Elementary cellular automaton
Encyclopedia
In mathematics
and computability theory
, an elementary cellular automaton is a one-dimensional cellular automaton
where there are two possible states (labeled 0 and 1) and the rule to determine the state of a cell in the next generation depends only on the current state of the cell and its two immediate neighbors. As such it is one of the simplest possible models of computation. Nevertheless, there is an elementary cellular automaton (rule 110, defined below) which is capable of universal computation
.
proposed a scheme, known as the Wolfram code
, to assign each rule a number from 0 to 255 which has become standard. Each possible current configuration is written on order, 111, 110, ... , 001, 000, and the resulting state for each of these configurations is written in the same order and interpreted as the binary representation of an integer. This number is taken to be the rule number of the automaton. For example, 110 written in binary is 011011102. So rule 110 is defined by the transition rule:
For example, if the definition of rule 110 is reflected through a vertical line, the following rule (rule 124) is obtained:
Rules which are the same as their mirrored rule are called amphichiral. Of the 256 elementary cellular automata, 64 are amphichiral.
The second such transformation is to exchange the roles of 0 and 1 in the definition. The result of applying this transformation to a given rule is called the complementary rule.
For example, if this transformation is applied to rule 110, the following rule (rule 137) obtained:
There are 16 rules which are the same as their complementary rules.
Finally, the previous two transformations can be applied successively to a rule to obtain the mirrored complementary rule. For example, the mirrored complementary rule of rule 110 is rule 193. There are 16 rules which are the same as their mirrored complementary rules.
Of the 256 elementary cellular automata, there are 88 which are inequivalent under these transformations.
with a simple form. The following rules are notable:
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It has the closed form expression
Note that rule 156 generates the same sequence.
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It has the closed form expression
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Note that rules 58, 114, 122, 178, 186, 242 and 250 generate the same sequence.
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It has the closed form expression
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modulo 2 and interpreting them as integers in binary, which can be graphically represented by a Sierpinski triangle
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modulo 2 and interpreting them as integers in base 4. Note that rules 18, 26, 82, 146, 154, 210 and 218 generate the same sequence.
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This has generating function
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It has the closed form expression
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Note that rule 252 generates the same sequence.
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It has the closed form expression
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Note that rule 254 generates the same sequence.
Rule 73 is Class 2 because any time there are two consecutive 1s surrounded by 0s, this feature is preserved in succeeding generations. This effectively creates walls which block the flow of information between different parts of the array. There are a finite number of possible configurations in the section between two walls so the automaton must eventually start repeating inside each section, though the period may be very long if the section is wide enough. These walls will form with probability 1 for completely random initial conditions. However, if the condition is added that the lengths of runs of consecutive 0s or 1s must always be odd, then the automaton displays Class 3 behavior since the walls can never form.
Rule 54 is Class 4, but it remains unknown whether it is capable of universal computation. Interacting structures form, but structures that are useful for computation have yet to be found.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
and computability theory
Computability theory (computer science)
Computability is the ability to solve a problem in an effective manner. It is a key topic of the field of computability theory within mathematical logic and the theory of computation within computer science...
, an elementary cellular automaton is a one-dimensional cellular automaton
Cellular automaton
A cellular automaton is a discrete model studied in computability theory, mathematics, physics, complexity science, theoretical biology and microstructure modeling. It consists of a regular grid of cells, each in one of a finite number of states, such as "On" and "Off"...
where there are two possible states (labeled 0 and 1) and the rule to determine the state of a cell in the next generation depends only on the current state of the cell and its two immediate neighbors. As such it is one of the simplest possible models of computation. Nevertheless, there is an elementary cellular automaton (rule 110, defined below) which is capable of universal computation
Turing completeness
In computability theory, a system of data-manipulation rules is said to be Turing complete or computationally universal if and only if it can be used to simulate any single-taped Turing machine and thus in principle any computer. A classic example is the lambda calculus...
.
The numbering system
There are 8 = 23 possible configurations for a cell and its two immediate neighbors. The rule defining the cellular automaton must specify the resulting state for each of these possibilities so there are 256 = 28 possible elementary cellular automata. Stephen WolframStephen Wolfram
Stephen Wolfram is a British scientist and the chief designer of the Mathematica software application and the Wolfram Alpha computational knowledge engine.- Biography :...
proposed a scheme, known as the Wolfram code
Wolfram code
Wolfram code is a naming system often used for one-dimensional cellular automaton rules, introduced by Stephen Wolfram in a 1983 paper and used in his book A New Kind of Science....
, to assign each rule a number from 0 to 255 which has become standard. Each possible current configuration is written on order, 111, 110, ... , 001, 000, and the resulting state for each of these configurations is written in the same order and interpreted as the binary representation of an integer. This number is taken to be the rule number of the automaton. For example, 110 written in binary is 011011102. So rule 110 is defined by the transition rule:
| current pattern | 111 | 110 | 101 | 100 | 011 | 010 | 001 | 000 |
|---|---|---|---|---|---|---|---|---|
| new state for center cell | 0 | 1 | 1 | 0 | 1 | 1 | 1 | 0 |
Reflections and complements
Although there are 256 possible rules, many of these are trivially equivalent to each other up to a simple transformation of the underlying geometry. The first such transformation is reflection through a vertical axis and the result of applying this transformation to a given rule is called the mirrored rule. These rules will exhibit the same behavior up to reflection through a vertical axis, and so are equivalent in a computational sense.For example, if the definition of rule 110 is reflected through a vertical line, the following rule (rule 124) is obtained:
| current pattern | 111 | 110 | 101 | 100 | 011 | 010 | 001 | 000 |
|---|---|---|---|---|---|---|---|---|
| new state for center cell | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 0 |
Rules which are the same as their mirrored rule are called amphichiral. Of the 256 elementary cellular automata, 64 are amphichiral.
The second such transformation is to exchange the roles of 0 and 1 in the definition. The result of applying this transformation to a given rule is called the complementary rule.
For example, if this transformation is applied to rule 110, the following rule (rule 137) obtained:
| current pattern | 111 | 110 | 101 | 100 | 011 | 010 | 001 | 000 |
|---|---|---|---|---|---|---|---|---|
| new state for center cell | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 |
There are 16 rules which are the same as their complementary rules.
Finally, the previous two transformations can be applied successively to a rule to obtain the mirrored complementary rule. For example, the mirrored complementary rule of rule 110 is rule 193. There are 16 rules which are the same as their mirrored complementary rules.
Of the 256 elementary cellular automata, there are 88 which are inequivalent under these transformations.
Single 1 histories
One method used to study these automata is to follow its history with an initial state of all 0s except for a single cell with a 1. When the rule number is even it makes sense to interpret state at each time, t, as an integer expressed in binary, producing a sequence a(t) of integers. In many cases these sequences have simple, closed form expressions or have a generating functionGenerating function
In mathematics, a generating function is a formal power series in one indeterminate, whose coefficients encode information about a sequence of numbers an that is indexed by the natural numbers. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general...
with a simple form. The following rules are notable:
Rule 28
The sequence generated is 1, 3, 5, 11, 21, 43, 85, 171, ... . This is the sequence of Jacobsthal numbers and has generating function.It has the closed form expression
Note that rule 156 generates the same sequence.
Rule 50
The sequence generated is 1, 5, 21, 85, 341, 1365, 5461, 21845, ... . This has generating function.It has the closed form expression
.Note that rules 58, 114, 122, 178, 186, 242 and 250 generate the same sequence.
Rule 54
The sequence generated is 1, 7, 17, 119, 273, 1911, 4369, 30583, ... . This has generating function.It has the closed form expression
.Rule 60
The sequence generated is 1, 3, 5, 15, 17, 51, 85, 255, ... . This can be obtained by taking successive rows of Pascal's trianglePascal's triangle
In mathematics, Pascal's triangle is a triangular array of the binomial coefficients in a triangle. It is named after the French mathematician, Blaise Pascal...
modulo 2 and interpreting them as integers in binary, which can be graphically represented by a Sierpinski triangle
Sierpinski triangle
The Sierpinski triangle , also called the Sierpinski gasket or the Sierpinski Sieve, is a fractal and attractive fixed set named after the Polish mathematician Wacław Sierpiński who described it in 1915. However, similar patterns appear already in the 13th-century Cosmati mosaics in the cathedral...
.
Rule 90
The sequence generated is 1, 5, 17, 85, 257, 1285, 4369, 21845, ... . This can be obtained by taking successive rows of Pascal's trianglePascal's triangle
In mathematics, Pascal's triangle is a triangular array of the binomial coefficients in a triangle. It is named after the French mathematician, Blaise Pascal...
modulo 2 and interpreting them as integers in base 4. Note that rules 18, 26, 82, 146, 154, 210 and 218 generate the same sequence.
Rule 94
The sequence generated is 1, 7, 27, 119, 427, 1879, 6827, 30039, ... . This can be expressed as.This has generating function
.Rule 102
The sequence generated is 1, 6, 20, 120, 272, 1632, 5440, 32640, ... . This is simply the sequence generated by rule 60 (which is its mirror rule) multiplied by successive powers of 2.Rule 150
The sequence generated is 1, 7, 21, 107, 273, 1911, 5189, 28123, ... . This can be obtained by taking the coefficients of the successive powers of (1+x+x2) modulo 2 and interpreting them as integers in binary.Rule 158
The sequence generated is 1, 7, 29, 115, 477, 1843, 7645, 29491, ... . This has generating function.Rule 188
The sequence generated is 1, 3, 5, 15, 29, 55, 93, 247, ... . This has generating function.Rule 190
The sequence generated is 1, 7, 29, 119, 477, 1911, 7645, 30583, ... . This has generating function.Rule 220
The sequence generated is 1, 3, 7, 15, 31, 63, 127, 255, ... . This is the sequence of Mersenne numbers and has generating function.It has the closed form expression
.Note that rule 252 generates the same sequence.
Rule 222
The sequence generated is 1, 7, 31, 127, 511, 2047, 8191, 32767, ... . This is every other entry in the sequence of Mersenne numbers and has generating function.It has the closed form expression
.Note that rule 254 generates the same sequence.
Random initial state
A second way to investigate the behavior of these automata is to examine its history starting with a random state. This behavior can be better understood in terms of Wolfram Classes. Wolfram gives the following examples as typical rules of each class.- Class 1: Cellular automata which rapidly converge to a uniform state. Examples are rules 0, 32, 160 and 250.
- Class 2: Cellular automata which rapidly converge to a repetitive or stable state. Examples are rules 4, 108, 218 and 232.
- Class 3: Cellular automata which appear to remain in a random state. Examples are rules 22, 30, 126, 150, 182.
- Class 4: Cellular automata which form areas of repetitive or stable states, but also form structures that interact with each other in complicated ways. An example is rule 110. Rule 110 has been shown to be capable of universal computation.
Unusual cases
In some cases the behavior of a cellular automaton is not immediately obvious. For example, for Rule 62, interacting structures develop as in a Class 4. But in these interactions at least one of the structures is annihilated so the automaton eventually enters a repetitive state and the cellular automaton is Class 2.Rule 73 is Class 2 because any time there are two consecutive 1s surrounded by 0s, this feature is preserved in succeeding generations. This effectively creates walls which block the flow of information between different parts of the array. There are a finite number of possible configurations in the section between two walls so the automaton must eventually start repeating inside each section, though the period may be very long if the section is wide enough. These walls will form with probability 1 for completely random initial conditions. However, if the condition is added that the lengths of runs of consecutive 0s or 1s must always be odd, then the automaton displays Class 3 behavior since the walls can never form.
Rule 54 is Class 4, but it remains unknown whether it is capable of universal computation. Interacting structures form, but structures that are useful for computation have yet to be found.