Tables of CellularAutomaton Properties

1 9 86

Introduction

Thi s appendix gives tables of propert ies of one-dimensional cellular automata with

two possible values at each site (k = 2), and with rules depending on nearest neigh­

bour s (r = I). These cellular automata are some of the simplest that can be con­structed. Yet they are already capable of a grea t diversity of highly complex be­

haviour. The tables in this appendix attempt to capture some of this behaviour, both

pictorially and numeri cally.

There are 256 possible rules for k = 2, r = 1 cellular automata. Table 1 givesforms for these rules, together with simple equiva lences among them.

Tables 2 and 3 show pattern s produced by evo lution according to all possible in­

equivalent rules, starting from "typical" disordered or random initial conditions. Sev­

eral general classes of qualitative behaviour are seen (see pages 115-1 57 in this book):

1. A fixed, homogeneous, state is eventually reached (e.g . rules 0, 8, 136).

2. A pattern consisting of separated periodic regions is produ ced (e.g. rules 4,

37, 56,73) .

3. A chao tic, aperiodic, pattern is prod uced (e.g . rules 18,45, 146).

4. Complex, localized structures are generated (e.g . rule 110). (This behaviour

is clearly visible in the pictures of table 15.)

Much of the data in this appendi x can be understood in term s of this classificat ion .

The pattern s produced with a particular rule by evo lution from different disorderedinitial states are qualitatively similar. Nevertheless, changes in initial conditions can

lead to detailed changes in the configurations produ ced . Table 4 shows the pattern of

Originally pub lished in Theory and Applications oj Cellular AUlOmara, World Scientific Publish ing Co. Ltd., pages485-5 57 ( 1986).

S13

Wolfram on Cellular Automata and Complexity

differences produced by single-site changes in initial conditions. For class I rules,the changes always die out. For class 2 rules, they may persist, but remain localized.Class 3 rules, however, show "instability": small changes in initial conditions canlead to an ever-expanding region of differences. "Information" on the initial statethus propagates, typically at a fixed speed, through the cellular automaton. In class 4cellular automata, such informa tion transmission occurs irregularly, through motionof specific localized structures.

Table 6 gives the values of some statistical quantities which characterize someof the behaviour seen in tables 2, 3 and 4. The definitions of entropies and Lya­punov exponents for cellular automata (see pages 115-157 in this book) are closelyanalogous to those for conventional continuous dynamical systems.

Tables 2, 3, 4 and 6 concern the generic behaviour of cellular automata with"typical" disordered initial conditions. The generation of complexity in cellularautomata is however perhaps more clear ly illustrated by evolution from particular,simple, initial conditions, as in table 5. With such initial conditions, some cellularautomaton rules yield simple or regular patterns. But other rules yield highly complexpatterns, which seem in many respects random.

Tables 2 through 6 suggest that many different k = 2, r = 1 cellular automataexhibit similar behaviour. Table I gives some simple equivalences between rules.Table 7 gives equivalences arising from more complex transformations. Often dif­ferent regions in a cellular automaton will form "domains" which show differentequivalences.

Table 8 gives further relations between rules, in the form of factorizat ions whichexpress one rule as compositions of others.

An important feature of cellular automata is their capability for "self organiza­tion". Even starting from arbitrary disordered or random initial conditions, their timeevolution can pick out particular "ordered" states. Tables 9 through II give mathe­matical characterizations of the sets of configurations that can occur in the evolutionof k = 2, r = I cellular automata. Table 9 concerns blocks of site values which arefiltered out by the cellular automaton evolution.

The complete set of configurations produced after any finite number of time stepscan be described in terms of regular formal languages (see pages 159- 202 in thisbook). Tables 10 and 11 give the values of quantities which characterize the certainaspects of the "complexity" of these languages.

The behaviour of class 3 and 4 cellular automata often seems to be so complexthat its outcome cannot be determined except by essentially performing a directsimulation. Tables 10 and I I may provide some quantitative basis for this supposition.Table 12 gives a more direct measure of the difficulty of computing the outcome ofcellular automaton evolution in the context of a simple computational model involvingBoolean functions.

The results for most of the tables here are for cellular automata on lattices withan infinite number of sites. Tables 13 and 14 give some of the more complete results

514

Tablesof CellularAutomaton Properties (19861

that can be obtained for cellular automata on finite lattices (or with spatially periodicconfigurations). Table 13 shows fragments of the state transition diagram s whichdescribe the global evolution of finite cellular automata . Table 14 plots some of theiroverall properties.

Many of the k = 2, r = 1 cellular automata show highly complex behaviour. Suchbehaviour is probably most evident in rule 110. Table 15 gives some propertie s ofthe particle-like structures which are found in this rule. One suspects that with ap­propriate combin ations of these structures, it should be possible to perform universalcomputation.

The final table shows patterns produced by reversible generalizations of the stan­dard k = 2, r = 1 cellular automata. Qualitatively similar behaviour is again seen.

It is remarkable that with such simple construction, the k = 2, r = I cellularautomata can show such complex behaviour. The tables in this appendix give somefirst attempts at charac terizing and quantifying this behaviour. Much, however, stillremains to be done .

515

Wolfram on Cellular Automata and Complexity

Table 1: Rule Forms and Equivalences

rulenumber I equivalent rules

hex IIboolean expression dep min

dec binary conj ref! c.r.

0 00000000 ) 0 - - - 255 0 255 01 00000001 (a_IaOal) ••• 127 1 127 I2 00000010 ) (a-IaOal) ••• 191 16 247 23 00000011 3 (a-laO) ..- 63 17 119 34 00000100 04 (a-Iaoad ••• 223 4 223 45 00000101 05 (a-l ad .-. 95 5 95 56 00000110 06 (a_laOal) + (a-IaOal) ••• 159 20 215 67 00000111 07 (a_Ial) + (a-laO) ••• 31 21 87 78 00001000 08 (a_I aoad ••• 239 64 253 89 00001001 09 (a-l aoail + (a- Iaoa l) ••• III 65 125 9

10 00001010 Oa (a_1 a l) .-. 175 80 245 10II 00001011 Ob (a-l ao) + (a_Iad ••• 47 81 117 I I12 00001100 Oc (a-laO) ..- 207 68 221 1213 00001101 Od (a-lail + (a_lao) ••• 79 69 93 1314 00001110 Oe (a-l ao) + (a_Iad ••• 143 84 213 1415 00001111 Of (a_I) 0-- 15 85 85 1516 00010000 10 (a-laOa l) ••• 247 2 191 2I7 00010001 II (aOal) - .. 119 3 63 318 00010010 12 (a _1 aoal) + (a_IaOa l) ••• 183 18 183 1819 00010011 13 (aoad + (a_lao) ••• 55 19 55 1920 00010100 14 (a- l aoal) + (a- IaOal) ••• 215 6 159 621 00010101 15 (aOal) + (a_I al) ••• 87 7 3.1 722 00010110 16 (a-l aoal) + (a_I aOa l) + (a_1 aoal) ••• 151 22 151 2223 0001011 1 17 (aoad + (a_lad + (a-laO) ••• 23 23 23 2324 00011000 18 (a- l aOal) + (a-l aoal) ••• 231 66 189 2425 00011001 19 (a- Iaoal) + (aoa l) ••• 103 67 61 2526 00011010 la (a-laoad + (a_lad ••• 167 82 181 2627 000110 11 Ib (aoal) + (a-l ad ••• 39 83 53 2728 00011100 Ie (a-laoad + (a_lao) ••• 199 70 157 2829 0001 1101 Id (aoad + (a-laO) ••• 71 71 29 2930 00011110 Ie (a-l aoad + (a-l ao) + (a_I a l) 0 •• 135 86 149 3031 00011111 If (aoad + (a-I) ••• 7 87 21 732 00100000 20 (a-laoal) ••• 251 32 251 3233 00100001 21 (a_laOal) + (a- Iaoail ••• 123 33 123 3334 00100010 22 (aOal) - .. 187 48 243 3435 00100011 23 (a-l ao) + (aoal) ••• 59 49 115 3536 00100100 24 (a-l aoail + (a- Iaoal) ••• 219 36 219 3637 00100101 25 (a_IaOal) + (a-l ail ••• 91 37 91 3738 00100110 26 (a- l aoad + (aOal) ••• 155 52 21 1 3839 00100111 27 (a- lad + (aOal) ••• 27 53 83 2740 00101000 28 (a-laoal) + (a- Iaoal) ••• 235 96 249 4041 00101001 29 (a- laOal) + (a-laoad + (a-l aoa l ) ••• 107 97 121 4142 00101010 2a (aoad + (a-l al) ••• 171 112 241 4243 0010101 1 2b (a-l ao) + (aOal) + (a_Ia l) ••• 43 113 113 4344 00101100 2c (a-l aoal) + (a-l ao) ••• 203 100 217 4445 00101101 2d (a-l aoal) + (a-l al) + (a-l ao) 0 •• 75 101 89 4546 00101110 2e (a-l ao) + (aOal) ••• 139 116 209 46

516

Tables of Cellular Automaton Properties (1 9861

rulenumber equivalent rulesboolean expression dep min

dec binary hex conj ref! c.r.

47 00101111 2f (aoad + (a-d ••• I I 117 81 I I48 00110000 30 (a_lao) ..- 243 34 187 3449 00110001 31 (aoal) + (a_1 ao) ••• 115 35 59 3550 00110010 32 (a_Iao)+ (aOal) ••• 179 50 179 5051 001100 11 33 (ao) - 0- 51 51 51 5152 00110100 34 (a_1 aoal) + (a_1 ao) ••• 211 38 155 3853 00110101 35 (a_1 al) + (a_Iao) ••• 83 39 27 2754 001101 10 36 (a_1 aoal) + (a_Iao)+ (aoad . 0. 147 54 147 5455 0011011 1 37 (a- lad + (ao) ••• 19 55 19 1956 00111000 38 (a_IaOal) + (a_laO) ••• 227 98 185 5657 00111001 39 (a_Iaoal) + (aOal) + (a- lao) . 0. 99 99 57 5758 00111010 3a (a- lao) + (a_Ial) ••• 163 114 177 5859 001110 11 3b (a- Iad+(ao) ••• 35 115 49 3560 00111100 3c (a_I ao)+ (a_1 ao) 00- 195 102 153 6061 00111101 3d (a_1 ad + (a_1 ao) + (a_Iao) ••• 67 103 25 2562 00111110 3e (a_Ial) + (a_lao) + (a-lao) ••• 131 11 8 145 6263 00111111 3f (ao) + (a-I) ..- 3 119 17 364 01000000 40 (a-Iaoad ••• 253 8 239 865 01000001 41 (a_IaOal) + (a_Iaoal) ••• 125 9 I II 966 01000010 42 (a_I aOa l ) + (a-IaOal) ••• 189 24 231 2467 01000011 43 (a_Iaoal) + (a_1 ao) ••• 61 25 103 2568 01000100 44 (aoal ) - .. 221 12 207 1269 01000101 45 (a_Ia]) + (aoad ••• 93 13 79 1370 01000110 46 (a_1 aoa,) + (aoal) ••• 157 28 199 2871 01000111 47 (aoal) + (a_Iao) ••• 29 29 71 2972 01001000 48 (a-Iaoa l) + (a_Iaoal) ••• 237 72 237 7273 01001001 49 (a_Iaoal) + (a-Iaoad + (a_1aoal) ••• 109 73 109 7374 01001010 4a (a-Iaoad + (a-lad ••• 173 88 229 7475 01001011 4b (a_Iaoal) + (a_Iao) + (a_1 01) 0 •• 45 89 101 4576 01001100 4c (aOal)+(a_IaO) ••• 205 76 205 7677 01001101 4d (a_Ial) + (aoal) + (a- laO) ••• 77 77 77 7778 01001110 4e (aoal) +(a-Iad ••• 141 92 197 7879 010011 11 4f (aoal) + (a_I) ••• 13 93 69 1380 01010000 50 (a- lad .-. 245 10 175 1081 01010001 51 (aoad+(a- Iad ••• 11 7 11 47 II82 01010010 52 (a_1 aoal) + (0-1al) ••• 181 26 167 2683 010100 11 53 (a_Ial) + (a_1 ao) ••• 53 27 39 2784 01010100 54 (a-Ial)+(aoad ••• 21 3 14 143 1485 01010101 55 (ad --0 85 15 15 1586 01010110 56 (a_IaOal) + (a_Ial) + (aoad ••0 149 30 135 3087 01010111 57 (a_lao) + (al) ••• 21 31 7 788 01011000 58 (a-Iaoad+(a-Ial) ••• 229 74 173 7489 01011001 59 (a_Iaoal ) + (aoad + (a_lad ••0 101 75 45 4590 01011010 5a (a_lad+(a_Ia l) 0-0 165 90 165 9091 01011011 5b (a_laO) + (a-lad + (a_l ad ••• 37 91 37 3792 01011100 5c (a_1 al) + (a_1 00) ••• 197 78 141 7893 01011101 5d (a- laO) + (a l ) ••• 69 79 13 1394 01011110 5e (a- lao)+ (a-la d + (a_lad ••• 133 94 133 9495 01011111 5f (a ]) + (a_]) .-. 5 95 5 5

517

Wolfram on Cellulor Automoto ond Complex ity

rule number equivalent rulesboolean expression dep min

dec binary hex conj ref c.r.

96 01100000 60 (a_Iaoad + (a_1aoal) ••• 249 40 235 4097 01100001 61 (a_Iaoal ) + (a_Iaoal ) + (a_1aoal) ••• 121 41 107 4198 01100010 62 (0- 1aoa]) + (aoad ••• 185 56 227 5699 01100011 63 (a_Iaoa]) + (a_1 ao) + (aoal) . 0. 57 57 99 57

100 01100100 64 (a-l aoa]) + (aoa]) ••• 217 44 203 44101 01100101 65 (a_1aoal) + (a_1 a l) + (aoal) •• 0 89 45 75 45102 01100110 66 (aoal) + (aoa] ) -OD 153 60 195 60103 01100111 67 (a_1ao) + (aoal) + (aoaI ) ••• 25 61 67 25104 01101000 68 (a_1 aoal ) + (a_1 aoal) + (a_Iaoad ••• 233 104 233 104105 01101001 69 (a_Iaoa I ) + (a_1aoal ) + (a_]aoa] ) + (a_1 aoal) 000 105 105 105 105106 01101010 6a (a_Iaoal) + (aoal) + (a_Ia l) •• 0 169 120 225 106107 01101011 6b (a_1aoal) + (a_Iao) + (aoal) + (a_Iad ••• 41 121 97 41108 01101100 6c (a_Iaoal) + (aoal) + (a_Iao) . 0. 201 108 201 108109 01101101 6d (a_1aoal) + (a_Ia l) + (aoal) + (a- lao) ••• 73 109 73 73110 01101110 6e (a_lao) + (aoal) + (aoal) ••• 137 124 193 110II I 01101111 6f (aoal) + (aoa l) + (a_I) ••• 9 125 65 911 2 01110000 70 (a- lad + (a_lao) ••• 241 42 171 42113 01110001 71 (aoal) + (a_1al) + (a_1 ao) ••• 113 43 43 43114 011 10010 72 (a_1 al ) + (aoa]) ••• [n 58 163 58115 01110011 73 (a_1 al) + (ao) ••• 49 59 35 3511 6 011 10100 74 (aoad + (a_lao) ••• 209 46 139 46117 01110101 75 (a_lao) + (a l) ••• 81 47 11 11118 011 10110 76 (a_1ao) + (aoal ) + (aoal) ••• 145 62 131 62119 01110111 'n (a ]) + (ao) - .. 17 63 3 3120 01111000 78 (a_1 aoaI) + (a_1 al) + (a_1ao) 0 •• 225 106 169 106121 01111001 79 (a_1 aoal) + (aoal) + (a_Ia l) + (a-l ao) ••• 97 107 41 41122 01111010 7a (a_lao) + (a-l ad + (a- lad ••• 161 122 161 122123 01111011 7b (a_1al) + (a_1al) + (ao) ••• 33 123 33 33124 01111100 7c (a_Ia l ) + (a_lao) + (a_lao) ••• 193 110 137 110125 01111101 7d (a_1 ao) + (a_1«o) + (aI) ••• 65 I II 9 9126 01111110 7e (a_Ial) + (aoal ) + (a_1ao) ••• 129 126 129 126127 0111111 1 7f (ad + (ao) + (a-d ••• I 127 1 1128 10000000 80 (a-I aoal ) ••• 254 128 254 128129 10000001 81 (a_Iaoal ) + (a_1aoal) ••• 126 129 126 126130 10000010 82 (a-I aoad + (a_laoal) ••• 190 144 246 130131 10000011 83 (a_1aoad + (a_Iao) ••• 62 145 118 62132 10000100 84 (a_]aoad + (a_]aoal ) ••• 222 132 222 132133 10000101 85 (a_1 aoal) + (a_Ia l) ••• 94 133 94 94134 10000110 86 (a_1 aoal) + (a_Iaoal ) + (a_IaoaI) ••• 158 148 214 134135 10000111 87 (a_Iaoal ) + (a_1al) + (a_Iao) 0 •• 30 149 86 30136 10001000 88 (aoad - .. 238 192 252 136137 10001001 89 (a_1 aoal) + (aoal ) ••• 110 193 124 110138 10001010 8a (a_Iad + (aoa l ) ••• 174 208 244 138139 10001011 8b (a_Iao) + (aoal ) ••• 46 209 116 46140 10001100 8c (a_lao) + (aoal) ••• 206 196 220 140141 10001101 8d (a-l ad + (aoal) ••• 78 197 92 78142 10001110 8e (a_Iao) + (a_1a]) + (aoal) ••• 142 212 212 142143 10001111 8f (aoal) + (a- d ••• 14 213 84 14144 10010000 90 (a_1aoal) + (a_1aoal) ••• 246 130 190 130145 10010001 91 (a_Iaoal) + (aoal) ••• 11 8 131 62 62

518

Tables of Cellular Automaton Properties (19861

rule number equivalent rulesboolean expression dep min

dec binary hex conj ref! c.r.

146 10010010 92 (a-Iaoal) + (a-laoad + (a_1 aoad ••• 182 146 182 146147 10010011 93 (a-Iaoa l) + (0001)+ (a-lao) . 0. 54 147 54 54148 10010100 94 (a_100(1) + (0_1 aoal) + (a_1 aoal) ••• 214 134 158 134149 1001 0101 95 (a_1aoa l) + (0001) + (0-1( 1) •• 0 86 135 30 30150 1001011 0 96 (a_100( 1)+ (0_1aoal) + (0- 1aoa l) + (a_1aoal) 000 150 150 150 150151 10010111 97 (a_1aoal ) + (aoad + (0- 1( 1)+ (0- 1(0) ••• 22 151 22 22152 10011000 98 (a_Iaoad + (aoa l) ••• 230 194 188 152153 10011001 99 (00121 )+ (aoad - 0 0 102 195 60 60154 10011010 9a (a_1aoad + (0- 1a l) + (aoad •• 0 166 210 180 154

, 155 10011011 9b (a-lao) + (12001) + (aoal ) ••• 38 211 52 38156 10011100 9c (a_1 00.11) + (0- 1ao) + (aoal ) . 0. 198 198 156 156157 10011101 9d (a- l ao) + (0001) + (aoal ) ••• 70 199 28 28158 100ll11O ge (a_1 00(1)+ (12_1ao) + (12-1a l) + (aoal) ••• 134 214 148 134159 100ll11l 9f (120.1 1)+ (aoal) + (a-I) ••• 6 215 20 6160 10100000 aO (a-lad .-. 250 160 250 160161 10100001 a1 (a-Iaoal)+(a- Iad ••• 122 161 122 122162 10100010 a2 (aoad + (a_Ia,) ••• 186 176 242 162163 10100011 a3 (0-1( 0) + (a_1 ad ••• 58 177 114 58164 10100100 a4 (a-Iaoad + (a-lad ••• 218 164 218 164165 10100101 a5 (a-lad + (a_Ial) 0-0 90 165 90 90166 10100110 a6 (a-Iaoad + (aoad + (a-lad •• 0 154 180 210 154167 10100111 a7 (0-1( 0) + (a-lad + (a_1a l) ••• 26 181 82 26168 10101000 a8 (a_1 ad + (aoal) ••• 234 224 248 168169 . 10101001 a9 (0-100.11)+ (a_1 ad + (aoal) •• 0 106 225 120 106170 10101010 aa (ad - -0 170 240 240 170171 10101011 ab (0-1( 0) + (al) ••• 42 241 11 2 42172 10101100 ac (a_lao) + (a- lad ••• 202 228 216 172173 10101101 ad (a-lao) + (0-1( 1)+ (a_Ia l) ••• 74 229 88 74174 10101110 ae (a-lao) + (al) ••• 138 244 208 138175 10101111 af (a- I) + (a l) .-. 10 245 80 10176 10110000 bO (a- lao) + (a- la d ••• 242 162 186 162177 10110001 bl (aoad + (a_1a l) ••• 114 163 58 58178 10110010 b2 (a- lao) + (aoal ) + (a_1 a l ) ••• 178 178 178 178179 10110011 b3 (a_1 al) + (00) ••• 50 179 50 50180 10110100 b4 (a-laOal) + (a- laO ) + (a_la l ) 0 •• 210 166 154 154181 10110101 b5 (a-l ao) + (0-1.11 ) + (a_1al) ••• 82 167 26 26182 101 10110 b6 (0- 1aoal ) + (a- lao) + (aoal) + (a_1a l) ••• 146 182 146 146183 10110111 b7 (a_lad + (a_la d + (00) ••• 18 183 18 18184 10111000 b8 (a-l ao) + (aoal) ••• 226 226 184 184185 10111001 b9 (a_I(0) + (0001)+ (aoal) ••• 98 227 56 56186 10111010 ba (a-lao) + (a l) ••• 162 242 176 162187 lOll 101I bb (00) + (al) - .. 34 243 48 34188 10111100 be (a_1al) + (a_1 ( 0) + (0_1 ao) ••• 194 230 152 152189 lOll I 101 bd (.10 .11)+ (a_lad + (0_1 ao) ••• 66 231 24 24190 lO ll I110 be (a_1( 0) + (0-1ao) + (al) ••• 130 246 144 130191 10111111 bf (00) + (a- I) + (al) ••• 2 247 16 2192 11000000 cO (a_lao) ..- 252 136 238 136193 11000001 c l (.1-1.1001)+ (a_lao) ••• 124 137 11 0 ll O194 11 000010 c2 (12-1 aoad + (a_Iao) ••• 188 152 230 152195 11000011 c3 (a-lao) + (a- lao) 00 - 60 153 102 60

519

Wo lfram on Cellular Automata and Complexity

rule number equivalent rulesboolean expression dep min

dec binary hex conj ref c.r.

196 11000100 c4 (aOal) + (a-lao) ••• 220 140 206 140197 11000101 c5 (a-Iad+(a-Iao) ••• 92 141 78 78198 11000110 c6 (a_Iaoal) + (aOal) + (a_Iao) . 0. 156 156 198 156199 11000111 c7 (a-lad + (a-laO) + (a_lao) ••• 28 157 70 28200 11001000 c8 (a-l ao) + (aoal) ••• 236 200 236 200201 11001001 c9 (a_Iaoal) + (a_Iao) + (aoad . 0. 108 201 108 108202 11001010 ca (a_Iao) + (a_Ial) ••• 172 216 228 172203 11001011 cb (a_lad + (a-lao) + (a_lao) ••• 44 217 100 44204 11001100 cc (ao) -0- 204 204 204 204205 11001101 cd (a-Iad+(ao) ••• 76 205 76 76206 11001110 ce (a_Ial) + (ao) ••• 140 220 196 140207 11001111 cf (a-I) + (ao) ..- 12 221 68 12208 11010000 dO (a-Iad+(a-lao) ••• 244 138 174 138209 11010001 dl (aOal) + (a-l ao) ••• 116 139 46 46210 11010010 d2 (a_laoad + (a-l ad + (a-lao) 0 •• 180 154 166 154211 11010011 d3 (a- lal) + (a_I ao) + (a-l ao) ••• 52 155 38 38212 11010100 d4 (a_Iad + (aoal) + (a_Iao) ••• 212 142 142 142213 11010101 d5 (a_Iao) + (ad ••• 84 143 14 14214 11010110 d6 (a_Iaoal) + (a_lal) + (aoal) + (a_Iao) ••• 148 158 134 134215 11010111 d7 (a_Iao) + (a_I ao) + (al) ••• 20 159 6 6216 11011000 d8 (a-l al) + (aoad ••• 228 202 172 172217 11011001 d9 (a_Iao) + (aoal) + (aoal) ••• 100 203 44 44218 11011010 da (a_lao) + (a-l ad + (a_lal) ••• 164 218 164 164219 11011011 db (aoal) + (a_I al) + (a_Iao) ••• 36 219 36 36220 11011100 dc (a_Ial) + (ao) ••• 196 206 140 140221 11011101 dd (ad + (ao) - .. 68 207 12 12222 11011110 de (a_Ial) + (a-Ial) + (ao) ••• 132 222 132 132223 11011111 df (al) + (a-I) + (ao) ••• 4 223 4 4224 11100000 eO (a_lao) + (a_lad ••• 248 168 234 168225 11100001 el (a-l aoal) + (a_Iao) + (a_Ial) 0 •• 120 169 106 106226 11100010 e2 (a_lao) + (aoal) ••• 184 184 226 184227 11100011 e3 (a_lad + (a-lao) + (a-lao) ••• 56 185 98 56228 11100100 e4 (aoal)+(a_lal) ••• 216 172 202 172229 11100101 e5 (a-l ao) + (a_I al) + (a_I al) ••• 88 173 74 74230 11100110 e6 (a_I ao) + (aoal) + (aoad ••• 152 188 194 152231 11100111 e7 (a_Ial) + (aoal ) + (a_Iao) ••• 24 189 66 24232 11101000 e8 (a_Iao) + (a_Ial) + (aoal) ••• 232 232 232 232233 11101001 e9 (a_I aOal) + (a-l ao) + (a_Ial) + (aoal) ••• 104 233 104 104234 11101010 ea (a-lao) + (ad ••• 168 248 224 168235 11101011 eb (a_Iao) + (a_lao) + (a l) ••• 40 249 96 40236 11101100 ec (a_Ial) + (ao) ••• 200 236 200 200237 11101101 ed (a_Iad + (a_lal) + (ao) ••• 72 237 72 72238 11101110 ee (ao) + (al) - .. 136 252 192 136239 11101111 ef (a-I) + (ao) + (al) ••• 8 253 64 8240 11 110000 fO (a-d 0-- 240 170 170 170241 11110001 fl (aOal) + (a_I) ••• 112 171 42 42242 I II 10010 f2 (aoal) + (a_I) ••• 176 186 162 162243 11 110011 f3 (ao) + (a-I) ..- 48 187 34 34244 11 110100 f4 (aOal) + (a-d ••• 208 174 138 138

520

Tables of C ellular Automa ton Prope rties (19 86)

rule num ber equivalent rulesboolean expression dep min

dec binary hex conj ref! c.r.

245 11110 101 f5 (al)+(a_l) .-. 80 175 10 10246 11110 110 f6 (aoa,) + (aoal) + (a_I) ••• 144 190 130 130247 111 1011 1 f7 (a l) + (aO) + (a_I) ••• 16 19 1 2 2248 11111000 f8 (aoa l) + (a_I) ••• 224 234 168 168249 1111100 1 f9 (aoal ) + (aoal) + (a_I) ••• 96 235 40 40

250 11111010 fa (a- I ) + (a l ) .-. 160 250 160 16025 1 11111011 fb (ao) + (a_I) + (a l) ••• 32 25 1 32 32252 11111100 fc (a_I) + (ao) ..- 192 238 136 136253 1111110 1 fd (al ) + (a_I) + (ao) ••• 64 239 8 8254 11111110 fe (a_I) + (ao) + (al) ••• 128 254 128 128255 111111 11 ff 1 - - - 0 255 0 0

Forms of rules and equivalences between rules.

The table lists all 256 possible rules for k = 2, r = l one-dimensional cellularautomata. Such cellular automata consist of a line of sites, each with value 0 or I.At each time step, the value a j of a site at position i is updated acco rding to the rule

This table lists the 223 = 256 possible choices of ¢ .Each digit in the binary representation of the rule number gives the value of ¢ for

a particular set of (a i_I ' a j , a i+ I) . The digit corresponding to the coefficient of 2/1 inthe rule number gives the value of ¢( n2 , n I ' no)' where n = 4n 2 + 2n I + no' Thusthe leftmost digit in the binary rep resentation of the rule number gives ¢( I, I , I), thenext gives ¢( I , 1, 0), and so on, down to ¢(O, 0, 0).

The table also gives the decimal and hexadecimal representations of the rulenumbers.

Each ¢ can be considered a Boolean function of three variables, say a_I ' ao anda l • The table gives the minimal disjunctive normal form representations for theseBoolean functions. Boolean multiplication and addition are used (correspondingto AND and OR operations). Bar denotes complementation. In each case, theexpression with the min imal number of components, using only these operati ons, isgiven.

The column labelled "dep" gives the dependence of ¢(a_l , ao' a l ) on each of thea_i' ao and aI' The symbol - indicates no change in ¢ when the corresponding aj

is changed. The symbol 0 deno tes linear dependence of ¢ on the corresponding aj :

wheneve r aj changes , ¢ also changes. The symbol . denotes arbitrary dependenceof ¢. Rules such as 90 in which only 0 and - dependence occ urs, are called additive,

and can be represented as linear functions modulo two.For each rule, the table gives rules equivalent under simple transformations.

"conj" denotes conjugation: interchange of the roles of 0 and I. "refl" denotes reftec-

521

Wolfram on Cellul ar Automata and Complex ity

tion. Rules invariant under reflection are symmetric. " c.r," denotes the combinedoperation of conjugation and reflection.

Many of the properties considered in this Appendix are unaffected by these trans­formations. The rules form equivalence classes under these transformations, and it isusually convenient to consider only the minimal (lowest-numbered) representativesof each class, as given by the last column in the table.

In some cases, further equivalences between rules can be used. Table 7 gives oneimportant set of such further equivalences.

Some special rules are:

51 complement170 left shift204 identity240 right shift

Table by Lyman P. Hurd (Mathematics Department, Princeton University). (Boolean expres­

sions by S. Wolfram.)

522

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Wolfram on Cellular Automata and Complexity

,. -.-- it~ 11mHrU I ~ 7J ( 0 100' 00')

~ ~ ~ ~ ~ ~ ~ ~~~~ ~~ ~~

mlr!~II!il!I!••TT -_.r u le 33 (00 ' 0000') ru le 34 (00100010) rule 35 (00100011 ) r u le 36 ( 00100, ee)

1111il1~ ., ...._..,. r'jrule 37 (00 100' 0 1) r u l e 38 ( 00 100 110 ) rul e 40 (0010' 00 0) ru le 41 ( 0010 100l)

•• -•• rn[ .~ • 11I1I .. . _~

r ul e 42 (0010'010) rule 43 (0010'011 ) rule 44 (00' 01100 ) rule 45 ( 001 0 1101)

• 111r ul e 46 (00'01 110 ) rul e 50 (001'0010 ) r u l e 51 ( 001 10011) r ul e 54 (00',0,10 )

• • IErule 56 (00 " ' 000) rule 57 ( 00111 001) rule 58 ( 0011101 0 ) ru le 60 (00"1100 )

•rule 61 ( 00 11110 1) ru l e 62 (00,,1 ,'0) rul e 72 (01001000 )

P'~r u l e 74 ( 0 ' 00 10 10 ) ru le 76 ( 0, 00, , 00) r ul e 77 ( 0' 00" 0 ') rul e 78 (0 , 00",0 )

524

Tables of Ce llular Automaton Proper ties (1986)

II ~~~II~""""""" ' n'¥ rl~ '0" · · ·w··y~y ~• •• •• I ~"' .~ p~r:'\-. . .:...

. .' ", ' '. "0: .:f.~

. ~ '. ~~"~ <:;'''"+.~ . . • .

. _ ._ ';;~_-I:ffR ...

rul e 90 (010"01 0) ru le 94 ( 01 0 11 110) rule 104 ( 01 10 1000 ) r u l e 105 ( 01 101 001)

~ .. ~~ ~ ~~ ~ III Iiir u l e 106 (0 "010 10) rule ' 08 ( 0 " 0 1100) rule 110 (01 101 110) ru le '22 ( 0 " " 0 10)

IB U ' U ' ~ . -- I'--r ul e 12 6 ( 0 '11"10) rul e ' 28 (100000 00) rul e ' 30 (1 00000' 0) rule 13 2 ( 100001 00)

~ ".n'T.r rrrru le 134 (10000110) r ule ' 36 ( 1000'000) ru l e '38 ( ' 00010' 0 ) r u l e '40 (' 000 1100)

••••••• 1~~~~ _i •• ,ii~11~ ~.ru l e 142 ( 10001 110) ru le 146 ( ' 00 ' 00 10) r u1. 150 ( 100' 0 11 0) r ul e '52 ( 10011000 )

III ~ ~ ~~~~ ~ ....._...,.r ul e ' 54 ( 10011 01 0) rule 156 ( , 00 ', , 00) ru l e 160 ( 10 100000) ru le 162 (1 0' 000 10)

nrl .. ··""·~~ rrr rr u l e '64 (1 0' 00 100) rule 168 ( 101 0' 000) r u le 170 (10 10 10 '0 ) r ule 172 ( ' 0 ' 0 1100)

525

Wolfram on Cellular Automata and Complexity

rul e 178 ( 101 10010)

IIIIrul e 204 (11 00 1100)

rul e 184 " 0 11 10 00)

III r "r ul e 232 ("1 0' 000 )

ru l. 188 (1 0 1111 00 ) ru l . 200 (1 100 ' 000 )

Pattern s generated by evolution from disordered initial states.

Each picture is for a different ru le. All the "minimal representative" rules of ta­ble 1 are included. (Other rules have patterns equ ivalent to those of their minimalrepresentatives.)

Sites with values 1 and 0 are represented respect ively by black and white square s.The initial configuration is at the top of each picture. The values of sites in it arechosen randomly to be 0 or 1 with probability 1/2. Successive lines are obtained byapplica tions of the cellular automaton rule.

These pictures show the evo lution of cellular automata with 80 sites for 60 timesteps. Period ic bound ary conditions were imposed on the edges.

Different specific initial configurations for a particular rule almost always yieldqualitatively similar patterns. Different rules are however seen to give a wide varietyof diffe rent kinds of patterns.

526

Tablesof CellularAutomaton Properties (19861

Table 3: Blocked Patterns from Disordered States

------ ...-- 11111111~.~

r u l. S (SSSSSSSS ) ru l. 1 (SSSSSSS I ) r u l. 2 ( SSSSSSl S ) r ul e 3 (SSSSSSI1 )

'-lIr- ' 111111~~rule 4 ( SSSSS ISS) r u l . 5 ( SSSSS IS I) ru le 6 (Se e e e l l e ) rule 7 ( e e e e e n i)'HU_'_. ~ ., ,r u le 8 ( e e e e i e e e ) r u l.9 ( e e eel eSl) r u l e l e ( e e e e l e l e ) r u l e 11 (Se eel e l1)

~ I -1 111 11 ~ ~ .ru le 12 (SSeSll ee) r ul e 13 ( e e eell el) r u l e 14 (eeSel l1S) r u le 15 ( e e e S l l 11 )

r~ii'''r1 1111111 . 11111rule 18 ( Se e l Sel S) ru le 19 ( e e S l e e l1) r u l e 22 (eeSle l1S ) ru le 23 (000 10 111)

~•• ~\ .r u t e 24 ( 000 11000 ) ru le 25 ( 00e l1 e 0 1) rul e 26 ( 000 1101 e ) r u l.27 (00e1 10 11)

'11111111' I1111 ....T._.ru le 28 (S0e l 11ee) r u l . 29 ( 000 111e l) r ul e 30 ( e0el l 11S) rul e 32 ( 0el s ees s)

527

Wo lfram on Cellular Automata and Complexity

rnl~ ~ .~ In---rut e 33 ( 00 100001) rcl e 3 4 ( 001 000 10) rut e 35 ( 001 000 11) r ut e 36 ( 001 00 100)

~- ml~] ~ -- y "" , -" ~" ~: Ir u l e 37 ( 001 00 10 1) rul e 38 (001 0011 0 ) r ut e 40 ( 00 10 1000) rul e 41 ( 00 101 00 1)

~ ' ---- 1-~~ 1 -

r u l e 42 (001 0 ' 01 0) r u fe 43 ( 00' 01 0 11) r u t e 44 ( 00' 011 00) rul . 45 ( 001 0 1101)

~ ~ -I I I Tr ut e 46 ( 00' 01 110) r uf e 50 ( 00 11 00' 0) r u l.5' ( 00 11 00 11) r ut e 54 ( 0011 011 0)

~ 1iI· .•rule 56 (00'11 000 ) rul e 57 (0 0111 00') r ut e 58 ( 00 111 0 10) rul e 60 ( 00111 100 )

• .-- - - ~ n-~iUru l.6 l ( 00 1111 01) rul . 62 ( 00 11 1" 0) rul .72 (0'001000 ) r ut e 73 ( 0' 001 00 ')

~ - 1- 1111 "l Ull"rul e 7 4 ( 0 100 ' 0 10 ) r ut e 76 (01 0011 00) rul . 77 ( 01 0011 0 ') rut e 78 ( 0' 00 1110)

528

Tobles of Cellular Automoton Properties (19861

ru l e 90 ( 01 0 110 10 ) ru le 94 ( 0 10 11 110) rule '04 ( 0 110 1000 ) ru l e 105 ( 0 110 100 1)

•

1':1 .. ~ - ~ '~ I fhrl' ,.. • . '",''''' " ...• ~~ ., ~ ' > , , -

, ~'11 }k'\ .~ . " . . .1~'~ ., i. 'it . '

. . '. ' ~l'~ ' . ., 11/1/; . .e .) l/flh~' ~ ' .

rul e 106 ( 011010 10) ru l e 108 ( 0 110 110e ) r ul e 110 (0" 01110) ru l e 122 (01111010 )•.._..._-~ I U '

r ut e ' 26 ( 01 111 110) r u l e 128 (' 00 00 000) r ut e 130 ( 10000 0 10) r ut e 132 ( 10 000 10 0 )

~ -.Rrul e '34 (10000110 ) rul e 136 ( ' 0001000) r u t e 138 ( 1000 10 10) r u l e 1+0 ( 100 011 00)

- !~l" " j (8~rul e 1+2 ( 1000"' 0) r u Ie '46 (' 00 10 0 10) r ule 15 0 ( 100 ' 01 10) r ul e 15 2 (1 00 110 00)

__ UIUI ··.. ··· ·~ru le 154 ( 100 110 10) ru l e 156 (1001 110 0 ) ru l e 160 (1 0 100 000) r ut e 162 ( 10 1000 10 )

YU'r m ·W~ V · ~ - ~.T'Til

ru le 1604 ( 10 100100) ru le 168 (1 0 10 1000) r u l e 170 (1 01 0 10 10) rul e 17 2 ( 10 10 1 100 )

529

Wolfram on Cellula r Automata and Complexity

II nlll ···~ Ir II111111 111111

r ul e 178 ( ' 8" 081 0)

ru l e 28 4 ( 118 011 8 0)

r u t e 184 (1 811 18 00)

ru l e 232 (11 10 ' 008)

rul e ' 88 (1 01 11188) rvt e 28 0 (1 100' 000)

Blocks in patterns generated by evolution from disord ered initial states.

The pictures in this table are analogous to those in table 2, but show only everyother site in both space and time. Certain features become clearer in this "blocked"representation.

It is common for cellular automata to exhibit several "phases". The blockedrepresentation often makes differences between these phases visible.

530

Tables ofCellular Automaton Properties (19861

Table 4: DiHerence Patterns

I / \rule e (00000000) ru Ie 1 ( 0000000 1) rul. 2 (000000'0) ru Ie J (000000" )

) \rule 4- ( 00000 ' 00) ru le 5 ( 00000 ' 0 ' ) r u l . 6 ( 00000 " 0 ) r u Ie 7 ( 00000 ' " ), / -,ru I . 8 (0000 ' 000) r u I . 9 (0000' 00 ' ) r u l e 10 ( 0000' 0 ' 0) rul e 1 1 ( 0000' 011 )

'\ <,rul . ' 2 ( 0000" 00) ru l e 13 (000011 01) r u fe '4 ( 0000 1" 0) r ul e 15 ( 00001111 )

A A Iru l e 18 (000' 00' 0) r u l e '9 ( 000 ' 0011 ) r ul . 22 ( 000 ' 011 0) rul e 23 ( 000' 0' 11)

~ < / \r ul e 24 (00011 _) ru le 25 (000 1'001) r ul e 26 ( 00 011 0 ' 0) ru le 27 ( 00011 0 11 )

~rul e 28 ( 00 011 '00) r u I e 29 (00011 '0') r u l . 30 (_'11 '0 ) r ul e 32 (00' 00000)

I (r ul. 33 ( 00' 00001 ) r u l. 34 (00 '00010 ) ru le 35 ( 00' 00 011 ) ru l. 36 (00' 00' 00)

fJ / Arul . 37 (00' 00' 01) r ul . 38 (00 ' 0011 0) rul . 40 ( 00' 0' 000) ru l. 4 ' (00 ' 01001)

531

Wolfram on Cellular Automata and Comp lex ity

/ ) 1 Aru l e 42 ( 00 10 10 10) ru l e 43 ( 001 01 0 11) rul e .4 ( 00 10 1100 ) ru l e 45 ( 00 10 110 1)

/ I ~r u I. 4 6 (00 10 11 10) ru l e 59 (00 1100 10) r ul e S1 ( 0011 00 11) r u l e 54 (00 110 110)

-, ; / ~rul e 56 ( 00 11 1000) rul e 57 ( 001 1100 1) ru l e 58 ( 00 11 10 10) r u l e 60 (00 111100)

A ~ tJru l e 6 1 (001 111 0 1) ru l e 62 ( 00 111 110) r u l e 72 (01 001 000) ru le 7J ( 01 001 001)

/ Iru le 74 ( 0 100 10 10 ) r u l e 76 ( 01 001 100) r u I. 77 (01 001 10 1) r ul e 78 (01 00111 0)

A I I Aru l e 90 ( 01 01 10 18) r ul e 94 ( 0 10 1111 0) r u l e 10 4 (0 1101008 ) ru le 105 ( 01 18 1001)

/ 11 Arule 10 6 ( 0 110 101 0 ) r u Ie 108 (0 110 1100) rule 110 (01181110) ru l e 122 (8 1111810)

A,

/ru l e 126 ( 0 1111 110 ) rul e 128 ( 18 000000) r u le 138 ( 1000001 0 ) ru l e 132 (, 8_ , 00)

/ru le 13 4 (100001 10 ) ru le 136 (1 0881 000) r u le 138 ( 10001010 ) ru l. 148 (1 0001 108 )

532

Tables of Cellular Automaton Properties 11 9861

( A A -,ru le 142 ( 1000 111 0) rule 146 ( 100 100 10) rule 150 (10010110 ) r u l e '52 ( 100 11000)

/ I /r ul e 154 ( 10011 0 10 ) rule 156 ( 100 11 100) r u le 160 ( 10 100000) r u le 16 2 ( 10 100010 )

r / /

r u l e 164 ( 10 100 100) rule 168 ( 10 101 000 ) rule 170 ( 10 10 10 10) rule 172 ( 10 10 1100)

\ /r u l e 178 ( 1011 001 0) ru le 184 ( 10 111000) rule '88 ( 10 11 1100) r u l e 200 (11001000 )

r ul e 204 (11 0011 00) r u le 23 2 ( 11 10 1000)

Differences in patterns produced by evolution from disordered states resultingfrom changes in single initial site values.

The evolution of small perturbations made in the initial configurations for all the"minimal representative" rules of table I are given. In each case, an initial configura­

tion was chosen in which sites had value 0 or I with prob abilit y 1/ 2, and the patternobtained by evo lution according to the cellular automaton rule was found. Thenthe value of the centre site in the initial con figuration was complemented, and theresulting patte rn obtained by cellular automaton evolution was found. The picturesshow as black squares the site values that differed between the patterns found withthese initial configurations. Evolution for 40 time steps is shown.

In some cases, the differences die out, or remain localized , with time. In othercases, the differences grow. The left and right grow th speeds correspond to the leftand right Lyapunov exponents AL and AR , given in table 6.

For some rules (such as 18), initial perturbations on some configurations maygrow, but on others may die out. The pictures show results from a particular trial.

533

Wolfram on Cellular Automata and Complex ity

Table 5: Patterns from Single Site Seeds

/ruI. e (e0eeeeee) r ul e , (eeeeeee1) r u f e 2 (eeeeee, e) rul e 3 (eee0e011)

/rul. 4 (eeeeeiee) r ul e 5 (eeeee, e,) rul e 6 (eeeeeI10) r uf e 7 (eeeeetn)

/ru l e 8 ( 0 0001 000) rul e 9 (eeee, ee1) r ut e ,e (eeeele10) r ul e 11 ( 0000 10 11)

/rul. 12 (eeeell ee) rcre ' 3 (eeeell e1) r ut e 14 (eeeell 1e) rut e '5 (eeee111 1)

r ul e 18 (eeel ee1e) r ut e '9 (eee, ee11) r ul e 22 (eeelell e) r ul e 23 (e0e, ell 1)

-, lila" Arul . 24 (eeell eee) r u I. 25 (eee, , ee,) r u l e 26 (eeel le, e) rul. 27 (eeel1el 1)

~ • Aru I. 28 (eee, 1tee ) ru l e 29 (eeell1 el) r ule 3e (eeer t i te) rul. 3, (eaer t t t i)

/ru le 32 (ee, eeeee) rule 33 (eel eeee1) rul. 34 (ee, eee, e) rul e 35 (ee, eeel l )

/rut e 36 (ee, eel ee) ru I e 31 (ee, ee, el ) rul. 38 (ee, ee" e) rul. 39 (eeleell 1)

534

ru l . 40 ( 001 010 00 ) rul. 41 ( 00 10 100 1)

/ru l . 42 ( 001 01 01 0 )

Tables of Cellular Automaton Properties 119861

rut e 43 (0e l e 10 11)

.. /rul e 44 ( 001 011 00)

rul . 50 (00 1100 10)

r ut e 45 ( 00101 10 1)

rul . 51 (00 1100 11)

r ul e 46 (0 0 10111 0)

r ul . 5 4 ( 00 110 110 )

rul .47 ( 00 10 1111)

r u l e 55 (00 110 111)

ru l e 56 ( 0011 1000 )

~ru le 60 (001111 00 )

rul . 72 ( 01 001 000 )

r u l . 76 (01001100)

rul . 90 ( 01 01 101 0)

rul . 10 4 (011 01_)

r u t e 57 (0e l l1 e01)

rul .61 ( ee 111 10 1)

" ~a~~~~~~~E~ "~rul . 73 ( 0 100 100 1)

r u l . 77 ( 01 00 110 1)

r ul. 91 (010 110 11)

ru l . 105 (011 el eel)

rul . 58 (0e111 01 0 )

ru l . 62 ( 00 111110 )

/r u l. 74 ( 01 00 101 0)

ru l. 78 (0 1001110)

ru l e 94 ( 01 01111 0)

/ru l . 106 (011 01 eI 0)

r cr e 59 ( e 0 11 10 11)

r ul e 63 ( 00 111111)

r u t e 75 ( 0 1001 011)

1111rul e 79 (0 100 1111)

ru l . 95 ( el e l111 1)

ru l . 107 ( 0 1101 011)

535

Wolfram on Cellular Automata and Comp lexi ty

r u f e 110 ( 01 10 1110 ) ru le 1' 1 (0110 111 1)

Arul e 126 ( 01 11 1110) rul . 12 7 ( 01111111)

/ -rut e 130 (1 000001 0) ru l. 13 1 (1 000001 1)

/ -ru l. 134 ( 10000 110 ) r ul e 135 (1 0000 111)

/ -rul . 138 (1 0001 01 0) r ul . 139 (1 0001 011)

/ -rut e 142 (1 00011 10) rul . 143 (1 0001111)

Arul e 150 (1 00 1011 0) rul e 15 1 (1 ee1811')

.,11111111111111.

rul . 147 (1 001 0011)

rul . 141 (1 000 1101)

rul . 137 (1 000 100 1)

r ule 109 (0 1101 10 1)

rut e 133 ( 10000 10 1)

r ule 129 (1 0000001)

r ul . 123 ( 0 1111 0 11)

III.

rul e 140 (1 00011 00)

rul e 14 6 (1 001 0010)

ru l . 136 (1 000 1000)

r u Ie 108 (01 101 100)

r ul . 128 ( 10000000)

rul. 132 ( 10000 100)

rul . 122 (01111010 )

~ •• Ar ut e 152 (10011000 ) ru l. 153 ( 100 1100 1) ru l . 15 4 (1 08 118 10 ) r ul . 155 (1 ee1181 1)

rul e 156 (100 111 00 ) ru l. 157 (1 08111 01) rul . 158 ( 1001' 110 ) ru l . 159 ( 1eell 11 1)

536

ru l. 168 (18188008)

r ul . 16. (10 108100 )

rul. 161 (1 0 10088 1)

rul. 165 ( 18 100 18 1)

/rul e 162 (1018801 0)

/ru l. 166 (10188110)

Tables of Cellular Automaton Properties (19861

r ul e 163 ( 10108811)

r ul e 167 (10100111)

r u l . 168 ( 101 0 1000)

r ul e 172 ( 101 011 00 )

r u l e 178 (1 011 001 8)

-,r u le 184 (1 01 11000)

r c t e 188 ( 10 11 1100 )

r uf e 169 (1 01 8 188 1)

r ul e 173 (10101181)

ru l. 179 ( 10 1100 11)

rul e 185 ( 1011 100 1)

r u l e 189 ( 10 11 110 1)

r ul e 170 (10181818)

/rule 174 (10181110)

rul. 182 (10118118)

rut e 186 ( 10 1110 18 )

r ul e 198 ( 101 11110)

rul. 171 ( 10101011)

r ut e 175 ( 18 10 111 1)

rul. 183 (10110111)

r ul e 187 (10111011)

r ct e 191 ( 10 11111 1)

••r u l e 200 ( 1100 188 0 )

r u t e 284 ( 11001 108)

r ul e 20 1 (11 001 00 1)

r cl e 20 5 ( 11001101 )

r ut e 202 ( 1100 10 10)

r ul e 206 ( 11001 110)

r ut e 203 (11001011 )

••rul e 207 (1 1001111 )

537

Wolfram on Cellula r Automata and Complexi ty

rul. 218 (118 118 18 )

ru l.232 ( 11101000)

ru l . 236 (111 811 00 )

rul. 258 ( 11 11 10 18)

rut e 219 ( 1181 18 11)

rul e 233 (11181 081)

r ul e 237 ( 1118 118 1)

ru l.251 ( 11 1110 11)

r u l e 222 ( 118 11118)

/r u t e 234 ( 111 0 18 18)

r ul e 238 (111 8 11 18 )

rul e 254 ( 11 11 ' 118 )

ru l e 22 3 (11 8 111 11)

rul e 235 ( 1110 1011)

r ul e 239 ( 111 01111)

r et e 255 ( 111111 11)

rul . 18 ( 8801 88 18 )

ru l . 45 (88 18 110 ')

538

ru le 38 (808 ' ,1 , 0)

ru le 73 (8 1881 881)

ru t e 185 ( 8 118 188 1)

rul e 158 (18818118)

Tables of Cellular Automaton Properties (1986)

ru t e 118 (81 181118)

rul. 169 (18181881)

Patterns generated by evolution from configurations containing a singlenonzero site.

The first part ofthe table shows picture s for all distinct rules. Since the initial configu­ration is not invariant under complementation, rules which differ by complementationcan produce different patterns, and are shown separately. Only the minimal repre­sentative is shown for rules related by reflection. In all cases, the pattern s correspondto evolution for 38 time steps.

Many rules are seen to yield equivalent patterns. The results of table 7 can oftenbe used to deduce these equivalences.

Some rules (such as l22 ) yield asymptotically homogeneous pattern s. Others(such as 90 and ISO) yield asymptotically self similar or fractal patterns. (Thefractal dimensions of the patterns obtained from rules 90 and ISO are respectively

log2 3 "" 1.59 and log2(1 + .)5) "" 1.69.) But some rules (such as 30 and 73) yieldirregular patterns which show no periodic or almost periodi c behaviour. The secondpart of the table gives some of the distinct pattern s obtained by evolution for 360time steps. Note that the structure on the right of the pattern generated by rule 110eventually dies out, leaving an essentially periodic structure.

539

Wolfrom on Cellula r Automata and Complexity

Table 6: Statistical Properties

density hex) AL AR h(1)hI'

h lmin lI' I' I'

0 0 0 - - 0 0 0

I 1/8 .43536 0 0 0 0 0

2 1/8 .48752 I -I hex) I (x) 0I' ' I'

3 1/4 .70 121 - 1/2 1/2 h~X )/2 h~X )/ 2 0

4 1/8 .51771 0 0 0 0 0

5 7/ 16 .702± .00 1 0 0 0 0 0

6 .241 ± .00 1 <.573 ± .00 1 I - I hex) h~X ) 0I'

7 .469 ± .00 1 <.502 ± .00 1 - 1/2 1/2 h~X )/2 h~X )/2 0

8 0 0 - - 0 0 0

9 .41O± .OOI <.264 ± .002 -I I hex) heX) 0I' I'

10 1/4 .68872 I - I h~X ) 0(x)

hI'

I I 1/2 <.567 ± .00 1 -I I hex) hex) 0I' I'

12 1/4 .68872 0 0 0 0 0

13 .437 ± .00 1 .378 ± .001 0 0 0 0 0

14 1/2 0 (- I , I ) (I , - I ) 0 0 0

15 1/2 1 - I I 1.0 1.0 0

18 1/4 1/2 I I 0.5 1.0 1.0

19 1/2 .6235 1 0 0 0 0 0

22 .35095 ± .00002 <.795 ± .00 1 .7660 ± .0002 .7660 ± .0002 .744± .003 <.9146 ± .0007 <.9 146 ± .0007

23 1/2 .599± .001 0 0 0 0 0

24 3/16 .5508 1 - I I hex) hex) 0I' I'

25 .447 ± .00 1 <.180 ± .00 1 -1 /2 1/2 h~X )/2 h~X)/2 0

26 .386 ± .001 <.790 ± .00 1 I - I heX) h~X ) 0I'

27 .53 1 ± .00 1 <.800 ± .001 -1 / 2 1/2 h~X )/2 h;:)/ 2 0

28 1/2 .500 ± .OO I 0 0 0 0 0

29 1/2 .86742 0 0 0 0 0

30 1/2 I .2428 ± .0002 I 1 <1.15436 <.76314 1

32 0 0 - - 0 0 0

33 .396 ± .OO ! <.637 ± .00 1 0 0 0 0 0

34 1/4 .68872 I - I hex) hex) 0I' I'

35 .375 ± .00 1 <.645 ± .00 1 - 1/2 1/2 h~' )/2 h~X )/2 0

36 1/16 .32483 0 0 0 0 0

37 .384± .00 1 .506 ± .00 1 0 0 0 0 0

38 9/32 .73733 1 -! hex) h;: ) 0I'

40 0 0 - - 0 0 0

4 1 .372 ± .OO I <.360 ± .00 1 - I I h~X ) hex) 0I'

42 3/8 .85684 I - I heX) hex) 0I' I'

43 1/2 0 (-1 , I) (I , -I ) 0 0 0

44 .167± .OO ! .528 ± .001 0 0 0 0 0

45 1/2 I . 1724 ± .0003 I I < 1.13036 <.673893

46 3/8 .5508 ! I - I hex) hex) 0I' I'

50 1/2 .60 1 ± .(0 ) 0 0 0 0 0

5 1 1/2 I 0 0 0 0 0

540

Tobles of C ellular Automoton Properties 119861

density h (x ) AL AR h (l )h I'

h lminlI' I' I'

54 .49 ± .0 1 <.2720 ± .0005 .553 ± .002 .553 ± .002 <.250 ± .002 <.250 ± .002 <.250 ± .002

56 .376 ± .00 1 <.589 ± .00 1 -1 I (x ) I (x ) 0h I' ' I'

57 1/2 0 (- 1, I ) ( I , -I ) 0 0 0

58 .625± .00 1 <.332 ± .00 1 1 - I I (x ) (x) 0' I' h I'

60 1/2 I 0 I 1 2 2

62 .644 ± .002 <.262 ± .00 1 0 0 0 0 0

72 1/8 .32483 0 0 0 0 0

73 .463 ± .00 1 <.7 14 ± .00 t 0 0 0 0 0

74 .318 ± .00 1 <.629 ± .00 1 1 - I h (x ) h (x ) 0I' I'

76 3/8 .85060 0 0 0 0 0

77 1/2 .599 ± .001 0 0 0 0 078 .562 ± .00 1 .377 ± .00 1 0 0 0 0 0

90 1/2 I 1 1 I 2 2

94 .584 ± .00 1 <.562 ± .001 0 0 0 0 0

104 .068 ± .00 1 .208 ± .00 1 0 0 0 0 0

105 1/2 I 1 I I 2 2

106 1/2 I 1 - .1335 ± .0006 I <1.06985 <.46 1366

108 5/16 .78025 0 0 0 0 0

110 4/7 0 (.26- 5) (-.27- 0.) 0 0 0

122 1/2 1/2 1 1 0.5 1.0 1.0126 1/2 1/2 1 1 0.5 1.0 1.0

128 0 0 - - 0 0 0

130 .167 ± .00 1 .525 ± .00 1 1 - I h~x ) h( x ) 0I'

132 1/8 .599 ± .00 1 0 0 0 0 0134 .292 ± .00 1 <.533 ± .00 1 1 - I h (x ) (x')

0I' h I'

136 0 0 - - 0 0 0138 3/8 .806 ± .001 I - I h~X ) h (x ) 0I'

140 1/4 <.678 ± .00 1 0 0 0 0 0

142 1/2 0 (- 1, 1) (1, - 1) 0 0 0

146 1/4 1/2 1 1 0.5 1.0 1.0

150 1/2 I 1 I I 2 2

152 .185 ± .00 1 .5 15 ± .00 1 - I 1 I (x ) (x)0I I' h I'

154 1/2 I 1 - I I 1 0

156 1/2 .502 ± .00 1 0 0 0 0 0

160 0 0 - - 0 0 0

162 .333 ± .00 1 .667 ± .00 1 I - I h (x ) (x) 0I' h I'

164 .083 ± .00 1 .389±.00 1 0 0 0 0 0

168 0 0 - - 0 0 0

170 1/2 -1 1 I 1.0 1.0 0

172 1/8 .485 ± .00 1 0 0 0 0 0

178 1/2 .599 ± .00 1 0 0 0 0 0

184 1/2 0 (- 1, I ) (I , -1 ) 0 0 0

200 3/8 .70 121 0 0 0 0 0204 1/2 I 0 0 0 0 0

232 1/2 .599 ± .00 1 0 0 0 0 0

541

Wolfram on Cellular Automata and Complexity

Statistical properties of evolutio n from disordered states.

Results are give n for all the "minimal representative" rules of table I . In all cases,initial configurations were used in which each site has value 0 or I with probability

1/2. Some properties of some rules remain unchan ged with different kinds of initialconfigurations.

Rational numbers, or numbers without errors, are quoted whenever analyticalarguments yield exac t results. In a few cases, the rigour of these arguments may be

subject to question.The column labelled "density" gives the asymptotic density of nonzero sites.

For some, but not all, rules this depend s on the initial density, here taken to be1/ 2. For most rules, the relaxation to the final density appears to be approx imatelyexponential. For some rules (such as 18), in which particle-like exc itations undergorandom annihilation, the relaxation may be like ( -1 / 2, or slower. Rule 110 shows

part icularly slow relaxation.The co lumn labelled hf l give s estimates for the asymptot ic spatial measure en­

tropy, as defined in pages I L5-157 in this book. Thi s quantity gives a measure ofthe "information content" of cellular automa ton configurations. It is comp uted bybreaking the configuration into blocks of sites, say of Length X, then eva luating the

quanti ty -t L Pi log, Pi ' where the sum runs over a1l2x possible blocks, which aretaken to occur with proba bilities Pi ' hf l is the limit of this quantity as X ~ 00. Thevalues decrease monotonically with X, allowi ng upper bound s on the X ~ 00 limitto be der ived from finite X results. Where errors are quoted, the values or bounds onhf l give n in the table were obtained after 400 time steps, with blocks up to lengthX = II considered. (More accurate results were obtained for rules 22 and 54.) Fitsto values obtained as a function of X suggest that the exact hf l for rules 22 and 54may in fact be zero .

The definition of hf ) implies that it achieves its maximal value of I only when

all possible sequences of site values occur with equal probability, so that each sitehas value 0 or I with independent probabilit y I / 2. h~X ) = 0 if only a finite number ofcomplete cellular automaton configurations can occur.

Results for h~r ) give n without errors in the table were obtained by explicit con­struction of prob abil istic regular languages which represent the sets of configurationsproduced by cellular automaton evo lution, as in table II .

The quanti ties AL and AR are left and right Lyapunov ex ponents, which measurethe rate of information transmission. They give the slopes of the left and rightboundarie s of the difference patterns illustrated in table 4. Thus they measure therate at which perturb ations in cellular automaton configurations spread to the left andright.

The notation - indicates that almost all changes in init ial confi gurations die out,

so that the AL ,R are not defined.

542

Tablesof Cellular Automaton Properties 11986)

The notation (- 1, 1) indicates that the information propagation direction canalternate, typically as progressively more distant particle-like structures from theinitial configuration are encountered. There is probab ly no definite infinite size limitfor the AL•R in such cases .

Rule 110 shows highly complex information transmission propert ies, associatedwith the particle-like structures of table 15. The values of AL •R given in the tablefor this case are possible bounds associated with the fastest and slowest-movingparticle-like structures.

The quantity ht ) is the temporal measure entropy, which measures the informationcontent of time sequences of values of individual sites. It is evaluated by applyingthe same procedure as for h~) but to sequences of values of a single site attained onmany successive time steps. It can be shown (see pages 115- 157 in this book) thath (t ) < (A + A ) h (x)J1 - L R J1 '

The quantities h~) and ht )measure respectively the information content of spatialand temporal sequences that are one site wide. The quantity hJ1 gives the entropyassociated with spacetime patches of sites of arbitrary width. (Nevertheless, formany rules, the exact value of hJ1 is in fact obtained from patches of width I or 2.) Ingeneral, hJ1 ~ 2ht ), and ht ) -s hJ1 ~ (AL + AR)h1

X) .

The quantity hJ1 is evaluated by considering spacet ime patches of sites that extendin the time direction. The last column of the table uses a generalization in whichthe patches can extend in any spacetime direction. It gives the minimum valuehJ1 obtained as a function of direction. (The actual bounds given in the table wereobtained from vertical or diagona l patches; other directions may yield stricter bounds.)

Table by Peter Grassberger (Physics Department, University ojWuppertal).

543

Wolfram on Cellula r Automata and Complexity

Table 7: Blocking Transformation Equivalences

0 0: 0010(1 )

I 0: 11 10 (2)200: 00 II (2)204: 000 III (2)

2 34: 00 10 (2)170: 000 100 (3)0: 1000 1100 (4)

3 0: I I 10 (2)240: 00 11 (4)

4 204: 00 10 (1)0: 00 11 (1)

5 200: 00 10 (2)204: 000100 (2)0: I I I 110 (2)51: 00010 11010 (1)

6 184: 00 10 (2)34: 00 11 (2)170: 0000 1000 (4)128: 0100 1100 (4)240: 1000 1010 (4)85: 10000 11000 (5)0: 11000 11100 (10)

7 192: 00 10 (2)0: 000 100 (2)240: 000 II I (6)

8 0: 00 10 (1)

9 0: 0010 1110 (2)170: 1000 0010 (6)34: 1000 00 11 (6)204: 01000001100000 (5)240: 0000000o 11010000 (8)

10 34: 00 10 (2)170: 000 100 (3)0: 0 10 110 (3)

I I 240: 00 11 (2)0: 010 110 (3)15: 000 111 (3)128: 1100 100 1 (4)170: 1100100 1001100 (7)

12 204: 00 10(1 )0: 1001 10 (1)

13 192: 00 10 (2)0: 100110 (1)204: 10100 10010 (1)

14 240: 10 0 1 (2)34: 00 11 (2)15: 0 10101 (3)0: 1100 1000 (4)170: 0000 1100 (4)128: 11000110 (4)

544

15 240: 00 10 (2)15: 110001 (3)

18 90: 00 10 (2)204: 11000 10 100 (2)0: 10100 11100 (2)

19 51: 00 11 ( I)0: 11 10 (2)204: 00 11 (2)

22 146: 00 10 (2)90: 0000 1000 (4)0: 11011000 11111000 (4)

23 5 1: 00 11 (1)128: 00 10 (2)204: 00 11 (2)0: 000 100 (2)

24 48: 00 10 (2)240: 000 100 (3)0: 100 01 1 (3)

25 0: 1101000 1111000 (7)240: 0000000 1101000 (14)

26 90: 00 10 (2)85: 010 110 (3)170: 100 100 10 1100 (6)0: 10110100 10111100 (8)240: 11100100 100 11100 (16)

27 48: 11 10 (4)85: 0 10110 (3)240: 000 100 (6)0: 0 100 1100 (8)170: 100100 10 1100 (6)

28 192: 00 10 (2)200: 1001 (2)51: 100 110 (1)204: 100110 (2)0: 1010 1100 ( I)

29 204: 00 10 (2)200: 1001 (2)51: 100 110 ( I)0 : 1010 1100 ( I)

30

32 0: 00 11 ( I)128: 00 10 (2)

33 132: 00 10 (2)200: 00 11 (2)0: 11 1 100 (2)204: 000 I II (2)128: 0000 10 10 (4)

34 170: 00 10 (2)0: 100 110 (3)

35 240: 00 II (4)0: 100 11 0 (3)170: 10100 10010 (5)

36 0: 00 11 (I )

4: 00 10 (2)204: 000 100 (1)

37 200: 00 11 (2)0: 11 1111 00 (2)204: 0000 11 11 (2)170: 100000 11 0000 (6)240: 0 10000 11 0000 (6)128: 010000 111000 (6)

38 34: 00 10 (2)85: 100 110 (3)170: 0000 1000 (4)0: 11 00 1110 (4)

40 128: 00 10 (2)0: 00 11 (2)170: 110 10 10110 (5)

4 1 148: 00 10 (2)184: 0000 1000 (4)176: 0000 1010 (4)170: 11 010 10110 (5)240 : 00000000 10000000 (8)128: 10000000 10100000 (8)0: 11111000 11001000 (8)136: 0 1010000 10101101 (8)

42 170: 00 10 (2)34: 00 11 (2)0: 11 00 11 10 (4)

43 170: 1001 (2)240: 00 11 (2)15: 000 111 (3)0: 1001 1000 (4)128: 1100 0110 (4)

44 12: 00 10 (2)204: 000 100 (I )

0: 1000 11 00 ( I)

45

46 34: 00 11 (2)0: 110 100 (3)170: 000 110 (3)

50 51: 10 0 1 (1)128: 00 10 (2)204: 10 01 (2)0: 1010 1000 (2)

5 1 5 1: 10 01 (I )

204: 00 10 (2)

Tables of Cell ular Automaton Proper ties (19 86 )

54 50: 00 10 (2)51: 1000 00 10 (2)128: 0000 I000 (4)204: 1000 00 10 (4)170: 1000 1110 (4)240: 00 10 1110 (4)0: 0000 10 111010 (4)

56 240: 00 10 (2)128: 10 01 (2)184: 0 10 101 (3)0: 1010 0110 (2)34: 1010 1101 (4)170: 11010 10110 (5)

57 128: 10 01 (2)184: 0 1010 1 (3)0: 10 10 0 110 (2)48 : 1010 0 100 (4)34: 10 10 110 1 (4)240: 10 100 100 10 (5)170: 11010 10110 (5)

58 128: 00 10 (2)0: 101 100 (3)240: 1100 1110 (8)170: 11010 10110 (5)

60 60: 00 10 (2)

62 240: 11 00 1110 (8)204: 11000 11 0 10 (3)0: 111110 100000 (3)

72 0: 00 10 (I )

4: 00 II (2)204: 000 110 (1)

73 204: 1100 0 110 (2)5 1: 11 000 11010 ( I)0: 101101 0000 (2)

74 34: 00 10 (2)170: 000 100 (3)0: 10000 10100 (5)85: 1110000 1101000 (7)

76 204: 00 10 ( I)0: 1010 1110 (I )

77 204: 1001 ( 1)128: 00 10 (2)0: 1010 1000 (1)

78 0: 101 100 ( I)204: 11010 10110 (1)

90 90: 00 10 (2)

94 90: 00 11 (2)0: 1010 1110 (1)204: 1010 0 101 (2)5 1: 1001011110 (I )

136: 111100 11 0110 (6)192: 110110011 110 (6)

545

Wolfram on Cellular Automato and Complexity

104 128: 00 10 (2)4: 00 II (2)0: 000 100 (I )

204: 0000 1100 (1)

105 150: 00 10 (2)

106 170: 00 10 (2)108 76: 00 10 (2)

204: 000 100 (1)51: 10100 11100 (I )

0: 10010 11110 (2)

110 0: 110100 101100 (9)240: 11000 100110 (9)170: 10011000 11111000 (16)

122 128: 00 10 (2)90: 00 II (2)0: 1010 1000 (2)204: 1110011110 (2)

126 90: 00 11 (2)204: 11100 11110 (2)0: 0 1110 10001 (2)

128 0: 00 10 (I )

128: 00 II (2)

130 34: 00 10 (2)170: 000 100 (3)0: 1000 1100 (4)128: 1000 1111 (4)

132 204: 00 10 (I )128: 00 I I (2)0: 000 110 (I )

134 184: 00 10 (2)162: 00 II (2)170: 0000 1000 (4)128: 0100 1100 (4)240: 1000 1010 (4)85: 10000 11000 (5)0: 100000 10 1100 (6)

136 0: 00 10 (I)136: 00 II (2)

138 34: 00 10 (2)170: 00 11 (2)0: 010 110 (3)

140 204: 00 10 (I )

0: 100 110 (I )

142 240: 10 0 1 (2)170: 00 II (2)15: 0 10 101 (3)0: 1100 1000 (4)128: 1100 OlIO (4)

546

146 90: 00 10 (2)204: 11000 10100 (2)0: 100 101 1110 (2)

150 150: 00 10 (2)

152 48: 00 10 (2)240: 000 100 (3)136: 1000 1I1I (4)0: 0 1000 11000 (5)

154 90: 00 10 (2)85: 0 10 110(3)170: 1100 0110 (4)

156 192: 00 10 (2)200: 10 01 (2)136: 10 II (2)51: 100 110 (I )204: 100 110 (2)0: 1010 1100 ( I)

160 128: 00 10 (2)0: 000 100 (1)

162 170: 00 10 (2)128: 10 I I (2)0: 100 110 (3)

164 90: I I 10 (2)128: 00 II (2)204: 000100(1 )0: ()()()() 1100 (I )

168 128: 00 10 (2)136: 00 II (2)170: 10 I I (2)0: 000 100 (1)

170 170: 00 10 (2)

172 34: I I 10 (2)204: 000 100 (I )

170: 110 III (3)0: 1000 1100 (I )

178 5 1: 10 01 (I)128: 00 10 (2)204: 10 01 (2)0: 1010 1000 (2)

184 240: 00 10 (2)128: 1001 (2)170: 10 I I (2)184: 0 10 101 (3)0: 10 10 0 110 (2)

200 0: 00 10 ( I)204: 00 II (I )

204 204: 00 10 ( I)

232 204: 00 II (I )

128: 00 10 (2)0: 000 100 ( I)

Tables ofCellular Automaton Properties (1986)

Equivalences between rules under blocking transformations.

When only particular blocks of site values occur, the evolution of one cellular au­tomaton rule (say R) may be equivalent to that of anoth er (say R' ). Thu s for example,the evolution under rule 1 of configurations consisting of the blocks 000 and IIIis equivalent to evolution under rule 204 in which 000 is replaced by 0, and II Iis replaced by 1. (Two time steps in evolution according to rule I are necessary toreproduce one time step of evolution according to rule 204 .) Since rule 204 is theidentity, this implies that configurations consisting only of the blocks 000 and I I Imust be periodic under rule 1 (with period 2).

In general, one may con sider replacing site values 0 and I in evo lution acco rding torule R by block s Bo and B1• In some cases, the resultin g evolution may correspondto T time steps of another rule R'. Evolution acco rding to rule R' can thus be"simulated" by evolution according to rule R, under the blocking transformation

o --) Bo' 1 --) B i - Such blocking transform ations can be considered analogo us toblock spin transformation s in the renormalization group approach.

The table gives possible simulations for all the "minimal representative" rules oftable 1. The notation R': Bo B1 (T) indicates simulation of rule R' by replacing 0with the block Bo, and 1 with BI; T steps of rule R are needed to reproduce one stepof rule R' evolution.

The table includ es all simulations for block lengths up to 8. The blocks Boand B1

are always assumed distinct. Only one representative set of blocks is given for eachsimulation. (Thus for example, only the blocks 00 and 10 are given for the simulationof rule 90 by rule 18; the blocks 00 and 01 would also suffice.) Simulations withblock length 1 are not included; these correspond to transformations given in table 1.No simulations are found for rules 30 and 45 up to block length 8.

Many rules are seen to be equ ivalent under blocking transformations to simplerules, such as 204 (the identit y), 170 (left shift), 240 (right shift), 51 (complemen­tation) and O. Equi valence is also often found to the additiv e rules 90 and 150. Animportant property of all these simple rules is that they simulate themselves underblocking transformations. This has the consequence that patterns generated by theserules are self similar. Fractal patterns are thus produ ced by evolution according torules 90 and 150 from single site seeds, as shown in table 5.

The simulations given in the table occur when only particular blocks occur inthe configuration of a cellular automaton. In disordered configurations, all possibleblocks can occur. But since a cellular automaton under most rules is irreve rsible,only a subset of blocks may occur after a sufficiently long time. Often the subsetof blocks that occur is, at least approximately, the blocks which correspond to aparticular simulation. In this case, the behaviour of one cellular automato n may beconsidered "attracted" to that of another.

It is common to find "domains" in which only particular blocks occur. Withineach such domain, the evolution may correspond to that of a simpler rule. The

547

Wolfram on Cellu lar Automata and Comple x ity

domains are separated by walls or "defects" , whose behaviour is not reproduced bythe simpler rule. In some cases, the defects remain stationary; in others, they executerandom walks, and, for example, annihilate in pairs. In the latter cases , the sizes ofdomains grow slowly with time.

While a large subset of possible initial configurations for a cellular automaton maybe attracted to a particular form of behaviour, there are usually some special initialstates (typically occurrin g among disordered states with probabili ty zero), for whichvery different behaviour occurs. Such special initial states may for example consistof blocks which yield a simulation to which the rule is not generically attracted.

The blocking transformations considered in the table represent one form of trans­formation between rules. Many others can also be considered. A general class, whichincludes the blocking transformations of the table, are those transformations whichcan be carried out by arbitrary finite state machines.

The blocking transformations used in the table have the property that they reducethe total number of sites. This is a consequence of the fact that the blocks used arealways taken not to overlap . An alternati ve approach is to perform replacementsfor overlapping blocks, thus obtaining configurations with the same number of sites.An example of such a replacement is 00 ~ 0, 01 ~ I , I0 ~ 1, II ~ O. For somerules, the resulting transformed configurations show evolution according to otherk = 2, r = I cellular automaton rules. Rules related in this way must have thesame global properties, and must yield for example the same entropies. The minimalrepresentative rules from table I equivalent under such transformations are:

15,240 24023,232 13243, 21 2 18451,204 20477, 178 22285, 170 170105,1 50 15011 3,1 42 226

Main table by John Milnor (Institute fo r Advanced Study). (Original program byS. Wolfram.) Second table by Peter Grassberger.

548

Tables of C ellular Automaton Proper ties (1986)

Table 8: Factorizations into Compositions of Simpler Rules

¢ ¢t ¢20 0 0

0 120 480 600 1920 2040 2400 252

17 034 034 19251 068 068 19285 0

102 011 9 0136 0153 0170 0187 0187 3204 0221 0221 3238 0255 0255 3255 12255 15255 48255 51255 60255 63

¢ ¢t ¢'2I 17 192

238 32 17 48

238 123 17 240

51 192204 3238 15

8 119 48136 12

12 34 4834 24051 48

204 12221 12221 15

15 51 240204 15

18 17 60238 60

19 17 252238 63

24 102 48153 12

34 34 1234 20485 48

170 12221 48221 51

36 102 192153 3

¢ ¢t (h46 34 60

34 252221 60221 63

51 51 20485 240

170 15204 51

60 51 60102 240153 15204 60

72 11 9 60136 60

90 102 60153 60

126 102 252153 63

128 11 9 3136 192

136 85 311 9 51136 204170 192

170 85 51170 204

200 11 9 63136 252

204 51 5185 15

170 240204 204

Factorizations into compositions of simpler rules.

The 256 rules in table I are stated as functions of three site values ¢(a_/, ao> at). Ofthese, 48 depend only on two of the site values. Some other rules can be formedfrom compositions of these simpler rules. This table lists rules which can be formedby compositions according to

where - indicates that the value is irrelevant. Only minimal representative rules fromtable I are included. In each case, all possible compo sitions are listed. Note thatmost of the compositions do not commute.

Table by Erica l en (Los Alamos National Laboratory).

549

Wolfram on Cellula r Automata and Complexity

Table 9: Lengths of Distinct Blocks of Sites NewlyExcluded at Time t

rul e t = I 2 3 4

0 1 - - -I 3 - - -

2 2 - - -

3 3 - - -4 2 - - -5 5 - - -6 3 6 7 77 4 5 5 68 2 I - -9 4 7 9 9

10 3 - - -II 3 5 7 912 2 - - -13 4 4 6 614 3 5 7 915 - - - -18 3 II 12 1319 3 3 - -

22 8 7 11 923 5 6 7 824 2 3 - -25 5 6 8 826 4 10 8 II27 4 6 6 928 3 6 6 829 4 - - -

30 - - - -32 2 4 6 833 4 7 6 634 2 - - -35 4 6 7 936 3 2 - -37 9 8 9 838 4 3 - -

40 3 4 5 74 1 5 9 8 942 3 - - -

43 5 7 9 I I44 4 4 6 645 - - - -46 3 3 - -

50 3 5 9 115 1 - - - -

54 5 9 9 756 3 4 6 8

550

rule t -I 2 3 4

57 6 5 5 758 4 5 5 660 - - - -62 5 7 8 7

72 3 3 - -73 6 6 7 1474 4 6 6 776 3 - - -77 5 6 7 878 4 4 6 590 - - - -94 5 7 II II

104 8 8 8 7105 - - - -

106 - - - -108 5 4 - -

110 5 10 11 II122 5 7 8 10126 3 12 13 14128 3 5 7 9130 4 6 7 10132 4 5 6 7134 5 6 6 8136 3 4 5 6138 3 - - -

140 4 5 6 7142 5 7 9 II146 6 6 8 8150 - - - -152 5 5 6 6154 - - - -156 6 7 7 9160 5 7 9 II162 4 6 8 10164 9 9 8 9168 4 5 6 7170 - - - -172 4 5 6 7178 5 6 7 8184 4 6 8 10200 3 - - -204 - - - -

232 5 6 7 8

Tables of Cellula r Automaton Properties (1986)

Lengths of distinct blocks of sites newly excluded at time t.

Most cellular automaton rules are irreversible , so that even starting from all possibleinitial configurations, only a subset of configurations can occur after t time steps. Inthis subset of configurations, only certain blocks of site values can occur. The subsetcan be specified by giving the blocks which are excluded. In some cases (such as rule128), the number of distinct excluded blocks is finite; in other cases, it is countabl yinfinite. Irreversibility leads to an increase in the size of the set of excluded blockswith time.

The table gives the lengths of the shortest blocks which are newly excluded afterexactly t time steps. Such blocks can occur in configurations up to time t - I, butcannot occur at time t or after. The lengths L (t ) of the shortest blocks newly excludedat time t obey the inequality (see pages 159-202 in this book) L (t ) ~ L (t - I) - 2.

The notation - in the table indicates that no blocks are newly excluded at a par­ticular time step. This implies that the rule has reached a stable set of configurations,which can occur after any number of steps. It should be noted, however, that thistable takes no account of the probabilities with which different configurations mayoccur.

Table by Lyman P. Hurd (Mathematics Department, Princeton University). (Originalprogram by S. Wolfram.)

551

Wolfram on Cellular Automata and Complexity

Table 10: Regular Language Complexities

1 - I 1 - 2 1- 3 1 -4 1- 5 I > 5 00

0 1 [1] 1 [1] 1 [1]1 4 [6] 4 [6] 4 [6]2 3 [4] 3 [4] 3 [4]3 3 [5] 3 [5] 3 [5]4 2 [3] 2 [3] 2 [3]5 9 [15] 9 [15] 9 [15]6 9 [16] 13 [22] 22 [37] 26 [44] 31 [52]7 4 [7] 7 [12] 12 [2 1] 14 [24] 16 [27]8 3 [4] I [ I] 1 [I] I [ I]9 9 [16] 22 [40] 44 [80] 106[198] 266 [500]

10 4 [6] 4 [6] 4 [6]11 3 [5] 7 [12] 10 [17] 12 [20] 14 [23]12 2 [3] 2 [3] 2 [3]13 6 [II ] 10 [17] 12 [19] 14 [2 1] 16 [23]14 3 [5] 7 [12] 10 [17] 12 [20] 14 [23]15 1 [2] 1[2] 1[2]16 3 [4] 3 [4] 3 [4]17 3 [5] 3 [5] 3 [5]18 5 [9] 47 [91] 143 [270]19 3 [5] 5 [8] 5 [8] 5 [8]

20 10 [17] 2 1 [37] 32 [57] 37 [65] 50 [89]2 1 4 [7] 9 [16] 12 [21] 14 [24] 16 [27]22 15 [29] 280 [551] 4506 [8963]23 11 [20] 15 [26] 19 [32] 23 [38] 27 [44]24 2 [3] 3 [4] 3 [4] 3 [4]25 6 [ II ] 26 [50] 55 [106] 114 [220] 333 [649]26 13 [25] 92 [179] 2238 [4454]27 10 [18] 14 [25] 18 [32] 21 [37] 24 [42]28 3 [5] 8 [14] 10 [17] 11 [18] 12 [19]29 4 [7] 4 [7] 4 [7]

30 I [2] 1[2] 1 [2]32 2 [3] 5 [7] 7 [9] 9 [11] 11 [13] 21+ 1[21+ 3] 4 [6]33 5 [9] II [20] 26 [47] 40 [68] 4 1 [68]34 2 [3] 2 [3] 2 [3]35 4 [7] 7 [13] 9 [16] 10 [18] 12 [21]36 3 [5] 3 [4] 3 [4] 3 [4]37 15 [29] 194 [376] 870 [1698] 3735 [7290]38 5 [9] 5 [8] 5 [8] 5 [8]

40 10 [17] 12 [19] 15 [22] 18 [25] 2 1 [28]41 14 [27] 128 [250] 1049 [2069]42 3 [5] 3 [5J 3 [5]43 9 [16] 13 [22] 17 [28] 21 [34] 25 [40]44 4 [7] II [20] 18 [32] 23 [40] 27 [46]45 1 [2] 1 [2] 1 [2]46 3 [5] 5 [8] 5 [8] 5 [8]48 2 [3] 2 [3] 2 [3]49 4 [7] 6 [10] 7 [I I] 9 [14] 10 [15]

552

Tables of Cellular Automaton Properties (19861

/- 1 1 - 2 1 - 3 1-4 1 - 5 1 > 5 00

50 3 [5) 8 [14) 10 [17) 12 [20) 14 [23)5 1 I [2] 1 [2] 1 [2]52 4 [7) 5 [9] 5 [9) 5 [9]

53 10 [18] 15 [25] 17 [28] 2 1 [33] 23 [36]54 9 [16] 17 [32] 94 [179] 675 [13 16]56 3 [5) 5 [9] 7 [12] 9 [15] 11 [18]57 II [20] 15 [27] 15 [26] 24 [42] 32 [55]58 10 [18] 20 [35] 33 [55] 55 [88] 76 [122]

60 1 [2) 1[2] 1 [2)6 1 5 [9] 16 [30] 40 [76] 94 [ 177] 185 [350)62 5 [9) 2 1 [39) 6 1 [114) 8 1 [150) 129 [240)64 3 [4) I [I ) I [ I) 1 [ I)65 9 [15) 20 [35) 42 [75) 88 [157) 220 [40 1)66 2 [3) 3 [4) 3 [4) 3 [4)68 2 [3) 2 [3) 2 [3)69 5 [8) 10 [17) 12 [19) 14 [23) 16 [25)

70 3 [5) 8 [14) 9 [15) 11 [19) 11 [19)72 5 [9) 5 [8) 5 [8) 5 [8)73 15 [29) 82 [155) 390 [757) 1443 [2796)74 13 [25) 45 [85) 66 [123) 69 [125) 75[ 135]76 3 [5) 3 [5) 3 [5)77 I I [20) 15 [26) 19 [32) 23 [38) 27 [44)78 10 [18) 15 [27) 18 [30) 20 [34) 22 [36)

80 4 [6) 4 [6) 4 [6)8 1 3 [5) 7 [11) 9 [14) I I [16) 13 [19]82 13 [25) 167 [33 1) 3 134 [6257]84 3 [5) 7 [12] 9 [14] II [17] 13 [19)85 1 [2) 1 [2) 1[2]86 1 [2) 1 [2) 1[2]88 13 [25) 63 [117] 114 [2 10] 117 [21 3] 1288 [2 106)89 1 [2] 1 [2] 1 [2]

90 1 [2] 1 [2) 1 [2)92 10 [18] 14 [23] 18 [29] 18 [27] 22 [33]94 15 [29] 230 [455] 3904 [77 60]96 9 [16] I I [17] 14 [20] 17 [23] 20 [26)97 14 [27] 99 [195] 626 [1237]98 3 [5] 4 [6] 6 [9] 8 [12] 10 [15]

100 5 [9] 11 [19) 17 [29] 18 [29] 22 [34)102 1 [2] 1 [2] 1 [2)104 15 [29] 265 [525 ) 2340 [4647] 1394 [2675] 1542 [29 13]105 1 [2] 1 [2) 1[2)106 1 [2) 1[2] 1 (2)108 9 [16) II (19) 11 [19] I I [19)

110 5 [9) 20 [38) 160 [3 12] 1035 [2037]112 3 [5] 3 [5] 3 (5)113 9 [16) 13 [22) 17 [28) 2 1 [34] 25 [40]114 10 [18] 20 [35) 33 [56] 50 [82] 72 [115]116 3 [5) 5 [8] 5 [8) 5 [8]118 5 [9) 16 [29) 49 [92] 74 [139] 95 [175]

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Wo lfram on Cellula r Automato and Complexity

1 =1 1 =2 1 =3 1= 4 I=S I >S 00

120 1 [2] 1 [2] 1 [2]122 IS [29] 179 [347] S088 [9933]124 S [9] 20 [38] 208 [407] 13S6 [2672]126 3 [S] 13 [23] 107 [198] 2867 [S476]

128 4 [6] 6 [8] 8 [10] 10 [12] 12 [14] .21 + 2[21+ 4] 3 [S]

130 9 [I S] 14 [2 1] 18 [2S] 22 [29] 26 [33]132 S [9] 7 [12] 9 [I S] II [18] 13 [21]134 14 [27] 44 [82] 99 [182] 12S [224]136 3 [S] 4 [6] S [7] 6 [8] 7 [9] 1+2[1+4] 3 [S]138 3 [S] 3 [S] 3 [S]

140 4 [7] S [9] 6 [11] 7 [13] 8 [ I S]142 9 [16] 13 [22] 17 [28] 2 1 [34] 2S [40]144 9 [16] 16 [28] 20 [34] 24 [40] 28 [46]146 IS [29] 92 [177] I S87 [3126]148 14 [27] 68 [127] 113 [209] 188 [347]

ISO 1 [2] 1 [2] 1[2 ]IS2 6 [11] 20 [37] 30 [SS] 32 [S9] 36 [6S]IS4 1 [2] 1 [2] 1[2]IS6 11 [20] 20 [3S] 24 [42] 28 [47] 34 [S8]

160 9 [ IS] 16 [24] 2S [3S] 36 [48] 49 [63] (I + 2)2[(1 + 2)(1 + 4)] 9 [ IS]162 S [8] 7 [10] 9 [12] 11 [14] 13 [16]164 IS [29] 116 [227] 667 [1310] 1214 [2363]168 4 [7] S [8] 6 [9] 7 [10] 8 [11] 1+3 [1 +6] 3 [S]

170 1 [2] 1 [2] 1 [2]172 10 [18] 11 [20] 12 [22] 13 [24] 14 [26]176 6 [11] 8 [14] 10 [17] 12 [20] 14 [23]178 11 [20] IS [26] 19 [32] 23 [38] 27 [44]

180 I [2] 1 [2] 1 [2]184 4 [7] 6 [10] 8 [13] 10 [16] 12 [19]188 S [9] 14 [2S] 21 [36] 2S [43] 33 [S6]

192 3 [S] 4 [6] S [7] 6 [8] 7 [9]196 4 [7] S [8] 6 [9] 7 [10] 8 [I I]

200 3 [S] 3 [S] 3 [S]204 1 [2] I [2] 1 [2]208 3 [S] 3 [S] 3 [S]

212 9 [16] 13 [22] 17 [28] 2 1 [34] 2S [40]216 10 [18] 11 [19] 12 [20] 13 [2 1] 14 [22]

224 4 [7] S [8] 6 [9] 7 [10] 8 [I I] 1+3[1+6] 3 [S]

232 I I [20] IS [26] 19 [32] 23 [38] 27 [44]

240 1 [2] 1[2] 1[2]

Regular language complexities.

The set of configurations that can appear after t steps in the evo lution of a one­dimensional cellular automaton can be shown to form a regular formal language (seepages 159-202 in this book). Possible configurations thus correspond to possiblepaths through a finite graph which represents the grammar for the regular language.

554

Tablesof CellularAutomaton Properties 11986)

The table gives the min imum number of nodes in the graphs for such grammars; thenumber of arcs is given in brackets in each case. The notation . indicates that theregular language is the same as at the preceding time step.

Entries in the last column of the table give sizes of graphs for regular languagesrepresenting limiting sets of states that can be reached after any number of steps.

The size of a regular grammar gives a measure of the "complexity" of the setof configurations it describes. Notice that the grammar specifies merely whichconfigurations can possibly occur; it does not account for the probabilities of diffe rentconfigurations.

The graphs used for the table represent possible sequences of site values that occurin configuratio ns read from left to right. Ru les related by reflection may in generalyield different regular languages. The table thus includes minim al representativesfor all rules from table I not related by complementation.

Entries in the table for t :s; 5 that have been left blank were not found. Theyare probably ~ 20000 . The growth of regular language complex ities is bounded by22'" - l.

For some rules, it has been possible to find explicit forms for the regular languagesproduced after any number of time steps. Form ulae for complexities in these casesare listed in the table. In many cases, it is however suspected that the limiting setdoes not form a regular language, and may in fact be non-recursive.

Table by Lyman P. Hurd (Mathematics Department , Princeton University). (Originalprogram by S. Wolfram.)

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Wo lfram on Cellula r Automato and Complex ity

Table 11 : Measure Theoretical Complexities

rule 1 = 1 t =2 t = 30 0 0 0I 0.822 3 0.8223 0.82232 0.7356 0.7356 0.73563 0.9003 0.9003 0.90034 0.3768 0.3768 0.37685 1.8005 1.8005 1.80056 1.783 1.964 1.9687 1.2707 1.670 1.9338 0.7356 0 09 1.9 135 2.598 3.303

10 1.1247 1.1247 1.124711 1.0434 1.3442 1.97312 0.5623 0.5623 0.562313 1.4756 1.7879 1.71314 0.9026 1.3607 1.92715 0 0 016 0.7356 0.7 356 0.735617 0.9003 0.900 3 0.900318 0.9026 2. 129 3.93319 0.9026 1.1539 1.153920 1.7756 2.759 3.0592 1 1.2707 1.8266 1.93 122 2.59 1 4.601 6.21323 1.9862 2.153 2.33024 0.5623 0.9216 0.92 1625 1.643 2.665 3.23126 2.244 2.659 2.94527 1.666 2.128 2.39228 0.8305 1.6009 1.664529 1.2652 1.2652 1.265 230 0 0 032 0.3768 0.4957 0.234633 1.2930 1.941 2.52934 0.5623 0.5623 0.562335 1.1034 1.748 2.00636 0.9003 0.4634 0.463437 2.518 4.435 5.41038 1.298 1.256 1.25640 1.775 1.547 1.1274 1 2.332 4.134 5.47142 0.9003 0.9003 0.900 343 1.9584 2.269 2.48344 0.900 3 1.574 1.74845 0 0 046 0.5623 0.92 16 0.921648 0.562 3 0.5623 0.562349 1.251 2 1.451 1.45550 0.8305 1.589 1.7755 1 0 0 0

556

rule 1 = I 1 =2 1=352 0.9003 0.92 16 0.92 1653 1.906 2.057 2.0 1654 1.7707 2.609 3.92 156 0.8305 1.3503 1.73257 2.086 2.190 1.94658 1.9 132 2.132 2.10660 0 0 06 1 1.430 2.065 3.07662 1.34 1 2.406 3.72064 0.7356 0 065 1.5310 2.268 2.97066 0.5623 0.92 16 0.92 1668 0.5623 0.5623 0.562369 1.0562 1.790 1.84270 0.8305 1.7802 1.81572 0.900 3 0.4634 0.463473 2.604 3.685 4.47374 2.461 2.7 13 2.75576 0.8305 0.8305 0.830577 1.9862 2.153 2.33078 1.7553 2.111 2.02980 1.1247 1.1247 1.12478 1 1.0434 1.5837 1.71682 2.460 3.823 5.37584 0.8992 1.740 1.98485 0 0 086 0 0 088 2.244 1 3.08 1 3.60589 0 0 090 -0 0 092 1.91 32 1.768 1.73594 2.599 3.682 5.3 1196 1.782 1.3689 0.99597 2.49 1 3.470 4.84698 0.83 05 0.8932 0.979

100 1.298 1.667 1.632102 0 0 0104 2.59 1 4.379 4.969105 0 0 0106 0 0 0108 1.7707 1.5093 1.5093110 1.344 2.435 3.407112 0.9003 0.900 3 0.900 3113 1.957 2.266 2.482114 1.754 2.344 2.825116 0.5623 0.92 15 0.9215118 1.342 1.945 2.827120 0 0 0122 2.600 4.307 5.981

Tables of Cellular Automaton Properties (1986 )

rule 1 = 1 1=2 1 =3124 1.343 2.321 3.933126 0.9003 2.049 3.9 14128 0.8223 0.457 0.1986130 1.533 1.263 1.025132 1.292 1.459 1.637134 2.496 3.050 3.010136 0.9003 0.8223 0.641138 1.0434 1.0434 1.0434140 1.2512 1.5808 1.7551142 1.957 2.266 2.482144 1.9 13 1.988 2.056146 2.604 3.742 5.350148 2.328 3.589 3.8 15150 0 0 0152 1.644 2.626 3.03 1154 0 0 0156 2.083 2.506 2.563160 1.805 1.633 1.281162 1.0562 0.8654 0.7355

rule 1= 1 1=2 1-3164 2.520 3.343 3.353168 1.2707 1.3676 1.369170 0 0 0172 1.909 2.080 2.242176 1.4757 1.723 1.863180 0 0 0184 1.2652 1.575 1.788188 1.433 1.962 1.733192 0.9003 0.8113 0.64 15196 1.1034 1.0302 0.9 170200 0.9003 0.9003 0.9003204 0 0 0208 1.0434 1.0434 1.04342 12 1.9584 2.269 2.4832 16 1.667 2.033 2.065224 1.2707 1.368 1.369232 1.9862 2.153 2.330240 0 0 0

Measures of the information content of regular grammars for sets ofconfigurations genera ted by evolution from disordered initial states.

This table gives values of a probabili stic analog ue of the regular language complexityof table 10, in which the nodes of regular language graphs are weighted with theprobabili ties that they are vis ited.

Starting from a diso rdered state in which all possible configurations occur withequal probabil ity, irreversible cellular automaton evo lution can lead to ensembles inwhich different config urations occur with different probabilities. These ensemblescan be descri bed by probabilistic analogues of regular languages.

All the configurations that can occur after t steps correspond to possible pathsthrough the standard regular language graphs of table 10. To account for the differentprobabil ities of different configurations, one may weight the nodes of the graphaccordi ng to the probabi lities Pi that they are visited. In terms of these probabi lities,one may then compute a measure theoretical complexity - L Pi log, Pi' where thesum runs over all nodes in the regular language graph. The table gives estimatedvalues for this quantity. The last digit in each estimate is subject to statistical errors.

Table and concept by Peter Grassberger (Physics Department, University ofWupper­tal) .

557

Wo lfra m on Cellular Automata an d Complexity

Table 12: Iterated Rule Expression Sizes

rule 1 = I 1 = 2 1 = 3 1= 4 1 = 50 0(0) 0 (0) 0 (0) 0(0) 0 (0)I I (I) 6 (4) I (I) 6 (4) I (I )2 I (I) 1 ( I) I (I ) I (I) ) -/ 1 (1)3 I (I) 3 (2) I (I) 3 (2) I (I )4 I ( I) I ( I) I (I ) 1 ( I) 1 ( I)5 I (I ) 2 (2) 1 (1) 2 (2) I ( I)6 2 (2) 4 (4) 13 (10) 25 (2 1) 110 (50)7 2 (2) 7 (4) 6 (6) 18 (8) 10 (10)8 I ( I) 0 (0) 0 (0) 0 (0) 0 (0)9 2 (2) 5 (5) 18 (11) 43 (31) 138 (53)

10 I (I) 1 ( I) I (1) I (I ) 1 (I)II 2 (2) 5 (4) 10 (6) 26 (12) 50 (16)12 I (1) I ( I) I (I) 1 ( I) I (I)13 2 (2) 6 (4) 9 (5) 13 (7) 17 (8)14 2 (2) 5 (5) 17 (10) 51 (24) 144 (48)15 I (I ) I ( I) I ( I) I ( I) 1 ( I)18 2 (2) 4 (4) 18 (18) 35 (26) 140 (108)19 2 (2) 8 (5) 3 (3) 8 (5) 3 (3)22 3 (3) 7 (7) 27 (26) 80 (62) 308 (206)23 3 (3) 8 (5) 7 (7) 33 (9) I I (II )24 2 (2) 4 (4) 4 (4) 4 (4) 4 (4)25 2 (2) 4 (4) 8 (8) 20 (16) 42 (27)26 2 (2) 6 (6) 2 1 (17) 56 (43) 192 (100)27 3 (2) 4 (4) 7 (5) 7 (6) 15 (7)28 2 (2) 4 (4) I I (7) 15 I I) 30 (12)29 3 (2) 4 (4) 3 (2) 4 (4) 3 (2)30 3 (3) 9 (7) 23 (17) 76 (4 1) 185 (105)3 1 2 (2) 6 (4) 7 (6) 12 (10) 11 (9)32 I ( I) 1 (I) I (I ) 1 ( I) I (I )33 2 (2) 7 (7) 12 (12) 44 (23) 38 (24)34 I (I ) I (I) I ( I) I (1) I ( I)35 2 (2) 4 (3) 8 (4) 13 (8) 30 ( I I)36 2 (2) 2 (2) 2 (2) 2 (2) 2 (2)37 2 (2) 8 (7) 25 (17) 75 (47) 238 (109)38 2 (2) 4 (3) 4 (4) 4 (3) 4 (4)39 3 (2) 3 (3) 6 (5) 5 (5) 15 (7)40 2 (2) 3 (3) 5 (5) 8 (8) 13 (13)41 3 (3) 8 (8) 26 (24) 92 (69) 283 (2 18)42 2 (2) 2 (2) 2 (2) 2 (2) 2 (2)43 3 (3) 7 (6) 24 (12) 62 (27) 176 (55)44 2 (2) 3 (3) 4 (4) 8 (5) 10 (6)45 3 (3) 9 (8) 24 (20) 72 (53) 2 19 (118)46 3 (2) 6 (4) 6 (4) 6 (4) 6 (4)47 2 (2) 5 (4) 13 (6) 28 (8) 64 (16)50 2 (2) 6 (4) 15 (6) 3 1 (8) 64 (10)5 1 I (I) I (I ) I ( I) I ( I) I ( I)54 3 (3) 7 (6) 18 (15) 59 (38) 165 (85)55 2 (2) 5 (3) 5 (5) 5 (3) 5 (5)

558

rule 1 - I 1 - 2 1 - 3 1 -4 1 - 556 2 (2) 6 (4) 14 (9) 38 (20) lO3 (45)57 3 (3) 7 (6) 17 (12) 41 (23) 130 (50)58 2 (2) 6 (5) 15 (10) 34 (18) 80 (32)59 2 (2) 4 (4) 9 (8) 14 (10) 34 (17)60 2 (2) 2 (2) 8 (8) 2 (2) 8 (8)6 1 3 (3) 4 (4) 14 (10) 21 (17) 60 (30)62 3 (3) 6 (6) 20 (12) 56 (27) 137 (48)63 2 (2) 2 (2) 2 (2) 2 (2) 2 (2)72 2 (2) 2 (2) 2 (2) 2 (2) 2 (2)73 3 (3) 8 (7) 36 (20) 90 (46) 276( 118)74 2 (2) 5 (5) 13 ( I I) 30 (22) 77 (45)75 3 (3) 8 (8) 24 (20) 8 1 (52) 24 1 (118)76 2 (2) 2 (2) 2 (2) 2 (2) 2 (2)77 3 (3) 7 (5) 14 (7) 32 (9) 57 (11)78 2 (2) 5 (4) 8 (7) 20 (10) 2 1 (12)79 2 (2) 7 (4) 9 (5) 13 (6) 20 (7)90 2 (2) 2 (2) 8 (8) 2 (2) 8 (8)"9 1 3 (3) 9 (6) 26 (19) 82 (47) 255 (107)94 3 (3) 8 (8) 26 (19) 106 (46) 276 ( lO6)95 2 (2) 2 (2) 2 (2) 2 (2) 2 (2)

104 3 (3) 6 (6) 15 (14) 27 (26) 49 (45)lO5 4 (4) 4 (4) 16 (16) 4 (4) 256 (256)lO6 3 (3) 5 (5) 25 (2 1) 46 (37) 192 (126)lO7 4 (4) 10 (9) 37 (28) 108 (70) 390 (210)lO8 3 (3) 5 (5) 9 (8) 5 (5) 9 (8)lO9 4 (4) 10 (8) 3 1 (20) 9 1 (54) 268 (118)11 0 3 (3) 7 (6) 15 (15) 40 (28) 95 (60)II I 3 (3) 7 (6) 21 (14) 57 (25) 139 (56)122 3 (3) 9 (8) 27 (20) 88 (48) 264 (136)123 3 (3) 8 (8) 22 (13) 51 (28) 8 1 (30)126 3 (3) 8 (8) 22 (19) 103 (67) 22 1 (116)127 3 (3) 3 (3) 3 (3) 3 (3) 3 (3)128 I ( I) I (I ) I ( I) 1 ( I) I ( I)129 2 (2) 9 (8) 26 (20) 93 (78) 250 (120)130 2 (2) 2 (2) 5 (4) 5 (4) 8 (6)131 2 (2) 6 (5) 13 (10) 36 (25) 88 (45)132 2 (2) 3 (3) 4 (4) 5 (5) 6 (6)133 2 (2) 8 (7) 23 (17) 74(4 1) 2 16(1 11)134 3 (3) 6 (6) 20 (17) 46 (34) 174 (90)135 3 (3) 9 (7) 22 (17) 66 (4 1) 202 (107)136 I ( I) I ( I) 1 ( I) I (I) I ( I)137 2 (2) 8 (6) 14 (14) 39 (25) 111(60)138 2 (2) 2 (2) 2 (2) 2 (2) 2 (2)139 2 (2) 5 (4) 6 (4) 6 (4) 6 (4)140 2 (2) 2 (2) 2 (2) 2 (2) 2 (2)141 2 (2) 7 (4) 10 (6) 22 (8) 28 (9)142 3 (3) 7 (6) 18 (12) 52 (27) 151 (55)143 2 (2) 5 (4) 12 (8) 32 (15) 86 (34)

Tablesof Cellular Automaton Properties119861

rule 1= I 1= 2 1 = 3 1=4 1-5146 3 (3) 6 (6) 29 (29) 6 1 (48) 224 ( 193)147 3 (3) 8 (7) 2 1 ( 18) 69 (39) 207 (79)150 4 (4) 4 (4) 16 (16) 4 (4) 256 (256)151 4 (4) 10 (7) 33 (29) 104 (63) 372 (20 1)152 2 (2) 4 (4) 7 (7) 12 (I I) 20 (19)153 2 (2) 2 (2) 8 (8) 2 (2) 8 (8)154 3 (3) 4 (4) 28 (15) 6 (6) 42 ( 19)155 3 (3) 7 (4) 8 (5) 8 (4) 8 (5)156 3 (3) 3 (3) 14 (8) 12 (8) 43 (13)157 3 (3) 7 (4) I I (7) 24 (8) 23 (10)158 4 (4) 12 (8) 34 (20) 106 (37) 330 (92)159 3 (3) 9 (7) 18 (14) 63 (24) 139 (55)160 I ( I ) I ( I) I (1) I (I) I (I )161 2 (2) 6 (6) 20 (18) 65 (43) 236 (140 )162 2 (2) 3 (3) 4 (4) 5 (5) 6 (6)163 2 (2) 5 (5) 15 (9) 32 (15) 73 (25)164 2 (2) 5 (4) 10(10) 16 (13) 27 (2 1)165 2 (2) 2 (2) 8 (8) 2 (2) 8 (8)166 3 (3) 4 (4) 24 (15) 6 (6) 32 (19 )167 3 (3) 8 (7) 26 (19) 82 (43) 2 18 ( 104)168 2 (2) 5 (4) 10 (8) 23 (16) 49 (32)169 3 (3) 6 (5) 28 (2 1) 76 (37) 244 (124 )170 I (I ) I (I) I (I ) I (I ) I ( I)171 3 (2) 3 (2) 3 (2) 3 (2) 3 (2)172 2 (2) 4 (3) 6 (4) 7 (5) 9 (6)173 3 (3) 10 (7) 32 (14) 70 (27) 206 (46)174 2 (2) 2 (2) 2 (2) 2 (2) 2 (2)175 2 (2) 2 (2) 2 (2) 2 (2) 2 (2)178 3 (3) 7 (5) 16 (7) 32 (9) 65 ( I I)179 2 (2) 4 (4) 8 (6) 15 (8) 3 1 ( 10)182 4 (4) II (9) 47 (33) 103 (55) 466 (162)183 3 (3) 7 (7) 24 (22) 55 (22) 203 (69 )

rule 1 -I 1- 2 1-3 1-4 1 - 5184 2 (2) 5 (5) 15 (13) 48 (37) 161 (1 11)185 3 (3) 5 (5) 24 ( 10) 5 1 (22) 149 (45)186 3 (2) 8 (3) 20 (4) 43 (5) 88 (6)187 2 (2) 2 (2) 2 (2) 2 (2) 2 (2)188 3 (3) 6 (5) 24 (10 ) 39 ( 15) 125 (23)189 3 (3) 10 (5) 8 (5) 8 (5) 8 (5)190 4 (3) 5 (4) 23 (6) 2 1 (7) 9 1 (9)191 3 (3) 3 (3) 3 (3) 3 (3) 3 (3)200 2 (2) 2 (2) 2 (2) 2 (2) 2 (2)20 1 3 (3) 10 (5) 23 ( 10) 10 (5) 23 ( 10)202 2 (2) 4 (4) 7 (6) I I (8) 16 (10)203 3 (3) 7 (5) 17 (9) 4 1 (13 ) 66 (19 )204 1 (I) I (I) I (I ) I (I ) I (I )205 3 (2) 3 (2) 3 (2) 3 (2) 3 (2)206 3 (2) 5 (3) 8 (4) I I (5) 15 (6)207 2 (2) 2 (2) 2 (2) 2 (2) 2 (2)2 18 3 (3) 9 (6) 40 ( 16) 92 (24) 158 (38)219 3 (3) 10 (5) 10 (5) 10 (5) 10 (5)222 4 (3) 10 (5) 19 (7) 3 1 (9) 46 (I I)223 3 (3) 3 (3) 3 (3) 3 (3) 3 (3)232 3 (3) 8 (5) 19 (7) 33 (9) 58 (I I)23 3 4 (4) 10 (7) 39 (20) 112 (34) 307 (63)234 2 (2) 6 (4) 17 (7) 44 ( 12) 106 (2 1)235 4 (3) I I (6) 28 ( 10) 62 (14) 134 (19)236 3 (2) 3 (2) 3 (2) 3 (2) 3 (2)237 4 (3) 7 (5) 7 (5) 7 (5) 7 (5)238 2 (2) 4 (3) 6 (4) 9 (5) 12 (6)239 3 (3) I (I ) 1 (1) 1 (I ) I (I )250 2 (2) 3 (3) 4 (4) 5 (5) 6 (6)25 1 3 (3) 6 (5) 10 (7) 19 (9) 28 (I I )254 4 (3) 11 (5) 24 (7) 45 (9) 76 (I I)255 I ( I) I (I ) I ( I) I ( I) I (I)

Sizes of Boolean expressions representing function s corresponding to iterationsof cellular automaton rules.

Cellu lar automaton rules with k =4and r = I can be expressed as Boolean functionsof three variables, as in table 1. Iterations of these rules for t steps correspond tofunctions of 2t + I variables, which may be expressed as Boolean expressions.

The minimal Boolean expressions obtained after one step were given in table I.This table gives the numbers of terms in Boolean expressions obtained after t timesteps. An increase in these numbers potentially reflects increasing difficulty ofcomputing the outcome of more steps of cellular automaton evolution.

The first number in each case gives the number of prime implicants in the corre­sponding Boolean expression. The possible values of a set of n Boolean variab lescorrespond to the vertices of a Boolean n-cube. The cases in which a Boolean func­tion has value I then correspond to a region of the Boolean n-cube. The number

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of prime impli cants is essentially the number of hyperplanes of various dimensionswhich must be combined to form this region .

Boolean expressions can conveniently be stated in a disjunctive normal form(DNF), in which they are writt en as a disjunction (OR) of conjunctions (ANDs). Thenumb er of prime implicants gives an upper bound on the number of terms needed in

such a form.Notice that complementation of a functi on has no simple effect on its DNF

express ion. As a result , the table includes minimal representatives for all rules fromtable 1 not related by reflection.

The general problem of finding an absolu tely minimal DNF representation for afunction appears to be computationally intractable. The table gives in parenthesesthe numbers of terms in minimal DNF expression s found by the espresso computerprogram (R. Rudell , Computer Science Department , University of California, Berke­ley, 1985) which incorporates kno wn algebraic and heur istic techn iques. In mostcases, the result s given are prob ably absolutely minimal.

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Table 13: Finite Lattice State Transition Diagrams

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State transition diagrams for cellular automata on finite size lattices.

A k = 2 cellular automaton on a finite latt ice with N sites has a total of 2N possiblestates. The complete evolution of such a cellular automaton can be represented by afinite diagram which shows the possible transition s between these states. Each nodein the diagram corresponds to a complete configuration or state of the finite cellularautomaton. A directed arc leads from each such node to its successor under one timestep of cellular automaton evolution. The possible time sequences of configurationsin the complete evolution of the cellular automaton then correspond to possible pathsthrough the directed graph thus formed .

After a time of at most 2N steps, a finite cellular automaton must always entera cycle, periodi cally visiting a fixed set of states. In general, the complete statetransition diagram contains a number of distinct cycles.

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Tablesof Cellular Automaton Properties (19 861

The table shows the fragment of the state transition diagram associated with thelongest cycle, for all inequi valent k =2, r = I rules. Results are given for lattices ofsizes N = 9, N = IO and N = II. In all cases, the latti ces are taken to have periodicboundary conditions, as if their sites were arrranged in a circle.

The table also gives the length s and mult ipliciti es of all the cycles for each rule.(The notation used is g x L , representin g g cycles of length L. ) otice that the statetrans ition diagram frag ments assoc iated with different cycles of the same length maynot be identical. When there are several cycles of maximal length , the fragmentshown is the one invo lving the largest total numb er of states.

State transition diagram fragments have the general form of cycles fed by trees.The cellular automaton always reaches the cycle after a sufficiently long time. Thetrees represent transients, and contain states which can occur only after a limitednumber of time steps. Such transient phenomena are a manifestation of irreve rsibilityin the cellular automaton evo lution.

Some finite cellular automata, such as rule 13, are revers ible, so that their statetransition diagrams contain no transient s, and all states are on cycles.

In some other cases, such as rule 90, highly regular state transition diag rams areobtained, containing for example only balanced trees (see pages 159- 202 in thisbook). Many rules, however, yield complicated state transition diagrams.

In the pictures given, individual nodes are not indicated. Nevertheless, the arcsjoi ning nodes are all of equal length in a particular diagram. The overall scale ofeach diagram can be deduced from the total cycle length given.

The constra int of equal length in some cases forces arcs to intersect in the diag ram.In some cases , there are dense areas containing large numbers of arcs. For highlyirreversible rules, such as rule 0, large numbers of arcs converge on a single node,and appear essentially as a filled black circle .

Notice that the results given here and in table 14 for cellular automata on finitelatt ices with periodic boundary conditions also apply to infinite cellular automata inwhich only spatia lly periodic configurations are considered.

Table by Holly Peck (Los Alamos Na tional Laboratory).

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Table 14: Global Properties for Finite Lattices

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Tables of Ce llular Automaton Properties 11986)

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Global properties of cellular automata on finite lattices.

This table gives some properties of state transition diagram s for finite cellular au­tomata. Each picture shows values plotted as a function of latt ice size N , varying

from 3 to 16. (Periodic boundary conditions are assumed for the cellular automatonevolution.)

The quantities shown are as follows . (In all cases, values for integer sizes N areshown joined by lines.)

"Mean transient length" represents the average number of steps necessary for anyparticular state to evolve to a cycle. "Maximum transient length" gives the maximumnumber of steps needed.

"Mean cycle length" gives the average length of the cycle on to which anyparticular state evo lves. (Each cycle is thus weighted in the average with the numberof states which evo lve to it.) "Maximum cycle length" gives the longest cycle foreach value of N. Some such cycles are shown in table 13.

"Fraction on cycles" (given in logarithmic form) represent s the fraction of all 2N

possible states which appear on cycles, and thus can occ ur after a long time. Thisquantity is related to the set (topological) entropy for invariant set of the evolution.

The last picture for each rule gives the total number of distinct cycles (in loga­rithmic form). This can be considered as the number of possible distinct attractorsfor the evo lution.

Table by Holly Peck (Los Alamos Nationa l Laboratory). (Original program byS. Wolfram.)

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Table 15: Structures in Rule 110

Tables of Cellular Automaton Properties (19861

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Structures in rule 110.

The previous two pages show patterns produ ced by evolution according to rule 110,starting from a disordered initial configuration. The first picture shows all sites ona size 400 lattice. The second picture shows every other site in space and time on a

size 800 lattice.The configurations produced after many steps can be represented in terms of

particle-like structures superimposed on a periodi c background. The background isfound to have spatial period 14 and temporal period 7, and corres ponds to repeti­tions of the block B = 10011011111000. The configurations are then of the form. . . BBBBPBBB , where the particles P that have been found so far are :

veloci ty

-6/ 12-2/4

-14/42- 8/ 30-4/15-4/36-8/20

0/70/70/7

2/ 102/ 102/3

p

1000110011101111111000111110001110000 111011111111110001001 100110001 I 1111000000001110111111 1100011110000 I 10001111111 100010001 10001001101 1111111000111010110001110100011011 111000111000

The "velocity" is written as (spatial period)/(temporal period).One may speculate that the behaviour of rule 110 is sophisticated enough to

support universal computation.

Table of particles by Doug Lind (Mathematics Department, University a/Washington,

Seattle).

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Table 16: Patterns Generated by Second-Order Rules

1111111Itlll11111. 1II~IIr ul . 8 ( 88888888) r u l. , ( 8888 888 1) r u l. 2 ( 8888881 8) rut e 3 (888 00011)

111111111 fllll\ljlr ul .4 (00000' 8 0) ru l. 5 (0000 0' 01) rul .6 (0 00001 10) rul .7 (00000 " 1)

11111111'1 ":" I~:" II'I' 1<11"1'" ," 1"11:' I! :1 i 1

1in ~ i :' ' ::: L ;~ ~ l "

! ,I 1 t.~ ;: g -: : ~< _, 4> ~: ~ I , : ~ :' ~ :

rul .8 (00881 888) rut e 9 (8888'881) r u l . 18 (8888'818 ) rut e " (8888 1811)

11!l lrl l 'I '~ f ' I 'Wllf.:i:-ll-". III! ;Ilill :~;i::"~~~; ':1! _~: Ii :i~ : l~:;~ :", .~ i

rul e 12 ( 88 0811 88) rul . 13 ( 888811 81) r u t e ' 4 ( 888 8111 8) rut e 15 ( 88881111)

IIIIBI II i ~ ~ '111'~i11r u l. 18 ( 8881 881 8) r ut e 19 ( 888 ' 88 '1) r u fe 22 ( 888 1811 8 ) r ut e 23 (8881 81'1)

ru l . 24 (88811 888) r uf e 25 (88811 881) rul e 26 (88811 8' 8) ru l. 27 (888" 811)

I~ ' I' ·.. lor; "

r :r j r .. ' .. , ' " :.r; ~ ~ r:

p .. .. I ..... r,. ..

rul e 28 ( 88 0111 88) rul.29 (888' 1181 ) rul. 38 (88811"8) ru l.32 ( 88 108 088 )

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Tables of Cellulor Automaton Properties (1986)

• .• • •. .. I'll -- b ! - ~ - _ .I. ~ -ru l e 33 ( ee 1eeee l ) rut e 34 ( ee l e ee 1e) r u t e 35 ( ee1 eee l1 ) r u l e 36 ( ee 1ee l ee )

•...... . . :.il .··: . 4,.• . .A ~ . : -, r

,', . ' . . 1 . . .,' •.. .. ~

.. .' '. , : ~ I .~~ ' " ' ;- . ... . . . ~ . ~ ..

r u t e 37 ( ee l ee 1e 1) rul e 36 ( ee1 ee1 1e) r u l e 4e (eel e1eee) r ul .41 ( ee 1e 1ee 1j

••iIIIEr u l. 42 (ee l e l e1 e) r ul e 43 (ee 1e1 ell) rul. 44 ( ee 1e1 1ee) r ul e 45 ( ee 1e 11e 1)

••••r ul e 46 ( ee 1e 111e) ru l.5e ( eel lee 1e ) r u l . 51 (e e 11ee11) r ut e 54 ( ee 11e 11e )

• ' ..-.,'"-,,,~-, .. . _- . ~ .

rul e 56 (ee111eee ) rul.57 (ee111ee1) rul.56 (ee11 1e 1e) rut e se (ee n t iee )

_ _ r r1111=li

I · ~j~:(.f ~- ~ !hllll! t -~r~r u l . 61 ( eel111 e1 ) rul . 62 ( ee 11111e ) ru l .72 ( e l eel eee ) r vl e 73 ( e 1ee l ee 1)

Illliiill irlll ll· ·I~~rl'·· :;~~I~ll·~j ~1,~i~~I~~':i: ~ ::: :li i ~':. " '::~'" , ~~~. ~: .~M~~:i: :::: :: i ~ ',:~ '~ " -: '; %~" ~~~ ~':2: :::; :: a ~ , ::. ~~:' ~ , ~,~ ~..J.!1>! +"! .,.

r u l e 74 ( 0 180 10 10) rul e 76 ( 0 10011 00) ru l e 77 ( 01 e 01 10l) r u le 78 ( 0 10011 10)

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Wolfram on Cellular Automata a nd Complexity

rut e 90 ( 0'0110' 0 )

III.r u l e 94 ( 0 101 111 0) rul e 104 ( 0 " 01 000) r ut e ' 05 (0 110 100 1)

•. . "'I· · ~I· · • .,. . • .. .<" ~ ,...., ~ ~ '. .

" '. ~ . .. r .. .

~ /, ',. ..'. ':,(",,0 ._ . I,..,... .

r u l e 106 ( 0 1101 0 10) r u fe 108 ( 011 011 00) r u l e 110 (0,10 ', , 0) eu fe ' 22 (0 1, ', 01 0 )

• 1I11I111I 11111m Ifillll!IIIr ot e '26 ( 0 111 111 0) rut e 128 (1 0000000) rvl e 130 ( 1000001 0) r u t e 132 (1 0000 100)

1111111II~~Inll II1III!I!I!1111: !I~ ~~.Hili Ili~lil '~' ill

,u l . 134 ( 10000 110) r ut e 136 ( ' 000 ' 000) rul. 138 (1000101 0) rul e 140 (1 0001 100 )

rcre 142 (1 0001" 0) r u t e 14 6 ( 1001 00 10) r ul . 15 0 (1 00 10 110) , u '. 152 (1 00" 000)

I,11111il ll· .. ' ... : : :: ~

. ~ :1. .~ ~ . ~ • ]I:

rul e 154 ( 1001 10 10) r ut e 156 ( ' 00 11 100) rul e 160 ( 10 ' 00000) ru l e 162 (1 0' 000 10)

~.~0'1 II'~;< ~,-~IIJ~It~ ~I , __ n ~I.I

r u t e 164 ( 10 100 100) r ul e 168 ( '01 01 000) rul e 170 (' 0 101 0 10) r ul e 172 ( 10 10 1100)

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•"" .. ' :' . .'•. ~. . . ,t . I . ( >. [ •

. ' " .. - .

. ' . _ _. _. t

Tables of Cellular Automaton Properties (1986)

1\11111\1111\rule 178 (,e" ee,e) r u le 184 (1 0 11 1000 ) r ut e ' 88 (,e" " ee) r u le 2ee (" ee, eee)

r ul e 2e4 (" ee" ee) r u le 232 (",e,eee)

~ •r ut e e (eeeeeeee) ru l e 1 (eeeeeee, ) ru I. 2 (eeeeeele) r u t e 3 (eeeeeel ' )

~h~

'u l . 4 (eeeeeree) r ul e 5 (eeeee,el ) ' ul . 6 (eeeee" e) , u l .7 (eeeee" , )

I~~?>·~.--- ~ ••~L::':':':':::'~. .. .... .... -,u l . 8 (eeeel eee) r u I. 9 (eeee,ee, ) rul e , e (eeee,e, e) r u l e 11 (eeee, e" )

Iii ~ IIJWlli§rul e 12 (eeee" ee) rut e '3 (eeee" eq r ut e 14 (eeeel " e) rule '5 (eeee" ,, )

'u l . '8 (eeelee,e) rule 19 (eee,ee,,) rut e 22 (eeele" e) r ut e 23 (eee,e" ,)

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ru le 24 ( 000 1 ' 000 ) ru l e 25 ( 000 " 00') r u l e 26 ( 000 1 ' 0 ' 0) r u l e 27 ( 000 110 11)

~ _I.~ A.. ---,~~r u Ie 28 (00011100 ) ru le 29 ( 000111 0 ') r u l e 30 ( 000 11 110) rule 32 ( 00 ' 00000 )

• •rule 33 ( 00' 00001)

=--=ru l e 37 ( 00' 00 ' 01)

r u le 42 ( 00 ' 0 10'0)

rul e 46 (00' 0 11'0 )

r u le 56 (00" ' 000)

ru le 3 4 ( 00' 000 ' 0 )

ru l e 38 ( 00 ' 00 110)

ru l e 43 ( 00 10 ' 011)

ru l e 50 ( 00 11001 0 ).,rul e 57 ( 00 ' 1100 ')

r u l e 35 ( 00 ' 000 11)

ru le 40 ( 00' 0 ' 000)

rul e" ( 00 ' 0 1100)

r ul e 5' ( 00" 00 11)

r ule 58 ( 00 11 ' 0' 0 )

r u l e 36 ( 00 ' 00' 00)

ru l e 4 ' ( 00' 0' 00')

ru l e 45 (00 10 110 ' )

A.ru l e 54 ( 00 110 110 )

r u l e 60 ( 00 111 ' 00)

r ul e 6 ' ( 00 11 11 01)

r u l e 74 (0 ' 00 10' 0)

ru l e 62 ( 00 11"' 0)

ru l e 76 ( 0' 00 1100)

r u le 72 ( 0' 001 000)

r ul e 77 ( 0 ' 0011 0 1)

ru l e 73 ( 0 ' 00 ' 00 ' )

r ul e 78 ( 0' 00111 0)

r u l e 90 ( 0' 0 110' 0 )

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ru l e 94 ( 0' 0 11 11 0 ) r u l e 104 ( 0 110' 000) rul e '05 ( 0 110 ' 00 1 )

Tables af Cellula r Automaton Properties (1986)

~ ~ Ar ul e 186 (81 18 18 18 ) r ule lee ( 811 811 88) ,u l . 11 8 (8 118 1118) ru le 122 (81 I 1181 8)

A ~'u l. 126 (81111118) ru l e 128 ( 10888888 ) r ut e 138 ( 1808801 8 ) ru I, 132 ( 1888 0108 )

~ A• •r ut e 134 ( 18888 118 ) r ut e 136 (1 8081 888) r ut e 138 (1800 1010) r u l e 140 ( 10001 100 )

A AI ~ ~ ~ \

r u l e 142 ( 1000 11 10 ) ru le 146 ( 100 100 10 ) r ul e 150 (10010 110) r ut e 152 (10011000)

A ~ A..~~~~~~~ru le 154 ( 100 ' 10'0 ) rule 156 ( 10011 100) ru le 160 ( 10 100000 ) ru t e 162 ( 10 1008 18)

~r ul e 154 ( 101 88 188 ) r u Ie 168 ( 18 18 188 0) r ul e 170 ( 101 81018) r u I, 172 (18181100)

A ~ ~ruI, 178 ( 18 1188 18) ru le 184 ('811 1888) ru le 188 ( 18 111188) ru le zee (1 188 18 08)

rut e 284 (110811 88) r ul e 232 (1118 188 8)

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Patterns generated by second-order reversible rul es.

This table shows patterns produced by second-order generalizations of the k = 2,r = I cellular automata considered above . The rules are of the form

a (t+l ) ="'(a (l ) a (l ) a (I») + a (l -I ) mod 2I If' I - I ' f ' 1+1 I '

where ¢J is a standard k =2, r = I function, as listed in table I. Such rules determinethe configuration at time t + I in term s of the configurations both at time t and timet - I . The rules have the special feature that they are reversible: given configurationsat times t and t + I, the configuration at time t - I can be deduced uniquely accordingto the rule

(I- I) _ "'( (I ) (I) (I) ) + (1+ 1) d 2a j - 'f' a j _ l , a j , a i+1 a j mo .

The first set of pattern s were generated with disordered initial configurations attimes 0 and - I . In the second set of patterns, the configurations at time 0 and -Iwere both taken to be I.

The forms of behaviour produced by these reversible rules are qualitatively similarto those from standard k = 2, r = I cellular automata, shown in tables 2 and 5.Evolution to a homogeneous, fixed, pattern is however impossible for reversiblesystems.

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