Cellular Automaton

A cellular automaton (plural: cellular automata) is a discrete model studied in computability theory and mathematics. It consists of an infinite, regular grid of cells, each in one of a finite number of states. The grid can be in any finite number of dimensions. Time is also discrete, and the state of a cell at time t is a function of the state of a finite number of cells called the neighborhood at time t-1. These neighbors are a selection of cells relative to some specified, and does not change (Though the cell itself may be in its neighborhood, it is not usually considered a neighbor). Every cell has the same rule for updating, based on the values in this neighbourhood. Each time the rules are applied to the whole grid a new generation is produced. One example of a cellular automaton (CA) would be an infinite sheet of graph paper, where each square is a cell, each cell has two possible states (black and white), and the neighbors of a cell are the 8 squares touching it. Then, there are 29 = 512 possible patterns for a cell and its neighbors. The rule for the cellular automaton could be given as a table. For each of the 512 possible patterns, the table would state whether the center cell will be black or white on the next time step. This is an example of a two dimensional cellular automaton. See Conway's Game of Life for the most popular CA of this form. 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, often called a configuration. 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. 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 edges are usually handled with a toroidal arrangement: when you go off the top, you come in at the corresponding position on the bottom, and when you go off the left you come in on the right (This essentially simulates an infinite periodic tiling). This can be visualized as taping the left and right edges together to form a tube, then taping the top and bottom edges of the tube together to form a torus (doughnut shape). Universes of other dimensions are handled similarly. This is done in order to solve boundary problems with neighborhoods. For example, with a 1-dimensional cellular automaton, like the examples below, the neighborhood of a cell xi t—where t is the time step (vertical), and i is the index (horizontal) in one generation—is xi-1t-1, xit-1, xi+1t-1, there are obviously going to be problems when a neighbourhood on a left border is going to reference the upper left cell as part of its neighborhood, which it cannot, since it is not in the cellular space!

History of Cellular Automata

Cellular automata were invented by Stanislaw Ulam at Los Alamos laboratory in the 1940's. John von Neumann - Ulam's colleague at Los Alamos - who was, at that time, working on a study of self-replicating systems realized the potential of CA to function as a simplified model of the physics of our universe. In the impossibility to design a self-replicating machine in the physical world, (see Von Neumann machine) Neumann implemented an algorithmic self-replicating system within the universe of a two-dimensional CA. The result was a universal copier and constructor (UCC) working within a CA with a small neighborhood (only cells that touch are neighbors), and with 29 states per cell. Neumann proved mathematically that a particular pattern would make endless copies of itself within the given cellular universe. In the 1970s a two-state, two dimensional cellular automaton named Game of Life became very widely known, particularly among the early computing community. Invented by John Conway, and popularized by Martin Gardner in a Scientific American article, its rules are as follows: If a black cell has 2 or 3 black neighbors, it stays black. If a white cell has 3 black neighbors, it becomes black. In all other cases, the cell becomes white. Despite the simplicity of the rule, an impressive diversity of behavior is achieved, fluctuating between apparent randomness and order. One of the most apparent features of the Game of Life is the frequent occurrence of gliders, which are 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. See Conway's Game of Life for more details. Possibly because it was viewed as a largely recreational topic, little follow-up work was done outside of investigating the particularities of the Game of Life and a few related rules. In 1969, however, 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 just the output of a deterministic computation on a giant cellular automaton. This was the first book on what today is called digital physics. In 1983 Stephen Wolfram published the first of a series of papers systematically investigating a very basic but essentially unknown class of cellular automata, which he terms elementary cellular automata (see below). The unexpected complexity of the behavior of these simple rules—and the failure of mathematical methods to meaningfully describe them—lead Wolfram to suspect that complexity in nature may be due to similar mechanisms, and also is not amenable to traditional mathematical analysis. Additionally, during this period Wolfram formulated the concepts of intrinsic randomness and computationally irreducibility, and suggested that rule 110 may be universal—a fact proved as part of the development of his later book. Wolfram left academia in the mid-late 1980s to create Mathematica, which he then used to extend his earlier results to a broad range of other simple, abstract systems. In 2002 he published his results in the 1280-page text A New Kind of Science, which extensively argued that the discoveries about cellular automata are not isolated facts but are robust and have significance for all disciplines of science. Despite much confusion in the press and academia, the book did not argue for a fundamental theory of physics based on cellular automata, and although it did describe a few specific physical models based on cellular automata, it also provided models based on qualitatively different abstract systems.

The Simplest Cellular Automata

The simplest nontrivial CA would be one-dimensional, with two possible states per cell, and a cell's neighbors defined as the cell on either side of it. A cell and its two neighbors forms a neighborhood of 3 cells, so there are 23=8 possible patterns for a neighborhood. So, there are 28=256 possible rules. These 256 CAs are generally referred to using a standard naming convention invented by Wolfram. The name of a CA is a number which, in binary, gives the rule table. For example, these are tables defining the "rule 30 CA". A table completely defines a CA rule. For example, the rule 30 table says that if three adjacent cells in the CA currently have the pattern 100 (left cell is on, middle and right cells are off), then the middle cell will become 1 (on) on the next time step. The rule 110 CA says the opposite for that particular case. If the columns are written in order as shown in the table above (the reverse order of counting in binary), then the number of the rule can be read off in binary. The bottom line in the first table has the bits 00011110, which is the binary representation of the number 30, so that CA is called the "rule 30 CA". Similarly, the rule 110 CA derives its name from 01101110, which is the binary representation of the decimal number 110. A number of papers have analyzed and compared these 256 CAs. The rule 30 and rule 110 CAs are particularly interesting. Rule 30 generates randomness despite the lack of anything that could reasonably be considered random input. Stephen Wolfram proposed using its center column as a pseudorandom number generator (PRNG), and despite occasional claims to the contrary, it passes every standard test for randomness, and Wolfram uses this rule in the Mathematica product for creating random integers. In particular in the 1990s a cryptography survey book claimed that rule 30 was equivalent to a linear feedback shift register (LFSR) but in fact the claim was about rule 90. Although Rule 30 produces randomness on many input patterns, there are also an infinite number of input patterns that result in repeating patterns. The trivial example of such a pattern is the input pattern only consisting of zeros. A less trivial example, found by Matthew Cook, is any input pattern consisting of infinite repetitions of the pattern '00001000111000', with repetitions optionally being separated by six ones. 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, Cook proved in 1994 that these structures were rich enough to support 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 published before the publication of A New Kind of Science. In 2004, Cook's proof was finally published in Wolfram's journal Complex Systems , over ten years after Cook came up with it.

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