A Note on Elementary Cellular Automata Classification
AA Note on Elementary Cellular AutomataClassification
Genaro J. Mart´ınezJune 17, 2013 ∗ Departamento de Ciencias e Ingenier´ıa de la Computaci´on,Escuela Superior de C´omputo, Instituto Polit´ecnico Nacional, M´exico, D.F. International Centre of Unconventional Computing,University of the West of England, BS16 1QY Bristol, United Kingdom [email protected]
Abstract
We overview and compare classifications of elementary cellular au-tomata, including Wolfram’s, Wuensche’s, Li and Packard, communica-tion complexity, power spectral, topological, surface, compression, lat-tices, and morphological diversity classifications. This paper summarisesseveral classifications of elementary cellular automata (ECA) and com-pares them with a newly proposed one, that induced by endowing ruleswith memory.
Keywords: elementary cellular automata, classification, memory.
Contents ∗ Accepted by publish in
Journal of Cellular Automata . a r X i v : . [ n li n . C G ] J un ECA classifications versus memory classification 10 d -spectrum classification (2003) . . . . . 134.6 Communication complexity classification (2004) . . . . . . . . . . 144.7 Topological classification (2007) . . . . . . . . . . . . . . . . . . . 164.8 Power spectral classification (2008) . . . . . . . . . . . . . . . . . 174.9 Morphological diversity classification (2010) . . . . . . . . . . . . 184.10 Distributive and non-distributive lattices classification (2010) . . 194.11 Topological dynamics classification (2012) . . . . . . . . . . . . . 204.12 Expressivity analysis (2013) . . . . . . . . . . . . . . . . . . . . . 214.13 Normalised compression classification (2013) . . . . . . . . . . . . 224.14 Surface dynamics classification (2013) . . . . . . . . . . . . . . . 234.15 Spectral classification (2013) . . . . . . . . . . . . . . . . . . . . 234.16 Bijective and surjective classification (2013) . . . . . . . . . . . . 244.17 Creativity classification (2013) . . . . . . . . . . . . . . . . . . . 25 This paper describes several ECA classifications. Consequently, we can comparequickly some basic properties and use these relations for future classifications.In this respect, a remarkable and important result was published by Culik IIand Yu in 1988, who proved that such classifications are undecidable [9]. How-ever, novel and recombined techniques continue reporting more approaches [39].Also, we will compare each classification with a recent one, memory classifica-tion .Our motivation begins with a classification of ECA rules, but composed withmemory functions (for details about cellular automata with memory (CAM),please see [2, 3]).We know from previous studies that ECA composed with memory (ECAM)yield another automaton with uniform, periodic, chaotic, or complex behaviour.In particular, previous analysis has shown that chaotic ECA rules composedwith memory are capable of displaying complex behaviour (for ECA rule 30 see[17], for ECA rule 45 see [18], and for ECA rule 126 see [20]). Of course, thecomposition produces a new rule, but with elements of the original ECA rule.This way, memory functions help to ‘discover’ hidden information in dynamicalsystems from simple functions (or rules), and “transform” simple and chaoticrules to complex rules or vice versa [19]. Another approach to get complex ruleswas displayed by Gunji’s analysis, deriving complex rules from other CA rules,using intermediate layer lattices [13]. 2 lass I : leads to uniform behaviour,Figure 1: ECA rule 32.
Class II : leads to periodic behaviour,Figure 2: ECA rule 10.3 lass III: leads to chaotic behaviour,Figure 3: ECA rule 90.
Class IV : leads to complex behaviour.Figure 4: ECA rule 54 (filtered).An ECA is the most basic one-dimensional CA representation. ECA havetwo states in its alphabet Σ = { , } and two close neighbours with respect toone central cell. Thus, a central cell x i may take an element of the alphabet4 ule equivalent rules rule equivalent rules0 255 56 98, 185, 2271 127 57 992 16, 191, 247 58 114, 163, 1773 17, 63, 119 60 102, 153, 1954 223 62 118, 131, 1455 95 72 2376 20, 159, 215 73 1097 21, 31, 87 74 88, 173, 2298 64, 239, 253 76 2059 65, 111, 125 77 -10 80, 175, 245 78 92, 141, 19711 47, 81, 117 90 16512 68, 207, 221 94 13313 69, 79, 93 104 23314 84, 143, 213 105 -15 85 106 120, 169, 22518 183 108 20119 55 110 124, 137, 19322 151 122 16123 - 126 12924 66, 189, 231 128 25425 61, 67, 103 130 144, 190, 24626 82, 167, 181 132 22227 39, 53, 83 134 148, 158, 21428 70, 157, 199 136 192, 238, 25229 71 138 174, 208, 22430 86, 135, 149 140 196, 206, 22032 251 142 21233 123 146 18234 48, 187, 243 150 -35 49, 59, 115 152 188, 194, 23036 219 154 166, 180, 21037 91 156 19838 52, 155, 211 160 25040 96, 235, 249 162 176, 186, 24241 97, 107, 121 164 21842 112, 171, 241 168 224, 234, 24843 113 170 24044 100, 203, 217 172 202, 216, 22845 75, 89, 101 178 -46 116, 139, 209 184 22650 179 200 23651 - 204 -54 147 232 - Table 1: 88 equivalent ECA rules.5nd update simultaneously on a configuration, this update is done accordinglyto an evolution rule ϕ . Hence, we have that an evolution rule can be expressedas : ϕ ( x ti − r , . . . , x ti , . . . , x tx + r ) → x t +1 i , where r represents the number of left andright neighbours, and x ∈ Σ. This way, the evolution function in ECA domainis ϕ ( x ti − , x ti , x tx +1 ) → x t +1 i . If k = | Σ | then we can calculate the number ofneighbours and evolution rules. k r +1 determines the number of neighboursand k k r +1 the number of different evolution rules. Therefore, for ECA domain( k = 2 , r = 1) there are 256 different evolution rules [37].In 1983, Wolfram establishes a classification in CA [34], with four classes thatdescribe global behaviour from random initial conditions. Figures 1–4 illustrateeach class with a particular evolution rule. All evolutions evolve on a ring of637 cells to 318 generations, using the same random initial condition at 50% ofcells in state zero or one.Some researchers found that CA rules have equivalences. The earliest wasWalker and Ashby in 1966 [32]. Next was Martin, Odlyzko, and Wolfram in1984 [24]Li and Packard also found that ECA rules have equivalences [15]. Conse-quently, a similar behaviour can be projected and organised in set of rules, thusECA rule-space has 88 equivalent sets of rules, as shown in table 1. Recentlysuch a property was restudied and obtained across topological properties in [12]. This study considers two main analyses: That of Wolfram and Wuensche’sclassifications.
In his seminal book “Theory and Applications of Cellular Automata”, Wolframhad included an extended number of properties in one-dimensional CA, dis-played in “Tables of Cellular Automata Properties” section [36]. Here we havereviewed and updated such a classification with the
WolframAlpha engine. Wolfram’s classification is shown in Table 2. This classification establishesthat from random initial conditions a CA will reach in a long time, one of thefollowing classes of global behaviour.
Class I : evolve to uniform behaviour;
Class II : evolve to periodic behaviour;
Class III : evolve to chaotic behaviour;
Class IV : evolve to complex behaviour. WolframAlpha computational knowledge engine . class I class II
65 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 19, 23,24, 25, 26, 27, 28, 29, 33, 34, 35, 36, 37, 38, 42,43, 44, 46, 50, 51, 56, 57, 58, 62, 72, 73, 74, 76,77, 78, 94, 104, 108, 130, 132, 134, 138, 140, 142,152, 154, 156, 162, 164, 170, 172, 178, 184, 200,204, 232. class III
11 18, 22, 30, 45, 60, 90, 105, 122, 126, 146, 150. class IV chaotic rules relate to developing cryptography, randomnumber generators, and fields of attraction. However, the so called class IV or complex rules have captured most attention given their potential for computa-tional universality, and their applications in artificial life by the simulations ofparticles, waves, mobile self localizations, or gliders. Their capacity to containintrinsically complex systems [38, 35, 16, 1, 31, 14]. This kind of discrepancybetween chaotic rules, and complex rules capable of computational universality,are discussed in the CA literature (for details please see [25, 26]).
Wuensche and Lesser did a detailed analysis on basin of attraction fields and ruleclusters of equivalences in ECA, published in their book “The Global Dynamicsof Cellular Automata” written in 1992 [33].In this book, Wuensche recognises three main transformations that relateequivalent ECA rules expressed in rule clusters . Such transformations are: re-flection, negation, and complementation. Moreover, these equivalences can beexplored quickly and automatically for other CA with free software
DiscreteDynamics Lab [41].Rule clusters classification displays 88 equivalences (as is shown in Table 1)in ECA, and besides establishes basically three kind of symmetries: symmetric,semi-asymmetric, and full-asymmetric set of rules [33].Also, these equivalences become refined in that some transformations (re-fection, negation, or complement) produce identity or another transformation,causing a rule cluster of 4 equivalent rules and their 4 equivalent compliments,to collapse. Where reflection produces the same rule, the rule is called sym-metric (otherwise asymmetric) – but the reflection algorithm allows for semi- Complex Cellular Automata Repository http://uncomp.uwe.ac.uk/genaro/Complex_CA_repository.html . Discrete Dynamics Lab (DDLab) . symmetric
36 0, 1, 4, 5, 18, 19, 22, 23, 32, 33, 36, 37,50, 51, 54, 72, 73, 76, 77, 90, 94, 104,105, 108, 122, 126, 128, 132, 146, 150,160, 164, 178, 200, 204, 232. semi-asymmetric
32 2, 3, 6, 7, 8, 9, 12, 13, 26, 27, 30, 34, 35,38, 40, 41, 44, 45, 58, 62, 74, 78, 106,110, 130, 134, 136, 140, 154, 162, 168,172. full-asymmetric
20 10, 11, 14, 15, 24, 25, 28, 29, 42, 43, 46,57, 60, 138, 142, 152, 156, 170, 184.Table 3: Wuensche’s equivalences relation.asymmetric and fully-asymmetric rules.
In this section, we propose a classification based in memory functions. The fullanalysis of this classification can be found in [19]. Here, we will merely comparethis classification with the other ones.
Memory classification on ECA is investigated in “Designing Complex Dynamicswith Memory” [19]. A ECAM is a ECA composed with a memory function, thenew rule open a new and extended domain of rules based in the ECA domain[17].Basically, if you select an ECA rule and compose this rule with a memoryfunction (in our analysis we have considered three basic functions : majority,minority, and parity). Hence, we will achieve a new rule with a base of aECA rule. Of course, a number of features from the original ECA rule willbe more evident on its ECAM generalisation. Therefore, the memory functionwill determine if the original ECA rule preserves the same class (respect toWolfram’s classes) or if it changes to another class.Following this simple principle, we know now that ECA rules composed withmemory can be classified in three classes: strong , because the memory functions are unable to transform one classto another; moderate , because the memory function can transform the rule to anotherclass and conserve the same class as well;8 eak , because the memory functions do most transformations and therule changes to another different class quickly.This way, the next table displays the ECA classification based in memoryfunctions.classificationtype num. rules strong
39 2, 7, 9, 10, 11, 15, 18, 22, 24, 25, 26, 30, 34,35, 41, 42, 45, 46, 54, 56, 57, 58, 62, 94, 106,108, 110, 122, 126, 128, 130, 138, 146, 152,154, 162, 170, 178, 184. moderate
34 1, 3, 4, 5, 6, 8, 13, 14, 27, 28, 29, 32, 33, 37,38, 40, 43, 44, 72, 73, 74, 77, 78, 104, 132,134, 136, 140, 142, 156, 160, 164, 168, 172. weak
15 0, 12, 19, 23, 36, 50, 51, 60, 76, 90, 105, 150,200, 204, 232.Table 4: ECAM’s classification relation.
Memory classification presents a number of interesting properties.We have ECA rules which composed with a particular kind of memory areable of reach another class including the original dynamic. The main feature isthat, at least, this rule with memory is able to reach every different class. Ruleswith this property are called universal ECAM (5 rules). universal ECAM:
22, 54, 146, 130, 152.Particularly, all these UECAM are classified strong in ECAM’s classification. strong:
22, 54, 146, 130, 152. moderate: - weak: -On the other hand, we have ECA that when composed with memory are ableto yield a complex ECAM but with elements of the original ECA rule. Theyare called complex ECAM (44 rules). complex ECAM:
6, 9, 10, 11, 13, 15, 22, 24, 25, 26, 27, 30, 33, 35, 38, 40,41, 42, 44, 46, 54, 57, 58, 62, 72, 77, 78, 106, 108, 110, 122,126, 130, 132, 138, 142, 146, 152, 156, 162, 170, 172, 178,184. 9nd they can be particularised in terms of ECAM’s classification, as follows: strong:
9, 10, 11, 15, 22, 24, 25, 26, 30, 35, 41, 42, 46, 54, 57, 58,62, 106, 108, 110, 122, 126, 130, 138, 146, 152, 162, 170,178, 184. moderate:
6, 13, 27, 33, 38, 40, 44, 72, 77, 78, 132, 142, 156, 172. weak: -It is remarkable that none of the rules classified in weak class is able to reachcomplex behaviour. These set of rules are strongly robust to any perturbationin terms of ECAM’s classification.
In these sections, we will compare several ECA classifications reported in CAliterature all along the CA-history versus memory classification.
Wolfram’s classification in “Universality and complexity in cellular automata”,establishes four classes: { uniform (class I), periodic (class II), chaotic (class III), complex (class IV) } For details please see [37, 38]. class I:
0, 8, 32, 40, 128, 136, 160, 168. strong: moderate:
8, 32, 40, 136, 160, 168. weak: class II:
1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 19, 23, 24, 25,26, 27, 28, 29, 33, 34, 35, 36, 37, 38, 42, 43, 44, 46, 50, 51,56, 57, 58, 62, 72, 73, 74, 76, 77, 78, 94, 104, 108, 130, 132,134, 138, 140, 142, 152, 154, 156, 162, 164, 170, 172, 178,184, 200, 204, 232. strong:
2, 7, 9, 10, 11, 15, 24, 25, 26, 34, 35, 42, 46, 56, 57, 58, 62,94, 108, 130, 138, 152, 154, 162, 170, 178, 184. moderate:
1, 3, 4, 5, 6, 13, 14, 27, 28, 29, 33, 37, 38, 43, 44, 72, 73,74, 77, 78, 104, 132, 134, 140, 142, 156, 164, 172. weak:
12, 19, 23, 36, 50, 51, 76, 200, 204, 232.10 lass III:
18, 22, 30, 45, 60, 90, 105, 122, 126, 146, 150. strong:
18, 22, 30, 45, 122, 126, 146. moderate: - weak:
60, 90, 105, 150. class IV:
41, 54, 106, 110. strong:
41, 54, 106, 110. moderate: - weak: - Li and Packard’s classification in “The Structure of the Elementary CellularAutomata Rule Space”, establishes five ECA classes: { null, fixed point, periodic, locally chaotic, chaotic } .For details please see [15]. null:
0, 8, 32, 40, 128, 136, 160, 168. strong: moderate:
8, 32, 40, 136, 160, 168. weak: fixed point:
2, 4, 10, 12, 13, 24, 34, 36, 42, 44, 46, 56, 57, 58, 72, 76, 77,78, 104, 130, 132, 138, 140, 152, 162, 164, 170, 172, 184,200, 204, 232. strong:
2, 10, 24, 34, 42, 46, 56, 57, 58, 130, 138, 152, 162, 170,184. moderate:
4, 13, 44, 72, 77, 78, 104, 132, 140, 164, 172. weak:
12, 36, 76, 200, 204, 232. periodic:
1, 3, 5, 6, 7, 9, 11, 14, 15, 19, 23, 25, 27, 28, 29, 33, 35, 37,38, 41, 43, 50, 51, 74, 94, 108, 131(62), 134, 142, 156, 178. strong:
7, 9, 11, 15, 25, 35, 41, 62, 94, 108, 178. moderate:
1, 3, 5, 6, 14, 27, 28, 29, 33, 37, 38, 43, 74, 134, 142, 156. weak:
19, 23, 50, 51. locally chaotic:
26, 73, 154. strong:
26, 154. moderate: weak: - 11 haotic:
18, 22, 30, 45, 54, 60, 90, 105, 106, 132, 129(126), 137(110),146, 150, 161(122). strong:
18, 22, 30, 45, 54, 106, 122, 126, 110, 122, 146. moderate: - weak:
60, 90, 105, 150.
Wuensche’s equivalences in “The Global Dynamics of Cellular Automata”, es-tablishes three ECA kinds of symmetries: { symmetric, semi-asymmetric, full-asymmetric } .For details please see [33]. symmetric:
0, 1, 4, 5, 18, 19, 22, 23, 32, 33, 36, 37, 50, 51, 54, 72, 73,76, 77, 90, 94, 104, 105, 108, 122, 126, 128, 132, 146, 150,160, 164, 178, 200, 204, 232. strong:
18, 22, 54, 108, 122, 126, 128, 146, 178. moderate:
1, 4, 5, 32, 33, 72, 73, 77, 104, 132, 160, 164. weak:
0, 19, 23, 36, 50, 51, 76, 90, 105, 150, 200, 204. semi-asymmetric:
2, 3, 6, 7, 8, 9, 12, 13, 26, 27, 30, 34, 35, 38, 40, 41,44, 45, 58, 62, 74, 78, 106, 110, 130, 134, 136, 140,154, 162, 168, 172. strong:
2, 7, 9, 26, 30, 34, 35, 41, 45, 58, 62, 106, 110, 130,154, 162. moderate:
3, 6, 8, 13, 27, 38, 40, 44, 74, 78, 134, 136, 140, 168,172. weak: full-asymmetric:
10, 11, 14, 15, 24, 25, 28, 29, 42, 43, 46, 57, 60, 138,142, 152, 156, 170, 184. strong:
10, 11, 15, 24, 25, 42, 46, 57, 138, 152, 170, 184. moderate:
14, 28, 29, 43, 142, 156. weak: chain rules:
30, 45, 106, 154. strong:
30, 45, 106, 154. moderate: - weak: - 12 .4 Index complexity classification (2002) Index complexity in “A Nonlinear Dynamics Perspective of Wolframs New Kindof Science. Part I: Threshold of Complexity”, establishes three ECA classes: { red ( k = 1 ), blue ( k = 2 ), green ( k = 3 ) } .For details please see [8]. red ( k = 1 ):
0, 1, 2, 3, 4, 5, 7, 8, 10, 11, 12, 13, 14, 15, 19, 23, 32, 34,35, 42, 43, 50, 51, 76, 77, 128, 136, 138, 140, 142, 160, 162,168, 170, 178, 200, 204, 232. strong:
2, 7, 10, 11, 15, 34, 35, 42, 128, 138, 162, 170, 178. moderate:
1, 3, 4, 5, 8, 13, 14, 32, 43, 77, 136, 140, 142, 160, 168. weak:
0, 12, 19, 23, 50, 51, 76, 200, 204, 232. blue ( k = 2 ):
6, 9, 18, 22, 24, 25, 26, 28, 30, 33, 36, 37, 38, 40, 41, 44,45, 54, 56, 57, 60, 62, 72, 73, 74, 90, 94, 104, 106, 108, 110,122, 126, 130, 132, 134, 146, 152, 154, 156, 164. strong:
9, 18, 22, 24, 25, 26, 30, 41, 45, 54, 56, 57, 62, 94, 106, 108,110, 122, 126, 130, 146, 152, 154. moderate:
6, 28, 33, 37, 38, 40, 44, 72, 73, 74, 104, 132, 134, 156, 164. weak:
36, 60, 90. green ( k = 3 ):
27, 29, 46, 58, 78, 105, 150, 172, 184. strong:
46, 58, 184. moderate:
27, 29, 78, 172. weak: d -spectrum classification (2003) Density parameter with d -spectrum in “Experimental Study of Elementary Cel-lular Automata Dynamics Using the Density Parameter”, establishes three ECAclasses: { P, H, C } .For details please see [11]. 13 :
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 19, 23, 24,25, 27, 28, 29, 32, 33, 34, 35, 36, 37, 38, 40, 42, 43, 44, 50,51, 56, 57, 58, 62, 72, 74, 76, 77, 78, 104, 108, 128, 130,132, 134, 136, 138, 140, 142, 152, 156, 160, 162, 164, 168,170, 172, 178, 184, 200, 204, 232. strong:
2, 7, 9, 10, 11, 15, 24, 25, 34, 35, 42, 56, 57, 58, 62, 108,128, 130, 138, 152, 162, 170, 178, 184. moderate:
1, 3, 4, 5, 6, 8, 13, 14, 27, 28, 29, 32, 33, 37, 38, 40, 43, 44,72, 74, 77, 78, 104, 132, 134, 136, 140, 142, 156, 160, 164,168, 172. weak:
0, 12, 19, 23, 36, 50, 51, 76, 200, 204, 232. H:
26, 41, 54, 73, 94, 110, 154. strong:
26, 41, 94, 110, 154. moderate: weak: - C:
18, 22, 30, 45, 60, 90, 105, 106, 122, 126, 146, 150. strong:
18, 22, 30, 45, 106, 122, 126, 146. moderate: - weak:
60, 90, 105, 150.
Communication complexity classification in “Cellular Automata and Commu-nication Complexity”, establishes three ECA classes: { bounded, linear, other } .For details please see [10]. bounded:
0, 1, 2, 3, 4, 5, 7, 8, 10, 12, 13, 15, 19, 24, 27, 28, 29, 32,34, 36, 38, 42, 46, 51, 60, 71(29), 72, 76, 78, 90, 105, 108,128, 130, 136, 138, 140, 150, 154, 156, 160, 162(missing),170, 172, 200, 204. strong:
2, 7, 10, 24, 34, 42, 46, 108, 128, 130, 138, 154, 162, 170. moderate:
1, 3, 4, 5, 8, 13, 15, 27, 28, 29, 32, 38, 72, 78, 136, 140, 156,160, 172. weak:
0, 12, 19, 36, 51, 60, 76, 90, 105, 150, 200, 204.14 inear:
11, 14, 23, 33, 35, 43, 44, 50, 56, 58, 77, 132, 142, 152, 168,178, 184, 232. strong:
11, 35, 56, 58, 152, 178, 184. moderate:
14, 33, 43, 44, 77, 132, 142, 168. weak:
23, 50, 232. other:
6, 9, 18, 22, 25, 26, 30, 37, 40, 41, 45, 54, 57, 62, 73, 74, 94,104, 106, 110, 122, 126, 134, 146, 164. strong:
9, 18, 22, 25, 26, 30, 41, 45, 54, 57, 62, 94, 106, 110, 122,126, 146. moderate:
6, 37, 40, 73, 74, 104, 134, 164. weak: -Additionally, bound class can be refined in other four subclasses. bounded by additivity:
15, 51, 60, 90, 105, 108, 128, 136, 150,160, 170, 204. strong:
15, 51, 108, 128, 170. moderate: weak:
60, 90, 105, 150, 204. bounded by limited sensibility:
0, 1, 2, 3, 4, 5, 8, 10, 12,19, 24, 29, 34, 36, 38, 42,46, 72, 76, 78, 108, 138,200. strong:
2, 10, 24, 34, 42, 46, 108,138. moderate:
1, 3, 4, 5, 8, 29, 38, 72, 78. weak:
0, 12, 19, 36, 76, 200. bounded by half-limited sensibility:
7, 13, 28, 140, 172. strong: moderate:
13, 28, 140, 172. weak: - bounded for any other reason:
27, 32, 130, 156, 162. strong: moderate:
27, 32, 156. weak: -15 .7 Topological classification (2007)
Topological classification in “A Nonlinear Dynamics Perspective of Wolfram’sNew Kind of Science. Part VII: Isles of Eden”, establishes six ECA classes: { period-1, period-2, period-3, Bernoulli σ t -shift, complex Bernoulli-shift, hyperBernoully-shift } .For details please see [7]. period-1:
0, 4, 8, 12, 13, 32, 36, 40, 44, 72, 76, 77, 78, 94, 104, 128,132, 136, 140, 160, 164, 168, 172, 200, 204, 232. strong:
94, 128. moderate:
4, 8, 13, 32, 40, 44, 72, 77, 78, 104, 132, 136, 140, 160, 164,168, 172. weak:
0, 12, 36, 76, 200, 204, 232. period-2:
1, 5, 19, 23, 28, 29, 33, 37, 50, 51, 108, 156, 178. strong: moderate:
1, 5, 28, 29, 33, 37, 156. weak:
19, 23, 50, 51. period-3: strong: moderate: - weak: - Bernoulli σ t -shift:
2, 3, 6 , 7, 9, 10, 11, 14, 15, 24, 25, 27, 34, 35,38, 42, 43, 46, 56, 57, 58, 74, 130, 134, 138,142, 152, 162, 170, 184. strong:
2, 7, 9, 10, 11, 15, 24, 25, 34, 35, 42, 46, 56,57, 58, 130, 138, 152, 162, 170, 184. moderate:
3, 6, 14, 27, 38, 43, 74, 134, 142. weak: - complex Bernoulli-shift:
18, 22, 54, 73, 90, 105, 122, 126, 146, 150. strong:
18, 22, 122, 126, 146. moderate: weak:
90, 105, 150. hyper Bernoully-shift:
26, 30, 41, 45, 60, 106, 110, 154. strong:
26, 30, 41, 45, 110, 154. moderate: - weak:
60. 16 .8 Power spectral classification (2008)
Power spectral classification in “Power Spectral Analysis of Elementary CellularAutomata”, establishes four ECA classes: { category 1: extremely low power density, category 2: broad-band noise, category3: power law spectrum, exceptional rules } .For details please see [27]. category 1 extremely low power density:
0, 1, 4, 5, 8, 12, 13, 19,23, 26, 28, 29, 33, 37,40, 44, 50, 51, 72, 76, 77,78, 104, 128, 132, 133(94),136, 140, 156, 160, 164,168, 172, 178, 200, 232. strong:
26, 94, 128, 178. moderate:
1, 4, 5, 8, 13, 28, 29, 33,37, 40, 44, 72, 77, 78, 104,132, 136, 140, 156, 160,164, 168, 172. weak:
0, 12, 19, 23, 50, 51, 76,200, 232. category 2 broad-band noise:
2, 3, 6, 7, 9, 10, 11, 14, 15, 18, 22, 24,25, 26, 27, 30, 34, 35, 38, 41, 42, 43,45, 46, 56, 57, 58, 60, 74, 90, 105, 106,129(126), 130, 134, 138, 142, 146, 150,152, 154, 161(122), 162, 170, 184. strong:
2, 7, 9, 10, 11, 15, 18, 22, 24, 25, 26, 30,34, 35, 41, 42, 45, 46, 56, 57, 58, 106,122, 126, 130, 138, 146, 152, 154, 162,170, 184. moderate:
3, 6, 14, 27, 38, 43, 74, 134, 142, . weak:
60, 90, 105, 150. category 3 power law spectrum:
54, 62, 110. strong:
54, 62, 110. moderate: . weak: . exceptional rules:
73, 204. strong: - moderate: weak: .9 Morphological diversity classification (2010) Morphological diversity classification in “On Generative Morphological Diver-sity of Elementary Cellular Automata”, establishes five ECA classes: { chaotic, complex, periodic, two-cycle, fixed point, null } .For details please see [4]. chaotic:
2, 10, 18, 22, 24, 26, 30, 34, 42, 45, 56, 60, 73, 74, 90, 94,105, 106, 126, 130, 138, 150, 152, 154, 161(122), 162, 170,184. strong:
2, 10, 18, 22, 24, 26, 30, 34, 42, 56, 94, 106, 122, 126, 130,138, 152, 154, 162, 170, 184. moderate:
73, 74. weak:
60, 90, 105, 150. complex:
54, 110. strong:
54, 110. moderate: - weak: - periodic:
18, 26, 60, 90, 94, 154. strong:
18, 26, 94, 154. moderate: - weak:
60, 90. two-cycle:
1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 19, 23, 24, 25,27, 28, 29, 33, 34, 35, 36, 37, 38, 42, 43, 44, 46, 50, 51, 56,58, 74, 76, 106, 108, 130, 132, 134, 138, 140, 142, 152, 156,162, 164, 170, 172, 178, 184, 204. strong:
2, 7, 9, 10, 11, 15, 24, 25, 34, 35, 42, 46, 56, 58, 106, 108,130, 138, 152, 162, 170, 178, 184. moderate:
1, 3, 4, 5, 6, 13, 14, 27, 28, 29, 33, 37, 38, 43, 44, 74, 132,134, 140, 142, 156, 164, 172. weak:
12, 19, 23, 36, 50, 51, 76, 204.18 xed point:
0, 2, 4, 8, 10, 11, 12, 13, 14, 24, 32, 34, 36, 40, 42, 43, 44,46, 50, 56, 57, 58, 72, 74, 76, 77, 78, 104, 106, 108, 128,130, 132, 136, 138, 140, 142, 152, 160, 162, 164, 168, 170,172, 178, 184, 200, 204, 232. strong:
2, 10, 11, 24, 34, 42, 46, 56, 57, 58, 106, 108, 128, 130, 138,152, 162, 170, 178, 184. moderate:
4, 8, 13, 14, 32, 40, 43, 44, 72, 74, 77, 78, 104, 132, 136,140, 142, 160, 164, 168, 172. weak:
0, 12, 36, 50, 76, 200, 204, 232. null:
0, 8, 32, 40, 72, 104, 128, 136, 160, 168, 200, 232. strong: moderate:
8, 32, 40, 72, 104, 136, 160, 168. weak:
0, 200, 232.
Distributive and non-distributive lattices classification in “Inducing Class 4 Be-havior on the Basis of Lattice Analysis”, establishes four ECA classes: { class 1, class 2, class 3, class 4 } .For details please see [13]. class 1:
0, 32, 128, 160, 250(160), 254(128). strong: moderate:
32, 160. weak: class 2:
4, 36, 50, 72, 76, 94, 104, 108, 132, 164, 178, 200, 204,218(164), 232, 236(200). strong:
94, 108, 178. moderate:
4, 72, 104, 132, 164. weak:
36, 50, 76, 200, 204, 232. class 3:
18, 22, 54, 122, 126, 146, 150, 182(146). strong:
18, 22, 54, 122, 126, 146. moderate: - weak: lass 4: strong: moderate: . weak: . Topological classification in “A Full Computation-Relevant Topological Dy-namics Classification of Elementary Cellular Automata”, establishes four ECAclasses: { equicontinuous, almost-equicontinuous, sensitive, sensitive positively expansive } .For details please see [30, 6]. equicontinuous:
0, 1, 4, 5, 8, 12, 19, 29, 36, 51, 72, 76, 108, 200, 204. strong: moderate:
1, 4, 5, 8, 29, 72. weak:
0, 12, 19, 36, 51, 76, 200, 204. almost-equicontinuous:
13, 23, 28, 32, 33, 40, 44, 50, 73, 77, 78,94, 104, 128, 132, 136, 140, 156, 160,164, 168, 172, 178, 232. strong:
94, 128, 178. moderate:
13, 28, 32, 40, 73, 77, 78, 104, 132, 136,140, 156, 160, 164, 168, 172. weak:
23, 50, 232. sensitive:
2, 3, 6, 7, 9, 10, 11, 14, 15, 18, 22, 24, 25, 26, 27, 30, 34,35, 37, 38, 41, 42, 43, 45, 46, 54, 56, 57, 58, 60, 62, 74, 106,110, 122, 126, 130, 134, 138, 142, 146, 152, 154, 162, 170,184. strong:
2, 7, 9, 10, 11, 15, 18, 22, 24, 25, 26, 30, 34, 35, 41, 42, 45,46, 54, 56, 57, 58, 62, 106, 110, 122, 126, 130, 138, 146, 152,154, 162, 170, 184. moderate:
3, 6, 14, 27, 37, 38, 43, 74, 134, 142. weak: sensitive positively expansive:
90, 105, 150. strong: - moderate: - weak:
90, 105, 150.20lso, this classification can be refined into three sub-classes: weakly periodic,surjective, and chaotic (in the sense of Denavey). weakly periodic:
2, 3, 10, 15, 24, 34, 38, 42, 46, 138, 170. strong:
2, 10, 15, 24, 34, 42, 46, 138, 170. moderate:
3, 38. weak: - surjective:
15, 30, 45, 51, 60, 90, 105, 106, 150, 154, 170, 204. strong:
15, 30, 45, 154, 170. moderate: - weak:
51, 60, 90, 105, 150, 204. chaotic (in the sense of Denavey):
15, 30, 45, 60, 90, 105, 106, 150,154, 170. strong:
15, 30, 45, 106, 154, 170. moderate: - weak:
60, 90, 105, 150.
This is a classification by the evolution of a configuration consisting of an iso-lated one surrounded by zeros, that is a bit different from conventional ECAclassifications previously displayed. In “Expressiveness of Elementary CellularAutomata”, we can see five ECA kinds of expressivity: {
0, periodic patterns, complex, Sierpinski patterns, finite growth } .For details please see [28].
0, 7, 8, 19, 23, 31, 32, 40, 55, 63, 72, 104, 127, 128, 136,160, 168, 200, 232. strong:
7, 128. moderate:
8, 32, 40, 72, 104, 136, 160, 168. weak:
0, 19, 23, 200, 232. periodic patterns:
13, 28, 50, 54, 57, 58, 62, 77, 78, 94, 99, 109, 122,156, 178. strong:
54, 57, 58, 62, 94, 122, 178. moderate:
13, 28, 73, 77, 78, 156. weak:
50. 21 omplex:
30, 45, 73, 75, 110. strong:
30, 45, 110. moderate: weak: - Sierpinski patterns:
18, 22, 26, 60, 90, 105, 126, 146, 150, 154. strong:
18, 22, 26, 126, 146, 154. moderate: - weak:
60, 90, 105, 150. finite growth:
1, 2, 3, 4, 5, 6, 9, 10, 11, 12, 14, 15, 24, 25, 27, 29, 33, 34,35, 36, 37, 38, 39, 41, 42, 43, 44, 46, 47, 51, 56, 59, 71, 74,76, 103, 106, 107, 108, 111, 130, 132, 134, 138, 140, 142,152, 162, 164, 170, 172, 184, 204. strong:
2, 9, 10, 11, 15, 24, 25, 34, 35, 41, 42, 46, 56, 106, 108, 130,152, 162, 170, 184. moderate:
1, 3, 4, 5, 6, 14, 27, 29, 33, 37, 38, 43, 44, 74, 140, 142, 164,172. weak:
12, 36, 51, 76, 204.
Normalised compression classification in “Asymptotic Behaviour and Ratios ofComplexity in Cellular Automata Rule Spaces”, establishes two ECA classes: { C , , C , } .For details please see [42]. C , :
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 19, 23, 24,25, 26, 27, 28, 29, 32, 33, 34, 35, 36, 37, 38, 40, 42, 43, 44,46, 50, 51, 56, 57, 58, 72, 74, 76, 77, 78, 104, 108, 128, 130,132, 134, 136, 138, 140, 142, 152, 154, 156, 160, 162, 164,168, 170, 172, 178, 184, 200, 204, 232. strong:
2, 7, 9, 10, 11, 15, 24, 25, 26, 34, 35, 42, 46, 56, 57, 58, 108,128, 130, 138, 152, 154, 170, 178, 184. moderate:
1, 3, 4, 5, 6, 8, 13, 14, 27, 28, 29, 32, 33, 37, 38, 40, 43, 44,72, 74, 77, 78, 104, 132, 134, 136, 140, 142, 156, 160. weak:
0, 12, 19, 23, 36, 50, 51, 76, 200, 204, 232.22 , :
18, 22, 30, 41, 45, 54, 60, 62, 73, 90, 94, 105, 106, 110, 122,126, 146, 150. strong:
18, 22, 30, 41, 45, 54, 62, 94, 106, 110, 122, 126, 146. moderate: weak:
60, 90, 105, 150.
Expressivity classification in “Emergence of Surface Dynamics in ElementaryCellular Automata”, establishes three ECA classes: { type A, type B, type C } .For details please see [23]. type A:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 19, 23, 24,25, 27, 28, 29, 32, 33, 34, 35, 36, 37, 38, 40, 42, 43, 44, 46,50, 51, 56, 57, 58, 72, 74, 76, 77, 78, 94, 104, 108, 128, 130,132, 134, 136, 138, 140, 142, 152, 156, 160, 162, 164, 168,170, 172, 178, 184, 200, 204, 232. strong:
2, 7, 9, 10, 11, 15, 24, 25, 34, 35, 42, 46, 56, 57, 58, 128,130, 152, 170, 178, 184. moderate:
1, 3, 4, 5, 6, 8, 13, 14, 27, 28, 29, 32, 33, 37, 38, 40, 43, 44,72, 74, 77, 78, 104, 108, 132, 134, 136, 140, 142, 156, 160,164, 168, 172. weak:
0, 12, 19, 23, 36, 50, 51, 76, 200, 204, 232. type B:
18, 22, 26, 30, 41, 45, 60, 90, 105, 106, 122, 126, 146, 150,154. strong:
18, 22, 26, 30, 45, 106, 122, 126, 146, 154. moderate: - weak:
18, 22, 26, 30, 45, 106, 122, 126, 146, 154. type C:
54, 62, 73, 110. strong:
54, 62, 110. moderate: weak: - Spectral classification in “A Spectral Portrait of the Elementary Cellular Au-tomata Rule Space”, establishes four ECA classes: { DE/SFC, DE/SFC SFC, EB, S } .23or details please see [29]. DE/SFC:
0, 1, 2, 5, 6, 7, 8, 9, 10, 11, 12, 14, 19, 22, 23, 24, 25, 26,27, 29, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 46, 50,54, 56, 57, 58, 62, 72, 73, 74, 76, 77, 94, 104, 108, 110, 128,130, 132, 134, 136, 138, 140, 142, 152, 160, 162, 164, 168,172, 178, 184, 200, 232. strong:
2, 7, 9, 10, 11, 22, 24, 25, 26, 29, 34, 35, 41, 42, 46, 54, 56,57, 58, 62, 94, 108, 110, 128, 130, 138, 152, 162, 178, 184. moderate:
1, 5, 6, 8, 14, 27, 32, 33, 37, 38, 40, 43, 44, 72, 73, 74, 77,104, 132, 134, 136, 140, 142, 160, 164, 168, 172. weak:
0, 12, 19, 23, 36, 50, 76, 200, 232.
DE/SFC SFC:
3, 4. strong: - moderate:
3, 4. weak: - EB:
13, 18, 28, 78, 122, 126, 146, 156. strong:
18, 122, 126, 146. moderate:
13, 28, 78, 156. weak: - S:
15, 30, 45, 51, 60, 90, 105, 106, 150, 154, 170, 204. strong:
15, 30, 45, 106, 154, 170. moderate: - weak:
51, 60, 90, 105, 150, 204.
In this section, we have just bijective and surjective classification (personalcommunication, Harold V. McIntosh and Juan C. Seck Tuoh Mora): { bijective, surjective } .For details please see [21, 22]. bijective:
15, 51, 170, 204. strong:
15, 170. moderate: - weak:
51, 204. 24 urjective:
30, 45, 60, 90, 105, 106, 150, 154. strong:
30, 45, 106, 154. moderate: - weak:
60, 90, 105, 150.
Creativity classification in “On Creativity of Elementary Cellular Automata”,establishes four ECA classes: { creative, schizophrenic, autistic savants, severely autistic } .For details please see [5]. creative:
3, 5, 11, 13, 15, 35. strong:
11, 15, 35. moderate:
3, 5, 13. weak: - schizophrenic:
9, 18, 22, 25, 26, 28, 30, 37, 41, 43, 45, 54, 57, 60, 62, 73,77, 78, 90, 94, 105, 110, 122, 126, 146, 150, 154, 156. strong:
9, 18, 22, 25, 26, 30, 41, 45, 54, 57, 62, 110, 122, 126, 146,152, 154. moderate:
28, 37, 43, 73, 77, 78, 156. weak:
60, 90, 105. autistic savants:
1, 2, 4, 7, 8, 10, 12, 14, 19, 32, 34, 42, 50, 51, 76,128, 136, 138, 140, 160, 162, 168, 170, 200, 204. strong:
2, 7, 10, 34, 42, 128, 138, 162, 170. moderate:
1, 4, 8, 14, 32, 136, 140, 160, 168. weak:
12, 19, 50, 51, 76, 200, 204. severely autistic:
23, 24, 27, 29, 33, 36, 40, 44, 46, 56, 58, 72, 74, 104,106, 108, 130, 132, 142, 152, 164, 172, 178, 184, 232. strong:
24, 46, 56, 58, 106, 108, 130, 152, 178, 184. moderate:
27, 29, 33, 40, 44, 72, 74, 104, 132, 142, 164, 172. weak:
23, 36, 232.
Acknowledgement
Thanks to several authors of all previous classifications for discussion and im-prove this note. 25 eferences [1] A. Adamatzky (Ed.) (2002)
Collision-Based Computing , Springer.[2] R. Alonso-Sanz (2009)
Cellular Automata with Memory , Old City Publish-ing.[3] R. Alonso-Sanz (2011)
Discrete Systems with Memory , World ScientificSeries on Nonlinear Science, Series A.[4] A. Adamatzky & G. J. Mart´ınez (2010) On generative morphological di-versity of elementary cellular automata,
Kybernetes submitted to Complex Systems (arXiv:1305.2537).[6] G. Cattaneo, M. Finelli & L. Margara (2000) Investigating topological chaosby elementary cellular automata dynamics,
Theoretical Computer Science
Int.J. of Bifurcation and Chaos
Int.J. of Bifurcation and Chaos
Complex Systems Theoretical Computer Science
Discrete Mathematics and TheoreticalComputer Science , 15–166.[12] J. Guan, S. Shen, C. Tang & F. Chen (2007) Extending Chua’s global equiv-alence theorem on Wolfram’s New Kind of Science,
Int. J. of Bifurcationand Chaos
Complex Systems
Physica D
Complex Systems
The Origins of Order: Self-Organization and Selec-tion in Evolution , Oxford University Press, New York.[17] G. J. Mart´ınez, A. Adamatzky, R. Alonso-Sanz & J. C. Seck-Tuoh-Mora(2010) Complex dynamic emerging in Rule 30 with majority memory,
Com-plex Systems
Int. J. ofBifurcation and Chaos in preparation .[20] G. J. Mart´ınez, A. Adamatzky, J. C. Seck-Tuoh-Mora & R. Alonso-Sanz(2010) How to make dull cellular automata complex by adding memory:Rule 126 case study,
Complexity
Physica D One Dimensional Cellular Automata , Luniver Press.[23] J. C. Seck-Tuoh-Mora, J. M. Marin, G. J. Mart´ınez & N. H. Romero,Emergence of Surface Dynamics in Elementary Cellular Automata,
Com-munications in Nonlinear Science and Numerical Simulation , submitted.[24] O. Martin, A. M. Odlyzko & S. Wolfram (1984) Algebraic properties ofcellular automata,
Communications in Mathematical Physics
Journal ofCellular Automata
Irre-ducibility and Computational Equivalence , H. Zenil (Ed.), Springer, chapter17, 237–259.[27] S. Ninagawa (2008) Power Spectral Analysis of Elementary Cellular Au-tomata,
Complex Systems
Int. J. Modern Physics C
Irreducibility and Compu-tational Equivalence , H. Zenil (Ed.), Springer, chapter 16, 211–235.2730] M. Sch¨ule & R. Stoop (2012) A full computation-relevant topological dy-namics classification of elementary cellular automata,
Chaos
Cellular Automata Machines , The MITPress.[32] C. C. Walker & W. R. Ashby (1966) On temporal characteristics of behaviorin certain complex systems,
Kybernetik
The Global Dynamics of Cellular Au-tomata , Santa Fe Institute Studies in the Sciences of Complexity, Addison-Wesley Publishing Company.[34] S. Wolfram (1983) Statistical Mechanics of Cellular Automata,
ReviewModern Physics Phys-ica D Theory and Applications of Cellular Automata , WorldScientific Press, Singapore.[37] S. Wolfram (1994)
Cellular Automata and Complexity: Collected Papers ,Addison-Wesley Publishing Company.[38] S. Wolfram (2002)
A New Kind of Science , Wolfram Media, Inc., Cham-paign, Illinois.[39] A. Wuensche (1999) Classifying Cellular Automata Automatically,
Com-plexity
Parallel Processing Letters
Exploring Discrete Dynamics , Luniver Press, 2011.[42] H. Zenil & E. V. Zapata, Asymptotic Behaviour and Ratios of Complexityin Cellular Automata Rule Spaces,