell and its Moore neighborhood, there are 512 (= 2<sup>9</sup>) possible patterns. For each of the 512 possible patterns, the rule table would state whether the center cell will be black or white on the next time interval. [[Conway's Game of Life]] is a popular version of this model. 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.<ref name=kier-seybold-cheng-15/> The general equation for such a system of rules is ''k''<sup>''k''<sup>''s''</sup></sup>, where ''k'' is the number of possible states for a cell, and ''s'' is the number of neighboring cells (including the cell to be calculated itself) used to determine the cell's next state.<ref name=bialynick-9>{{Harvnb|Bialynicki-Birula, Bialynicka-Birula|2004|p=9|ref=bialynick}}</ref> Thus, in the two dimensional system with a Moore neighborhood, the total number of automata possible would be 2<sup>2<sup>9</sup></sup>, or {{val|1.34|e=154}}.
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| 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''.<ref name=schiff-41>{{Harvnb|Schiff|2011|p=41|ref=schiff}}</ref> 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.
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| [[Image:Torus.png|thumb|right|A [[torus]], a toroidal shape]]
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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 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 is done in order to solve boundary problems with neighborhoods, but another advantage of this system is that it is easily programmable using [[modular arithmetic]] functions. For example, in a 1-dimensional cellular automaton like the examples below, the neighborhood of a cell ''x<sub>i</sub><sup>t</sup>'' is {''x''<sub>''i''−1</sub><sup style="margin-left:-2ex;">''t''−1</sup>, ''x''<sub>''i''</sub><sup>''t''−1</sup>, ''x''<sub>''i''+1</sub><sup style="margin-left:-2ex;">''t''−1</sup>}, where ''t'' is the time step (vertical), and ''i'' is the index (horizontal) in one generation.
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| ==History==
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| [[Image:John von Neumann ID badge.png|left|thumb|upright|[[John von Neumann]], [[Los Alamos National Laboratory|Los Alamos]] ID badge]]
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| [[Stanislaw Ulam]], while working at the [[Los Alamos National Laboratory]] in the 1940s, studied the growth of crystals, using a simple [[Lattice model (physics)|lattice network]] as his model.<ref name=pickover>{{cite book | author=Pickover, Clifford A. | title=The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics | page=406 | year=2009| publisher=Sterling Publishing Company, Inc | isbn=978-1402757969}}</ref> At the same time, [[John von Neumann]], Ulam's colleague at Los Alamos, was working on the problem of [[self-replicating system]]s.<ref name=schiff-1>{{Harvnb|Schiff|2011|p=1|ref=schiff}}</ref> Von Neumann's initial design was founded upon the notion of one robot building another robot. This design is known as the kinematic model.<ref>John von Neumann, “The general and logical theory of automata,” in L.A. Jeffress, ed., Cerebral Mechanisms in Behavior – The Hixon Symposium, John Wiley & Sons, New York, 1951, pp. 1–31.</ref><ref>John G. Kemeny, “Man viewed as a machine,” Sci. Amer. 192(April 1955):58–67; Sci. Amer. 192(June 1955):6 (errata).</ref> 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 read a paper entitled "The general and logical theory of automata" at the [[Hixon Symposium]] in 1948.<ref name=schiff-1/> Ulam was the one who suggested using a ''discrete'' system for creating a reductionist model of self-replication.<ref name=schiff-3>{{Harvnb|Schiff|2011|p=3|ref=schiff}}</ref><ref name=ilachinski-xxix>{{Harvnb|Ilachinski|2001|p=xxix|ref=ilachinski}}</ref> [[Nils Aall Barricelli]] performed many of the earliest explorations of these models of artificial life.
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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.<ref name=bialynick-8>{{Harvnb|Bialynicki-Birula, Bialynicka-Birula|2004|p=8|ref=bialynick}}</ref> Thus was born the first system of cellular automata. Like Ulam's lattice network, [[Von Neumann cellular automata|von Neumann's cellular automata]] are two-dimensional, with his self-replicator implemented algorithmically. The result was a [[Von Neumann universal constructor|universal copier and constructor]] working within a cellular automaton with a small neighborhood (only those cells that touch are neighbors; for von Neumann's cellular automata, only [[orthogonal]] cells), and with 29 states per cell.<ref name=wolfram-876>{{Harvnb|Wolfram|2002|p=876|ref=wolfram}}</ref> Von Neumann gave an existence proof 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.<ref name=wolfram-876/> This design is known as the [[tessellation]] model, and is called a [[von Neumann universal constructor]].<ref name=TSRA>{{cite book | author=von Neumann, John | coauthors=Burks, Arthur W. | title=Theory of Self-Reproducing Automata | year=1966 | publisher=[[University of Illinois Press]]}}</ref>
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| Also in the 1940s, [[Norbert Wiener]] and [[Arturo Rosenblueth]] developed a model of excitable media with some of the characteristics of a cellular automaton.<ref name="Wiener-Rosenblueth">{{cite journal | last1 = Wiener | first1 = N. | last2 = Rosenblueth | first2 = A. | year = 1946 | title = The mathematical formulation of the problem of conduction of impulses in a network of connected excitable elements, specifically in cardiac muscle | url = | journal = Arch. Inst. Cardiol. México | volume = 16 | issue = | page = 205 | ref = harv }}</ref> 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 continuous, and wave fronts are curves.<ref name="Wiener-Rosenblueth"/><ref name="reshodko">{{Cite journal|last1=Letichevskii|first1=A. A. |last2= Reshodko|first2=L. V.|title=N. Wiener's theory of the activity of excitable media |journal=Cybernetics|volume=8|year=1974|pages=856–864}}</ref> A true cellular automaton model of excitable media was developed and studied by J. M. Greenberg and S. P. Hastings in 1978; 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.<ref>{{cite journal | doi = 10.1038/355349a0 | last1 = Davidenko | first1 = J. M. | last2 = Pertsov | first2 = A. V. | last3 = Salomonsz | first3 = R. | last4 = Baxter | first4 = W. | last5 = Jalife | first5 = J. | year = 1992 | title = Stationary and drifting spiral waves of excitation in isolated cardiac muscle | url = | journal = Nature | volume = 355 | issue = 6358| pages = 349–351 | pmid = 1731248 |bibcode = 1992Natur.355..349D | ref = harv }}</ref>
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| In the 1960s, cellular automata were studied as a particular type of [[dynamical system]] and the connection with the mathematical field of [[symbolic dynamics]] was established for the first time. In 1969, [[Gustav A. Hedlund]] compiled many results following this point of view<ref>{{cite journal | doi = 10.1007/BF01691062 | last1 = Hedlund | first1 = G. A. | authorlink = Gustav A. Hedlund | year = 1969 | title = Endomorphisms and automorphisms of the shift dynamical system | url = http://www.springerlink.com/content/k62915l862l30377/ | journal = Math. Systems Theory | volume = 3 | issue = 4| pages = 320–3751 | ref = harv }}</ref> 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 function|continuous]] [[endomorphisms]] of [[shift space]]s.
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| In 1969, German computer pioneer [[Konrad Zuse]] published his book ''[[Calculating Space]]'', 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 ''[[digital physics]]''.<ref name=schiff-182>{{Harvnb|Schiff|2011|p=182|ref=schiff}}</ref>
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| In the 1970s a two-state, two-dimensional cellular automaton named Game of Life became widely known, particularly among the early computing community. Invented by [[John Horton Conway|John Conway]] and popularized by [[Martin Gardner]] in a ''[[Scientific American]]'' article,<ref>{{cite journal | last1 = Gardner | first1 =Martin | year = 1970 | title = Mathematical Games: The fantastic combinations of John Conway's new solitaire game "life" | url = http://www.ibiblio.org/lifepatterns/october1970.html | journal = Scientific American | issue = 223 | pages = 120–123 | ref = harv }}</ref> its rules are as follows: If a cell has two black neighbors, it stays the same. If it has three black neighbors, it becomes black. In all other situations it becomes white. 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]].<ref>Paul Chapman. Life universal computer. http://www.igblan.free-online.co.uk/igblan/ca/ November 2002</ref> 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.<ref name=Wainwright-16>{{Harvnb|Wainwright|2010|p=16|ref= Adamatzky}}</ref>
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| [[Stephen Wolfram]] independently began working on cellular automata in mid 1981 after considering how complex patterns seemed to be formed in nature in violation of the [[Second Law of Thermodynamics]].<ref name=wolfram-880/> His investigations were initially spurred by an interest in modelling systems such as [[neural network]]s.<ref name=wolfram-880/> He published his first paper in ''[[Reviews of Modern Physics]]'' investigating ''[[elementary cellular automaton|elementary cellular automata]]'' ([[Rule 30]] in particular) in June 1983.<ref name=reviews/><ref name=wolfram-880>{{Harvnb|Wolfram|2002|p=880|ref=wolfram}}</ref> The unexpected complexity of the behavior of these simple rules led Wolfram to suspect that complexity in nature may be due to similar mechanisms.<ref name=wolfram-880/> His investigations, however, led him to realize that cellular automata were poor at modelling neural networks.<ref name=wolfram-880/> Additionally, during this period Wolfram formulated the concepts of intrinsic [[randomness]] and [[computational irreducibility]],<ref name=wolfram-881>{{Harvnb|Wolfram|2002|p=881|ref=wolfram}}</ref> and suggested that [[rule 110]] may be [[Turing-complete|universal]]—a fact proved later by Wolfram's research assistant [[Matthew Cook]] in the 1990s.<ref>{{cite journal | author=Mitchell, Melanie | authorlink=Melanie Mitchell | title=Is the Universe a Universal Computer? | work=[[Science (magazine)|Science]] | date=4 October 2002 | volume=298 | issue=5591 | pages=65–68 | doi=10.1126/science.1075073}}</ref>
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| In 2002 Wolfram published a 1280-page text ''[[A New Kind of Science (book)|A New Kind of Science]]'', which extensively argues that the discoveries about cellular automata are not isolated facts but are robust and have significance for all disciplines of science.<ref name=wolfram-1-7>{{Harvnb|Wolfram|2002|pp=1–7|ref=wolfram}}</ref> Despite confusion in the press,<ref>{{cite news | url=http://www.nytimes.com/2002/06/09/books/review/09JOHNSOT.html | author=Johnson, George | title='A New Kind of Science': You Know That Space-Time Thing? Never Mind | work=[[The New York Times]] | publisher=[[The New York Times Company]] | date=9 June 2002 | accessdate=22 January 2013}}</ref><ref>{{cite news | url=http://www.economist.com/printedition/displayStory.cfm?Story_ID=1154164 | title=The Science of Everything | work=[[The Economist]] | date=30 May 2002 | accessdate=22 January 2013}}</ref> the book did not argue for a fundamental theory of physics based on cellular automata,<ref name=wolfram-ch10>{{Harvnb|Wolfram|2002|pp=433–546|ref=wolfram}}</ref> and although it did describe a few specific physical models based on cellular automata,<ref name=wolfram-ch3>{{Harvnb|Wolfram|2002|pp=51–114|ref=wolfram}}</ref> it also provided models based on qualitatively different abstract systems.<ref name=wolfram-ch4>{{Harvnb|Wolfram|2002|pp=115–168|ref=wolfram}}</ref>
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| ==Classification==
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| Wolfram, in ''A New Kind of Science'' 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 type of patterns for specific rules, Wolfram's classification was the first attempt to classify the rules themselves. In order of complexity the classes are:
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| *Class 1: Nearly all initial patterns evolve quickly into a stable, homogeneous state. Any randomness in the initial pattern disappears.<ref name=ilachinski-12>{{Harvnb|Ilachinsky|2001|p=12|ref=ilachinski}}</ref>
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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 local.<ref name=ilachinski-12/>
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| *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 indefinitely.<ref name=ilachinski-12/>
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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.<ref name=ilachinski-13>{{Harvnb|Ilachinsky|2001|p=13|ref=ilachinski}}</ref> Class 2 type stable or oscillating structures may be the eventual outcome, but the number of steps 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 conjectured that many, if not all class 4 cellular automata are capable of universal computation. This has been proven for Rule 110 and Conway's game of Life.
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| These definitions are qualitative in nature and there is some room for interpretation. According to Wolfram,
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| "...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 cellular automata: there are occasionally rules...that show some features of one class and some of another."<ref name=wolfram-231>{{Harvnb|Wolfram|2002|p=231|ref=wolfram}}</ref> Wolfram's classification has been empirically matched to a clustering of the compressed lengths of the outputs of cellular automata.<ref>{{cite journal|last=Zenil|first=Hector|title=Compression-based investigation of the dynamical properties of cellular automata and other systems|journal=Complex Systems|year=2010|volume=19|issue=1|url=http://www.complex-systems.com/pdf/19-1-1.pdf}}</ref>
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| There have been several attempts to classify cellular automata in formally rigorous classes, inspired by the 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; membership in these proved to be [[Undecidable problem|undecidable]].<ref name="DelormeMazoyer1998">{{cite book|editor=M. Delorme, J. Mazoyer|title=Cellular automata: a parallel model|url=http://books.google.com/books?id=dGs87s5Pft0C&pg=PA239|year=1998|publisher=Springer|isbn=978-0-7923-5493-2|page=239|chapter=Topological chaos and CA|author=G. Cattaneo, E. Formenti, L. Margara}}</ref><ref name="Voorhees1996">{{cite book|author=Burton H. Voorhees|title=Computational analysis of one-dimensional cellular automata|url=http://books.google.com/books?id=WcZTQHPrG68C&pg=PA8|year=1996|publisher=World Scientific|isbn=978-981-02-2221-5|page=8}}</ref><ref name="Garzon1995">{{cite book|author=Max Garzon|title=Models of massive parallelism: analysis of cellular automata and neural networks | year=1995 | publisher=Springer | isbn=978-3-540-56149-1 | page=149}}</ref>
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| Wolfram's class 2 can be partitioned into two subgroups of stable (fixed-point) and oscillating (periodic) rules.<ref name=li-packard>{{cite journal | url=http://www.complex-systems.com/pdf/04-3-3.pdf | author1=Li, Wentian | authorlink1=Wentian Li | author2=Packard, Norman | title=The structure of the elementary cellular automata rule space | journal=Complex Systems | pages=281–297 | volume=4 | year=1990 | accessdate=January 25, 2013}}</ref>
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| ===Reversible===
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| {{main|Reversible cellular automaton}}
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| A cellular automaton is said to be ''reversible'' if for every current configuration of the cellular automaton there is exactly one past configuration ([[preimage]]).<ref name=kari-379>{{Harvnb|Kari, Jarrko|1991|p=379|ref=gutowitz}}</ref> If one thinks of a cellular automaton as a function mapping configurations to configurations, reversibility implies that this function is [[bijective]].<ref name=kari-379/> 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 [[Curtis–Hedlund–Lyndon theorem]], a topological characterization of cellular automata.<ref>{{Cite journal|first=D.|last=Richardson|title=Tessellations with local transformations|journal=J. Computer System Sci.|volume=6|year=1972|pages=373–388|issue=5|doi=10.1016/S0022-0000(72)80009-6|ref=harv}}</ref><ref>{{Cite book|title=Cellular Automata in Hyperbolic Spaces – Tome I, Volume 1|first=Maurice|last=Margenstern|publisher=Archives contemporaines|year=2007|isbn=978-2-84703-033-4|page=134|url=http://books.google.com/books?id=wGjX1PpFqjAC&pg=PA134|ref=harv}}</ref> For cellular automata in which not every configuration has a preimage, the configurations without preimages are called ''[[Garden of Eden (cellular automaton)|Garden of Eden]]'' patterns.<ref name=schiff-103>{{Harvnb|Schiff|2011|p=103|ref=schiff}}</ref>
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| For one-dimensional cellular automata there are known algorithms for deciding whether a rule is reversible or irreversible.<ref>Serafino Amoroso, Yale N. Patt, ''Decision Procedures for Surjectivity and Injectivity of Parallel Maps for Tessellation Structures''. J. Comput. Syst. Sci. 6(5): 448–464 (1972)</ref><ref>{{Cite journal|journal=Complex Systems|volume=5|year=1991|pages=19–30|title=De Bruijn Graphs and Linear Cellular Automata|first=Klaus|last=Sutner|url=http://www.complex-systems.com/pdf/05-1-3.pdf|ref=harv}}</ref> However, for cellular automata of two or more dimensions reversibility is [[undecidable problem|undecidable]]; 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 [[Jarkko Kari]] is related to the tiling problem by [[Wang tile]]s.<ref>{{Cite journal|first=Jarkko|last=Kari|authorlink=Jarkko Kari|title=Reversibility of 2D cellular automata is undecidable|journal=Physica D|volume=45|year=1990|pages=379–385|doi=10.1016/0167-2789(90)90195-U|bibcode = 1990PhyD...45..379K|ref=harv}}</ref>
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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 [[thermodynamics]]. 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.<ref>{{Cite journal|title=On the circuit depth of structurally reversible cellular automata|first=Jarkko|last=Kari|authorlink=Jarkko Kari|journal=Fundamenta Informaticae|volume=38|year=1999|pages=93–107|ref=harv}}</ref><ref>{{Cite journal|title=Representing reversible cellular automata with reversible block cellular automata|last=Durand-Lose|first=Jérôme|journal=Discrete Mathematics and Theoretical Computer Science|volume=AA|year=2001|pages=145–154|url=http://www.dmtcs.org/dmtcs-ojs/index.php/proceedings/article/download/264/855|ref=harv}}</ref>
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| ===Totalistic===
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| A special class of cellular automata are ''totalistic'' 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 ''t'' depends only on the ''sum'' of the values of the cells in its neighborhood (possibly including the cell itself) at time ''t''−1.<ref name=wolfram-60>{{Harvnb|Wolfram|2002|p=60|ref=wolfram}}</ref><ref name="cadu">{{Cite book|title=Cellular automata: a discrete universe|first=Andrew|last=Ilachinski|publisher=World Scientific|year=2001|isbn=978-981-238-183-5|pages=44–45|url=http://books.google.com/books?id=3Hx2lx_pEF8C&pg=PA4|ref=harv}}</ref> If the state of the cell at time ''t'' does depend on its own state at time ''t''−1 then the cellular automaton is properly called ''outer totalistic''.<ref name="cadu"/> [[Conway's Game of Life]] is an example of an outer totalistic cellular automaton with cell values 0 and 1; outer totalistic cellular automata with the same [[Moore neighborhood]] structure as Life are sometimes called [[{{Not a typo|Life-like}} cellular automaton|{{Not a typo|life-like}} cellular automata]].<ref>The phrase "{{Not a typo|life-like}} cellular automaton" dates back at least to {{harvtxt|Barral|Chaté|Manneville|1992}}, 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 {{harvtxt|Adamatzky|2010}}. See: {{Cite journal|title=Collective behaviors in a family of high-dimensional cellular automata|first1=Bernard|last1=Barral|first2=Hugues|last2=Chaté|first3=Paul|last3=Manneville|journal=Physics Letters A|volume=163|issue=4|year=1992|pages=279–285|doi=10.1016/0375-9601(92)91013-H|ref=harv|bibcode = 1992PhLA..163..279B }}</ref><ref name=eppstein-72-73>{{Harvnb|Eppstein|2010|pp=72–73|ref=Adamatzky}}</ref>
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| ===Related automata===
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| There are many possible generalizations of the cellular automaton concept.
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| [[Image:Oscillator.gif|right|frame|A cellular automaton based on hexagonal cells instead of squares (rule 34/2)]]
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| One way is by using something other than a rectangular (cubic, ''etc.'') grid. For example, if a plane is [[hexagonal tiling|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]].<ref>http://www.newscientist.com/article/dn22134-first-gliders-navigate-everchanging-penrose-universe.html</ref>
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| Also, rules can be probabilistic rather than deterministic. Such cellular automata are called [[probabilistic cellular automata]]. A probabilistic rule gives, for each pattern at time ''t'', the probabilities that the central cell will transition to each possible state at time ''t''+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."
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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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| 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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| There are ''[[continuous automaton|continuous automata]]''. These are like totalistic cellular automata, but instead of the rule and states being discrete (''e.g.'' a table, using states {0,1,2}), continuous functions are used, and the states become continuous (usually values in [[unit interval|[0,1]]]). The state of a location is a finite number of real numbers. Certain cellular automata can yield diffusion in liquid patterns in this way.
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| [[Continuous spatial automata]] 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 [[reaction-diffusion]] textures, differential equations proposed by [[Alan Turing]] to explain how chemical reactions could create the stripes on [[zebra]]s and spots on leopards.<ref>{{cite journal | last = Murray | first = J.|title = Mathematical Biology II |publisher = [[Springer Science+Business Media|Springer]] | ref = harv}}</ref> When these are approximated by cellular automata, they often yield similar patterns. MacLennan [http://www.cs.utk.edu/~mclennan/contin-comp.html] considers continuous spatial automata as a model of computation.
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| There are known examples of continuous spatial automata which exhibit propagating phenomena analogous to gliders in the Game of Life.<ref>Pivato, M: "RealLife: The continuum limit of Larger than Life cellular automata", ''Theoretical Computer Science'', 372 (1), March 2007, pp. 46–68</ref>
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| ==Elementary cellular automata==
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| {{Main|Elementary cellular automaton}}
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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 to be the adjacent cells on either side of it. A cell and its two neighbors form a neighborhood of 3 cells, so there are 2<sup>3</sup>=8 possible patterns for a 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<sup>8</sup>=256 possible rules.<ref name=bialynick-9/> These 256 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 255. A number of papers have analyzed and compared these 256 cellular automata. The [[rule 30]] and [[rule 110]] 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. Each row of pixels represents a generation in the history of the automaton, with ''t''=0 being the top row. Each pixel is colored white for 0 and black for 1.
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| {{Div col begin|2}}
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| [[File:CA rule30s.png|thumb|Rule 30]]<br>
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| '''[[Rule 30]] cellular automaton'''
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| {| class="wikitable" style="text-align: center"
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| |-
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| ! current pattern
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| | 111 || 110 || 101 || 100 || 011 || 010 || 001 || 000
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| |-
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| ! new state for center cell
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| | 0 || 0 || 0 || 1 || 1 || 1 || 1 || 0
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| |}
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| [[File:CA rule110s.png|thumb|Rule 110]]<br>
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| '''[[Rule 110]] cellular automaton'''
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| {| class="wikitable" style="text-align: center"
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| |-
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| ! current pattern
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| | 111 || 110 || 101 || 100 || 011 || 010 || 001 || 000
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| |-
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| ! new state for center cell
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| | 0 || 1 || 1 || 0 || 1 || 1 || 1 || 0
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| |}
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| {{Div col end}}
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| Rule 30 exhibits ''class 3'' behavior, meaning even simple input patterns such as that shown lead to chaotic, seemingly random histories.
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| Rule 110, like the Game of Life, exhibits what Wolfram calls ''class 4'' 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 ''[[A New Kind of Science (book)|A New Kind of Science]]'', as a research assistant to Wolfram in 1994, [[Matthew Cook]] proved that some of these structures were rich enough to support [[Universal Turing Machine|universality]]. This result is interesting because rule 110 is an extremely simple one-dimensional system, and one which is 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 [[Santa Fe Institute]] 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 to be announced before the publication of ''A New Kind of Science''.<ref name=giles>{{cite journal | author=Giles, Jim | title=What Kind of Science is This? | work=[[Nature (magazine)|Nature]] | issue=417 | pages=216–218 | year=2002}}</ref> In 2004, Cook's proof was finally published in Wolfram's journal ''[[Complex Systems (journal)|Complex Systems]]'' (Vol. 15, No. 1), over ten years after Cook came up with it. Rule 110 has been the basis over which some of the smallest universal Turing machines have been built.<ref>{{cite journal | url=http://www.nybooks.com/articles/archives/2002/oct/24/is-the-universe-a-computer/?pagination=false | author=Weinberg, Steven | title=Is the Universe a Computer? | work=[[The New York Review of Books]] | publisher=Rea S. Hederman | date=October 24, 2002 | accessdate=October 12, 2012}}</ref>
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| ==Rule space==
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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 [[vertex (geometry)|vertices]] of the 8-dimensional unit [[hypercube]]. This unit hypercube is the cellular automaton rule space. For next-nearest-neighbor cellular automata, a rule is specified by <math>2^5=32</math> 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 [[edge (geometry)|edge]] of the [[hypercube]]. This rule-to-rule distance is also called the
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| [[Hamming distance]].
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| Cellular automaton rule space allows us to ask the question concerning whether rules with similar dynamical behavior are ``close" to each. 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 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-1's, thus located in one region of the space, whereas class 3 rules tend to have higher proportion (50%) of bit-1's.<ref name=li-packard />
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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.<ref>{{cite journal|journal=Physica D|volume=45|issue=1-3|year=1990|pages=77–94|title= Transition phenomena in cellular automata rule space | author=Wentian Li, Norman Packard, Chris G Langton|doi=10.1016/0167-2789(90)90175-O|url= |bibcode = 1990PhyD...45...77L }}</ref> This observation is the foundation for the phrase [[edge of chaos]], and is reminiscent of the [[phase transition]] in [[thermodynamics]].
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| ==Biology==
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| [[Image:Textile cone.JPG|thumb|left|''[[Conus textile]]'' exhibits a cellular automaton pattern on its shell.<ref name=coombs/>]]
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| Some biological processes occur—or can be simulated—by cellular automata.
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| Patterns of some [[seashell]]s, like the ones in ''[[Cone snail|Conus]]'' and ''[[Cymbiola]]'' genus, are generated by natural cellular automata. The [[pigment]] cells reside in a narrow band along the shell's lip. Each cell [[secretion|secretes]] pigments according to the activating and inhibiting activity of its neighbor pigment cells, obeying a natural version of a mathematical rule.<ref name=coombs>{{citation | url=http://www.maths.nott.ac.uk/personal/sc/pdfs/Seashells09.pdf | author=Coombs, Stephen |title=The Geometry and Pigmentation of Seashells | pages=3–4 | date=February 15, 2009 | accessdate=September 2, 2012}}</ref> The cell band leaves the colored pattern on the shell as it grows slowly. For example, the widespread species ''[[Conus textile]]'' bears a pattern resembling Wolfram's [[rule 30]] cellular automaton.<ref name=coombs/>
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| Plants regulate their intake and loss of gases via a cellular automaton mechanism. Each [[Stomata|stoma]] on the leaf acts as a cell.<ref>{{cite journal | doi = 10.1073/pnas.0307811100 | last1 = Peak | first1 = West | last2 = Messinger | first2 = Mott | year = 2004 | title = Evidence for complex, collective dynamics and emergent, distributed computation in plants | url = http://www.pnas.org/cgi/content/abstract/101/4/918 | journal = Proceedings of the National Institute of Science of the USA | volume = 101 | issue = 4| pages = 918–922 |bibcode = 2004PNAS..101..918P | ref = harv | pmid=14732685 | pmc=327117}}</ref>
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| Moving wave patterns on the skin of [[cephalopod]]s can be simulated with a two-state, two-dimensional cellular automata, each state corresponding to either an expanded or retracted [[chromatophore]].<ref>http://gilly.stanford.edu/past_research_files/APackardneuralnet.pdf</ref>
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| Threshold automata have been invented to simulate [[neuron]]s, and complex behaviors such as recognition and learning can be simulated.<ref name=ilachinski-275>{{Harvnb|Ilachinsky|2001|p=275|ref=ilachinski}}</ref>
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| [[Fibroblast]]s bear similarities to cellular automata, as each fibroblast only interacts with its neighbors.<ref>{{cite book|title=Disordered Systems and Biological Organization|year=1986|pages=374–375|author=Yves Bouligand|authorlink=Fibroblasts, Morphogenesis and Cellular Automata}}</ref>
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| ==Chemical types==
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| The [[Belousov–Zhabotinsky reaction]] is a spatio-temporal chemical oscillator which can be simulated by means of a cellular automaton. In the 1950s [[A. M. Zhabotinsky]] (extending the work of [[B. P. Belousov]]) discovered that when a thin, homogenous layer of a mixture of [[malonic acid]], 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 ''[[Scientific American]]'',<ref>A. K. Dewdney, The hodgepodge machine makes waves, Scientific American, p. 104, August 1988.</ref> [[A. K. Dewdney]] discussed a cellular automaton<ref>M. Gerhardt and H. Schuster, A cellular automaton describing the formation of spatially ordered structures in chemical systems, Physica D 36, 209–221, 1989.</ref> which was developed by Martin Gerhardt and Heike Schuster of the University of Bielefeld (West Germany). This automaton produces wave patterns resembling those in the Belousov-Zhabotinsky reaction.
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| ==Applications==
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| ===Computer processors===
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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 [[tessellation]], 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.<ref>{{cite book | author=Muhtaroglu, Ali | title=Cellular Automaton Processor Based Systems for Genetic Sequence Comparison/Database Searching | chapter=4.1 Cellular Automaton Processor (CAP) | pages=62–74 | publisher=Cornell University |date=August 1996}}</ref> One such cellular automaton processor array configuration is the [[systolic array]]. Cell interaction can be via electric charge, magnetism, vibration ([[phonons]] at quantum scales), or any other physically useful means. This can be done in several ways so no wires are needed between any elements. This is very unlike processors used in most computers today, [[Von Neumann architecture|von Neumann designs]], which are divided into sections with elements that can communicate with distant elements over wires.
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| ===Cryptography===
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| [[Rule 30]] was originally suggested as a possible [[Block cipher]] for use in [[cryptography]]. Two dimensional cellular automata are used for [[random number generation]].<ref>{{cite journal | first1=M. | last1=Tomassini | first2=M. | last2=Sipper | first3=M. | last3=Perrenoud | title=On the generation of high-quality random numbers by two-dimensional cellular automata | work=IEEE Transactions on Computers | volume=49 | issue=10 | pages=1146–1151 | year=2000}}</ref>
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| Cellular automata have been proposed for [[public key cryptography]]. The [[one-way function]] 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. <!-- However, the designer of the rule can create it in such a way as to be able to easily invert it. Therefore, it is apparently a [[trapdoor function]], and can be used as a public-key cryptosystem. The security of such systems is not currently known. -->
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| ===Error correction coding===
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| CA have been applied to design error correction codes in the paper "Design of CAECC – Cellular Automata Based Error Correcting Code", by
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| [[D. Roy Chowdhury]], [[S. Basu]], [[I. Sen Gupta]], [[P. Pal Chaudhuri]]. The paper defines a new scheme of building SEC-DED codes using CA, and
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| also reports a fast hardware decoder for the code.
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| ==Modeling physical reality==
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| {{main|Digital physics|digital philosophy}}
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| As Andrew Ilachinski points out in his ''Cellular Automata'', many scholars have raised the question of whether the universe is a cellular automaton.<ref name=ilachinski-660>{{Harvnb|Ilachinsky|2001|p=660|ref=ilachinski}}</ref> Ilachinski argues that the importance of this question may be better appreciated with a simple observation, which can be stated as follows. Consider the evolution of [[rule 110]]: if it were some kind of "alien physics", what would be a reasonable description of the observed patterns?<ref name=ilachinski-661-662>{{Harvnb|Ilachinsky|2001|pp=661–662|ref=ilachinski}}</ref> If you didn't know how the images were generated, you might end up conjecturing about the movement of some particle-like objects (indeed, physicist [[James P. Crutchfield|James Crutchfield]] made a rigorous mathematical theory out of this idea proving the statistical emergence of "particles" from CA).<ref>J. P. Crutchfield, [http://csc.ucdavis.edu/~cmg/papers/CalcEmerg.pdf <nowiki>"The Calculi of Emergence: Computation, Dynamics, and Induction"</nowiki>], [[Physica (journal)#Physica D: Nonlinear Phenomena|Physica D]] 75, 11–54, 1994.</ref> Then, as the argument goes, one might wonder if ''our'' world, which is currently well described by [[particle physics|physics with particle-like objects]], could be a CA at its most fundamental level.
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| While a complete theory along this line is still to be developed, entertaining and developing this hypothesis led scholars to interesting speculation and fruitful intuitions on how can we make sense of our world within a discrete framework. [[Marvin Minsky]], the AI pioneer, investigated how to understand particle interaction with a four-dimensional CA lattice;<ref>M. Minsky, "Cellular Vacuum", International Journal of Theoretical Physics 21, 537–551, 1982.</ref> [[Konrad Zuse]]—the inventor of the first working computer, the [[Z3 (computer)|Z3]]—developed an irregularly organized lattice to address the question of the information content of particles.<ref>K. Zuse, "The Computing Universe", Int. Jour. of Theo. Phy. 21, 589–600, 1982.</ref> More recently, [[Edward Fredkin]] exposed what he terms the "finite nature hypothesis", i.e., the idea that "ultimately every quantity of physics, including space and time, will turn out to be discrete and finite."<ref>E. Fredkin, "Digital mechanics: an informational process based on reversible universal cellular automata", Physica D 45, 254–270, 1990</ref> Fredkin and Wolfram are strong proponents of a CA-based physics.
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| In recent years, other suggestions along these lines have emerged from literature in non-standard computation. Wolfram's ''[[A New Kind of Science]]'' considers CA to be the key to understanding a variety of subjects, physics included. The ''Mathematics Of the Models of Reference''—created by [[iLabs]]<ref>[http://www.ilabs.it/ iLabs]</ref> founder Gabriele Rossi and developed with Francesco Berto and Jacopo Tagliabue—features an original 2D/3D universe based on a new "rhombic dodecahedron-based" lattice and a unique rule. This model satisfies universality (it is equivalent to a Turing Machine) and perfect reversibility (a ''desideratum'' if one wants to conserve various quantities easily and never lose information), and it comes embedded in a first-order theory, allowing computable, qualitative statements on the universe evolution.<ref>[http://www.mmdr.it/defaultEN.asp F. Berto, G. Rossi, J. Tagliabue, The Mathematics of the Models of Reference], [[College Publications]], 2010</ref>
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| ==See also==
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| {{Wikipedia books|Cellular Automata}}
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| ===Specific rules===
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| * [[Brian's Brain]]
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| * [[Langton's ant]]
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| * [[Wireworld]]
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| * [[Rule 90]]
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| * [[Rule 184]]
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| * [[von Neumann cellular automata]]
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| * [[Nobili cellular automata]]
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| * [[Codd's cellular automaton]]
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| * [[Langton's loops]]
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| * [[CoDi]]
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| ===Problems solved===
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| * [[Firing squad synchronization problem]]
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| * [[Majority problem (cellular automaton)|Majority problem]]
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| ===See also===
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| * [[Automata theory]]
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| * [[Bidirectional traffic]]
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| * [[:Category:Cellular automata in popular culture|Cellular automata in popular culture]]
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| * [[Cyclic cellular automaton]]
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| * [[Excitable medium]]
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| * [[Quantum cellular automata]]
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| * [[Spatial decision support system]]
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| * [[Mirek's Cellebration]]
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| * [[Movable cellular automaton]]
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| ==Reference notes==
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| {{Reflist|2}}
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| ==References==
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| {{Refbegin|2}}
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| *{{cite book | title=Game of Life Cellular Automata|editor-first=Andrew | editor-last=Adamatzky | editor-link=Andrew Adamatzky | publisher=Springer | year=2010 | isbn=978-1-84996-216-2 | ref=Adamatzky}}
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| *{{cite book | first1=Iwo | last1=Bialynicki-Birula | first2=Iwona | last2=Bialynicka-Birula | title=Modeling Reality: How Computers Mirror Life | year=2004 | publisher=[[Oxford University Press]] | isbn=0198531001| ref=bialynick}}
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| *{{cite book | first1=Bastien | last1=Chopard | first2=Michel | last2=Droz | title=Cellular Automata Modeling of Physical Systems | year=2005 | publisher=[[Cambridge University Press]] | isbn=0-521-46168-5 | ref=chopard-droz}}
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| *{{cite book | editor=Gutowitz, Howard | title=Cellular Automata: Theory and Experiment | year=1991 | publisher=[[MIT Press]] | isbn=9780262570862 | ref=gutowitz}}
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| *{{cite book | author=Ilachinski, Andrew | title=Cellular Automata: A Discrete Universe | year=2001 | publisher=[[World Scientific]] | isbn=9789812381835 | ref=ilachinski}}
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| *{{cite book | first1=Lemont B. | last1=Kier | first2=Paul G. | last2=Seybold | first3=Chao-Kun | last3=Cheng | title=Modeling Chemical Systems using Cellular Automata | year=2005 | publisher=Springer | isbn=9781402036576 | ref=kier-seybold-cheng}}
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| *{{cite book | author=Schiff, Joel L. | title=Cellular Automata: A Discrete View of the World | year=2011 | publisher=Wiley & Sons, Inc | isbn=9781118030639 | ref=schiff}}
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| *{{cite book | author=Wolfram, Stephen | authorlink=Stephen Wolfram | title=A New Kind of Science | year=2002 | publisher=[[Wolfram Media]] | isbn=978-1579550080 | ref=wolfram}}
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| *[http://cafaq.com/ Cellular automaton FAQ] from the newsgroup comp.theory.cell-automata
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| *[[Alex Wissner-Gross|A. D. Wissner-Gross]]. 2007. ''[http://www.alexwg.org/publications/JCellAuto_4-27.pdf Pattern formation without favored local interactions]'', Journal of Cellular Automata 4, 27–36 (2008).
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| *[http://cell-auto.com/neighbourhood/index.html "Neighbourhood Survey"] (includes discussion on triangular grids, and larger neighborhood CAs)
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| * von Neumann, John, 1966, ''The Theory of Self-reproducing Automata'', A. Burks, ed., Univ. of Illinois Press, Urbana, IL.
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| *[http://cscs.umich.edu/~crshalizi/notebooks/cellular-automata.html Cosma Shalizi's Cellular Automata Notebook] contains an extensive list of academic and professional reference material.
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| *[http://www.stephenwolfram.com/publications/articles/ca/ Wolfram's papers on CAs]
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| * A.M. Turing. 1952. The Chemical Basis of Morphogenesis. ''Phil. Trans. Royal Society'', vol. B237, pp. 37–72. (proposes reaction-diffusion, a type of continuous automaton).
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| * Evolving Cellular Automata with Genetic Algorithms: A Review of Recent Work, Melanie Mitchell, James P. Crutchfeld, Rajarshi Das (In Proceedings of the First International Conference on Evolutionary Computation and Its Applications (EvCA'96). Moscow, Russia: Russian Academy of Sciences, 1996.)
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| * The Evolutionary Design of Collective Computation in Cellular Automata, James P. Crutchfeld, Melanie Mitchell, Rajarshi Das (In J. P. Crutch¯eld and P. K. Schuster (editors), Evolutionary Dynamics|Exploring the Interplay of Selection, Neutrality, Accident, and Function. New York: Oxford University Press, 2002.)
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| *The Evolution of Emergent Computation, James P. Crutchfield and Melanie Mitchell (SFI Technical Report 94-03-012)
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| *[http://www.wepapers.com/Papers/16352/files/swf/15001To20000/16352.swf Ganguly, Sikdar, Deutsch and Chaudhuri "A Survey on Cellular Automata"]
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| {{Refend}}
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| ==External links==
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| {{Wikibooks|Cellular Automata}}
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| {{Commons category|Cellular automata}}
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| *{{sep entry|cellular-automata|Cellular Automata|Francesco Berto & Jacopo Tagliabue}}
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| *[http://www.mirekw.com/ca/index.html Mirek's Cellebration] – 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.
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| *[http://www.collidoscope.com/modernca/ Modern Cellular Automata] – Easy to use interactive exhibits of live color 2D cellular automata, powered by Java applet. Included are exhibits of traditional, reversible, hexagonal, multiple step, fractal generating, and pattern generating rules. Thousands of rules are provided for viewing. Free software is available.
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| *[http://necsi.edu/postdocs/sayama/sdsr/java/ Self-replication loops in Cellular Space] – Java applet powered exhibits of self replication loops.
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| *[http://vlab.infotech.monash.edu.au/simulations/cellular-automata/ A collection of over 10 different cellular automata applets] (in Monash University's Virtual Lab)
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| *[http://www.sourceforge.net/projects/golly Golly] 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 which can demonstrate von Neumann type self-replication.
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| *[http://atlas.wolfram.com/TOC/TOC_200.html Wolfram Atlas] – An atlas of various types of one-dimensional cellular automata.
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| *[http://www.conwaylife.com/ Conway Life]
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| *[http://www.newscientist.com/article/mg20627653.800-first-replicating-creature-spawned-in-life-simulator.html First replicating creature spawned in life simulator]
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| *[http://www.mmdr.it/provaEN.asp ''The Mathematics of the Models of Reference''], featuring a general tutorial on CA, interactive applet, free code and resources on CA as model of fundamental physics
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| *[http://www.fourmilab.ch/cellab Fourmilab Cellular Automata Laboratory]
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| *[http://busyboxes.org Busy Boxes], a 3-D, reversible, [http://64.78.31.152/wp-content/uploads/2012/08/2stateRevCAin3D.pdf SALT]-architecture CA
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| *[http://uncomp.uwe.ac.uk/genaro/CA_repository.html Cellular Automata Repository] (CA researchers, historic links, free software, books and beyond)
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| {{Use dmy dates|date=May 2011}}
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| {{DEFAULTSORT:Cellular Automaton}}
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| [[Category:Cellular automata| ]]
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| [[Category:Systems theory]]
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| [[Category:Automata theory]]
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| [[Category:Dynamical systems]]
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| {{Link GA|es}}
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| {{Link GA|ja}}
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