Discrete Dynamics Lab

Update
May 2013

ddlabx11 New options for Automatic Derrida plots for sets of rules, equivalence classes and rule clusters.

Supersedes DDLabx11, Jan 2013 and includes further updates since the publication of Exploring Discrete Dynamics in May 2011


Automatically generated set of Derrida plots for
ECA rule-space, n=150. DDLab is able to filter out
and list the 88 equivalence or 48 rule-clusters of the
256 ECA rules. Its notable that there are only 14 independent plots.
DDLab has been updated at regular intervals since its release in 1995. Its precursor was the Atlas software included on diskette inside the back cover of "The Global Dynamics of Cellular Automata".
For a list and download of this and older versions click here.
Below are links to descriptions of previous updates,
Jan 2013
June 2012
Nov 2005
Dec 2003
July 2001
Feb 1999
Sept 1997

download ddlabx11 for Linux, Mac, Cygwin and DOS
summary of updates (EDD#x.x.x refers to the relevant chapter or section in "Exploring Discrete Dynamics")

48 rule clusters of the 256 v2k3 ECA
there are 88 equivalence classes
Automatic output in the terminal:

#: s n r nr D
1: 0 255 0 255 -inf
    255 0 255 0 -inf (-nan)
2: 1 127 1 127 -0.457
    254 128 254 128 -0.377 (-0.080)
3: 2 191 16 247 -0.445
    253 64 239 8 -0.426 (-0.020)
4: 3 63 17 119 -0.036
    252 192 238 136 0.014 (-0.050)
5: 4 223 4 223 -0.428
    251 32 251 32 -0.423 (-0.005)
6: 5 95 5 95 -0.014
    250 160 250 160 -0.021 (0.007)
7: 6 159 20 215 0.572
    249 96 235 40 0.559 (0.013)
8: 7 31 21 87 0.324
    248 224 234 168 0.302 (0.023)
9: 9 111 65 125 0.509
    246 144 190 130 0.575 (-0.066)
10: 10 175 80 245 -0.034
    245 80 175 10 0.007 (-0.041)
11: 11 47 81 117 0.304
    244 208 174 138 0.328 (-0.024)
12: 12 207 68 221 -0.023
    243 48 187 34 -0.042 (0.018)
13: 13 79 69 93 0.287
    242 176 186 162 0.315 (-0.029)
14: 14 143 84 213 0.324
    241 112 171 42 0.299 (0.025)
15: 15 15 85 85 -0.000
    240 240 170 170 -0.000 (0.000)
16: 18 183 18 183 0.586
    237 72 237 72 0.577 (0.008)
17: 19 55 19 55 0.302
    236 200 236 200 0.304 (-0.002)
18: 22 151 22 151 1.143
    233 104 233 104 1.136 (0.007)
19: 23 23 23 23 0.544
    232 232 232 232 0.582 (-0.038)
20: 24 231 66 189 0.588
    231 24 189 66 0.582 (0.006)
21: 25 103 67 61 0.800
    230 152 188 194 0.785 (0.015)
22: 26 167 82 181 0.784
    229 88 173 74 0.807 (-0.023)
23: 27 39 83 53 0.579
    228 216 172 202 0.574 (0.006)
24: 28 199 70 157 0.807
    227 56 185 98 0.794 (0.013)
25: 29 71 71 29 0.589
    226 184 184 226 0.601 (-0.011)
26: 30 135 86 149 1.003
    225 120 169 106 0.992 (0.012)
27: 33 123 33 123 0.576
    222 132 222 132 0.563 (0.013)
28: 35 59 49 115 0.297
    220 196 206 140 0.318 (-0.021)
29: 36 219 36 219 0.552
    219 36 219 36 0.564 (-0.011)
30: 37 91 37 91 0.774
    218 164 218 164 0.788 (-0.014)
31: 38 155 52 211 0.778
    217 100 203 44 0.800 (-0.022)
32: 41 107 97 121 1.150
    214 148 158 134 1.124 (0.026)
33: 43 43 113 113 0.577
    212 212 142 142 0.560 (0.016)
34: 45 75 101 89 0.983
    210 180 154 166 0.982 (0.002)
35: 46 139 116 209 0.583
    209 116 139 46 0.574 (0.009)
36: 50 179 50 179 0.306
    205 76 205 76 0.333 (-0.027)
37: 51 51 51 51 -0.000
    204 204 204 204 -0.000 (0.000)
38: 54 147 54 147 0.988
    201 108 201 108 0.968 (0.020)
39: 57 99 99 57 0.981
    198 156 156 198 0.971 (0.010)
40: 58 163 114 177 0.578
    197 92 141 78 0.566 (0.012)
41: 60 195 102 153 0.980
    195 60 153 102 0.986 (-0.006)
42: 62 131 118 145 0.798
    193 124 137 110 0.785 (0.013)
43: 73 109 73 109 1.153
    182 146 182 146 1.133 (0.021)
44: 77 77 77 77 0.571
    178 178 178 178 0.593 (-0.022)
45: 90 165 90 165 0.988
    165 90 165 90 0.983 (0.005)
46: 94 133 94 133 0.794
    161 122 161 122 0.815 (-0.022)
47: 105 105 105 105 1.555
    150 150 150 150 1.555 (0.000)
48: 126 129 126 129 0.569<
    129 126 129 126 0.578 (-0.009)

equivalence classes and rule clusters.

In EDD#18.1.1, the transform prompt for a single rule is extended to show the equivalence classes or rule clusters in the terminal.

Enter EquivClass-E for the equivalence class, giving,
110 n=137 r=124 nr=193 ...for example, an ECA rule v2k3,
where 110=start rule n=negative r=reflection nr=negative+reflection

Enter EquivClass-R for the rule cluster, giving,>

11539fa3 n=3a063577 r=541724ef nr=08db17d5 ...for example, rcode v2k5,
eeac605c n=c5f9ca88 r=abe8db10 nr=f724e82a ...the compiliments (shown in hex)

The method works for rcode, tcode and kcode.

Automatic Derrida plots for sets of rules, equivalence classes and rule clusters.

The main application of the Derrida plot (described in EDD#22) has been as an order-chaos measure for large RBN networks in the context of models of genetic regulatory networks, where the canalyzing inputs can be tuned to move the dynamics between order and chaos (EDD#15). However, the Derrida plot also provides Liapunov-like insights into CA rules. New options allow automatic plots of sets of rules in ascending decimal order, filtering out equivalent binary rcode and tcode, and listing equivalence classes and rule clusters.

Its of some interest to compare the Derrida plots of various CA rule spaces, for example the 256 elementary (ECA) rules (v2k3 rcode), or the 64 v2k5 totalistic rules (tcode or kcode). In the "Global Dynamics of Cellular Automata" (GDCA) it was shown that a rule has equivelent rules by negative, reflection and negative+reflection transformations, making an equivalence class were all behavior is strictly equivalent. A further complimentary transformation will group the rules into a rule cluster. Complimentary rules, though usually not equivalent, share some important behavior measures, including the Z-parameter and G-density (EDD#24.9), and the in-degree histogram (EDD#24.6). As set out in GDCA, the 256 elementary rules fall into 88 equivalence classes and 48 rule clusters. The 64 v2k5 totalistic rules fall into 36 equivalence classes and 20 rule clusters.

Experiment confirms, as expected, that the Derrida plots for the rules in an equivalence class will be the same (subject to sampling variation) as if plotting the same rule, and this turns out also to be true of the complimentary rules -- the rule cluster has the same Derrida properties. Its possible to automatically generate the Derrida plots for binary (v=2) rule-spaces according to equivalence classes or rule clusters, or for (v>=2) rules, counting up from the start rule in decimal. The procedure works for rcode in SEED-mode, and for kcode or tcode in TFO-mode. However, rule-tables must be within the limits in table 16.1 for expressing the rule as a decimal number.

The prompt in EDD#22.5 "Completing the Derrida Plot" is extended as follows,.

To automatically generate the Derrida plots, enter 1 for equivalence classes, 2 for rule clusters, or 3 for just a sequence of rules. In each case the initial rule, selected in EED#16 or revised with reset-r in section EDD#22.5 starts of the sequence, counting up, so to obtain an orderly and complete sequence of rules for an entire rule-space, the initial rule should be set to decimal rule 0 (zero). Add k to retain previous plots for a multiple automatic plot, for example, enter 1k, 2k or 3k. If k is not added each new automatic plot is displayed separately.

Starting with rule 0 for equivalence classes, just the lowest decimal rule will be plotted (as the representative rule) and the others skipped and filtered from the remaining list. For rule clusters, the compliment of the representative rule is also plotted, and its remaining equivalence class is also skipped and filtered from the remaining list. The rules making up equivalence classes or rule clusters are displayed in the terminal window (xterm) for Linux-like systems (Left panel), so this is a useful result in itself -- Derrida parameters can be minimised if this is the only data required.

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Last modified: May2013