Elementary cellular automaton

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. There is an elementary cellular automaton (rule 110, defined below) which is capable of universal computation, and as such it is one of the simplest possible models of computation.

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 = 2^2³ possible elementary cellular automata. Stephen Wolfram 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 in 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_d=01101110₂. So rule 110 is defined by the transition rule:

111	110	101	100	011	010	001	000	current pattern	P=(L,C,R)
0	1	1	0	1	1	1	0	new state for center cell	N_{110_d}=(C+R+CR+LC*R)%2

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:

111	110	101	100	011	010	001	000	current pattern	P=(L,C,R)
0	1	1	1	1	1	0	0	new state for center cell	N_{112_d+12_d=124_d}=(L+C+LC+LC*R)%2

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, we get the following rule

current pattern	000	001	010	011	100	101	110	111
new state for center cell	1	0	0	1	0	0	0	1

and, after reordering, we discover that this is rule 137:

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 (so that an input of 000 does not compute to a 1) 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 function with a simple form. The following rules are notable:

Rule 28

The sequence generated is 1, 3, 5, 11, 21, 43, 85, 171, ... (sequence A001045 in the OEIS). This is the sequence of Jacobsthal numbers and has generating function

{\frac {1+2x}{(1+x)(1-2x)}}

.

It has the closed form expression

a(t)={\frac {4\cdot 2^{t}-(-1)^{t}}{3}}

Rule 156 generates the same sequence.

Rule 50

The sequence generated is 1, 5, 21, 85, 341, 1365, 5461, 21845, ... (sequence A002450 in the OEIS). This has generating function

{\frac {1}{(1-x)(1-4x)}}

.

It has the closed form expression

a(t)={\frac {4\cdot 4^{t}-1}{3}}

.

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, ... (sequence A118108 in the OEIS). This has generating function

{\frac {1+7x}{(1-x^{2})(1-16x^{2})}}

.

It has the closed form expression

a(t)={\frac {22\cdot 4^{t}-6(-4)^{t}-4+3(-1)^{t}}{15}}

.

Rule 60

The sequence generated is 1, 3, 5, 15, 17, 51, 85, 255, ...(sequence A001317 in the OEIS). This can be obtained by taking successive rows of Pascal's triangle modulo 2 and interpreting them as integers in binary, which can be graphically represented by a Sierpinski triangle.

Rule 90

The sequence generated is 1, 5, 17, 85, 257, 1285, 4369, 21845, ... (sequence A038183 in the OEIS). This can be obtained by taking successive rows of Pascal's triangle 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, ... (sequence A118101 in the OEIS). This can be expressed as

a(t)={\begin{cases}1,&{\mbox{if }}t=0\\[5px]7,&{\mbox{if }}t=1\\[7px]{\dfrac {1+5\cdot 4^{n}}{3}},&{\mbox{if }}t{\mbox{ is even otherwise}}\\[7px]{\dfrac {10+11\cdot 4^{n}}{6}},&{\mbox{if }}t{\mbox{ is odd otherwise}}\end{cases}}

.

This has generating function

{\frac {(1+2x)(1+5x-16x^{4})}{(1-x^{2})(1-16x^{2})}}

.

Rule 102

The sequence generated is 1, 6, 20, 120, 272, 1632, 5440, 32640, ... (sequence A117998 in the OEIS). This is simply the sequence generated by rule 60 (which is its mirror rule) multiplied by successive powers of 2.

Rule 110

The sequence generated is 1, 6, 28, 104, 496, 1568, 7360, 27520, 130304, 396800, ... (sequence A117999 in the OEIS). Rule 110 has the perhaps surprising property that it is Turing complete, and thus capable of universal computation.[1]

Rule 150

The sequence generated is 1, 7, 21, 107, 273, 1911, 5189, 28123, ... (sequence A038184 in the OEIS). 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, ... (sequence A118171 in the OEIS). This has generating function

{\frac {1+7x+12x^{2}-4x^{3}}{(1-x^{2})(1-16x^{2})}}

.

Rule 188

The sequence generated is 1, 3, 5, 15, 29, 55, 93, 247, ... (sequence A118173 in the OEIS). This has generating function

{\frac {1+3x+4x^{2}+12x^{3}+8x^{4}-8x^{5}}{(1-x^{2})(1-16x^{4})}}

.

Rule 190

The sequence generated is 1, 7, 29, 119, 477, 1911, 7645, 30583, ... (sequence A037576 in the OEIS). This has generating function

{\frac {1+3x}{(1-x^{2})(1-4x)}}

.

Rule 220

The sequence generated is 1, 3, 7, 15, 31, 63, 127, 255, ... (sequence A000225 in the OEIS). This is the sequence of Mersenne numbers and has generating function

{\frac {1}{(1-x)(1-2x)}}

.

It has the closed form expression

a(t)=2\cdot 2^{t}-1

.

Note that rule 252 generates the same sequence.

Rule 222

The sequence generated is 1, 7, 31, 127, 511, 2047, 8191, 32767, ... (sequence A083420 in the OEIS). This is every other entry in the sequence of Mersenne numbers and has generating function

{\frac {1+2x}{(1-x)(1-4x)}}

.

It has the closed form expression

a(t)=2\cdot 4^{t}-1

.

Note that rule 254 generates the same sequence.

Images for rules 0-99

These images depict space-time diagrams, in which each row of pixels shows the cells of the automaton at a single point in time, with time increasing downwards. They start with an initial automaton state in which a single cell, the pixel in the center of the top row of pixels, is in state 1 and all other cells are 0.

Rule 0
Rule 1
Rule 2
Rule 3
Rule 4
Rule 5
Rule 6
Rule 7
Rule 8
Rule 9
Rule 10
Rule 11
Rule 12
Rule 13
Rule 14
Rule 15
Rule 16
Rule 17
Rule 18
Rule 19
Rule 20
Rule 21
Rule 22
Rule 23
Rule 24
Rule 25
Rule 26
Rule 27
Rule 28
Rule 29
Rule 30
Rule 31
Rule 32
Rule 33
Rule 34
Rule 35
Rule 36
Rule 37
Rule 38
Rule 39
Rule 40
Rule 41
Rule 42
Rule 43
Rule 44
Rule 45
Rule 46
Rule 47
Rule 48
Rule 49
Rule 50
Rule 51
Rule 52
Rule 53
Rule 54
Rule 55
Rule 56
Rule 57
Rule 58
Rule 59
Rule 60
Rule 61
Rule 62
Rule 63
Rule 64
Rule 65
Rule 66
Rule 67
Rule 68
Rule 69
Rule 70
Rule 71
Rule 72
Rule 73
Rule 74
Rule 75
Rule 76
Rule 77
Rule 78
Rule 79
Rule 80
Rule 81
Rule 82
Rule 83
Rule 84
Rule 85
Rule 86
Rule 87
Rule 88
Rule 89
Rule 90
Rule 91
Rule 92
Rule 93
Rule 94
Rule 95
Rule 96
Rule 97
Rule 98
Rule 99

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.[2]

Class 1: Cellular automata which rapidly converge to a uniform state. Examples are rules 0, 32, 160 and 232.
Class 2: Cellular automata which rapidly converge to a repetitive or stable state. Examples are rules 4, 108, 218 and 250.
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.[3]

Each computed result is placed under that results' source creating a two-dimensional representation of the system's evolution. The 88 inequivalent rules are as follows, evolved from random initial conditions:

Rule 0
Rule 1
Rule 2
Rule 3
Rule 4
Rule 5
Rule 6
Rule 7
Rule 8
Rule 9
Rule 10
Rule 11
Rule 12
Rule 13
Rule 14
Rule 15
Rule 18
Rule 19
Rule 22
Rule 23
Rule 24
Rule 25
Rule 26
Rule 27
Rule 28
Rule 29
Rule 30
Rule 32
Rule 33
Rule 34
Rule 35
Rule 36
Rule 37
Rule 38
Rule 40
Rule 41
Rule 42
Rule 43
Rule 44
Rule 45
Rule 46
Rule 50
Rule 51
Rule 54
Rule 56
Rule 57
Rule 58
Rule 60
Rule 62
Rule 72
Rule 73
Rule 74
Rule 76
Rule 77
Rule 78
Rule 90
Rule 94
Rule 104
Rule 105
Rule 106
Rule 108
Rule 110
Rule 122
Rule 126
Rule 128
Rule 130
Rule 132
Rule 134
Rule 136
Rule 138
Rule 140
Rule 142
Rule 146
Rule 150
Rule 152
Rule 154
Rule 156
Rule 160
Rule 162
Rule 164
Rule 168
Rule 170
Rule 172
Rule 178
Rule 184
Rule 200
Rule 204
Rule 232

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.[4]

Rule 73 is Class 2[5] 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[6] and also appears to be capable of universal computation, but has not been studied as thoroughly as Rule 110. Many interacting structures have been cataloged which collectively are expected to be sufficient for universality.[7]

References

Weisstein, Eric W. "Elementary Cellular Automaton". MathWorld.
Weisstein, Eric W. "Rule 30". MathWorld.
Weisstein, Eric W. "Rule 50". MathWorld.
Weisstein, Eric W. "Rule 54". MathWorld.
Weisstein, Eric W. "Rule 60". MathWorld.
Weisstein, Eric W. "Rule 62". MathWorld.
Weisstein, Eric W. "Rule 90". MathWorld.
Weisstein, Eric W. "Rule 94". MathWorld.
Weisstein, Eric W. "Rule 102". MathWorld.
Weisstein, Eric W. "Rule 110". MathWorld.
Weisstein, Eric W. "Rule 126". MathWorld.
Weisstein, Eric W. "Rule 150". MathWorld.
Weisstein, Eric W. "Rule 158". MathWorld.
Weisstein, Eric W. "Rule 182". MathWorld.
Weisstein, Eric W. "Rule 188". MathWorld.
Weisstein, Eric W. "Rule 190". MathWorld.
Weisstein, Eric W. "Rule 220". MathWorld.
Weisstein, Eric W. "Rule 222". MathWorld.

Cook, Matthew (2009-06-25). "A Concrete View of Rule 110 Computation". Electronic Proceedings in Theoretical Computer Science. 1: 31–55. doi:10.4204/EPTCS.1.4. ISSN 2075-2180.
Stephen Wolfram, A New Kind of Science p223 ff.
Rule 110 - Wolfram|Alpha
Rule 62 - Wolfram|Alpha
Rule 73 - Wolfram|Alpha
Rule 54 - Wolfram|Alpha
Martínez, Genaro Juárez; Adamatzky, Andrew; McIntosh, Harold V. (2006-04-01). "Phenomenology of glider collisions in cellular automaton Rule 54 and associated logical gates" (PDF). Chaos, Solitons & Fractals. 28 (1): 100–111. doi:10.1016/j.chaos.2005.05.013. ISSN 0960-0779.

External links

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[1] Cook, Matthew (2009-06-25). "A Concrete View of Rule 110 Computation". Electronic Proceedings in Theoretical Computer Science. 1: 31–55. doi:10.4204/EPTCS.1.4. ISSN 2075-2180.

[2] Stephen Wolfram, A New Kind of Science p223 ff.

[3] Rule 110 - Wolfram|Alpha

[4] Rule 62 - Wolfram|Alpha

[5] Rule 73 - Wolfram|Alpha

[6] Rule 54 - Wolfram|Alpha

[7] Martínez, Genaro Juárez; Adamatzky, Andrew; McIntosh, Harold V. (2006-04-01). "Phenomenology of glider collisions in cellular automaton Rule 54 and associated logical gates" (PDF). Chaos, Solitons & Fractals. 28 (1): 100–111. doi:10.1016/j.chaos.2005.05.013. ISSN 0960-0779.