Proof.

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

ð Þ iii <sup>∃</sup><sup>τ</sup> <sup>∈</sup> <sup>N</sup>, <sup>∀</sup><sup>v</sup> <sup>∈</sup> <sup>F</sup><sup>n</sup>

190 From Natural to Artificial Intelligence - Algorithms and Applications

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

probability to meet the conversion early on increases.

cannot be satisfied, for no ECA; see ([14], Thm. 2).

4.1. GAP: Graphs, algorithms, programming

symmetric or alternating group S2<sup>n</sup> or A2<sup>n</sup> , respectively.

4.1.1. GAP and the alternating group A2<sup>n</sup>

4. Bijective functions

Our results so far:

Theorem 1.

ð Þ iv <sup>∃</sup>τ<sup>0</sup> <sup>∈</sup> <sup>N</sup>, <sup>∀</sup><sup>τ</sup> <sup>≥</sup> <sup>τ</sup>0, <sup>∀</sup>v, w <sup>∈</sup> <sup>F</sup><sup>n</sup>

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

<sup>2</sup> , ∀w ∈ F<sup>n</sup>

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

ECA n ¼ 5 6 7 8 9 10 11 12 13 14 15 57 445 — 70 242 35 13 13 13 13 13 10 105 — — 570 — 14 — 6 — 6 — 8

Eventually, all conversions may happen at all times for some update scheme. This property

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

We first introduce the computer algebra system GAP and then give several examples.

i. The fair update rules for ECA-57 generate the full symmetric group S<sup>8</sup> for n ¼ 3.

ii. The fair update rules for ECA-57 generate the full alternating group A2<sup>n</sup> for n ¼ 4, …, 11.

GAP [19] is a system for computational discrete algebra, in particular computational group theory. We use GAP to decide, whether certain fair or unfair update rules generate the full

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

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

ASð Þ¼ v w:

ASð Þ¼ v w:

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

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

We checked ð Þii with GAP for certain sets of 5 fair rules for each of these sizes n, and GAP's function IsNaturalAlternatingGroup(G) returned true.

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 decide on the group generated by all rules.

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.
