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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 the same relative to each cell, and do 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. Each time the rules are applied 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 2 = 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. 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. 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 x_i -- where t is the time step (vertical), and i is the index (horizontal) in one generation -- is x_i-1, x_i, x_i+1, there are obviously going to be problems when a cell 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 John von Neumann in his study of self-replicating systems. As a simplified model of the physics of our universe, he designed a two-dimensional CA with a small neighborhood (only cells that touch are neighbors), and with 29 states per cell. Within that universe, he designed an initial pattern which acted like a self-replicating machine, and mathematically proved that it would make endless copies of itself.

Widespread interest in cellular automata started when John Conway invented the Game of Life. This is a two-dimensional CA, with two states per cell, and where each cell has 8 neighbors. The rule is simple: 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. The universe typically starts with only a few black cells. This CA was very simple, yet allowed the construction of patterns that appeared to move themselves across the grid of cells. Life became the most widely-studied CA in history, and research on it continues. See Conway's Game of Life for more details. {universal Turing machines, and self-replicating machines.}

In 2002, Stephen Wolfram published the book A New Kind of Science. It contains an analysis of one-dimensional cellular automata and proposes the thesis that physics should be modeled by cellular automata rather than by differential equations, on the grounds that cellular automata are simpler, yet still able to exhibit the complex phenomena seen in nature. It doesn't propose any particular cellular automaton for this.

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 2=8 possible patterns for a neighborhood. So, there are 2=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" and the "rule 110 CA" and a graphical representation of them starting from a 1 in the center of the image:

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.

The columns must be written in the given order (the reverse order of counting 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.

Wolfram proposed a pseudorandom number generator (PRNG) based on rule 30. Rule 30 is run on a finite universe whose cells have random initial states, and the output is the sequence of states for one particular cell. He argued that since the rule is nonlinear (the rules cannot be written as equations using just XOR and NOT), the PRNG should be much better than a simple linear feedback shift register (LFSR). The rule 30 PRNG would therefore be more useful for both statistical and cryptographic applications, so he used this generator as the PRNG for Mathematica. Unfortunately, the generator was later proved to be exactly equivalent to an LFSR, which renders it useless for cryptography and no better than an LFSR for statistical uses.

The rule 110 cellular automaton has interesting behavior, especially when the initial pattern is carefully chosen. Wolfram was the first to study it, and concluded that it obviously could not be Turing complete. Matthew Cook later proved it actually is Turing complete, and presented the proof at a Santa Fe Institute conference, but Wolfram suppressed its publication with a court order. A general overview of the proof is given in Wolfram's book. This is an interesting result, because it shows that even the simplest cellular automata have all the computational power of a universal Turing machine. It is not known which of these 256 simple CAs are Turing complete, other than rule 110 and the three other CAs trivially equivalent to it.

Reversible Cellular Automata

Sometimes it is possible to examine the state of a cellular automaton, and deduce the previous state. If the rules ensure that this is always possible, then the CA is called reversible. Given the rules, there is no general way to tell whether the CA is reversible. Jarkko Kari proved that this is undecidable for CAs with two or more dimensions. For CAs with one dimension, it is decidable.

For CAs that aren't reversible, there must exist patterns for which there is no previous state. These patterns are called Garden of Eden patterns. In other words, no pattern exists which will, in one step, become a Garden of Eden pattern.

Uses in Cryptography

Rule 30 was originally suggested as a possible stream cipher for use in cryptography. However, it was later broken, as described above.

Cellular automata have been proposed for public key cryptography. The one way function is the evolution of a finite CA whose inverse is hard to find. Given the rule, anyone can easily calculate future states, but it is 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 a trapdoor function, and can be used as a public-key cryptosystem. The security of such systems is not currently known.

Related Automata

There are many possible generalizations of the CA concept.

One way is by using something other than a rectangular (cubic, etc.) grid. For example, if a plane is tiled with equilateral triangles, those triangles could be used as the cells.

Also, the rules can be probabilistic rather than deterministic. The rule then gives, for each pattern at time t, the probability that the central cell will transition to each possible state at time t+1. Sometimes a simpler rule is used, such as, "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."

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.

The grid can be finite, so that patterns can "fall off" the edge of the universe.

In CA, 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 have a continuum of locations. The state of a location is a finite number of real numbers. Time is 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. When these are approximated by cellular automata, the CAs often yield similar patterns.

See also:

• Langton's ant

References

• Cellular automaton FAQ from the newsgroup comp.theory.cell-automata
• von Neumann, John, 1966, The Theory of Self-reproducing Automata, A. Burks, ed., Univ. of Illinois Press, Urbana, IL.
• Wolfram, Stephen, 1985, Cryptography with Cellular Automata, CRYPTO'85.
• http://www.santafe.edu/~shalizi/notebooks/cellular-automata.html
• Wolfram's papers on CAs
• 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).
• Jim Giles. 2002. What kind of science is this? Nature 417, 216 - 218. (discusses the court order that suppressed publication of the rule 110 proof).