Isotropic Cellular Automata: the DDLab iso-rule paradigm
IIsotropic Cellular Automata: the DDLab iso-rule paradigm
Andrew Wuensche ∗ Discrete Dynamics Lab.
Jos´e Manuel G´omez Soto † Universidad Aut´onoma de Zacatecas.Unidad Acad´emica de Matem´aticas. Zacatecas, Zac. M´exico.
Aug 2020
Abstract
To respect physics and nature, cellular automata (CA) models of self-organisation, emergence, computation and logical universality should beisotropic, having equivalent dynamics in all directions. We present a novelparadigm, the iso-rule, a concise expression for isotropic CA by the outputtable for each isotropic neighborhood group, allowing an efficient methodof navigating and exploring iso-rule-space. We describe new functions andtools in DDLab to generate iso-groups and iso-rules, for multi-value as wellas binary, in one, two and three dimensions. These methods include filing,filtering, mutating, analysing dynamics by input-frequency and entropy,identifying the critical iso-groups for glider-gun/eater dynamics, and auto-matically classifying iso-rule-space. We illustrate these ideas and methodsfor two dimensional CA on square and hexagonal lattices. keywords: DDLab, cellular automata, isotropy, iso-groups, iso-rules,glider-guns, logical universality, input-frequency, filtering, mutation.
Isotropy, rotational invariance, equivalence, lack of directional bias or prefer-ence, is assumed in all aspects of physics from quantum theory to cosmology,and must be inherited by living systems. Subject to the local gravitational,chemical and environmental context, evolution makes creatures with a kindof geometric symmetry, be it bilateral, multi-faceted, or fractal. In the sameway, emergent gliders in an isotropic cellular automata (CA) universe feature ahead/tail bilateral symmetry, with dynamics and interactions equivalent in alldirections. Indeed it would seem entirely unnatural if this were not the case, so itfollows that to comply with physics and nature, CA models of self-organisation, ∗ † [email protected], http://matematicas.reduaz.mx/ ∼ jmgomez a r X i v : . [ n li n . C G ] A ug a) the unslanted initial view (b) time-steps shifted 1/2 cell right Figure 1: v k space-time pattern of an isotropic CA, size 33, The rcodesize=64, The iso-rule (size=36): 140761540026777563655513706072505220, as agraphic: and in hexadecimal: 07 07 c6 c016 ff dc f5 b4 be 30 ea 8a 90. (a) is the initial view because even n-templates areskewed right (figure 6). (b) successive time-steps shifted 1/2 cell right to restoresymmetry[28, EDD:32.9.1]. emergence, computation and logical universality should be isotropic . But CArule-space in general is not isotropic. Rules in general are defined by rule-tablesof length v k where v =value-range and k =neighborhood size, with a rule-spaceof v v k . Isotropic rule-space, where symmetric initial patterns must conservesymmetry as time evolves — as from singleton seeds in figures 1 and 3 — makeup a tiny proportion of general rule-space.Although some rule categories, survival/birth, totalistic, reaction-diffusion,are isotropic by default, these iso-subsets can be transformed into a generalexpression of isotropic CA, where the “iso-groups” of equivalent neighborhoodtemplates (n-templates) by all possible spins and flips share the same output —figure 2 gives examples. iso-groups(102) v2k9(2d) iso-index=92>1 group(size4)=
335 359 461 485 (a) 2d v k (92/101) iso-groups(276) v3k7(hex2d) iso-index=224>1 group(size3)=
699 1667 1979 (b) 2d hex (224/275) iso-groups(171) v3k7(3d) iso-index=162>0 group(size3)= (b) 3d (162/170)
Figure 2:
Examples of iso-groups showing (group-index/max-index).
Especially significant are iso-rules analogous to Conway’s famous survival/birthgame-of-Life[3, 7], the first rule with logical gates constructed from glider/eaterdynamics made with the first glider-gun discovered by Gosper. Figure 4 illus-trates Life and other significant rules, including glider-gun iso-rules not basedon survival/birth where logical universality has been demonstrated. However, anisotropic rules can also be logically universal, for example the 1d rule 110[5],and the 2d X-rule[10] k hex 2d v k hex 2d v k s square 2d v k s square 2d v k cubic 3d v k cubic 3d Figure 3:
Examples of space-time pattern snapshots for v =4 2d and 3d isotropicCA from a singleton seed, a v> ife[3, 7]: p =30 Eppstein[6]: p =68 Sapin[15]: p =18(a) Conway’s survival/birth (s23/b3) game-of-Life[3, 7] and Eppstein’s s236/b3 rule[6].Sapin’s R-rule[15] evolved by genetic algorithm from iso-groups. Life and SapinR arelogically universal with glider streams stopped by eaters, Eppstein’s by head-on collisionsonly — lower panel.00 00 00 00 00 60 03 1c 61 c6 7f 86 a0 — Life04 89 86 1a 00 6d 23 1e 61 e6 7f 86 a0 — Eppstein11 34 1c 2c 52 36 7d 3b e0 f8 7e 0a a0 — SapinR LERVariant[12]: p =22 Precursor[10]: p =19 Sayab[11]: p =20(b) Three binary logically universal iso-rules[12] belong to a family with differentglider-guns. Gliders streams are stopped by eaters. The Variant and Precursor rules areclosely related differing by two outputs. The Sayab rule is a distant cousin differing fromthe Precursor by 33 outputs.24 c0 04 42 83 01 80 2c a4 29 04 e0 70 — Variant24 c0 04 42 83 01 80 24 a4 69 04 e0 70 — Precursor24 01 13 1a 14 20 50 2c 45 05 48 e0 50 — Sayab VPS Figure 4:
Significant binary ( v k ) glider-gun iso-rules on 3 × p =glider-gun period and firing frequency. Green trails denote motion. a) v3k3x1.vco, g1(hex)00a864 (b) v3k4t1.vco, g1(hex)2a945900 (c) v3k4x1.vco(hex)2282a1a4(d) v3k5x1.vco, g1(hex)004a8a2a8254xx (e) v3k6n6.vco, g16(hex)01059059560040xx (f) v3k7w1.vco, g1(hex)020609a2982a68aa64The Spiral rule[25] Figure 5:
Glider dynamics discovered in 3-value 2d k-totalistic rules, on n-templates[27] k =3 to 7, in figure 7. Cell values: 0=white, 1=red, 2=black. Greentrails indicate motion. Examples b, c, e, and f include glider-guns. The rules canbe loaded in DDLab by their file-names, in hexadecimal, or from the rule collectionsindex g(x). The dynamics emerge spontaneously, including the glider-gun in thelogically universal Spiral rule. The CA iso-rule notation provides a practical balance between a full lookup-table on which isotropy may be imposed and an abbreviated notation that mustbe isotropic — survival/birth or totalstic. The iso-rule notation is concise, butnot too concise. Insights can be gained into glider-gun mechanics by observ-ing iso-group activity, frequency and entropy. Iso-rules permit navigating andexploring iso-group mutants to establish their related families, and to discovernew significant iso-rules in iso-rule space.We describe new methods[31] for defining and automatically generating iso-rules on the basis of iso-groups with predefined n-templates in 1, 2 and 3 dimen-sions, and with value-ranges (colors) from 2 (binary) up to 8 values as in figure 1.The methods include editing, filing, filtering, mutating, analysing dynamics byinput-frequency and entropy, identifying the critical and neutral iso-groups forglider-gun/eater dynamics, and automatically classifying iso-rule-space. This is5een in the context of the superset of the general rule-table, and in iso-subsetsin a narrower sense, k-totalistic, t-totalistic, outer-totalistic, survival/birth andreaction-diffusion. General rule-tables and iso-subsets can be transformed intoiso-rules. Binary Moore neighborhood rules, and initial states, are compatiblewith “Golly”[8, 4].We present the ideas and methods mainly for 2d square and hexagonal exam-ples as in figures 4 and 5, but also include 1d and 3d. Glider-rules that featuregliders emerging spontaneously are readily found by classifying rule-space byinput-entropy variability[28, EDD:33], with examples in [22, 23, 9]. Glider-gunsmay also emerge spontaneously[23, 25, 11] but usually they are elaborately con-structed artifacts[7, 9, 10, 12]. A rule producing gliders, and preferably alsoeaters , makes a good starting point to build glider-guns and logically universaldynamics among its family of mutants.Mutant iso-rule-space can be navigated and explored with the program “Dis-crete Dynamics Lab” (DDLab)[29] — its many methods for studying space-timepatterns[28, EDD:23-30] and attractor basins[28, EDD:31-32] now apply to thenew iso-rule paradigm. DDLab is documented in the book “Exploring DiscreteDynamics”(EDD)[28], and we have usually included the relevant sections whenciting EDD. Both DDLab and EDD are updated and maintained online. The lattice geometry of a CA depends on the n-template, and there are a widerange of pre-defined n-templates in DDLab[28, EDD:10]. In figures 6, 7 and 8and we present those pre-defined n-templates where iso-groups and iso-rules arecomputed, and which themselves have a symmetric geometry.A CA “target” cell updates according to the values of its n-template. As canbe seen in figures 7 and 8, the target cell in some cases is not a member of then-template. The n-template is homogeneous throughout the network so requiresperiodic boundary conditions where each lattice boundary wraps around to itsopposite boundary resulting in a ring of cells in 1d, a torus in 2d, and a 3-torusin 3d. However, null boundary condition can also be imposed[28, EDD:31.3]. odd- k t0t1|cell 2| 0 cell=1 wiring=2 1 0 outwires=3 links:bi=3 self=3=33.3% t0t1|cell 4 | 0 cell=2 wiring=4 3 2 1 0 outwires=5 links:bi=10 self=5=20.0% t0t1|cell 6 | 0 cell=3 wiring=6 5 4 3 2 1 0 outwires=7 links:bi=21 self=7=14.3% t0t1|cell 8 |cell 0 cell=4 wiring=8 7 6 5 4 3 2 1 0 outwires=9 links:bi=36 self=9=11.1% t0t1|cell 10 |cell 0 cell=5 wiring=10 9 8 7 6 5 4 3 2 1 0 outwires=11 links:bi=55 self=11=9.1% —continues even- k t0t1|cell 1| 0 cell=1 wiring=1 0 outwires=2 links:bi=1 self=2=50.0% t0t1|cell 3 | 0 cell=2 wiring=3 2 1 0 outwires=4 links:bi=6 self=4=25.0% t0t1|cell 5 | 0 cell=3 wiring=5 4 3 2 1 0 outwires=6 links:bi=15 self=6=16.7% t0t1|cell 7 | 0 cell=4 wiring=7 6 5 4 3 2 1 0 outwires=8 links:bi=28 self=8=12.5% t0t1|cell 9 |cell 0 cell=5 wiring=9 8 7 6 5 4 3 2 1 0 outwires=10 links:bi=45 self=10=10.0% —continuesFigure 6:
1d size k n-templates, indexed k -1 to 0 from left to right, as definedin DDLab for odd and even k . Even- k n-templates are asymmetric, skewed to theright. However, they still support iso-rules as shown in figure 1 where the symmetryis preserved by shifting successive time-steps by / cell-space. Eaters are localised configurations that can stop a glider stream. Other important lo-calised configuration include reflectors, deflectors and oscillators. We have use “eaters” as ashorthand for all these.
2d cell=5,5=55 wiring=6,6 4,5 6,4 outwires=3 links:bi=0 self=0=0.0% (a) k =3
99 00
2d cell=5,5=55 wiring=6,6 5,5 4,5 6,4 outwires=4 links:bi=0 self=100=25.0%
32 10 (b) k =4t
99 00
2d cell=5,5=55 wiring=5,6 6,5 4,5 5,4 outwires=4 links:bi=200 self=0=0.0%
32 10 (c) k =4s
99 00
2d cell=5,5=55 wiring=5,6 6,5 5,5 4,5 5,4 outwires=5 links:bi=200 self=100=20.0%
43 2 10 (d) k =5
99 00
2d cell=5,5=55 wiring=6,6 5,6 6,5 4,5 6,4 5,4 outwires=6 links:bi=300 self=0=0.0% (e) k =6
99 00
2d cell=5,5=55 wiring=6,6 5,6 6,5 5,5 4,5 6,4 5,4 outwires=7 links:bi=300 self=100=14.3% (f) k =7
77 00
2d cell=4,4=36 wiring=5,5 4,5 3,5 5,4 3,4 5,3 4,3 3,3 outwires=8 links:bi=256 self=0=0.0% (h) k =8
77 00
2d cell=4,4=36 wiring=5,5 4,5 3,5 5,4 4,4 3,4 5,3 4,3 3,3 outwires=9 links:bi=256 self=64=11.1% (i) k =9 Figure 7:
2D n-templates ( k =3 to 9) as defined and indexed in DDLab[28, EDD:10],hexagonal or square. k =4 n-templates can be either. To achieve periodic boundaryconditions hexagonal n-templates require even lattice dimensions. The target cell iscentral even if not part of the n-template. For 3-value CA, glider rules are readilyfound for these n-templates as in figure 5. (a) k =6 (b) k =7 Figure 8:
3d n-templates ( k =6 or 7) as defined and indexed in DDLab[28, EDD:10].The target cell is central even if not part of the n-template. CA rules can be divided and defined according to a number of (possibly overlap-ping) types[28, EED:13]. These include the full rule-table (rcode), k-totalistic(kcode), t-totalistic (tcode), outer-totalistic, reaction-diffusion, survival/birth,and of course iso-rules.The most general rule type, rcode, can implement any logic including all thetypes listed above, and will form the basis for extracting iso-rules. Rcode is a listof the outputs of all v k possible neighborhoods depending only the value-range v and neighborhoods size k giving a rule-space of v v k , and is independent ofn-template geometry. The list order must be specified, and we follow Wolfram’sclassical convention[18, 19]; a descending order of neighborhood binary (or v -ary for v>
2) values from left to right which is also the rcode index, as in thisexample for binary k =3 where the decimal equivalent of the rcode string givesthe “Elementary Rule” number. In DDLab the neighborhoods are displayed vertically for compactness with a k index ( k -1 to 0, top down), making a so called “neighborhood matrix”, shownhere for binary k =3, and for k =5 where rcode is better expressed in hexadecimalrather than decimal, There are 2 =256 v k ule index - 7......0 31...... ........ ........ .......0: : : :2 - 11110000 4 - 11111111 11111111 00000000 00000000k-index 1 - 11001100 3 - 11111111 00000000 11111111 000000000 - 10101010 k-index 2 - 11110000 11110000 11110000 11110000-------- 1 - 11001100 11001100 11001100 11001100rcode 193 - 11000001 0 - 10101010 10101010 10101010 10101010-------- -------- -------- --------rcode (dec) 4276676736 - 11111110 11101000 11101000 10000000(hex) fee8e880 (majority rule) As a reference, the neighborhood matrix is displayed graphically prior toselecting, editing and transforming the rcode. The same matrix principles applyfor any values of v and k , as in the examples in figure 9,
40 31 15 040 31 15 0 (a) v k , =32, complete matrix
80 511 447 383 319 255 191 127 63 0 (b) v k , =512, complete matrix
60 2186 2159 2132 2105 2078 2051 2024 1997 1970 1943 1916 1889 1862 1835 (c) v k , =2187, left part only
40 3124 3099 3074 3049 3024 2999 2974 2949 2924 (d) v k , =3125, left part only Figure 9:
Examples of various v , k matrices showing rcode size. (a) v k rcode(32), 2d square iso-rule(12) (b) v k rcode(512), 2d square iso-rule(102)(c) v k rcode(2187), 2d hex iso-rule(276)(d) v k rcode(3225), 2d square iso-rule(600) Figure 10:
Examples of majority (voting) rules[28, EDD:16.7] for v , k in figure 9presented as bit/value strings (and sizes) as rcode, and transformed to 2d iso-rules.Only binary majority rcode with odd k is isotropic by default when initially selected. v =2 the central cell wins, for v> k rcode is isotropic.However, the transformation to an iso-rule also induces isotropy in the originalrcode. The iso-rule string is much shorter than the rcode-string as can be seenin table 1. Graphical string presentations can be rescaled, adjusted betweensingle or multiple rows, and allow various functions and manipulations with themouse and keyboard [28, EDD:16.4].An rcode is transformed to the iso-rule and its graphical string with a key-press, and further options will show iso-groups graphically with accompanyingdetails, (figure 11), or the the complete graphic of the iso-rule prototype neigh-borhoods (figure 12), in a separate window.
84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 (a) 3 sucessive v k
2d square (Moore) iso-groups for iso-indeces as shown (max=101) (b) 3 successive v k
2d hex iso-groups for iso-indeces as shown (max=275)
62 61 60 59 58 57 56 55 54 53 52 (c) 3 successive v k
3d iso-groups for iso-indeces as shown (max=171)
Figure 11:
Examples of 3 successive iso-groups shown graphically; their sizes dependon internal symmetries. (a) v k has 102 iso-groups, (b) and (c) are both v k but (b) is 2d hex with 276 iso-groups, and (c) is 3d with 172 iso-groups. a) 102 v k
2d square (Moore) neighborhood iso-group prototypes(b) 276 v k
2d hex neighborhood iso-group prototypes(c) 172 v k
3d neighborhood iso-group prototypes
Figure 12:
Examples of the complete set of iso-rule n-template prototypes showngraphically in descending order of their decimal equivalents from the top-left. (a) v k has 102 prototypes, (b) and (c) are both v k but (b) is 2d hex with 276prototypes, and (c) is 3d with 172 prototypes. .1 iso-rule advantages The iso-rule is arguably an improvement on previous isotropic CA notations, forexample by Sapin[15, 16] and Hensel , because the iso-rule is a simple lookup-table in a conventional order and is general, applying to a range of n-templatesizes k in 1d, square or hex 2d, and 3d, and extending beyond binary to a range ofvalues v . The iso-rule can be computed down by reducing a full CA lookup-tableor up by enhancing iso-subsets — totalistic, reaction-diffusion and survival/birth— so provide an intermediate granularity for mutation, bias, manipulation, orin a search by genetic algorithm[15]; isotropy is conserved whatever changes aremade.In DDLab, the iso-rule provides the basis for input-frequency/entropy, filter-ing and mutation in the same way as conventional full or totalistic CA lookup-tables: rcode, tcode or kcode. For glider/eater/glider-gun iso-rules, the input-frequency identifies both the critical and neutral iso-groups underlying dynam-ics. The methods for automatically classifying and examining rule-space basedon input-entropy variability[22, 23, 9] can be applied to iso-rules.The “negative”[28, EDD:18.5.2] of the iso-rule system gives equivalent iso-rules in terms of dynamics given a complimentary initial state (figure 16), whichwould reduce the effective size of iso-rule-space. The sizes of the iso-rule tables for the n-templates in figures 6, 7 and 8 depend on v , k , and the internal symmetries of the n-template so are difficult to calculateanalytically. The tables below give iso-group sizes computed algorithmically inDDLab.
1d k2 3 4 5 6 7 8 9 10-----------------------------------------2 | 3 6 10 20 36 72 136 272 5283 | 6 18 45 135 378 1134 3321 9963| 4 | 10 40 136 544 2080 8320v 5 | 15 75 325 1625 7875| 6 | 21 126 666 39967 | 28 196 1255 85758 | 36 288 20802d hex-k 2d square-k 3d3 4 6 7 4 5 8 9 6 7----------------- ----------------- -------2 | 4 8 13 26 2 | 6 12 51 102 | 2 | 10 203 | 10 30 92 276 3 | 21 63 954 2862 v 3 | 57 171| 4 | 20 80 430 1720 | 4 | 55 220 | 4 | 240 960v 5 | 35 175 1505 v 5 | 120 600 5 | 800| 6 | 56 336 | 6 | 231 13867 | 74 588 7 | 4068 | 120 960 8 | 666
Table 1:
Sizes of iso-rules lookup-tables (number of iso-groups) for 1d, 2d hex,2d square and 3d n-templates in figures 6, 7, and 8, for v, k within non-extendedlimits[28, EDD:7.1.1] in DDLab. The Hensel notation, which only applies to a binary Moore neighborhood (v2k9), isincorporated in DDLab[28, EDD:16.10.3] so that files for rules (and initial states) can beinterchanged[30] with “Golly” software[8] used in the game-of-Life community[4]. iso-subsets expressed as iso-rules Significant CA iso-subsets, rule types that are isotropic by default, includek-totalistic (kcode), t-totalistic (tcode), outer totalistic, reaction/diffusion, andsurvival/birth rules. These iso-subsets have their specific concise definitionsand rule-spaces which allow interesting coarse-grained mutations. However, bytransforming the iso-subsets to equivalent iso-rules finer-grained mutations be-come possible in a search for a wider range of significant rule families. Below,we define these iso-subsets and their rule-spaces . (a) hex 2d glider-gun (b) 3d glider-gun Figure 13:
Beehive rule glider-guns v k for (a) hex 2d shooting gliders in 6directions, and (b) 3d 40 × ×
40 shooting gliders in 4 directions. The 3d glider-gun emerges spontaneously, but localised structures (such as eaters) are absent.
Kcode rules are defined by a list of the outputs for all possible combinations ofvalue-frequencies in the neighborhood. Each combination is represented by astring of length v , shown vertically from v -1 down in this example for the v k k , so the last row of frequencies is redundant and could be omitted. In the spirit of Wolfram’s convention[18], the ordering of the combinationsdepend on their v -ary value, with the higher kcode index on the left. Kcoderules are independent of the n-template; they are isotropic because the positionsof values in are irrelevant. DDLab has three modes, SEED, FIELD and TFO[28, EDD:6.1]. TFO-mode (TotalisticForwards-Only) has advantages for these iso-subsets in the scope of v and k , but SEED-modeis required for automatic redefinition as iso-rules.
12n the example above for v k
6, outputs [0,1,2] are listed in reverse or-der of the kcode index, and can be expressed in decimal (if applicable) or inhexadecimal. DDLab automatically transforms kcode into its equivalent rcode,which can then be transformed to isotropic rcode and the iso-rule according tothe n-template. The 2d v k (92) 00220200000102000222020022222222222122220222212102222221022000220222022222212000022212022010(hex) 0a 20 01 20 2a 20 aa aa a9 aa 2a 99 2a a9 28 0a 2a 2a a9 80 2a 62 84(57) 202200001120002222222221210202200022220222122120021022010(hex) 02 28 01 60 2a aa a6 48 a0 2a 8a 9a 60 92 84 The size of a kcode-table, S k = ( v + k − / ( k ! × ( v − v, k as in the table below, -- k --2 3 4 5 6 7 8 9 10----------------------------------------2 | 3 4 5 6 7 8 9 10 113 | 6 10 15 21 28 36 45 55 66| 4 | 10 20 35 56 84 120 165 220 286v 5 | 15 35 70 126 210 330 495 715 1001| 6 | 21 56 126 252 462 792 1287 2002 30037 | 28 84 210 462 924 1716 3003 5005 80088 | 36 120 330 792 1716 3432 6435 11440 19448 Kcode where v =3 can be expressed as an ij -matrix based on the frequencyof 2s and 1s (0s are given by k − ( i + j ) so are not required). Such rules can bereinterpreted as conceptual discrete models of reaction-diffusion systems withinhibitor and activator reagents[1, 2]. Figure 2 gives examples. j i Beehive rule j i Spiral ruleTable 2:
The ij -matrices of (left) the v k Beehive rule [23, 1, 24] and (right)the v k Spiral rule [25, 2, 26], on 2d hex lattices. The output can be read offfrom the frequency of 2s (rows i ) and 1s (columns j ). Previous studies have shownthat a significant proportion of outputs are quasi-neutral — wildcards — with littleimpact on dynamics. These are shown in color: red for strong wildcards, blue forweak wildcards. .2 t-totalistic rules (tcode) Tcode rules are defined by a list of outputs for each possible total, the sum ofvalues in the neighborhood, and are useful for setting threshold functions. Tcodeis a subset of kcode because each total can include several kcode combinationsof value-frequencies, so tcode is also isotropic, but more directly because anyn-template’s rotation/reflection must give the same total. The size of a tcode-table S t = k ( v −
1) + 1.To set tcode each total is set out in reverse value order, S t -1 to 0, and isassigned an output [0,1, . . . , v -1], which is the tcode value-string. Here is anexample for the v k Tcode may be expressed in decimal (if applicable) or in hexadecimal. DDLabautomatically transforms tcode into its equivalent rcode, which can then betransformed to isotropic rcode and the iso-rule according to the n-template. Forthe majority rule (above) the rule tables for rcode (size 3225), and the iso-rule(size 600) for a 2d square n-template, are shown graphically in figure 10(d).For binary ( v =2) tcode depends on just the sum of 1s the neighborhood sotcode and kcode are identical, S t = S k , but for v ≥ =3 S t becomes progressivelysmaller than S k . Here is an example of binary tcode for the v k Outer-totalistic CA require v rules, one for each possible value of the centercell, so the size of the total string S o is v × S k for kcode, or v × S t for tcode. Themethod in DDLab works with any k , but makes most sense if the central cell isempty in the n-template.Binary ( v =2) life-like 2d CA can be defined by two k =8 tcode rules, witha total table size S o =18, whereas the equivalent S i =102 and S r =512. For thegame-of-Life two v k
22 00
2d cell=1,1=4 wiring=2,2 1,2 0,2 2,1 0,1 2,0 1,0 0,0 outwires=8 links:bi=36 self=0=0.0% k v and k . 14 .4 reaction-difusion rules (a) unfiltered (b) filtered Figure 14:
Snapshots of a hex 2d reaction-diffusion CA v k , with the iso-rule(size 1720) shown below. The threshold interval was set 1 to 4. The initial state60 ×
60 has a low density (0.01) of non-zero cells. (a) the emergent pattern, and (b)the pattern with the 3 most frequent iso-groups filtered, showing structures thatresemble glider-guns.
Reaction-Diffusion or excitable media dynamics [14], can be generated witha type of CA with 3 cell qualities: resting, excited, and refractory (or substrate,activator, and inhibitor). The rules are isotropic by default because they are ba-sically totalistic, depending on just totals of values in the neighborhood. Thereis usually one resting type, one excited type, and one or more refractory types.In DDLab these correspond to the values v =0, v =1, and v ≥
2, which cycle be-tween each other. A resting cell (0) remains as is until the number of excitedcells in its neighborhood falls within the threshold interval t , whereupon it be-comes excited (1). An excited cell (1) changes to the first refractory value (2)at the next time-step, then to the next refractory value (3) and so on, and thefinal ( v -2) refractory value changes back to resting (0), completing the followingclockwise cycle, resting(0)--->if within t threshold interval\ \\ (1) excited\ \(v-2)<----(3)<--(2) refractory The variables that need to be set to define a reaction-diffusion rule are v, k and the threshold interval within k . The number of refractory values is v -2.In DDLab, reaction-diffusion[28, EDD:13.8] can be set as rcode which allowstransformation to an iso-rule, or as outer-kcode which allows a greater range of v, k . 15he resulting dynamics, in 2d or 3d, can produce waves, spirals and relatedpatterns that can resemble the Belousov-Zhabotinsky reaction in a non-linearchemical medium and other types of excitable media. Filtering the wave-likepatterns by descending frequently of iso-groups can reveal dynamics reminiscentof glider-guns (figure 14). The filtering method[28, 32.11.5] is based on theinput-frequency histogram described in later sections.As well as the threshold interval, the dynamics are sensitive to the initialstate and its density of non-resting types (non-zero values)[28, 21.3] — usuallylow for best spiral-wave results. A survival/birth rule, including the game-of-Life, can be set in DDLab[28,EDD:16.1]. The rule is turned into rcode automatically and can then be trans-formed into an iso-rule. The rcode of the iso-rule can be transformed for anegative universe as in figure 16 by complimenting both neighbourhoods andoutputs[28, 18.5.3].Figure 15: Conway’s game-of-Life (s23/b3) shown as a 512 bit rcode in 8 rows.The diagonal symmetry in each 8x8 block is a necessary (but insufficient) indicationof isotropy but a useful visual clue for the general case of isotropic rcode for a binary v k
2d CA with a Moore neighborhood. Below the rcode is the 102 bit iso-rule— (hex) 00 00 00 00 00 60 03 1c 61 c6 7f 86 a0.
Figure 16:
The negative game-of-Life with a negative Gosper glider-gun. The iso-rule is(hex) 3e a7 a2 46 5b e2 df 7d f7 df ff ff ff. A negative universe also applies for v> =3 were black values are exchanged for white. Fredkin’s replicator. An initial pattern (the eye) re-emerges from appar-ent disorder as multiple copies at time-step 32. Partial eyes appear at timesteps 16and 24. Replication continues on a cycle of 4092 time-steps.
The survival/birth option is available for v ≥
2, and any k ≥ k =9 neighborhood, for 1d and 3d as well as 2d. For v ≥ v> v colors as in figure 18.Figure 18: The game-of-Life (s23/b3) applied to a v =3 CA. The algorithm inDDLab generates an equivalent rcode and iso-rule (size=2862). The dynamics isthe same as binary Life but with 2 colors + background. This example shows twodifferent color Gosper glider-guns, and added green dynamic trails. The sizes of rule-tables, or the amount of information required, S , to definedifferent rule or logic types, with a rule-space of v S . are summarised below,17code . . . S r = v k iso-rule . . . S i : according to v, k and n-template as in table 1kcode . . . S k = ( v + k − / ( k ! × ( v − . . . S t = k ( v −
1) + 1k-outer-totalistic . . . S ok = v × S k t-outer-totalistic . . . S ot = v × S t reaction-diffusion . . . S RD < S ot , ( v , k and threshold interval)survival/birth . . . S SB < S ot , (survival and birth totals)In general, S r >S i >S k >S t . S RD < S ot and S SB < S ot because both reaction-diffusion and survival/birth logic can be set within t-outer-totalistic rules. LifeEppsteinEppsteinrandominitialstateSapinR
Figure 19:
The input-frequency histograms (actual plots) in a moving window of100 time-steps, for the glider-guns of the following rules: Life, Eppstein (shown alsofrom a random initial state), and SapinR, all from figure 4(a). Pattern snapshotsare colored to match histogram colors. Note that apart from its glider-gun, theEppstein rule is chaotic. ariantPrecurSayab left : the same Sayabhistogram as abovebut on the alternativelog plot Figure 20: input-frequency histograms (actual plots) in a moving window of 100time-steps, for the glider-guns of the following rules: Variant, Precursor, and Sayabwith an alternative log plot, all from figure 4(b). Pattern snapshots are colored tomatch histogram colors. While iterating CA space-time patterns, DDLab is able to keep track of the fre-quency of rule-table lookups in a moving window of time-steps[28, EDD:32.12.3]by means of a dynamic “input-frequency” histogram (also know as “lookup-frequency”). This applies to any rule type. For complex rules the histogramreveals the key inputs (or lookups) that maintain gliders and glider-guns, as wellas those that are rarely or never visited. Figures 19, 20 show examples for iso-rules generated by the glider-guns shown alongside. The moving window can beany size but here it is set to 100 time-steps to allow the histogram to stabilise.The input-frequency is the proportion of each iso-group in this window, and isrepresented by bar height. The order of bars follows the iso-rule index, from all1’s (left) to all 0’s (right), and a missing bar denotes iso-groups that have notbeen visited.There are two types of histogram plots: either the actual frequency but (pos-sibly) amplified to show up small bars of rarely visited iso-groups while the mostvisited are subject to a maximum cut-off, or alternatively a log frequency plot19o clearly show all bars but still distinguishes between rare and frequent. Thereare also two types of space-time pattern presentation: cells colored according totheir actual values, or colored according to the histogram bar responsable fora given cell, as in these figures. These alternatives are toggled on-th-fly.Figures 19 and 20 show histograms for the the binary v k × frequency which will be employed in further examples.DDLab applies the input-frequency histogram for a number of supplemen-tary functions. Filtering permits cross referencing a particular iso-group indexwith its occurrence in the space-time pattern, as well as revealing structureswithin a repetitive background. The consequences of mutations can be moni-tored by flipping (and restoring) the output of random or selected iso-groupsto another value. The Shannon entropy of the histogram and its variability areapplied to automatically categorise rule-space between order, complexity andchaos. These function are discussed below. The 4 phases of the game-of-Life glider, moving North East. far left : colors by value withgreen dynamic trails. near left : colors correspondingto histogram colors.
Figure 21:
The log iso-histogram of the game-of-Life glider in a movingwindow of 10 time-steps showing the 20 bars (from a maximum of 102)that are responsible for the glider’s existence. When these are filtered asmarked by black blocks, the iso-group indeces, in ascending frequency order(63,56,31,27,22,21,18,15,14,13,10,9,7,6,5,4,3,2,1,0) can be viewed and amended ina separate window. Matching bar/cell colors cycle through 14 contrasting colors, which can be shuffled. Figure 22:
The v k emergent spirals rule with a log iso-histogram size 92 in amoving window of 100 time-steps. Space-time pattern cells are colored accordingto the histogram bar responsible for that cell. For the same time-step, 6 stages offiltering f1 to f6 (high to low frequency) are marked in black below the relevant bar,and shown as a filtered snapshots. Filtering f1 makes no difference to its appearancebecause white cells are the most frequent, represented by the rightmost black bar.In the dynamics of this rule gliders first emerge and gradually self-organise intostable spirals. The input-frequency histogram allows the progressive[28, EDD:32.11.5] (or targeted[28,EDD:32.16.7])) filtering and unfiltering of space-time patterns. This applies forany rule type according to frequency given by the height of histogram bars. It isdone on-the-fly by keyhits as space-time patterns iterate. Progressive filteringproceeds from high to low frequency, unfiltering from low to high. For eachfrequency filtered/unfiltered, a black block appears/disappears at the base ofthe relevant bar, and the corresponding cell disappears/reappears in the space-time pattern, whether colored by histogram colors as in figure 22, or by valueas in figure 14. Keyhits can remove the entire filter scheme or reverse thescheme for added flexibility. Bars can also be targeted to filter/unfilter, andmutated/restored described in section 8 below.When pattern colors correspond to histogram colors, filtering allows crossreferencing a particular iso-group index with its occurrence in the space-time21attern, so helps to reveal how complex structures, gliders, eaters and glider-guns are built and their sensitivity to mutation. The histogram in figure 22 is setfor a moving window of 100 time-steps to allow the bars to stabilise. However,if the pattern itself has largely stabilised and the moving window size is reduced(minimum one time-step), then the few structures that remain dynamic can bepicked out in the histogram by the bars that continue to oscillate.To determine the iso-groups responsible for any particular structure, say theglider in the game-of-Life, an isolated glider is run to generate its iso-histogram,which if fully filtered will provide the complete list of the responsible iso-groupsin descending frequency order, as in figure 21.
The input-frequency histogram allows interactive (or targeted) rule mutations[28,EDD:32.5.4] while watching their effects on space-tine patterns, to make/restoresingle mutations on-the-fly with keyhits, without the need to pause, in a sortof mutation game. Any number of mutations can be made in sequence, andrestored in reverse order to finally return to the start rule. When applied to aniso-rule, mutations conserve isotropy, of course.
The game-of-Life glider-gun. far left : colors by value withgreen dynamic trails. near left : colors correspondingto histogram colors.Blow up of the left lower cornershowing indicator blocks: filter(black) and mutation (red).The iso-rules (hex) are compared below:
00 00 00 00 00 60 03 1c 61 c6 7f 86 a0 ---game-of-Life.34 e6 e4 64 c0 60 03 1c 61 c6 7f 86 a0 ---after all 17 neutral mutations.
Figure 23:
The log iso-histograms of the Life glider-gun/eater. The all active barswere firstly filtered, then all 17 neutral bars were mutated. The glider-gun/eater ispreserved, though any other dynamics would be drastically altered. he SapinR glider-gun. far left : colors by value with greendynamic trails. near left : colors corresponding tohistogram colors.The iso-rules (hex) are compared below:
24 01 13 1a 14 20 50 2c 45 05 48 e0 50 ---SapinR.1a fe e9 c4 79 87 23 c3 4a 0d 48 e0 50 ---after all 14 neutral mutations.
Figure 24:
The log iso-histogram of the SapinR rule glider-gun/eaters. All ac-tive bars were firstly filtered, then all 14 neutral bars were mutated. The glider-gun/eaters are preserved, though other dynamics would be drastically altered. The Sayab glider-gun. far left : colors by value with greendynamic trails. near left : colors corresponding tohistogram colors.The iso-rules (hex) are compared below:
24 01 13 1a 14 20 50 2c 45 05 48 e0 50 ---Sayab.1a fe e9 c4 79 87 23 c3 4a 0d 48 e0 50 ---after all 52 neutral mutations.
Figure 25:
The log iso-histogram of the Sayab rule glider-gun. All active bars werefirstly filtered, then all 52 neutral outputs mutated. The glider-guns are preserved,though other dynamics would be drastically altered. he Beehive glider-gun with NULLboundary conditions. far left : colors by value with greendynamic trails. near left : colors corresponding tohistogram colors. above : the iso-histogram representing 92 iso-groups, 40 are active.The iso-rules (hex) are compared below:
8a 20 01 60 2a 20 aa aa a9 aa 2a 99 2a a9 28 0a 2a 2a 69 80 2a 62 84 --Beehive86 20 85 a0 1a 10 a4 a4 08 69 98 05 62 a8 20 06 02 26 66 80 0a a2 84 --52 neutral mutations left : The same Beehive rule glider-gun as ak-totalistic rule and histogram representing 28combinations of totals, The kcodes (hex) arecompared below:
All one-value mutation were explored in [24].
Figure 26:
The v k Beehive rule[23] glider-gun showing both its log iso- andk-totalistic histograms. Because the Beehive rule does not have eaters to constraingliders, boundary conditions were set to NULL. The histograms were firstly filteredthen neutral outputs randomly mutated, with each mutation making one of the twopossible changes because v =3. The glider-gun is preserved, though other dynamicswould drastically be altered. The mutation algorithm can operate in conjunction with on-the-fly filteringdescribed in section 7. The keyhit to mutate will preferentially select an un-filtered iso-group bar at random and assign a random value different from thecurrent output. For a binary rule the output is simply flipped. A red block isshown at the base of the bar — beside the black filter block if this is also active.When the latest mutation is restored its red block is removed.To observe the effects, the appearance of pattern filtering can be toggleoff/on with a keyhit while the histogram filtering scheme remains visible. Themutation game is most effective if all active bars have been filtered because amutation to an inactive iso-group will be neutral for self-contained dynamicssuch as a glider-gun/eater system. In the examples in figures 23 to 28 theinitial state is an isolated glider-gun, in most cases contained by eaters. Allactive bars are first filtered marking them with black blocks, Then mutations24 he Spiral glider-gun. far left : colors by value with greendynamic trails. near left : colors corresponding tohistogram colors. above : the iso-histogram representing 276 iso-groups, 46 are active.The iso-rules (hex) are compared below:
00 a6 58 a6 66 a6 6a aa a6 a8 02 90 08 96 28 02 92 08 98 a6 69 2a 66 65 9a 69 8a 9a a6 69 a266 6a 96 96 6a a9 a9 a6 66 a9 69 08 82 68 28 02 a6 a2 69 a6 66 66 99 a2 62 a6 98 8a 26 08 8a26 64 9a 40 2a 62 84 --Spiral98 08 06 21 15 50 94 45 59 50 52 60 51 58 94 49 44 95 25 60 96 84 89 10 44 94 16 00 50 85 1999 81 00 29 80 54 56 51 08 44 00 08 18 46 10 00 41 46 04 59 09 10 46 04 81 81 00 82 41 04 69a9 9a a5 00 18 02 84 ---233 neutral mutations. left : the k-totalistic histogram representing 36combinations of totals, The kcodes (hex) are com-pared below:
Figure 27:
The Spiral glider-gun showing both its log iso- and k-totalistic his-tograms. The histograms were firstly filtered then neutral outputs randomly mu-tated, with each mutation making one of the two possible changes because v =3.The glider-gun is preserved, though other dynamics would be altered. are made which automatically and randomly seeks out unfiltered inactive iso-groups represented by missing bars, marking them with red blocks. Becausethese are neutral mutations the relevant glider-gun system must be preserved.The mutation game is applied to the v k v k v k he Spiral 3d glider-gun. far left : colors by value. near left : colors corresponding tohistogram colors. above : the iso-histogram representing 171 iso-groups, 19 are active.The iso-rules (hex) are compared below:
2a 04 8a 01 22 04 06 56 04 95 a4 96 44 84 82 22 52 25 01 06 25 51 8a 61 64 15 59 56 19 21 0659 20 a1 60 18 02 45 01 12 00 02 84 --Spiral3d.04 29 59 98 45 59 59 29 02 6a 69 49 22 59 58 58 04 90 58 50 90 82 61 06 91 0a 22 28 62 4a 6984 65 26 89 86 51 82 58 8a 90 a2 84 ---after all 152 neutral mutations.
Figure 28:
The log iso-histogram of the Spiral3d glider-gun with iso-rule size=171— shorter than Spiral2d size 276. The 3d glider-gun emerges spontaneously, butstable structures (such as eaters) are absent. The histograms were firstly filteredthen neutral outputs mutated. The glider-guns are preserved, though other dynam-ics would be altered. Once all gaps have been mutated, further mutations will hit active barsdisrupting glider-guns, which can be retrieved with keyhits to unmutate andeventually (or immediately) reinstate the original rule. A keyhit can also rein-state the original glider-gun pattern.Mutations to active iso-groups can be done progressively by unfiltering theleast active, or a specific mutation index can be selected. Any mutation toan active iso-group will change the current space-time dynamics to a greateror lesser extent. If the change is interesting, a new glider or eater, the mu-tation can be retained. An undesirable change such as excessive disorder canbe repaired. Among functions in DDLab that can assist in these experimentsare on-the-fly keyhits for a random pattern, and for a random central block[28,EDD:32.8.1], respecting densities previously specified. Space-time patters canbe paused at any time to edit or save the current state or iso-rule and accessother functions[28, EDD:32.16].Exploring state-space genetically close to significant k-totalistic rules wasdone for the Beehive-rule[22] and the Spiral-rule[25], looking at all possiblesingle k-totalistic mutants[24, 26] and some significant alternative behaviourswere discovered. A finer grained search based on iso-rules is now possible.26 input-entropy and min-max variability ————–1 −→ ——————————time-steps −→ ——————————250entropy1entropy0 ———-———- ↑↓ min-max ← ignore initial time-steps Figure 29:
Top : 250 time-steps of Life from a random 40 ×
40 initial state with adensity of 30% shown as a type of axonometric projection where time-steps progressfrom left to right. Colors follow the iso-histogram.
Above : The entropy plot gener-ated simultaneously: x-axis time-steps 1 to 250, y-axis normalised Shannon entropy H N from 0 to 1. The min-max entropy is the greatest upslope after a short initialrun, 22 time-steps for the scatter-plots in figure 30. The entropy of the iso-histogram, the input-entropy, can be measured and plot-ted over time. The average entropy and its variability over a window of time-steps indicates the quality — ordered/complex/chaotic — of the dynamics, veryroughly as follows, order complexity chaosmean-entropy low medium highentropy-variability low high lowBoth the mean-entropy and entropy-variability are measured from a run oftime-steps starting from a random (but possible biased) initial state, discountinga short initial run to allow the the dynamics to settle into its typical behaviour.27he Shannon entropy of the input-frequency histogram (the actual plot, notlog ) measures its heterogeneity. The input-entropy H , at time-step t , for onetime-step ( w =1), is given by H t = − (cid:80) S − i =0 ( Q ti /n × log ( Q ti /n )), where Q ti isthe lookup frequency of neighborhood i at time t . S is the rule-table size and n is the CA lattice size. The normalised entropy H N is a value between 0 and1, H N = H t /log n used in the graphic display as in figure 29 and is usuallyaveraged over a small moving window (say w =10) of time-steps to smooth anotherwise jagged plot.The mean entropy is the average H N over a longer run of time-steps. Theentropy-variability known as min-max is the maximum up-slope found in a runof time-steps — the rise in entropy following a lower value.A high entropy variability can be produced by glider dynamics, becausecollisions create local chaos raising the entropy, from which gliders re-emergelowering entropy. The basic argument is that if the entropy continues to varysufficiently in typical dynamics, moving both up and down, then some kind oflarge scale structural interactions are unfolding. As well as glider dynamics,this might include competing zones of ordered domains, of order and chaos, ofdomains of competing chaos, or some combination of the above.Low entropy variability is a consequence of both steady chaos and steadyorder, especially when patterns stabilise or freeze. However, the mean entropyfor chaos high, for order low. In this way the quality of dynamics can bedistinguished.
10 automatically classifying rule-space
The two measures, mean entropy and min-max entropy variability, are appliedfor an automatic classification of rule-space by creating scatter plots in DDLabfor large samples of rules. The plots distribute rule types according to a mergingcontinuum of order/complexity/chaos on a 2d surface, and allow a targetedexamination of individual rules or rule sub-groups at characteristic locations onthe plot. Details and examples for creating, sorting, probing and interpretingthe scatter plots are provided in [28, EDD:33].Hitherto the scatter plots were based on full rule-tables[21, 22] even whenmade isotropic[9, 10], or were based on totalistic rules[23]. Now the scatter plotscan be based on the iso-rule paradigm as in figures 30. Further investigation ofthese recent plots will be held over to a subsequent paper.To construct the scatter plots in DDLab, random iso-rules and initial statesare generated but with biases in favour complex dynamics; in spite of this thedynamics captured is mostly chaotic. The bias criteria can follow known log-ically universal rules or some other conjecture. For each successive rule, thespace-time pattern is run from a set of random initial states. For each ini- Variability by min-max is preferable to the previously adopted[21, 22] standard deviationwhich gives a high value for monotonic entropy decrease, characteristic of a foreground patterngradually dying out, which would be misleading to identify complex dynamics. Min-max islow for dying out dynamics so this problem is avoided. a) orthogonal lattice v k , approx density 0/1: seed=70/30, iso-rule=70/30.(b) hexagonal lattice v k , approx density 0/1/2: seed=70/15/15, iso-rule=60,20,20. Figure 30:
Two mean-entropy/entropy-variability (min-max) scatter plots based ona 60 ×
60 2d lattice (a) v k as in Life and Sayab, and (b) v k as in the Spiralrule. Left : Each dot represents one or more rules falling within the squares of a256x256 grid where the x-axis is min-max entropy, and the y-axis is mean entropy,with dot colors indicating the pile-up frequency. This presentation allows rule-spaceto be probed to examine the rules and dynamics.
Right : with a z-axis showing log rule frequency. Measures were started after 22 initial time-steps and taken over thesubsequent 200 time-steps. The approximate densities of the random initial stateand the iso-rule are indicated. tial state, after a delay to allow the CA to settle into its typical behavior, thevariability of the input-entropy and the mean entropy are recorded. Then theaverage results from the set of initial states are plotted — the entropy vari-ability ( x -axis) against the mean entropy ( y -axis), and the data for each rule isappended to a file.Probing various locations of a sorted plot[28, EDD:33.6] with the pointerselects rules or rule patches which can then be listed and run in sequence tosee the dynamics, or scanned automatically in blocks of time-steps. Figure 31shows the locations of characteristic dynamical behaviours.29 (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:1)(cid:5)(cid:6)(cid:9)(cid:7)(cid:8)(cid:10)(cid:11)(cid:4)(cid:6)(cid:12)(cid:4)(cid:11)(cid:12)(cid:8)(cid:13)(cid:6)(cid:3)(cid:4)(cid:10)(cid:11)(cid:4)(cid:11)(cid:10)(cid:6)(cid:14)(cid:3)(cid:13)(cid:11) (cid:15)(cid:2)(cid:12)(cid:3)(cid:5)(cid:8)(cid:15)(cid:6)(cid:4)(cid:16)(cid:7)(cid:11)(cid:1)(cid:17)(cid:11)(cid:12)(cid:18) (cid:11)(cid:19)(cid:2)(cid:12)(cid:16)(cid:1)(cid:5)(cid:8)(cid:20)(cid:11)(cid:6)(cid:1)(cid:11)(cid:12)(cid:4)(cid:15)(cid:2)(cid:6)(cid:12)(cid:4)(cid:11)(cid:12)(cid:7)(cid:3)(cid:15)(cid:12)(cid:5)(cid:8)(cid:3)(cid:13)(cid:6)(cid:3)(cid:21)(cid:6)(cid:9)(cid:12)(cid:22)(cid:11)(cid:23)(cid:3)(cid:21)(cid:23)(cid:24)(cid:8)(cid:21)(cid:11)(cid:6)(cid:2)(cid:8)(cid:9)(cid:2)(cid:6)(cid:20)(cid:12)(cid:4)(cid:8)(cid:12)(cid:25)(cid:8)(cid:7)(cid:8)(cid:5)(cid:26)(cid:27)(cid:15)(cid:3)(cid:22)(cid:17)(cid:7)(cid:11)(cid:19)(cid:28)(cid:6)(cid:17)(cid:7)(cid:12)(cid:8)(cid:13)(cid:2)(cid:8)(cid:9)(cid:2)(cid:7)(cid:26)(cid:6)(cid:3)(cid:4)(cid:10)(cid:11)(cid:4)(cid:11)(cid:10)(cid:29)(cid:12)(cid:7)(cid:7)(cid:20)(cid:12)(cid:7)(cid:7)(cid:11)(cid:26)(cid:6)(cid:25)(cid:11)(cid:5)(cid:29)(cid:11)(cid:11)(cid:13)(cid:6)(cid:3)(cid:4)(cid:10)(cid:11)(cid:4)(cid:6)(cid:12)(cid:13)(cid:10)(cid:6)(cid:15)(cid:2)(cid:12)(cid:3)(cid:1) x y z Figure 31:
The typical shape of the x-y-z entropy-min-max scatter plot with char-acteristic dynamical behaviour found in different parts of the landscape and thebasis for a probing search avoiding the chaotic peak. The high variability complexplain contains gliders and interacting mobile structures but is unlikely to supportglider-guns because of the over-active dynamics and rare stable structure. This fig-ure is taken from our 2015 paper[9] for 93000 isotropic v k
2d rules but where theentropy was based on the full 512 rule-table instead of the shorter 102 iso-rule-tablein figure 30.
11 summary and discussion
Isotropy is arguably the proper canvas for CA logical universality to play outbased on glider-gun/eater dynamics, and we have investigated the few such ruleswe are aware of — the game-of-Life, Sayab, the Spiral rule, and others, some ofwhich we have discovered with earlier analogous methods[1, 2, 9, 10, 11, 12].Isotropic notations for binary CA exist such as the Hensel for Golly[8] andSapin’s[15, 16], but a general approach to encompass multi-value in one, twoand three dimensions was missing, so we have proposed the iso-rule paradigmin this paper. Iso-rules are based on a lookup-table of iso-groups, assemblies ofall rotated/reflected neigborhood configurations which can be examined graph-ically in DDLab. Iso-rules-tables are ordered in the spirit of Wolfram’s classicalconvention[18, 19].Iso-rules provide an intermediate granularity between isotropic rules basedon a full lookup-tables, and isotropic subsets — totalistic, reaction-diffusion andsurvival/birth rules. DDLab is able to convert these rule types to iso-rules, whichthen become subject to all DDlab’s other functions[28]. The input-frequency30istogram, filtering and mutation are significant functions for studying the low-level drivers of glider-gun/eater dynamics. Input-entropy and its variabilitydistinguish iso-rules according to ordered/complex/chaotic dynamics, and allowthe collection of large samples of iso-rules automatically, to classify iso-rule-space, and to search for new and interesting iso-rules. We have presented twosuch collections. With the tools in place these are trivial to generate overnighton a laptop. Future work will study these data in the search for novel glider-gun/eater dynamics and logically universally rules, motivated by the thoughtthat more examples would facilitate the search for underlying principles of self-organisation.
12 Acknowledgements
Figures and experiments were made with DDLab[29]. J.M. G´omez Soto ac-knowledges his residency at Discrete Dynamics Lab, and financial support fromthe Research Council of Mexico (CONACyT).
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