3. Patterns

2.2. ECA: Global update rules on the torus

188 From Natural to Artificial Intelligence - Algorithms and Applications

"� ⋯ �," the synchronous case, and have.

\ f g <; � <sup>n</sup>

ASi Meaning

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

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

� Cell ci fires simultaneously with cell ci�<sup>1</sup>

such that all cells fire exactly once during the execution of that sequence; see [3].

subset I ⊆ f g n � 1; n � 2;…; 1; 0 of indices may define the active cells.

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

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>

< Cell ci fires after cell ci�<sup>1</sup> > Cell ci fires before cell ci�<sup>1</sup>

are exactly 2<sup>n</sup> � 2 bijective fair rules, those from f g <; > <sup>n</sup>

<sup>∪</sup>f g �; > <sup>n</sup> ð Þ ð Þ <sup>∪</sup> �<sup>n</sup> f g with <sup>∣</sup>ASn<sup>∣</sup> <sup>¼</sup> <sup>3</sup><sup>n</sup> � <sup>2</sup><sup>n</sup>þ<sup>1</sup> <sup>þ</sup> 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

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

A fair update step (bijective or not) can be decomposed into a sequence of elementary steps

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

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),

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>

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.

<sup>i</sup> <sup>¼</sup> ECA ciþ<sup>1</sup>; ci ð Þ ; ci�<sup>1</sup> , i <sup>∈</sup> I, ci, i =∈ I:

\ <sup>&</sup>lt;n, <sup>&</sup>gt;<sup>n</sup> f g; see also ([10], 4.1]).

, with the meaning si ¼ 1, if cell ci

. The

2.2.1. Fair update schemes

ASn <sup>¼</sup> f g <; �; > <sup>n</sup>

2.2.2. Unfair update schemes

These cells fire simultaneously, hence c<sup>þ</sup>

fires, and si ¼ 0 if cell ci is inactive in this step.

We consider pattern conversions F<sup>n</sup> <sup>2</sup> ∋ v ↦ w ∈ F<sup>n</sup> <sup>2</sup> , where F<sup>n</sup> <sup>2</sup> can be identified with the set <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 ECA<sup>τ</sup> ASð Þ¼ <sup>v</sup> ECAAS ECA<sup>τ</sup>�<sup>1</sup> AS ð Þ<sup>v</sup> � � recursively for <sup>τ</sup> <sup>∈</sup> <sup>N</sup>, starting with ECA<sup>1</sup> AS ð Þ¼ v ECAAS ð Þv .

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.

> ð Þ<sup>o</sup> <sup>∃</sup><sup>v</sup> <sup>∈</sup> <sup>F</sup><sup>n</sup> <sup>2</sup> , ∀w ∈ 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> ASð Þ¼ v w:

Some v is mapped to every w by varying the rule AS and the required number of time steps. There are 44 ECAs doing this.

> ð Þ<sup>i</sup> <sup>∀</sup><sup>v</sup> <sup>∈</sup> <sup>F</sup><sup>n</sup> <sup>2</sup> , ∀w ∈ 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> ASð Þ¼ v w:

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)) doing this (checked for n ≤ 15).

$$(\text{ii)}\quad\forall\upsilon\in\mathbb{F}\_{2'}^{\upsilon}\quad\exists\tau\in\mathbb{F}\_{\prime}\quad\forall w\in\mathbb{N}\_{2'}^{\upsilon}\quad\exists AS\quad\in AS\_{\textnormal{n}}:\ \text{ }\mathit{ECA}\_{\text{AS}}^{\tau}(\upsilon)=w.$$

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


iii. The ðunfairÞ elementary update rules with exactly one active cell for ECA-57 generate the

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191

ð Þ ii; iii First observe that all these rules consist of an even number of transpositions. Hence, we will at most obtain the alternating group A2<sup>n</sup> . This group comprises those bijective functions on

We checked ð Þii with GAP for certain sets of 5 fair rules for each of these sizes n, and GAP's

For ð Þ iii we have the canonical set of n elementary unfair update rules, with exactly one cell active in each rule. This set generates all fair and unfair update rules and hence is sufficient to

Again, GAP shows that indeed the alternating group is generated, for n ¼ 4, …, 28. We used 144 GB of RAM, which was sufficient for <sup>n</sup> <sup>¼</sup> 28 but not so for <sup>n</sup> <sup>¼</sup> 29. □ We believe that, apart from the special case n ¼ 3, we always obtain the alternating group.

For every torus size <sup>n</sup> <sup>∈</sup> <sup>N</sup>, n <sup>≥</sup> 4, both the 2<sup>n</sup> � 2 fair update rules, as well as the <sup>n</sup> elementary unfair update rules with a single active cell, are a generating set for the full alternating group

generated by the elementary steps s ¼ 0001; 0010; 0100, and 1000, respectively.

gap> P00 := (16,1)(2,3)(4,5)(6,7)(10,11)(14,15);

gap> P01 := (16,2)(4,6)(5,7)(8,10)(12,14)(13,15);

gap> P02 := (16,4)(1,5)(8,12)(9,13)(10,14)(11,15);

gap> P03 := (16,8)(1,9)(2,10)(3,11)(5,13)(7,15);

, since GAP only uses numbers from N. P00 to P03 are the permutations

ð Þi By exhaustive generation of all 8! ¼ 40320 permutations on 000, 001 f ;…; 111g.

full alternating group A2<sup>n</sup> for n ¼ 4,…, 28.

<sup>0</sup>; <sup>1</sup>; …; <sup>2</sup><sup>n</sup> f g � <sup>1</sup> which have positive sign as a permutation.

decide on the group generated by all rules.

function IsNaturalAlternatingGroup(G) returned true.

Proof.

Conjecture.

on 2<sup>n</sup> elements, using ECA-57.

mjv@Panda �/GAP \$gap -b

We replaced 0 by 2<sup>n</sup>

true

gap> Size(G04); 10461394944000

4.1.2. Example for GAP usage with n ¼ 4

(1,16)(2,3)(4,5)(6,7)(10,11)(14,15)

(2,16)(4,6)(5,7)(8,10)(12,14)(13,15)

(1,5)(4,16)(8,12)(9,13)(10,14)(11,15)

(1,9)(2,10)(3,11)(5,13)(7,15)(8,16) gap> G04 := Group(P00,P01,P02,P03);; gap> IsNaturalAlternatingGroup(G04);

Table 1. Minimum time τ required to satisfy ð Þ iii :

ECA-57 realizes this for n ¼ 5, …, 15 (no further n has been considered).

ECA-105 realizes this for odd n ¼ 7; 9; 11; 13; 15 (no further n has been considered).

$$(\text{iii}) \quad \exists \tau \in \mathbb{N}, \quad \forall v \in \mathbb{F}\_{2'}^{\prime} \quad \forall w \in \mathbb{F}\_{2'}^{\prime} \quad \exists AS \quad \in AS\_{\mathfrak{n}} : \, \, ECA\_{\text{AS}}^{\dagger}(v) = w.$$

All v are mapped to all w at the same time, varying the rule AS. This is actually possible for the two survivors of property ð Þii ; see Table 1. The required time roughly decreases with growing n, since we have 3<sup>n</sup> � <sup>2</sup><sup>n</sup>þ<sup>1</sup> <sup>þ</sup> 1 rules to choose from for 2<sup>n</sup> patterns <sup>w</sup>. Thus for higher <sup>n</sup> the probability to meet the conversion early on increases.

$$(iv) \quad \exists \tau\_0 \in \mathbb{N} \quad \forall \tau \ge \tau\_0 \quad \forall v, w \in \mathbb{F}\_{2^\circ}^v \quad \exists AS \quad \in AS\_n : \quad ECA\_{AS}^\tau(v) = w.$$

Eventually, all conversions may happen at all times for some update scheme. This property cannot be satisfied, for no ECA; see ([14], Thm. 2).

For more details and results for QCAs consult Section 2 of [14] and Section 3 of [10].
