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Multiplicative ECA, code, documentation, JOSS review on Github

Multiplicative ECA, paper on arXiv

This project accomplishes several things. First is that complete sets exist for many combinations of parameters; every Elementary Wolfram Code has a set of permutations that allow for the composition of rings and fields as pointers to truth tables. Second is that since it permutes the log of the axis rather than the entire axis it is relatively efficient. Third is that it allows the extension of elementary automata to complex numbers. Overall it tightly integrates the binary ECA with permutations and complex numbers.

This is a general overview of solutions' output, using the doAdditiveWolframCode option in ECApostProcessing. There may be hundreds or thousands of solutions for a given rule, these are just 1 random solution of each normalization using random data for each. This site is meant to be an appendix to the repository, because I'm not going to put this giant dataset on git.

Normalization 5 tends to be the most interesting, or go by Class 3 or 4 rules

0-16 Logic gate truth tables as Wolfram codes

Gate: 0, False

ECA rules with left right black white symmetries

Column 1 = original, Column 2 = left-right, Column 3 = black-white, Column = left-right-black-white

0	255	0	255	Class: 1
1	127	1	127	Class: 2
2	191	16	247	Class: 2
3	63	17	119	Class: 2
4	223	4	223	Class: 2
5	95	5	95	Class: 2
6	159	20	215	Class: 2
7	31	21	87	Class: 2
8	239	64	253	Class: 1
9	111	65	125	Class: 2
10	175	80	245	Class: 2
11	47	81	117	Class: 2
12	207	68	221	Class: 2
13	79	69	93	Class: 2
14	143	84	213	Class: 2
15	15	85	85	Class: 2
16	247	2	191	Class: 2
17	119	3	63	Class: 2
18	183	18	183	Class: 3
19	55	19	55	Class: 2
20	215	6	159	Class: 2
21	87	7	31	Class: 2
22	151	22	151	Class: 3
23	23	23	23	Class: 2
24	231	66	189	Class: 2
25	103	67	61	Class: 2
26	167	82	181	Class: 2
27	39	83	53	Class: 2
28	199	70	157	Class: 2
29	71	71	29	Class: 2
30	135	86	149	Class: 3
31	7	87	21	Class: 2
32	251	32	251	Class: 1
33	123	33	123	Class: 2
34	187	48	243	Class: 2
35	59	49	115	Class: 2
36	219	36	219	Class: 2
37	91	37	91	Class: 2
38	155	52	211	Class: 2
39	27	53	83	Class: 2
40	235	96	249	Class: 1
41	107	97	121	Class: 4
42	171	112	241	Class: 2
43	43	113	113	Class: 2
44	203	100	217	Class: 2
45	75	101	89	Class: 3
46	139	116	209	Class: 2
47	11	117	81	Class: 2
48	243	34	187	Class: 2
49	115	35	59	Class: 2
50	179	50	179	Class: 2
51	51	51	51	Class: 2
52	211	38	155	Class: 2
53	83	39	27	Class: 2
54	147	54	147	Class: 4
55	19	55	19	Class: 2
56	227	98	185	Class: 2
57	99	99	57	Class: 2
58	163	114	177	Class: 2
59	35	115	49	Class: 2
60	195	102	153	Class: 3
61	67	103	25	Class: 2
62	131	118	145	Class: 2
63	3	119	17	Class: 2
64	253	8	239	Class: 1
65	125	9	111	Class: 2
66	189	24	231	Class: 2
67	61	25	103	Class: 2
68	221	12	207	Class: 2
69	93	13	79	Class: 2
70	157	28	199	Class: 2
71	29	29	71	Class: 2
72	237	72	237	Class: 2
73	109	73	109	Class: 2
74	173	88	229	Class: 2
75	45	89	101	Class: 3
76	205	76	205	Class: 2
77	77	77	77	Class: 2
78	141	92	197	Class: 2
79	13	93	69	Class: 2
80	245	10	175	Class: 2
81	117	11	47	Class: 2
82	181	26	167	Class: 2
83	53	27	39	Class: 2
84	213	14	143	Class: 2
85	85	15	15	Class: 2
86	149	30	135	Class: 3
87	21	31	7	Class: 2
88	229	74	173	Class: 2
89	101	75	45	Class: 3
90	165	90	165	Class: 3
91	37	91	37	Class: 2
92	197	78	141	Class: 2
93	69	79	13	Class: 2
94	133	94	133	Class: 2
95	5	95	5	Class: 2
96	249	40	235	Class: 1
97	121	41	107	Class: 4
98	185	56	227	Class: 2
99	57	57	99	Class: 2
100	217	44	203	Class: 2
101	89	45	75	Class: 3
102	153	60	195	Class: 3
103	25	61	67	Class: 2
104	233	104	233	Class: 2
105	105	105	105	Class: 3
106	169	120	225	Class: 4
107	41	121	97	Class: 4
108	201	108	201	Class: 2
109	73	109	73	Class: 2
110	137	124	193	Class: 4
111	9	125	65	Class: 2
112	241	42	171	Class: 2
113	113	43	43	Class: 2
114	177	58	163	Class: 2
115	49	59	35	Class: 2
116	209	46	139	Class: 2
117	81	47	11	Class: 2
118	145	62	131	Class: 2
119	17	63	3	Class: 2
120	225	106	169	Class: 4
121	97	107	41	Class: 4
122	161	122	161	Class: 3
123	33	123	33	Class: 2
124	193	110	137	Class: 4
125	65	111	9	Class: 2
126	129	126	129	Class: 3
127	1	127	1	Class: 2
128	254	128	254	Class: 1
129	126	129	126	Class: 3
130	190	144	246	Class: 2
131	62	145	118	Class: 2
132	222	132	222	Class: 2
133	94	133	94	Class: 2
134	158	148	214	Class: 2
135	30	149	86	Class: 3
136	238	192	252	Class: 1
137	110	193	124	Class: 4
138	174	208	244	Class: 2
139	46	209	116	Class: 2
140	206	196	220	Class: 2
141	78	197	92	Class: 2
142	142	212	212	Class: 2
143	14	213	84	Class: 2
144	246	130	190	Class: 2
145	118	131	62	Class: 2
146	182	146	182	Class: 3
147	54	147	54	Class: 4
148	214	134	158	Class: 2
149	86	135	30	Class: 3
150	150	150	150	Class: 3
151	22	151	22	Class: 3
152	230	194	188	Class: 2
153	102	195	60	Class: 3
154	166	210	180	Class: 2
155	38	211	52	Class: 2
156	198	198	156	Class: 2
157	70	199	28	Class: 2
158	134	214	148	Class: 2
159	6	215	20	Class: 2
160	250	160	250	Class: 1
161	122	161	122	Class: 3
162	186	176	242	Class: 2
163	58	177	114	Class: 2
164	218	164	218	Class: 2
165	90	165	90	Class: 3
166	154	180	210	Class: 2
167	26	181	82	Class: 2
168	234	224	248	Class: 1
169	106	225	120	Class: 4
170	170	240	240	Class: 2
171	42	241	112	Class: 2
172	202	228	216	Class: 2
173	74	229	88	Class: 2
174	138	244	208	Class: 2
175	10	245	80	Class: 2
176	242	162	186	Class: 2
177	114	163	58	Class: 2
178	178	178	178	Class: 2
179	50	179	50	Class: 2
180	210	166	154	Class: 2
181	82	167	26	Class: 2
182	146	182	146	Class: 3
183	18	183	18	Class: 3
184	226	226	184	Class: 2
185	98	227	56	Class: 2
186	162	242	176	Class: 2
187	34	243	48	Class: 2
188	194	230	152	Class: 2
189	66	231	24	Class: 2
190	130	246	144	Class: 2
191	2	247	16	Class: 2
192	252	136	238	Class: 1
193	124	137	110	Class: 4
194	188	152	230	Class: 2
195	60	153	102	Class: 3
196	220	140	206	Class: 2
197	92	141	78	Class: 2
198	156	156	198	Class: 2
199	28	157	70	Class: 2
200	236	200	236	Class: 2
201	108	201	108	Class: 2
202	172	216	228	Class: 2
203	44	217	100	Class: 2
204	204	204	204	Class: 2
205	76	205	76	Class: 2
206	140	220	196	Class: 2
207	12	221	68	Class: 2
208	244	138	174	Class: 2
209	116	139	46	Class: 2
210	180	154	166	Class: 2
211	52	155	38	Class: 2
212	212	142	142	Class: 2
213	84	143	14	Class: 2
214	148	158	134	Class: 2
215	20	159	6	Class: 2
216	228	202	172	Class: 2
217	100	203	44	Class: 2
218	164	218	164	Class: 2
219	36	219	36	Class: 2
220	196	206	140	Class: 2
221	68	207	12	Class: 2
222	132	222	132	Class: 2
223	4	223	4	Class: 2
224	248	168	234	Class: 1
225	120	169	106	Class: 4
226	184	184	226	Class: 2
227	56	185	98	Class: 2
228	216	172	202	Class: 2
229	88	173	74	Class: 2
230	152	188	194	Class: 2
231	24	189	66	Class: 2
232	232	232	232	Class: 2
233	104	233	104	Class: 2
234	168	248	224	Class: 1
235	40	249	96	Class: 1
236	200	236	200	Class: 2
237	72	237	72	Class: 2
238	136	252	192	Class: 1
239	8	253	64	Class: 1
240	240	170	170	Class: 2
241	112	171	42	Class: 2
242	176	186	162	Class: 2
243	48	187	34	Class: 2
244	208	174	138	Class: 2
245	80	175	10	Class: 2
246	144	190	130	Class: 2
247	16	191	2	Class: 2
248	224	234	168	Class: 1
249	96	235	40	Class: 1
250	160	250	160	Class: 1
251	32	251	32	Class: 1
252	192	238	136	Class: 1
253	64	239	8	Class: 1
254	128	254	128	Class: 1
255	0	255	0	Class: 1

ECA rules by the 88 left-right-black-white symmetry groups

0	255	0	255	Class: 1
1	127	1	127	Class: 2
2	191	16	247	Class: 2
3	63	17	119	Class: 2
4	223	4	223	Class: 2
5	95	5	95	Class: 2
6	159	20	215	Class: 2
7	31	21	87	Class: 2
8	239	64	253	Class: 1
9	111	65	125	Class: 2
10	175	80	245	Class: 2
11	47	81	117	Class: 2
12	207	68	221	Class: 2
13	79	69	93	Class: 2
14	143	84	213	Class: 2
15	15	85	85	Class: 2
18	183	18	183	Class: 3
19	55	19	55	Class: 2
22	151	22	151	Class: 3
23	23	23	23	Class: 2
24	231	66	189	Class: 2
25	103	67	61	Class: 2
26	167	82	181	Class: 2
27	39	83	53	Class: 2
28	199	70	157	Class: 2
29	71	71	29	Class: 2
30	135	86	149	Class: 3
32	251	32	251	Class: 1
33	123	33	123	Class: 2
34	187	48	243	Class: 2
35	59	49	115	Class: 2
36	219	36	219	Class: 2
37	91	37	91	Class: 2
38	155	52	211	Class: 2
40	235	96	249	Class: 1
41	107	97	121	Class: 4
42	171	112	241	Class: 2
43	43	113	113	Class: 2
44	203	100	217	Class: 2
45	75	101	89	Class: 3
46	139	116	209	Class: 2
50	179	50	179	Class: 2
51	51	51	51	Class: 2
54	147	54	147	Class: 4
56	227	98	185	Class: 2
57	99	99	57	Class: 2
58	163	114	177	Class: 2
60	195	102	153	Class: 3
62	131	118	145	Class: 2
72	237	72	237	Class: 2
73	109	73	109	Class: 2
74	173	88	229	Class: 2
76	205	76	205	Class: 2
77	77	77	77	Class: 2
78	141	92	197	Class: 2
90	165	90	165	Class: 3
94	133	94	133	Class: 2
104	233	104	233	Class: 2
105	105	105	105	Class: 3
106	169	120	225	Class: 4
108	201	108	201	Class: 2
110	137	124	193	Class: 4
122	161	122	161	Class: 3
126	129	126	129	Class: 3
128	254	128	254	Class: 1
130	190	144	246	Class: 2
132	222	132	222	Class: 2
134	158	148	214	Class: 2
136	238	192	252	Class: 1
138	174	208	244	Class: 2
140	206	196	220	Class: 2
142	142	212	212	Class: 2
146	182	146	182	Class: 3
150	150	150	150	Class: 3
152	230	194	188	Class: 2
154	166	210	180	Class: 2
156	198	198	156	Class: 2
160	250	160	250	Class: 1
162	186	176	242	Class: 2
164	218	164	218	Class: 2
168	234	224	248	Class: 1
170	170	240	240	Class: 2
172	202	228	216	Class: 2
178	178	178	178	Class: 2
184	226	226	184	Class: 2
200	236	200	236	Class: 2
204	204	204	204	Class: 2
232	232	232	232	Class: 2