Cellular automaton

A cellular automaton (CA) is a certain class of mathematical objects (or, more specifically, a mathematical structure) of which Conway's Game of Life is an example.

Informally, a cellular automaton consists of:

• A space of cells.
• A set of allowed states for each cell. An assignment of an state to every cell is called a “configuration” or “pattern” (the first term is more common in mathematical discussion and the later in informal discussions).
• A neighbourhood which defines which cells are considered to pass information to a given cell.
• A transition rule which specifies how given a cell and the states of its neighbours, a new state is produced.

The state of the cellular automaton evolves in discrete time, with the state of each cell at time t+1 being determined by the state of its neighbourhood at time t in accordance with the transition rule.

There are some variations on the above definition. It is common to require that there be a quiescent state (i.e., a state such that if the whole universe is in that state at generation 0 then it will remain so in generation 1). In Conway's Game of Life, the "OFF" state is quiescent, but the "ON" state is not. Other variations allow spaces other than Iⁿ, neighbourhoods that vary over space and/or time, probabilistic or other non-deterministic transition rules, and so on.

It is common for the neighbourhood of a cell to be the 3×...×3 hypercube centered on that cell, which is known as its Moore neighbourhood.

For a collection of cellular automata packaged for Golly, see [1].

Formal definition

Let us denote the set of integers by I and the length of any tuple x by |x|. For any tuples of integers x and y such that |x|=|y| we denote their element-wise addition by x+y.

A cellular automaton is a tuple (Iⁿ,S,N,f) such that the dimension n is at least 1, the set of states S is finite, the neighbourhood N is a tuple of elements of Iⁿ and f:S^|N|→S is the transition function.

A configuration of the cellular automaton (Iⁿ,S,N,f) is any function Iⁿ→S.

The global transition function F of the cellular automaton (Iⁿ,S,N,f) is a function F:Iⁿ→Iⁿ such that for any configuration c and element a∈Iⁿ we have F(c)(a)=f(a+N).

Let c be a configuration. When the cellular automata is clear from the context, then by cⁿ where n is a non-negative integer we denote the configuration Fⁿ(c) where F is the corresponding global transition function.

Let c and c' be configurations, then we say that c' is a translation of c if there exist an a∈Iⁿ such that for any x∈Iⁿ it holds that c'(a+x)=c(x). We say that the translation is proper if the condition holds for some n-tuple whose elements are not all 0.

For any configuration c, if there exists a n≥1 such that cⁿ = c we call c an oscillator. If c is an oscillator and n is the least positive integer such that cⁿ = c then we call n the period of c. If c is an oscillator with period 1 then we call c a still life. If c is an oscillator and is not a still life we call c a proper oscillator.

A glider is a configuration c such that cⁿ is a proper translation of c for some n>0.

Our definition of cellular automaton is a simple derivative of the one given in Codd (1968). The only difference in scope is that Codd only allows grids of dimension 2 and requires the presence of a quiescent state, that is, a state v₀ such that f(v₀,...,v₀)=v₀. Moreover, Codd only allows configurations in which finitely many cells are non-quiescent, while our definition of configuration allows any assignment of states to cells.

Generalizations and topological characterization

Our definition is enough to describe most of the cellular automata in which explicitly described configurations are studied. In particular it handles as expected the elementary cellular automata, CA on the Euclidean square tessellation (including all life-like CA and all non-totalistic life-like CA) and the Euclidean hexagonal tessellation. Furthermore, CA on the Euclidean triangular tessellation can be handled when proper care is taken. However, there exist objects which are cellular automata according to our informal description, but can not be defined as such in our formal definition. An example is cellular automata in hyperbolic tessellations (Margenstern, 2013). A very general definition of cellular automata is given in Moriceau (2011) and a further generalization is given in Wacker (2016). Interestingly, Wacker's definition uses amenable groups. Both amenable groups and cellular automata were introduced by the same man: John von Neumann.

A remarkable result, the Curtis-Hedlund-Lyndon-Richardson (CHLR) theorem, relates cellular automata theory with topology and dynamical system theory. An informal statement is given as follows: Let the set of states be given the discrete topology and the configuration space be given given the Tikhonov topology; then the global transition functions of the cellular automata over the configuration space are exactly those functions which are continuous and commute with the translations of the cellular space. In particular, the CHLR theorem holds for our definition of CA. Informally we state that the CHLR theorem holds for definitions of CA where the set of states is required to be finite, but if the set of states is allowed to be infinite, the set of allowable neighborhoods must be expanded or a further constraint must be imposed on continuous functions so that an analogous result holds (Ceccherini-Silberstein 2010; Sobottka 2015). Generally when a novel definition of cellular automaton is presented a corresponding version of the CHLR theorem is proved in the same treatise.

The CHLR theorem was demonstrated independently by Hedlund (1969) (in this paper credit is given to Curtis and Lyndon in addition to the author) and Richardson (1972). Neither of these papers use the term “cellular automaton”. Nonetheless the systems they study are, respectively, 2-dimensional deterministic cellular automata and n-dimensional non-deterministic cellular automata.

Self-replicating configurations

Informally, a configuration that creates a copy of itself after some number of generations is called a replicator.

John von Neumann investigated the possibility of building a self-replicating machine. He originally considered a mechanical approach, but decided that this was too hard to control. With the help of Stanislaw Ulam, he designed a new mathematical abstraction, the cellular automaton, in order to create a replicator. His CA was made in the late 1940s and is complex. It operates on the Von Neumann neighbourhood (a cell and its four orthogonally connected neighbours), and has 29 states. It is described in von Neumann, Burks (1966).

Subsequent to the von Neumann's cellular automaton new cellular automata have been produced capable of self-replication and universal computation with fewer states and using the same (von Neumann) neighborhood. Results include Codd (1968) with 8 states, Banks (1971) with 4 states and Serizawa (1987) with 3 states. As of 2017, Serizawa's CA appears to be the simplest (in terms of neighborhood size×number of states) cellular automaton known to have a non-trivial replicator.

Langton (1984) constructed a cellular automaton and a replicator. His cellular automaton is a simplification of Codd's. However, Langton's CA was not intended to support universal computation. The simplicity of the configuration is possible because of a novel method of self-replication. The replicator consists of a loop with circulating cell states that encode instructions to construct a new arm, then turn 90° and repeat until a new loop is formed. The new loop is then filled with the same circulating information and the connection between it and the loop that generated it is severed. The parent loop continues to build copies until no more space is available, then it becomes a still life.

Nehaniv (2002) presented a method that allows the transformation of conventional (synchronous) cellular automata to an equivalent asynchronous cellular automata (a case not covered by our definition above). In the same paper, a self-reproducing configuration is presented which is based on the described procedure applied to Langton's CA.

Conway's Game of Life is known to be universal, with 2 states and the Moore neighbourhood. Conway did not design Life for this purpose, unlike von Neumann's, Codd's, Banks' and Serizawa's rules, so it is purely coincidental that Life supports replicators. The Spartan universal computer-constructor could replicate, given a sufficient program tape. Between 2010 and 2017 much more efficient construction mechanisms have been developed that can be used for self-replication, but to date only a limited one-dimensional example has actually been completed (the linear propagator).

Formal definition

There are several possible non-equivalent definitions of self-replicators. Moore (1962) proposed a simple definition which does not attempt to exclude trivial cases (although Moore wrote “self-reproduction”, we assume that it synonymous with “self-replication”). However, his definitions of self-replicator is limited to the same cellular automata and configurations allowed by the definition of Codd (1968). Our definition is a slight generalization of Moore's definition to cover all the configurations allowed by our definition.

Not all of the self-replicators mentioned above meet our formal definition. Langton's self-replicating loop and the classical HighLife replicator meet our definition. Typically the configurations in which each copy of the initial configuration only generates one copy but does not returns to the initial state after finishing do not meet our definition.

Definition: Let (Iⁿ,S,N,f) be a cellular automaton. Let c be a configuration such that for some s∈S the set X={a∈Iⁿ|c(a)≠s} is finite. We say that c is a finite configuration, s is a background state of c and X is a support of c.

Theorem: Let c be a finite configuration, then there is a unique background state and support of c.

Let v and v' be background states of c. Let X be the set of all cells a such that c(a)≠v and respectively for X' and v'. The set Iⁿ\(X∪X') is non-empty because Iⁿ is infinite and X∪X' is finite. Let x∈Iⁿ\(X∪X'). By the properties of set union and complement we have x∈Iⁿ\X and x∈Iⁿ\X'; by definition of X and X' we have x=v and x=v'; therefore v=v'. Again from the definition of X and X' it follows that X=X'.

Definition: Let c be a finite configuration. We write bg(c) for the background state of c and we write su(c) for the support of c.

Definition: Let c and c' be finite configurations such that bg(c)=bg(c'). If su(c)∩su(c')=∅ we say that c is disjoint from c'. If su(c)⊂su(c') and for all a∈su(c) we have c(a)=c'(a) then we say that c' contains c.

Definition: Let c and c' be configurations. We say that c' contains at least n copies if c if there exist a set X of pairwise disjoint translations of c and for each x∈X, c contains x.

Definition: A configuration c is called a self-replicator if for any n≥1 there exist a t≥0 such that c_t contains at least n disjoint copies of c.

Definition: Let c and c be a configuration. Let n≥0 be an integer. If c' contains at least n copies of c and for any integer k>n it is not the case that c' contains at least k copies of c then we say that c' contains exactly n copies of c.

Life-like cellular automata

A cellular automaton is said to be Life-like if it meets the following four criteria:

• It has two dimensions (i.e., n=2).
• It has two states, usually called OFF and ON (i.e., |S|=2).
• The neighbourhood used is the Moore neighbourhood.
• The new state of a cell in the next generation can be expressed as a function of the number of cells in its neighbourhood that are in the ON state and the cell's own state; that is, the rule is outer totalistic (also called semitotalistic).

This class of cellular automata is named for Conway's Game of Life, the most famous cellular automaton, which Life-like cellular automata mimic. Many different terms are used to describe this class of cellular automata; it is also common to refer to it as the "Life family" or to simply use phrases like "similar to Life".

Rules

A common notation used to describe these automata is referred to as "S/B", which is known as its rule (or rulestring). S (for survival) is a list of all the numbers of ON cells that cause an ON cell to remain ON. B (for birth) is a list of all the numbers of ON cells that cause an OFF cell to turn on. If 0 is in the list, then blank regions of the universe will turn on in one generation. S/B is often referred to as the rule or rulestring of the given cellular automata.

As an example, the seeds rulestring is /2. All OFF cells that have exactly two adjacent ON cells will turn on in the next generation, while every ON cell dies in every generation, since the survival list is empty. The rulestring of Conway's Game of Life is 23/3.

Other notations are sometimes used to describe the rules of the given Life-like cellular automaton. The most common other format is "B{number list}/S{number list}", where the number lists are the numbers of neighbours that cause a dead cell to be born and cause an alive cell to stay alive, respectively. In this format, Conway's Game of Life would have the rulestring B3/S23.

There are 262144 (= 2¹⁸) distinct Life-like rules. Each rule has a complementary rule which behaves identically under on-off reversal; namely the rule in which birth occurs on all N except those for which 8-N is a survival condition in the original rule, and survival occurs on all N except those for which 8-N is a birth condition in the original rule. For example, the rule complementary to Conway's Life is 01234678/0123478. This however does not quite halve the number of effectively distinct rules, as there are 512 (= 2⁹) rules which are unaffected by on-off reversal.

Some straightforward inferences on the behavior of different kinds of rules can be made:

• In all rules where the lowest birth condition is 1 neighboring ON cell, all finite patterns grow at the speed of light in all directions. No still lifes, oscillators or spaceships are possible in these rules. Several have replicators, however. There are 65536 (= 2¹⁶) rules of the B1 type.
• All rules where the lowest birth condition is 2 neighboring ON cells are exploding or expanding in character; this is largely due to the fact that a domino at the corner of a pattern will give rise to a new domino, also located at the corner of the daughter pattern. Spaceships (such as the moon) and oscillators (such as the duoplet) do exist in many of these rules. There are 32768 (= 2¹⁵) rules of the B2 type.
• All rules where the lowest birth condition (if any) is 4 or more neighboring ON cells are stable in character, since no patterns ever grow beyond their initial bounding box. In particular, no spaceships can exist. There are 16384 (= 2¹⁴) rules of the B4+ type.
• In all rules where the lowest birth condition is 0 neighboring ON cells, and the highest survival condition is 8 neighboring ON cells, the vacuum is unstable and will be immediately filled (and remain filled) with ON cells; thus, there are no patterns that remain finite. All of these rules have distinct complementary rules, and they are not commonly studied on their own.
• This leaves 16384 rules in which the lowest birth condition is 3 neighboring ON cells, as well as 65536 rules in which the lowest birth condition is 0 neighboring cells, and 8 neighbors is not a survival condition. All chaotic rules must fall in either of these two areas of the rulespace. Most well-studied examples fall in the first one, since for long no commonly available software existed that could simulate the evolution of rules containing B0.

Non-totalistic rules

Main article: Non-totalistic Life-like cellular automaton

Various generalizations of Life-like cellular automata are possible. In non-totalistic rules, the transition function considers not just the number of cells in a given cell's neighborhood but also their alignment; for example, a cell might be born if bordered by three live cells in a row, but not by three live cells in other configurations. Non-totalistic rules are described using Hensel notation, an extension of B/S notation additionally describing allowed or forbidden configurations.

Well-known life-like cellular automata

Main article: List of Life-like cellular automata

The following table lists life-like cellular automata that are particularly well-known or well-studied.

Rulestring	Name	Description
B1357/S1357	Replicator	A rule in which every pattern is a replicator.
B1357/S02468	Fredkin	A rule in which, like Replicator, every pattern is a replicator.
B2/S	Seeds	An exploding rule in which every cell dies in every generation. It has many simple orthogonal spaceships, though it is in general difficult to create patterns that don't explode.
B2/S0	Live Free or Die	An exploding rule in which only cells with no neighbors survive. It has many spaceships, puffers, and oscillators, some of infinitely extensible size and period.
B3/S012345678	Life without death	An expanding rule that produces complex flakes. It also has important ladder patterns.
B3/S12	Flock	A rule which decays into small still lifes and oscillators
B3/S1234	Mazectric	An expanding rule that crystalizes to form maze-like designs that tend to be straighter (ie. have longer "halls") than the standard maze rule.
B3/S12345	Maze	An expanding rule that crystalizes to form maze-like designs.
B3/S23	Conway's Life	A chaotic rule that is by far the most well-known and well-studied. It exhibits highly complex behavior.
B36/S125	2x2	A chaotic rule with many simple still lifes, oscillators and spaceships. Its name comes from the fact that it sends patterns made up of 2x2 blocks to patterns made up of 2x2 blocks.
B36/S23	HighLife	A chaotic rule very similar to Conway's Life that is of interest because it has a simple replicator.
B368/S245	Move	A rule in which random patterns tend to stabilize extremely quickly. Has a very common slow-moving spaceship and slow-moving puffer.
B3678/34678	Day & Night	A stable rule that is symmetric under on-off reversal. Many patterns exhibiting highly complex behavior have been found for it.

References

• Banks, E. R. (1971) Information processing and transmission in cellular automata.
• Ceccherini-Silberstein, T.; Coornaert, M. (2010) Cellular Automata and Groups. DOI: 10.1007/978-3-642-14034-1. ISBN: 978-3-642-14033-4, 978-3-642-14034-1.
• Codd, E. F. (1968) Cellular Automata. No ISBN. WorldCat OCLC number: 637978742.
• Hedlund, G. A. (1969) Endomorphisms and Automorphisms of the Shift Dynamical System. DOI: 10.1007/BF01691062.
• Langton, C. G. (1984) Self-reproduction in cellular automata. DOI: 10.1016/0167-2789(84)90256-2.
• Margenstern, M. (2013) Small Universal Cellular Automata in Hyperbolic Spaces: A Collection of Jewels.
• Moore, E. F. (1962) Machine models of self-reproduction in Mathematical Problems in the Biological Sciences. DOI: 10.1090/psapm/014/9961.
• Moriceau, S. (2011) Cellular automata on a G-set. arXiv: 1105.5335 [math.DS].
• Nehaniv, C. E. (2002) Self-Reproduction in Asynchronous Cellular Automata in Proceedings 2002 NASA/DoD Conference on Evolvable Hardware. DOI: 10.1109/eh.2002.1029854.
• Richardson, D. (1972) Tessellation with Local Transformations. DOI: 10.1016/S0022-0000(72)80009-6.
• Serizawa, T. (1987) Three-State Neighbor Cellular Automata Capable of Constructing Self-Reproducing Machines. DOI: 10.1002/scj.4690180404.
• Sobottka, M.; Gonçalves, D. (2015) A note on the definition of siding block codes and the Curtis-Hedlund-Lyndon Theorem. arXiv: 1507.02180 [math.DS].
• von Neumann, J.; Burks, A. W. (1966) Theory of Self-Reproducing Automata. No ISBN. WorldCat OCLC number: 7298386.
• Wacker, S. (2016) Cellular Automata on Group Sets and the Uniform Curtis-Hedlund-Lyndon Theorem. DOI: 10.1007/978-3-319-39300-1_15. In Cellular Automata and Discrete Complex Systems. AUTOMATA 2016.

External links

Cellular automata rules lexicon at Mirek Wójtowicz's Cellebration page

• Cellular automaton at the Life Lexicon

Cellular Automaton at Wolfram Mathworld Totalistic Cellular Automaton at Wolfram Mathworld

• Rule Table Repository