2.1. Cellular automata: Neighborhoods and local update rules

We consider cellular automata (CA) on a torus or ring of n cells, that is index set Z=nZ, over the binary alphabet 0f g ; 1 . Cell index wraparound, that is, ci ¼ cj for i � jmodn, and the canonical cell names are cn�<sup>1</sup>, cn�<sup>2</sup>, …, c1, c0. We deal with elementary CAs (ECA) with three input cells, where the middle one is also the output cell.

The neighborhood (ci+1, ci, ci�1) can have eight different values from <sup>∈</sup> <sup>F</sup><sup>3</sup> <sup>2</sup>. Let k ≔ 4� ciþ<sup>1</sup> þ 2 � ci þ ci�<sup>1</sup> ∈ f g 0;…; 7 . Then ci is replaced by c<sup>þ</sup> <sup>i</sup> ≔ pk ∈ f g 0; 1 , where p0,…, p<sup>7</sup> are defined via Wolfram's rule [1] P<sup>7</sup> <sup>k</sup>¼<sup>0</sup> <sup>2</sup><sup>k</sup> pk ∈ f g 0;…; 255 .

The 2<sup>23</sup> ¼ 256 ECAs can be arranged into 88 equivalence classes under the symbolic symmetry 0/1 and the chiral symmetry left/right (ciþ<sup>1</sup> \$ ci�1); see ([14], Appendix A). It is sufficient to consider one member per class.

We considered quad CAs (QCAs) with four inputs and nonstandard neighborhoods in ([10], Section 1.2).

Local bijectivity requires that ECA ð Þ¼ a; 0; c 1� ECA ð Þ a; 1; c . This is equivalent to requiring that the hexadecimal digits of the rule be from 3, 6, 9, C.

Example: The behavior of the ECA with Wolfram rule 57 ¼ 00111001<sup>2</sup> ¼ 3916:

111 ↦ 0, 110 ↦ 0, 101 ↦ 1, 100 ↦ 1, 011 ↦ 1, 010 ↦ 0, 001 ↦ 0, 000 ↦ 1, in other words 0ci1 ↦ c<sup>þ</sup> <sup>i</sup> ¼ ci, for all other contexts we have 0ci0, 1ci0, 1ci1 ↦ c<sup>þ</sup> <sup>i</sup> ¼ ci.

#### 2.2. ECA: Global update rules on the torus

#### 2.2.1. Fair update schemes

We repeat the definition of asynchronicity rules from ([10], Section 2).

The set ASn of asynchronicity rules over Z=nZ consists in all words of length n over the alphabet f g <; �; > such that both "<" and ">" occur at least once. We also include the word "� ⋯ �," the synchronous case, and have.

sequence is used again in Section 5 and has been verified to coincide with OEIS A001610 up to

We also can define unfair bijective rules (full steps), where we first fix some subset of size

We next order adjacent active cells by the usual < , > signs. Hence, a run of r consecutive 1's (with wraparound) has 2<sup>r</sup>�<sup>1</sup> ways to fix the internal firing order. This order is independent of the other 1-runs, since the cells are separated by at least an inactive cell with ai ¼ 0. The

> #runs Y i¼1

where the pattern p avoids 0<sup>n</sup> and 1<sup>n</sup> and then the 1-runs in the pattern have lengths r1, r2, …,

For n ¼ 3, …, 7, we have 9 ¼ 3 þ 6, 30 ¼ 4 þ 10 þ 16, 90 ¼ 5 þ 15 þ 30 þ 40, 257 ¼ 6 þ 21þ 50 þ 84 þ 96, and 714 ¼ 7 þ 28 þ 77 þ 154 þ 224 þ 224, such unfair rules, respectively. The terms count how many patterns with k ¼ 1, 2, …, n � 1 active cells are feasible. These terms

<sup>2</sup> ∋ v ↦ w ∈ F<sup>n</sup>

<sup>0</sup>; <sup>1</sup>; …; <sup>2</sup><sup>n</sup> f g � <sup>1</sup> . From Definition ([10], Def. 3), by ECAASð Þ¼ <sup>v</sup> <sup>w</sup>, we mean that the elementary CA with rule ECA maps <sup>v</sup> <sup>∈</sup> f g <sup>0</sup>; <sup>1</sup> <sup>n</sup> to <sup>w</sup> <sup>∈</sup> f g <sup>0</sup>; <sup>1</sup> <sup>n</sup> via the asynchronicity scheme AS. We define

AS ð Þ<sup>v</sup> � � recursively for <sup>τ</sup> <sup>∈</sup> <sup>N</sup>, starting with ECA<sup>1</sup>

In [10, 14], we considered five universality properties ð Þo to ð Þ iv , where each v ↦ w makes use of a certain update rule AS applied several times. We only give a summary here. Property ð Þ iv is ruled out for any n ∈ N, while properties ð Þo to ð Þ iv have only been verified experimentally, for n ≤ 15.

Some v is mapped to every w by varying the rule AS and the required number of time steps.

All v are mapped to all w by varying the rule AS and the required number of time steps. There are 6 ECAs (rules 19, 23 (for n �= 0 mod 2), 37 (for n �= 0 mod 3), 41, 57, 105 (for n �= 0 mod 4))

All v are mapped to all w at the same time, which time may vary for v but not for w, for

<sup>2</sup> , where F<sup>n</sup>

<sup>2</sup> , <sup>∃</sup><sup>τ</sup> <sup>∈</sup> <sup>N</sup>, <sup>∃</sup>AS <sup>∈</sup> ASn : ECA<sup>τ</sup>

<sup>2</sup> , <sup>∃</sup><sup>τ</sup> <sup>∈</sup> <sup>N</sup>, <sup>∃</sup>AS <sup>∈</sup> ASn : ECA<sup>τ</sup>

<sup>2</sup> , <sup>∃</sup>AS <sup>∈</sup> ASn : ECA<sup>τ</sup>

2ri�<sup>1</sup>

2 Xn�1 p¼1

\ <sup>0</sup><sup>n</sup>; <sup>1</sup><sup>n</sup> f g, ai <sup>¼</sup> 1 meaning that cell ci is active.

http://dx.doi.org/10.5772/intechopen.80432

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Hard, firm, soft … Etherealware: Computing by Temporal Order of Clocking

<sup>2</sup> can be identified with the set

ASð Þ¼ v w:

ASð Þ¼ v w:

ASð Þ¼ v w:

AS ð Þ¼ v ECAAS ð Þv .

<sup>1</sup> <sup>≤</sup> <sup>k</sup> <sup>≤</sup> <sup>n</sup> � 1 of active cells by a word <sup>a</sup> from 0f g ; <sup>1</sup> <sup>n</sup>

number of bijective unfair rules on a torus of size n is

can be found in OEIS [18] as subsequences of A209697.

n ¼ 24.

considering wraparound.

We consider pattern conversions F<sup>n</sup>

ASð Þ¼ <sup>v</sup> ECAAS ECA<sup>τ</sup>�<sup>1</sup>

ð Þ<sup>o</sup> <sup>∃</sup><sup>v</sup> <sup>∈</sup> <sup>F</sup><sup>n</sup>

ð Þ<sup>i</sup> <sup>∀</sup><sup>v</sup> <sup>∈</sup> <sup>F</sup><sup>n</sup>

There are 44 ECAs doing this.

doing this (checked for n ≤ 15).

different rules AS .

ð Þii <sup>∀</sup><sup>v</sup> <sup>∈</sup> <sup>F</sup><sup>n</sup>

<sup>2</sup> , ∀w ∈ F<sup>n</sup>

<sup>2</sup> , ∀w ∈ F<sup>n</sup>

<sup>2</sup>, ∃τ ∈ F, ∀w ∈ N<sup>n</sup>

3. Patterns

ECA<sup>τ</sup>

$$AS\_n = (\{<, \equiv, >\} \, ^\nwarrow (\{<, \equiv\} ^\nwarrow (\equiv, >\} ^\nwarrow)) \cup \{\equiv ^\nu\} \text{ with } |AS\_n| = 3^n - 2^{n+1} + 2.$$

A rule AS ¼ AS<sup>n</sup>�<sup>1</sup>⋯ AS0 ∈ ASn defines the firing order as follows:


To ensure bijectivity, we must first have a locally bijective CA, and, furthermore, no two adjacent cells may fire simultaneously. Why this is so will be dealt with in Chapter 5. There are exactly 2<sup>n</sup> � 2 bijective fair rules, those from f g <; > <sup>n</sup> \ <sup>&</sup>lt;n, <sup>&</sup>gt;<sup>n</sup> f g; see also ([10], 4.1]).

A fair update step (bijective or not) can be decomposed into a sequence of elementary steps such that all cells fire exactly once during the execution of that sequence; see [3].

#### 2.2.2. Unfair update schemes

We now include unfair updates, where some cells may fire less often than others (even not at all).

We start with elementary steps (μ steps in [10]). During one elementary step, any nonempty subset I ⊆ f g n � 1; n � 2;…; 1; 0 of indices may define the active cells.

$$\text{These cells five simultaneously, hence } \mathbf{c}\_i^+ = \begin{cases} \text{ECA}(\mathbf{c}\_{i+1}, \mathbf{c}\_i, \mathbf{c}\_{i-1}), & \mathbf{i} \in I, \\ \mathbf{c}\_i, & \mathbf{i} \notin I. \end{cases}$$

We define elementary steps as words <sup>s</sup> <sup>¼</sup> sn�<sup>1</sup>⋯s1s<sup>0</sup> <sup>∈</sup> f g <sup>0</sup>; <sup>1</sup> <sup>n</sup> , with the meaning si ¼ 1, if cell ci fires, and si ¼ 0 if cell ci is inactive in this step.

A sequence of such elementary steps is upper indexed by the time step ð Þt .

A fair update rule consists in a number of elementary steps such that every cell fires exactly once. The fair rule as <sup>¼</sup> "<><><>" for <sup>n</sup> <sup>¼</sup> 6 can be decomposed into (sð Þ<sup>1</sup> <sup>¼</sup> <sup>010101</sup>, sð Þ<sup>2</sup> <sup>¼</sup> 101010), while the sequence (sð Þ<sup>1</sup> <sup>¼</sup> <sup>010001</sup>, sð Þ<sup>2</sup> <sup>¼</sup> 101010) is unfair, since cell <sup>c</sup><sup>2</sup> does not fire at all.

The number of bijective elementary steps is the number of words of length n over the alphabet f g 0; 1 , such that no adjacent 1's occur to ensure bijectivity. For n ¼ 3; 4; 5; 6 there are 3, 6, 10, 17 such steps, respectively. The sequence obeys the law nk ¼ nk�<sup>1</sup> þ nk�<sup>2</sup> þ 1. At least for n ¼ 3, …, 7, the sets are as follows: prepend a 0 to each pattern of length k � 1, prepend a 10 (a 01) to each pattern of size <sup>k</sup> � 1 terminating in 0 (in 1), and add the new pattern 10<sup>k</sup>�<sup>1</sup> . The sequence is used again in Section 5 and has been verified to coincide with OEIS A001610 up to n ¼ 24.

We also can define unfair bijective rules (full steps), where we first fix some subset of size <sup>1</sup> <sup>≤</sup> <sup>k</sup> <sup>≤</sup> <sup>n</sup> � 1 of active cells by a word <sup>a</sup> from 0f g ; <sup>1</sup> <sup>n</sup> \ <sup>0</sup><sup>n</sup>; <sup>1</sup><sup>n</sup> f g, ai <sup>¼</sup> 1 meaning that cell ci is active.

We next order adjacent active cells by the usual < , > signs. Hence, a run of r consecutive 1's (with wraparound) has 2<sup>r</sup>�<sup>1</sup> ways to fix the internal firing order. This order is independent of the other 1-runs, since the cells are separated by at least an inactive cell with ai ¼ 0. The number of bijective unfair rules on a torus of size n is

$$\sum\_{p=1}^{2^r-1} \prod\_{i=1}^{tmss} 2^{r\_i-1}$$

where the pattern p avoids 0<sup>n</sup> and 1<sup>n</sup> and then the 1-runs in the pattern have lengths r1, r2, …, considering wraparound.

For n ¼ 3, …, 7, we have 9 ¼ 3 þ 6, 30 ¼ 4 þ 10 þ 16, 90 ¼ 5 þ 15 þ 30 þ 40, 257 ¼ 6 þ 21þ 50 þ 84 þ 96, and 714 ¼ 7 þ 28 þ 77 þ 154 þ 224 þ 224, such unfair rules, respectively. The terms count how many patterns with k ¼ 1, 2, …, n � 1 active cells are feasible. These terms can be found in OEIS [18] as subsequences of A209697.
