A Class of Models with the Potential to Represent Fundamental Physics
  1. Introduction
  2. Basic Form of Models
  3. Typical Behaviors
  4. Limiting Behavior and Emergent Geometry
  5. The Updating Process for String Substitution Systems
  6. The Updating Process in Our Models
  7. Equivalence and Computation in Our Models
  8. Potential Relation to Physics
  9. Additional Material
  10. References
  11. Index
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technical introduction Typical Behaviors

3.12 Rules with Mixed Arity

So far essentially all the rules we have considered have “pure signatures” of the form mk nk for some arity k. Continued growth is never possible unless the right-hand side of a rule contains some relations with the same arity as appear on the left. But, for example, it is perfectly possible to have growth in rules with signatures like 12 2221. Such rules produce unary relations, which can serve as “markers” for the application of the rule, but cannot themselves affect how or where the rule is used:

GraphicsGrid[ Partition[ Labeled[ResourceFunction["WolframModelPlot"][ ResourceFunction["WolframModel"][#, {{0, 0}}, 4, "FinalState"], ImageSize -> 1.1 {150, 100}], RulePlot[ResourceFunction["WolframModel"][#], "RulePartsAspectRatio" -> 1, ImageSize -> Tiny]] & /@ {{{1, 2}} -> {{1, 2}, {1, 3}, {2}, {2}}, {{1, 2}} -> {{1, 3}, {2, 3}, {1}, {3}}, {{1, 2}} -> {{2, 1}, {1, 3}, {2}, {3}}, {{1, 2}} -> {{2, 2}, {1, 3}, {1}, {2}}, {{1, 2}} -> {{3, 1}, {1, 2}, {2}, {3}}, {{1, 2}} -> {{1, 3}, {2, 3}, {4, 3}, {2}}, {{1, 2}} -> {{3, 1}, {3, 1}, {2, 3}, {2}}, {{1, 2}} -> {{3, 2}, {3, 1}, {2, 1}, {1}}}, 4], ImageSize -> Full]

The 634 rules with signature 12 1312 all show very simple behavior (as do the 2212 rules with signature 12 131211), with not even trees being possible. But among the 7652 12 1322 rules there are not only many trees, but also closed structures such as:

GraphicsRow[ Prepend[Labeled[ ResourceFunction["WolframModelPlot"][ ResourceFunction["WolframModel"][#, {{0, 0}}, 8, "FinalState"], ImageSize -> 1.1 {150, 100}], RulePlot[ResourceFunction["WolframModel"][#], "RulePartsAspectRatio" -> 1, ImageSize -> Tiny]] & /@ {{{1, 2}} -> {{1, 3, 3}, {1, 3}, {2, 3}}, {{1, 2}} -> {{3, 1, 4}, {1, 3}, {2, 3}}, {{1, 2}} -> {{2, 3, 3}, {2, 3}, {3, 1}}}, Labeled[ResourceFunction["WolframModelPlot"][ ResourceFunction[ "WolframModel"][{{1, 2}} -> {{1, 1, 2}, {1, 3}, {3, 2}}, {{0, 0}}, 8, "FinalState"], ImageSize -> 1.1 {150, 100}, VertexCoordinates -> { 1 -> {7.544101589596744, 8.666365639455652}, 3 -> {10.839983001166226`, 6.495595368548078}, 6 -> {10.9637370388203, 9.437947253602175}, 7 -> {10.757540247406792`, 3.5135510833021395`}, 12 -> {9.646794564530998, 10.741475565171758`}, 13 -> {12.736718925745787`, 8.468006484957915}, 14 -> {12.57138492526735, 4.335468252718393}, 15 -> {9.352711584587713, 2.316456558512572}, 24 -> {8.544104974014536, 10.645916135567715`}, 25 -> {11.085816125263555`, 11.307996117909983`}, 26 -> {12.989747800777618`, 9.79072892959134}, 27 -> {12.910502791930698`, 7.401423601949318}, 28 -> {12.823527527127183`, 5.400687795057084}, 29 -> {12.730044252024754`, 2.9582019756264826`}, 30 -> {10.746310319718244`, 1.6219459485163572`}, 31 -> {8.270050447605167, 2.4811260509648063`}, 48 -> {8.054123844576646, 9.986737472938117}, 49 -> {9.20520688824398, 11.724165799243238`}, 50 -> {10.641960420980888`, 11.996921748138753`}, 51 -> {11.660097879295742`, 10.916060551547197`}, 52 -> {12.455881146369522`, 10.229353963556454`}, 53 -> {13.871178259788447`, 9.486357622799899}, 54 -> {13.896967833821769`, 8.080978738395144}, 55 -> {12.318869768891677`, 6.912919418264151}, 56 -> {12.299541931740222`, 5.969381129997366}, 57 -> {13.749884857846673`, 4.604381193887148}, 58 -> {13.613801954486703`, 3.1905784859799207`}, 59 -> {12.175238478002903`, 2.5786908356768814`}, 60 -> {11.345404537039927`, 1.9734311933592963`}, 61 -> {10.233355184942612`, 0.9624656711001867}, 62 -> {8.846844498777395, 1.347058330314055}, 63 -> {7.877513098919095, 3.1862322979610562`}, 96 -> {7.98834103989634, 9.381637670310939}, 97 -> {8.292422950043957, 10.873301436099807`}, 98 -> {8.81545994870674, 11.854766766386291`}, 99 -> {9.634854569940433, 11.887099441851982`}, 100 -> {10.256927161702494`, 11.903675546360553`}, 101 -> {11.207615640436874`, 12.35372009282974}, 102 -> {11.845633280099362`, 11.714486224396177`}, 103 -> {11.615933653995562`, 10.446349725262346`}, 104 -> {11.993608909254931`, 10.093890749315191`}, 105 -> {13.287132279912118`, 10.523629754621867`}, 106 -> {14.122787549551322`, 10.018903697949309`}, 107 -> {13.889778908791262`, 9.096117107742874}, 108 -> {13.962192608946498`, 8.501165776623402}, 109 -> {14.133837802481336`, 7.702157489442956}, 110 -> {13.248743820039738`, 7.1426385313789105`}, 111 -> {11.734716311311432`, 6.892186886353087}, 112 -> {11.79677231717904, 6.192097797523426}, 113 -> {13.201884059972004`, 5.655089980483216}, 114 -> {13.99997687163386, 4.972151384569982}, 115 -> {13.783208228768922`, 4.1822786075933385`}, 116 -> {13.662841521721495`, 3.5898611254490995`}, 117 -> {13.813041008281264`, 2.609693673096498}, 118 -> {12.974133040469166`, 2.207760200794086}, 119 -> {11.729630401802199`, 2.7504515234384104`}, 120 -> {11.33288577108337, 2.4334349906924584`}, 121 -> {11.480827547038972`, 1.1604206191335003`}, 122 -> {10.77889906885374, 0.5562902310544136}, 123 -> {9.843600512023079, 1.076519017365522}, 124 -> {9.262790550866267, 1.1609076874086397`}, 125 -> {8.456100180334724, 1.2356403563105527`}, 126 -> {8.055929012082807, 2.2228730041050864`}, 127 -> {7.890831526142487, 3.837955592739668}, 192 -> {7.933095891039079, 8.968259591153705}, 193 -> {8.356750229018724, 9.827264562344778}, 194 -> {8.040535495361695, 10.746782221248626`}, 195 -> {8.436821881940086, 11.364370138032697`}, 196 -> {8.613739786595575, 11.629466896066292`}, 197 -> {9.043974368209499, 12.562254632431172`}, 198 -> {9.532501740496809, 12.57048760335091}, 199 -> {9.824732170805015, 11.612512965166308`}, 200 -> {10.139425843203092`, 11.512188539060892`}, 201 -> {10.630959506725167`, 12.701726906501474`}, 202 -> {11.156920995874236`, 12.914077472053835`}, 203 -> {11.54296879334117, 12.32203660674515}, 204 -> {11.78873246961573, 12.111163234884815`}, 205 -> {12.297648884542651`, 11.778873784556236`}, 206 -> {12.137856089770438`, 11.154065452400047`}, 207 -> {11.392363300178964`, 10.072955365430804`}, 208 -> {11.67541966698753, 9.783355062044446}, 209 -> {12.66587421300373, 10.673656066023145`}, 210 -> {13.25523888998744, 10.946300205201908`}, 211 -> {13.726018207141811`, 10.543481525255563`}, 212 -> {13.979758547549999`, 10.218425018070272`}, 213 -> {14.725719458669683`, 10.070780026736372`}, 214 -> {14.644546313455248`, 9.514903316376039}, 215 -> {13.53903561024818, 8.936502350206167}, 216 -> {13.651719901184876`, 8.637034290852279}, 217 -> {14.69899487200969, 8.446661551149054}, 218 -> {14.807855452813474`, 7.949242781838116}, 219 -> {13.995044415104083`, 7.437182191586921}, 220 -> {13.747777745960537`, 7.253380485239571}, 221 -> {13.21587013915646, 6.854469604372164}, 222 -> {12.333438683191346`, 7.220696748797404}, 223 -> {11.321562532925979`, 6.83766713556012}, 224 -> {11.365083154144616`, 6.22425041746935}, 225 -> {12.542795352322369`, 6.104871924663508}, 226 -> {13.224435250411297`, 5.948651402307288}, 227 -> {13.673790860198473`, 5.477885430572527}, 228 -> {13.889398476060876`, 5.255090548679165}, 229 -> {14.656355473749457`, 4.632652694498461}, 230 -> {14.527123247749596`, 4.142809670047394}, 231 -> {13.448529019046607`, 4.111997370611543}, 232 -> {13.339006623588595`, 3.792654092351127}, 233 -> {14.375369415523553`, 3.091272300811752}, 234 -> {14.41513550107939, 2.530357073383152}, 235 -> {13.66527305854814, 2.3249527609810774`}, 236 -> {13.398484038263499`, 2.1379410094426237`}, 237 -> {12.939364615916324`, 1.7985238286648055`}, 238 -> {12.374599136930984`, 2.122327444722219}, 239 -> {11.438870448360184`, 3.0789965266947448`}, 240 -> {11.14518588138849, 2.806325492365179}, 241 -> {11.790331294330256`, 1.6744162274658332`}, 242 -> {11.93304125286494, 1.0758198069973322`}, 243 -> {11.397924917839095`, 0.762740087178158}, 244 -> {11.124338679276361`, 0.5652941797703978}, 245 -> {10.666338848699025`, 0.}, 246 -> {10.142419867971256`, 0.24722683165043727`}, 247 -> {9.782035197266126, 1.4692269802905928`}, 248 -> {9.475027547635344, 1.437249331896931}, 249 -> {9.102531454531304, 0.4754748416476536}, 250 -> {8.618072808874135, 0.5181232548036908}, 251 -> {8.236236650438025, 1.4223117966109207`}, 252 -> {8.151467751113227, 1.7079273273063658`}, 253 -> {7.813214498238482, 2.354742911863664}, 254 -> {8.226696672005438, 3.307701842931193}, 255 -> {7.8818624460383315`, 4.292005801517424}, 0 -> {7.315898410403255, 4.513634360544348}, 2 -> {4.020016998833773, 6.684404631451922}, 4 -> {4.102459752593207, 9.66644891669786}, 5 -> {3.8962629611796995`, 3.7420527463978246`}, 8 -> {5.507288415412287, 10.863543441487428`}, 9 -> {2.2886150747326486`, 8.844531747281607}, 10 -> {2.123281074254212, 4.711993515042085}, 11 -> {5.213205435469002, 2.438524434828242}, 16 -> {6.589949552394833, 10.698873949035193`}, 17 -> {4.113689680281755, 11.558054051483643`}, 18 -> {2.129955747975245, 10.221798024373516`}, 19 -> {2.0364724728728163`, 7.779312204942916}, 20 -> {1.9494972080693014`, 5.7785763980506815`}, 21 -> {1.8702521992223815`, 3.3892710704086593`}, 22 -> {3.774183874736444, 1.8720038820900164`}, 23 -> {6.315895025985464, 2.5340838644322847`}, 32 -> {6.982486901080905, 9.993767702038944}, 33 -> {6.013155501222604, 11.832941669685944`}, 34 -> {4.626644815057388, 12.217534328899813`}, 35 -> {3.514595462960072, 11.206568806640703`}, 36 -> {2.684761521997096, 10.60130916432312}, 37 -> {1.246198045513296, 9.98942151402008}, 38 -> {1.110115142153326, 8.575618806112852}, 39 -> {2.560458068259777, 7.210618870002634}, 40 -> {2.5411302311083226`, 6.267080581735849}, 41 -> {0.9630321661782304, 5.099021261604856}, 42 -> {0.9888217402115522, 3.693642377200101}, 43 -> {2.404118853630477, 2.950646036443546}, 44 -> {3.199902120704257, 2.2639394484528026`}, 45 -> {4.218039579019111, 1.1830782518612466`}, 46 -> {5.654793111756019, 1.4558342007567617`}, 47 -> {6.805876155423354, 3.193262527061883}, 64 -> {6.969168473857512, 9.342044407260332}, 65 -> {6.804070987917193, 10.957126995894914`}, 66 -> {6.403899819665275, 11.944359643689447`}, 67 -> {5.597209449133732, 12.01909231259136}, 68 -> {5.016399487976921, 12.103480982634478`}, 69 -> {4.08110093114626, 12.623709768945586`}, 70 -> {3.3791724529610274`, 12.0195793808665}, 71 -> {3.527114228916629, 10.746565009307542`}, 72 -> {3.1303695981978006`, 10.429548476561589`}, 73 -> {1.8858669595308335`, 10.972239799205914`}, 74 -> {1.0469589917187356`, 10.570306326903502`}, 75 -> {1.1971584782785047`, 9.5901388745509}, 76 -> {1.0767917712310773`, 8.99772139240666}, 77 -> {0.860023128366139, 8.207848615430017}, 78 -> {1.6581159400279954`, 7.524910019516784}, 79 -> {3.06322768282096, 6.987902202476573}, 80 -> {3.125283688688567, 6.287813113646913}, 81 -> {1.611256179960261, 6.037361468621089}, 82 -> {0.7261621975186632, 5.477842510557044}, 83 -> {0.8978073910535009, 4.678834223376597}, 84 -> {0.970221091208737, 4.083882892257126}, 85 -> {0.737212450448677, 3.161096302050691}, 86 -> {1.5728677200878813`, 2.656370245378133}, 87 -> {2.8663910907450685`, 3.0861092506848085`}, 88 -> {3.2440663460044377`, 2.733650274737654}, 89 -> {3.0143667199006376`, 1.4655137756038226`}, 90 -> {3.6523843595631256`, 0.8262799071702602}, 91 -> {4.603072838297505, 1.2763244536394467`}, 92 -> {5.225145430059566, 1.292900558148018}, 93 -> {6.044540051293259, 1.3252332336137087`}, 94 -> {6.567577049956043, 2.306698563900193}, 95 -> {6.87165896010366, 3.798362329689061}, 128 -> {6.978137553961668, 8.887994198482577}, 129 -> {6.633303327994561, 9.872298157068807}, 130 -> {7.0467855017615175`, 10.825257088136336`}, 131 -> {6.7085322488867725`, 11.472072672693635`}, 132 -> {6.623763349561974, 11.75768820338908}, 133 -> {6.241927191125864, 12.66187674519631}, 134 -> {5.757468545468695, 12.704525158352347`}, 135 -> {5.384972452364655, 11.742750668103069`}, 136 -> {5.077964802733874, 11.710773019709407`}, 137 -> {4.717580132028743, 12.932773168349563`}, 138 -> {4.193661151300974, 13.18}, 139 -> {3.735661320723638, 12.614705820229602`}, 140 -> {3.4620750821609043`, 12.41725991282184}, 141 -> {2.9269587471350587`, 12.104180193002668`}, 142 -> {3.069668705669743, 11.505583772534166`}, 143 -> {3.7148141186115087`, 10.37367450763482}, 144 -> {3.4211295516398152`, 10.101003473305255`}, 145 -> {2.4854008630690156`, 11.057672555277781`}, 146 -> {1.9206353840836758`, 11.381476171335194`}, 147 -> {1.4615159617365006`, 11.042058990557376`}, 148 -> {1.1947269414518598`, 10.855047239018923`}, 149 -> {0.44486449892060875`, 10.649642926616847`}, 150 -> {0.4846305844764469, 10.088727699188247`}, 151 -> {1.520993376411404, 9.387345907648873}, 152 -> {1.4114709809533927`, 9.068002629388456}, 153 -> {0.33287675225040303`, 9.037190329952605}, 154 -> {0.2036445262505424, 8.547347305501539}, 155 -> {0.9706015239391235, 7.9249094513208345`}, 156 -> {1.1862091398015266`, 7.702114569427473}, 157 -> {1.6355647495887027`, 7.231348597692712}, 158 -> {2.317204647677631, 7.075128075336492}, 159 -> {3.494916845855384, 6.9557495825306495`}, 160 -> {3.5384374670740204`, 6.3423328644398795`}, 161 -> {2.526561316808653, 5.959303251202596}, 162 -> {1.64412986084354, 6.325530395627836}, 163 -> {1.1122222540394624`, 5.926619514760429}, 164 -> {0.8649555848959167, 5.742817808413078}, 165 -> {0.052144547186525614`, 5.230757218161884}, 166 -> {0.1610051279903093, 4.733338448850946}, 167 -> {1.2082800988151234`, 4.542965709147721}, 168 -> {1.3209643897518202`, 4.243497649793833}, 169 -> {0.21545368654475183`, 3.665096683623961}, 170 -> {0.13428054133031608`, 3.1092199732636274`}, 171 -> {0.8802414524500008, 2.9615749819297275`}, 172 -> {1.1339817928581883`, 2.6365184747444363`}, 173 -> {1.604761110012559, 2.233699794798092}, 174 -> {2.1941257869962687`, 2.5063439339768543`}, 175 -> {3.1845803330124696`, 3.3966449379555534`}, 176 -> {3.4676366998210355`, 3.1070446345691956`}, 177 -> {2.722143910229562, 2.0259345475999524`}, 178 -> {2.562351115457348, 1.4011262154437638`}, 179 -> {3.0712675303842687`, 1.0688367651151847`}, 180 -> {3.317031206658829, 0.8579633932548489}, 181 -> {3.7030790041257635`, 0.2659225279461648}, 182 -> {4.229040493274832, 0.478273093498526}, 183 -> {4.720574156796907, 1.6678114609391077`}, 184 -> {5.035267829194984, 1.5674870348336913`}, 185 -> {5.327498259503191, 0.6095123966490892}, 186 -> {5.816025631790501, 0.617745367568828}, 187 -> {6.246260213404424, 1.5505331039337076`}, 188 -> {6.423178118059914, 1.8156298619673024`}, 189 -> {6.8194645046383044`, 2.4332177787513736`}, 190 -> {6.5032497709812755`, 3.3527354376552214`}, 191 -> {6.926904108960921, 4.211740408846294}}], RulePlot[ ResourceFunction[ "WolframModel"][{{1, 2}} -> {{1, 1, 2}, {1, 3}, {3, 2}}], "RulePartsAspectRatio" -> 1, ImageSize -> Tiny]]], ImageSize -> Full]

Previously we had only seen structures like the first one above in rules that depend on more than one relation. But as this illustrates, such structures can be produced even with just a single relation on the left:

{{x, y}} -> {{x, x, y}, {x, z}, {z, y}}
RulePlot[ResourceFunction[ "WolframModel"][{{x, y}} -> {{x, x, y}, {x, z}, {z, y}}]]
Append[ResourceFunction["WolframModelPlot"][#, "MaxImageSize" -> 100] & /@ ResourceFunction[ "WolframModel"][{{1, 2}} -> {{1, 1, 2}, {1, 3}, {3, 2}}, {{0, 0}}, 7, "StatesList"], ResourceFunction["WolframModelPlot"][ ResourceFunction[ "WolframModel"][{{1, 2}} -> {{1, 1, 2}, {1, 3}, {3, 2}}, {{0, 0}}, 8, "FinalState"], "MaxImageSize" -> 100, VertexCoordinates -> { 1 -> {7.544101589596744, 8.666365639455652}, 3 -> {10.839983001166226`, 6.495595368548078}, 6 -> {10.9637370388203, 9.437947253602175}, 7 -> {10.757540247406792`, 3.5135510833021395`}, 12 -> {9.646794564530998, 10.741475565171758`}, 13 -> {12.736718925745787`, 8.468006484957915}, 14 -> {12.57138492526735, 4.335468252718393}, 15 -> {9.352711584587713, 2.316456558512572}, 24 -> {8.544104974014536, 10.645916135567715`}, 25 -> {11.085816125263555`, 11.307996117909983`}, 26 -> {12.989747800777618`, 9.79072892959134}, 27 -> {12.910502791930698`, 7.401423601949318}, 28 -> {12.823527527127183`, 5.400687795057084}, 29 -> {12.730044252024754`, 2.9582019756264826`}, 30 -> {10.746310319718244`, 1.6219459485163572`}, 31 -> {8.270050447605167, 2.4811260509648063`}, 48 -> {8.054123844576646, 9.986737472938117}, 49 -> {9.20520688824398, 11.724165799243238`}, 50 -> {10.641960420980888`, 11.996921748138753`}, 51 -> {11.660097879295742`, 10.916060551547197`}, 52 -> {12.455881146369522`, 10.229353963556454`}, 53 -> {13.871178259788447`, 9.486357622799899}, 54 -> {13.896967833821769`, 8.080978738395144}, 55 -> {12.318869768891677`, 6.912919418264151}, 56 -> {12.299541931740222`, 5.969381129997366}, 57 -> {13.749884857846673`, 4.604381193887148}, 58 -> {13.613801954486703`, 3.1905784859799207`}, 59 -> {12.175238478002903`, 2.5786908356768814`}, 60 -> {11.345404537039927`, 1.9734311933592963`}, 61 -> {10.233355184942612`, 0.9624656711001867}, 62 -> {8.846844498777395, 1.347058330314055}, 63 -> {7.877513098919095, 3.1862322979610562`}, 96 -> {7.98834103989634, 9.381637670310939}, 97 -> {8.292422950043957, 10.873301436099807`}, 98 -> {8.81545994870674, 11.854766766386291`}, 99 -> {9.634854569940433, 11.887099441851982`}, 100 -> {10.256927161702494`, 11.903675546360553`}, 101 -> {11.207615640436874`, 12.35372009282974}, 102 -> {11.845633280099362`, 11.714486224396177`}, 103 -> {11.615933653995562`, 10.446349725262346`}, 104 -> {11.993608909254931`, 10.093890749315191`}, 105 -> {13.287132279912118`, 10.523629754621867`}, 106 -> {14.122787549551322`, 10.018903697949309`}, 107 -> {13.889778908791262`, 9.096117107742874}, 108 -> {13.962192608946498`, 8.501165776623402}, 109 -> {14.133837802481336`, 7.702157489442956}, 110 -> {13.248743820039738`, 7.1426385313789105`}, 111 -> {11.734716311311432`, 6.892186886353087}, 112 -> {11.79677231717904, 6.192097797523426}, 113 -> {13.201884059972004`, 5.655089980483216}, 114 -> {13.99997687163386, 4.972151384569982}, 115 -> {13.783208228768922`, 4.1822786075933385`}, 116 -> {13.662841521721495`, 3.5898611254490995`}, 117 -> {13.813041008281264`, 2.609693673096498}, 118 -> {12.974133040469166`, 2.207760200794086}, 119 -> {11.729630401802199`, 2.7504515234384104`}, 120 -> {11.33288577108337, 2.4334349906924584`}, 121 -> {11.480827547038972`, 1.1604206191335003`}, 122 -> {10.77889906885374, 0.5562902310544136}, 123 -> {9.843600512023079, 1.076519017365522}, 124 -> {9.262790550866267, 1.1609076874086397`}, 125 -> {8.456100180334724, 1.2356403563105527`}, 126 -> {8.055929012082807, 2.2228730041050864`}, 127 -> {7.890831526142487, 3.837955592739668}, 192 -> {7.933095891039079, 8.968259591153705}, 193 -> {8.356750229018724, 9.827264562344778}, 194 -> {8.040535495361695, 10.746782221248626`}, 195 -> {8.436821881940086, 11.364370138032697`}, 196 -> {8.613739786595575, 11.629466896066292`}, 197 -> {9.043974368209499, 12.562254632431172`}, 198 -> {9.532501740496809, 12.57048760335091}, 199 -> {9.824732170805015, 11.612512965166308`}, 200 -> {10.139425843203092`, 11.512188539060892`}, 201 -> {10.630959506725167`, 12.701726906501474`}, 202 -> {11.156920995874236`, 12.914077472053835`}, 203 -> {11.54296879334117, 12.32203660674515}, 204 -> {11.78873246961573, 12.111163234884815`}, 205 -> {12.297648884542651`, 11.778873784556236`}, 206 -> {12.137856089770438`, 11.154065452400047`}, 207 -> {11.392363300178964`, 10.072955365430804`}, 208 -> {11.67541966698753, 9.783355062044446}, 209 -> {12.66587421300373, 10.673656066023145`}, 210 -> {13.25523888998744, 10.946300205201908`}, 211 -> {13.726018207141811`, 10.543481525255563`}, 212 -> {13.979758547549999`, 10.218425018070272`}, 213 -> {14.725719458669683`, 10.070780026736372`}, 214 -> {14.644546313455248`, 9.514903316376039}, 215 -> {13.53903561024818, 8.936502350206167}, 216 -> {13.651719901184876`, 8.637034290852279}, 217 -> {14.69899487200969, 8.446661551149054}, 218 -> {14.807855452813474`, 7.949242781838116}, 219 -> {13.995044415104083`, 7.437182191586921}, 220 -> {13.747777745960537`, 7.253380485239571}, 221 -> {13.21587013915646, 6.854469604372164}, 222 -> {12.333438683191346`, 7.220696748797404}, 223 -> {11.321562532925979`, 6.83766713556012}, 224 -> {11.365083154144616`, 6.22425041746935}, 225 -> {12.542795352322369`, 6.104871924663508}, 226 -> {13.224435250411297`, 5.948651402307288}, 227 -> {13.673790860198473`, 5.477885430572527}, 228 -> {13.889398476060876`, 5.255090548679165}, 229 -> {14.656355473749457`, 4.632652694498461}, 230 -> {14.527123247749596`, 4.142809670047394}, 231 -> {13.448529019046607`, 4.111997370611543}, 232 -> {13.339006623588595`, 3.792654092351127}, 233 -> {14.375369415523553`, 3.091272300811752}, 234 -> {14.41513550107939, 2.530357073383152}, 235 -> {13.66527305854814, 2.3249527609810774`}, 236 -> {13.398484038263499`, 2.1379410094426237`}, 237 -> {12.939364615916324`, 1.7985238286648055`}, 238 -> {12.374599136930984`, 2.122327444722219}, 239 -> {11.438870448360184`, 3.0789965266947448`}, 240 -> {11.14518588138849, 2.806325492365179}, 241 -> {11.790331294330256`, 1.6744162274658332`}, 242 -> {11.93304125286494, 1.0758198069973322`}, 243 -> {11.397924917839095`, 0.762740087178158}, 244 -> {11.124338679276361`, 0.5652941797703978}, 245 -> {10.666338848699025`, 0.}, 246 -> {10.142419867971256`, 0.24722683165043727`}, 247 -> {9.782035197266126, 1.4692269802905928`}, 248 -> {9.475027547635344, 1.437249331896931}, 249 -> {9.102531454531304, 0.4754748416476536}, 250 -> {8.618072808874135, 0.5181232548036908}, 251 -> {8.236236650438025, 1.4223117966109207`}, 252 -> {8.151467751113227, 1.7079273273063658`}, 253 -> {7.813214498238482, 2.354742911863664}, 254 -> {8.226696672005438, 3.307701842931193}, 255 -> {7.8818624460383315`, 4.292005801517424}, 0 -> {7.315898410403255, 4.513634360544348}, 2 -> {4.020016998833773, 6.684404631451922}, 4 -> {4.102459752593207, 9.66644891669786}, 5 -> {3.8962629611796995`, 3.7420527463978246`}, 8 -> {5.507288415412287, 10.863543441487428`}, 9 -> {2.2886150747326486`, 8.844531747281607}, 10 -> {2.123281074254212, 4.711993515042085}, 11 -> {5.213205435469002, 2.438524434828242}, 16 -> {6.589949552394833, 10.698873949035193`}, 17 -> {4.113689680281755, 11.558054051483643`}, 18 -> {2.129955747975245, 10.221798024373516`}, 19 -> {2.0364724728728163`, 7.779312204942916}, 20 -> {1.9494972080693014`, 5.7785763980506815`}, 21 -> {1.8702521992223815`, 3.3892710704086593`}, 22 -> {3.774183874736444, 1.8720038820900164`}, 23 -> {6.315895025985464, 2.5340838644322847`}, 32 -> {6.982486901080905, 9.993767702038944}, 33 -> {6.013155501222604, 11.832941669685944`}, 34 -> {4.626644815057388, 12.217534328899813`}, 35 -> {3.514595462960072, 11.206568806640703`}, 36 -> {2.684761521997096, 10.60130916432312}, 37 -> {1.246198045513296, 9.98942151402008}, 38 -> {1.110115142153326, 8.575618806112852}, 39 -> {2.560458068259777, 7.210618870002634}, 40 -> {2.5411302311083226`, 6.267080581735849}, 41 -> {0.9630321661782304, 5.099021261604856}, 42 -> {0.9888217402115522, 3.693642377200101}, 43 -> {2.404118853630477, 2.950646036443546}, 44 -> {3.199902120704257, 2.2639394484528026`}, 45 -> {4.218039579019111, 1.1830782518612466`}, 46 -> {5.654793111756019, 1.4558342007567617`}, 47 -> {6.805876155423354, 3.193262527061883}, 64 -> {6.969168473857512, 9.342044407260332}, 65 -> {6.804070987917193, 10.957126995894914`}, 66 -> {6.403899819665275, 11.944359643689447`}, 67 -> {5.597209449133732, 12.01909231259136}, 68 -> {5.016399487976921, 12.103480982634478`}, 69 -> {4.08110093114626, 12.623709768945586`}, 70 -> {3.3791724529610274`, 12.0195793808665}, 71 -> {3.527114228916629, 10.746565009307542`}, 72 -> {3.1303695981978006`, 10.429548476561589`}, 73 -> {1.8858669595308335`, 10.972239799205914`}, 74 -> {1.0469589917187356`, 10.570306326903502`}, 75 -> {1.1971584782785047`, 9.5901388745509}, 76 -> {1.0767917712310773`, 8.99772139240666}, 77 -> {0.860023128366139, 8.207848615430017}, 78 -> {1.6581159400279954`, 7.524910019516784}, 79 -> {3.06322768282096, 6.987902202476573}, 80 -> {3.125283688688567, 6.287813113646913}, 81 -> {1.611256179960261, 6.037361468621089}, 82 -> {0.7261621975186632, 5.477842510557044}, 83 -> {0.8978073910535009, 4.678834223376597}, 84 -> {0.970221091208737, 4.083882892257126}, 85 -> {0.737212450448677, 3.161096302050691}, 86 -> {1.5728677200878813`, 2.656370245378133}, 87 -> {2.8663910907450685`, 3.0861092506848085`}, 88 -> {3.2440663460044377`, 2.733650274737654}, 89 -> {3.0143667199006376`, 1.4655137756038226`}, 90 -> {3.6523843595631256`, 0.8262799071702602}, 91 -> {4.603072838297505, 1.2763244536394467`}, 92 -> {5.225145430059566, 1.292900558148018}, 93 -> {6.044540051293259, 1.3252332336137087`}, 94 -> {6.567577049956043, 2.306698563900193}, 95 -> {6.87165896010366, 3.798362329689061}, 128 -> {6.978137553961668, 8.887994198482577}, 129 -> {6.633303327994561, 9.872298157068807}, 130 -> {7.0467855017615175`, 10.825257088136336`}, 131 -> {6.7085322488867725`, 11.472072672693635`}, 132 -> {6.623763349561974, 11.75768820338908}, 133 -> {6.241927191125864, 12.66187674519631}, 134 -> {5.757468545468695, 12.704525158352347`}, 135 -> {5.384972452364655, 11.742750668103069`}, 136 -> {5.077964802733874, 11.710773019709407`}, 137 -> {4.717580132028743, 12.932773168349563`}, 138 -> {4.193661151300974, 13.18}, 139 -> {3.735661320723638, 12.614705820229602`}, 140 -> {3.4620750821609043`, 12.41725991282184}, 141 -> {2.9269587471350587`, 12.104180193002668`}, 142 -> {3.069668705669743, 11.505583772534166`}, 143 -> {3.7148141186115087`, 10.37367450763482}, 144 -> {3.4211295516398152`, 10.101003473305255`}, 145 -> {2.4854008630690156`, 11.057672555277781`}, 146 -> {1.9206353840836758`, 11.381476171335194`}, 147 -> {1.4615159617365006`, 11.042058990557376`}, 148 -> {1.1947269414518598`, 10.855047239018923`}, 149 -> {0.44486449892060875`, 10.649642926616847`}, 150 -> {0.4846305844764469, 10.088727699188247`}, 151 -> {1.520993376411404, 9.387345907648873}, 152 -> {1.4114709809533927`, 9.068002629388456}, 153 -> {0.33287675225040303`, 9.037190329952605}, 154 -> {0.2036445262505424, 8.547347305501539}, 155 -> {0.9706015239391235, 7.9249094513208345`}, 156 -> {1.1862091398015266`, 7.702114569427473}, 157 -> {1.6355647495887027`, 7.231348597692712}, 158 -> {2.317204647677631, 7.075128075336492}, 159 -> {3.494916845855384, 6.9557495825306495`}, 160 -> {3.5384374670740204`, 6.3423328644398795`}, 161 -> {2.526561316808653, 5.959303251202596}, 162 -> {1.64412986084354, 6.325530395627836}, 163 -> {1.1122222540394624`, 5.926619514760429}, 164 -> {0.8649555848959167, 5.742817808413078}, 165 -> {0.052144547186525614`, 5.230757218161884}, 166 -> {0.1610051279903093, 4.733338448850946}, 167 -> {1.2082800988151234`, 4.542965709147721}, 168 -> {1.3209643897518202`, 4.243497649793833}, 169 -> {0.21545368654475183`, 3.665096683623961}, 170 -> {0.13428054133031608`, 3.1092199732636274`}, 171 -> {0.8802414524500008, 2.9615749819297275`}, 172 -> {1.1339817928581883`, 2.6365184747444363`}, 173 -> {1.604761110012559, 2.233699794798092}, 174 -> {2.1941257869962687`, 2.5063439339768543`}, 175 -> {3.1845803330124696`, 3.3966449379555534`}, 176 -> {3.4676366998210355`, 3.1070446345691956`}, 177 -> {2.722143910229562, 2.0259345475999524`}, 178 -> {2.562351115457348, 1.4011262154437638`}, 179 -> {3.0712675303842687`, 1.0688367651151847`}, 180 -> {3.317031206658829, 0.8579633932548489}, 181 -> {3.7030790041257635`, 0.2659225279461648}, 182 -> {4.229040493274832, 0.478273093498526}, 183 -> {4.720574156796907, 1.6678114609391077`}, 184 -> {5.035267829194984, 1.5674870348336913`}, 185 -> {5.327498259503191, 0.6095123966490892}, 186 -> {5.816025631790501, 0.617745367568828}, 187 -> {6.246260213404424, 1.5505331039337076`}, 188 -> {6.423178118059914, 1.8156298619673024`}, 189 -> {6.8194645046383044`, 2.4332177787513736`}, 190 -> {6.5032497709812755`, 3.3527354376552214`}, 191 -> {6.926904108960921, 4.211740408846294}}]]

The 44,686 rules with signature 12 2312 cannot even produce trees. Rules with signature 13 2312 can produce trees, as well as closed structures similar to those seen in 12 1322 rules.

A minimal way to add mixed arity to the left-hand sides of rule is to introduce unary relations—but the presence of these seems to inhibit the production of any more complex forms of behavior.

Looking at mixed binary and ternary left-hand sides, none of the 1,141,692 rules with signature 1312 1322 seem to produce even trees. But rules with signature 1312 2322 readily produce structures such as:

GraphicsGrid[ Partition[ ParallelMap[ Labeled[ResourceFunction["WolframModelPlot"][ ResourceFunction["WolframModel"][#[[1]], #[[2]], #[[3]], "FinalState"], ImageSize -> 1.1 {150, 100}], RulePlot[ResourceFunction["WolframModel"][#[[1]]], "RulePartsAspectRatio" -> 1, ImageSize -> Tiny]] &, {{{{1, 2, 3}, {3, 4}} -> {{2, 5, 3}, {3, 5, 4}, {1, 2}, {6, 2}}, {{0, 0, 0}, {0, 0}}, 200}, {{{1, 2, 3}, {2, 4}} -> {{5, 6, 4}, {4, 1, 6}, {2, 1}, {3, 6}}, {{0, 0, 0}, {0, 0}}, 40}, {{{1, 2, 3}, {4, 1}} -> {{1, 4, 5}, {6, 5, 3}, {5, 2}, {6, 1}}, {{0, 0, 0}, {0, 0}}, 13}, {{{1, 2, 3}, {4, 2}} -> {{4, 2, 2}, {2, 5, 6}, {5, 1}, {5, 4}}, {{0, 0, 0}, {0, 0}}, 14}, {{{1, 2, 3}, {2, 4}} -> {{5, 3, 3}, {4, 2, 5}, {4, 1}, {4, 5}}, {{0, 0, 0}, {0, 0}}, 12}, {{{1, 2, 3}, {1, 4}} -> {{1, 5, 4}, {5, 1, 2}, {3, 6}, {5, 6}}, {{0, 0, 0}, {0, 0}}, 10}, {{{1, 2, 3}, {3, 4}} -> {{5, 5, 5}, {5, 6, 4}, {3, 1}, {1, 5}}, {{0, 0, 0}, {0, 0}}, 16}, {{{1, 2, 3}, {2, 4}} -> {{5, 6, 4}, {4, 1, 6}, {2, 1}, {3, 6}}, {{0, 0, 0}, {0, 0}}, 50}}], UpTo[4]], ImageSize -> Full]

One can go on and look at rules with higher signatures, and probably the most notable finding is that—in keeping with the Principle of Computational Equivalence [1:12]—the overall behavior seen does not appear to change at all. Here are nevertheless a few examples of slightly unusual behavior found in 2312 3322 and 2312 4342 rules:

GraphicsGrid[ Partition[ ParallelMap[ Labeled[ResourceFunction["WolframModelPlot"][ ResourceFunction["WolframModel"][#[[1]], #[[2]], #[[3]], "FinalState"], ImageSize -> 1.1 {150, 100}], RulePlot[ResourceFunction["WolframModel"][#[[1]]], "RulePartsAspectRatio" -> 1, ImageSize -> Tiny]] &, {{{{1, 2, 3}, {3, 4, 5}, {6, 5}} -> {{7, 4, 8}, {8, 9, 6}, {10, 5, 9}, {4, 9}, {9, 5}}, {{0, 0, 0}, {0, 0, 0}, {0, 0}}, 300}, {{{1, 2, 3}, {4, 3, 5}, {3, 6}} -> {{4, 5, 1}, {4, 2, 7}, {5, 3, 8}, {1, 9}, {5, 4}}, {{0, 0, 0}, {0, 0, 0}, {0, 0}}, 250}, {{{1, 2, 3}, {1, 4, 5}, {3, 6}} -> {{7, 8, 7}, {7, 5, 6}, {9, 5, 5}, {1, 7, 4}, {7, 5}, {5, 10}, {11, 6}, {6, 9}}, {{0, 0, 0}, {0, 0, 0}, {0, 0}}, 100}, {{{1, 2, 3}, {4, 5, 6}, {2, 6}} -> {{7, 7, 2}, {6, 2, 8}, {8, 5, 7}, {8, 9, 3}, {1, 6}, {10, 6}, {5, 3}, {7, 11}}, {{0, 0, 0}, {0, 0, 0}, {0, 0}}, 16}}], UpTo[4]], ImageSize -> Full]
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