On patterns and dynamics of Rule 22 cellular automaton
Genaro J. Martinez, Andrew Adamatzky, Rolf Hoffmann, Dominique Deserable, Ivan Zelinka
OOn patterns and dynamics of Rule 22 cellularautomaton
Genaro J. Mart´ınez , , Andrew Adamatzky , Rolf Hoffmann Dominique D´es´erable , Ivan Zelinka
29 March 2020 ∗ Laboratorio de Ciencias de la Computaci´on, Escuela Superior de C´omputo,Instituto Polit´ecnico Nacional, M´exico. ( [email protected] ) Unconventional Computing Lab, University of the West of England, Bristol,United Kingdom. ( [email protected] ) Technische Universit¨at Darmstadt, Darmstadt, Hessen, Deutschland.( [email protected] ) Institut National des Sciences Appliqu´ees, Rennes, France.( [email protected] ) Fakulta Elektrotechniky a Informatiky, Technick´a Univerzita Ostrava,Czechia. ( [email protected] ) Abstract
Rule 22 elementary cellular automaton (ECA) has a 3–cell neighbor-hood, binary cell states, where a cell takes state ‘1’ if there is exactly oneneighbor, including the cell itself, in state ‘1’. In Boolean terms the cell-state transition is a xor function of three cell states. In physico–chemicalterms the rule might be seen as describing propagation of self-inhibitingquantities/species. Space-time dynamics of Rule 22 demonstrates non-trivial patterns and quasi-chaotic behavior. We characterize the phenom-ena observed in this rule using mean field theory, attractors, de Bruijndiagrams, subset diagrams, filters, fractals and memory.
Elementary cellular automata (ECA) [1, 2] are one–dimensional arrays of fi-nite state machines, or cells, which take states ‘0’ or ‘1’ and update their statedepending on their own current state and on the state of their two immedi-ate neighbors. Rule 22 ECA has a simple cell–state transition function: a cell ∗ Published in:
Complex Systems 28(2), 125-174, 2019 . a r X i v : . [ n li n . C G ] M a y akes state ‘1’ if exactly one of its neighbors, including the cell itself, is in state‘1’; otherwise, the cell takes state ‘0’. When perturbed at a single site the au-tomaton exhibits something similar to recurrent wave–fronts in excitable media,which develop into fractal structures of Sierpi´nski gasket [3, 4]. Due to countlessgeneration and annihilation of wave–fronts, a dynamics of Rule 22 is sometimescharacterized as chaotic [2] which rather reflects its unpredictability than anyrelation to noise or bifurcations. Most results of studies about Rule 22 were inalgebraic properties or statistical approximations of the automaton dynamics.Thus, Zabolitzky [5] reported results of an extended probabilistic analysis es-timating non-trivial behavior on very large arrays perturbed by configurationswith low densities of state ‘1’. He discovered critical properties that cannotbe reproduced when the automaton is perturbed by a random configuration.McIntosh provided a systematic analysis of small configurations emerging inRule 22 [6]; he proposed similarities with configurations observable in Conway’sGame of Life. A topological analysis linked to chaotic behavior of the rule canbe found in [7]. A one–dimensional cellular automaton CA( k, r ) is an array of cells x i where i ∈ Z . Each cell takes on a value from an alphabet S = { , , ..., k − } with k symbols. A chain of cells { x i } of finite length n represents a string or globalconfiguration c on Σ. The set of finite configurations is represented as Σ n . Anevolution is a sequence of configurations { c i } given by the mapping Φ : Σ n → Σ n and their global relation is provided by Φ( c t ) → c t +1 where t is a discrete timeand every global state of c is a sequence of cell states. Cells of each configuration c t are updated to the next configuration c t +1 simultaneously by a local transitionfunction ϕ : S r +1 → S as ϕ ( x ti − r , . . . , x ti , . . . , x ti + r ) → x t +1 i acting on a neighborhood of x i of length 2 r + 1. For (elementary) ECA(2 , ϕ : S → S becomes ϕ ( x ti − , x ti , x ti +1 ) → x t +1 i (1)and for Rule 22, its local cell–state transition is given by: ϕ R = (cid:26) , , , , , ,
000 (2)Rule 22 displays a typical chaotic global behavior from random initial condi-tions. Figure 1a shows the evolution with an initial condition starting with asingle cell in state ‘1’. A pattern growing is a fractal, similar to a Sierpi´nskigasket. Figure 1b shows a development from a random initial configuration witha density 0.5 of cells in state ‘1’. 2 a) (b)
Figure 1: Exemplar dynamics in ECA Rule 22. (a) Development from a singlecell in state 1. (b) Development from a random configuration with densityof 1-cells 0.5. Both space-time diagrams evolve on a ring of 600 cells for 350generations. Time evolves from top to bottom.
Rule 22 is considered as chaotic because:1. future configuration of the automaton is completely determined from itsinitial state because of the deterministic rule and synchronous updating,2. development of the automaton is sensitive to initial conditions (tiny per-turbation might lead to dramatic events),3. global transition graph has dense periodic orbits (attractors),4. configurations evolved can be characterized as random.We undertake an extensive and systematic analysis of Rule 22 using differentapproximations aiming at discovering an emergence of novel non-trivial patterns,periodic patterns, Garden of Eden configurations. These configurations arediscovered with the help of encoding initial conditions into regular expressions,de Bruijn diagrams, subset diagrams, cycle diagrams, fractals, and jump-graphs.We also show an effect of memory upon dynamics of Rule 22.
Mean field theory allows us to describe statistical properties of CA withoutanalyzing evolution spaces of individual rules [8, 9]. This approximation assumesthat elements of a set of states Σ are independent and not correlated with eachother in the rule’s evolution space. One can study probabilities of states in theneighborhood in terms of probability of a single state (the state in which theneighborhood evolves), thus a probability of the neighborhood–state is a productof the probabilities of each cell-state in the neighborhood. A polynomial on the3 a) (b)
Figure 2: Mean field curves for (a) GoL and (b) Rule 22.probabilities is derived and its curve can be used to classify the rules, as proposedby McIntosh in [9].
Using this approach we can construct a mean field polynomial for a two–dimensional CA with a semi–totalistic evolution rule: p t +1 = S max (cid:88) v = S min (cid:18) n − v (cid:19) p v +1 t q n − v − t + B max (cid:88) v = B min (cid:18) n − v (cid:19) p vt q n − vt (3)where n represents the number of cells in Moore’s neighborhood, v (resp. n − v )the number of occurrences of state ‘1’ (resp. ‘0’), p t (resp. q t ) the probability ofa cell being in state ‘1’ (resp. ‘0’) and with q t = 1 − p t . B and S are minimumand maximum of an interval for born and survival conditions in Conway’s Gameof Life (GoL), respectively. The GoL’s polynomial is the following: p t +1 = 84 p t q t + 56 p t q t . (4)The mean field curve F of Eq. 4 displayed in Fig. 2a shows three fixed points p t +1 = p t when crossing the identity. The first stable fixed point at the originguarantees its stable state, the second unstable point F = 0 . F = 0 .
37 indicates that GoL will converge almost surely to configurations withsmall densities of ‘1’.
For one dimension, we adjust Eq. 3 to a full local rule and not only a semi–totalistic one. All states of the neighborhood must be considered, thus p , q , n , and v have the same representation as above. But now the product will bewith the value of each neighborhood, whence, for a 1 d CA( k, r ) the mean field4olynomial p t +1 = k r +1 − (cid:88) j =0 ϕ j ( X ) p vt q n − vt which gives p t +1 = (cid:88) j =0 ϕ j ( X ) p vt q − vt (5)for ECA(2 ,
1) and where ϕ j ( X ) denotes the j –th transition of S in Eq. 1.Finally, the mean field polynomial for Rule 22 p t +1 = 3 p t q t = 3 p t (1 − p t ) (6)is deduced from (2).In Rule 22 state ‘1’ appears with probability = 0 .
375 (which is close tothe fixed stable point 0.37 of GoL). The mean field curve f of Eq. 6 displayed inFig. 2b shows a slope f (cid:48) (0) = 3 at the origin. Density is maximal at f (1 /
3) = ≈ .
444 before reaching the stable fixed point p t +1 = p t when crossing theidentity at p t = 1 −√ / ≈ . . It then crosses the inflection point f (2 /
3) = · f (1 /
3) with tangential slope f (cid:48) (2 /
3) = − f (1) = f (cid:48) (1) = 0 . Based on the mean field curves classification, Rule 22 is a chaotic ECA (Fig. 3).
Various scenarios of evolution are displayed in Fig. 4: • ( a ) From initial density d = 1 / d F = 0 . d to d F is not perceptible. ( a ) From initial density d = d F reaching spontaneously the fixed point. • ( b ) From initial density d = 0 . b ) From initial density d = 0 .
95 evolving towardsthe fixed point after a later phase transition delimited by a polygonalbroken line . There exists an interval between two thresholds d (cid:48) ≈ . d (cid:48)(cid:48) ≈ .
92 such that any (pseudo–)random initial distribution withdensity d (cid:48) < d < d (cid:48)(cid:48) converges almost surely towards fixed point d F . • ( c ) From initial density d = 0 .
97 evolving towards a dense pattern withobservable “backbones” after crossing the phase transition polygon; twobackbones arise from polygon vertices and their patterns are symmetricfrom either side. ( c ) From the same density d = 0 .
97 and another initialdistribution yielding a sparse pattern with backbones wherein no phasetransition line does appear. Phase transitions and critical exponents for Rule 22 leading to non–trivial long–rangeeffects were reported in [10, 11]. Asymptotic properties were described in [12]. Also observable in the “
Exactly 1 ” ECA [13] Fig.1d. a ) 400 generations ( a ) from initialdensity d = 1 / a ) from density d = d F reaching spontaneously the fixedpoint ( b ) 600 generations ( b ) from density d = 0 . d F afteran early phase transition ( b ) from density d = 0 .
95 evolving towards d F aftera later phase transition ( c ) 1000 generations from density d = 0 .
97 evolvingtowards ( c ) a dense pattern with thin backbones ( c ) a sparse pattern withlarge backbones. 7able 1: Statistical estimations of evolutions (%) in interval [ d (cid:48)(cid:48) ,
1] from sam-ples of 100 initial configurations for each density: ergodicity (ERG), disorderedsparse fractals (DSF), dense backbones (DBB), sparse backbones (SBB), van-ishing (VAN), rare periodic patterns (RPP). Bolded items reflect an irreversiblestate. Density
ERG
DSF
DBB
SBB
VAN RPP .
920 100 0 0 0 0 00 .
930 94 5 1 0 0 00 .
940 93 1 3 1 1 10 .
950 85 2 7 4 2 00 .
960 47 4 13 24 12 00 .
970 21 2 26 26 25 00 .
980 7 0 18 21 54 00 .
990 1 0 5 8 86 00 .
995 0 0 0 0 100 0Outside interval ] d (cid:48) , d (cid:48)(cid:48) [ sensitivity to initial conditions is high, with a positiveLyapunov exponent and a chaotic behavior. Depending on small perturbationsfrom initial configuration, four other evolutions are also possible: (i) disor-dered sparse fractals (ii) convergence towards fixed point d F but after a longperiod (Fig. 5) (iii) initial configuration vanishing at first step (iv) rare eventsof periodic patterns (Fig. 6). That is, six types of evolution altogether. Theirestimations of occurrence in interval [ d (cid:48)(cid:48) ,
1] are displayed in Tab. 1.Beyond the phase transition polygon in case of convergence towards fixedpoint d F , the evolution becomes ergodic, in the sense that the system has thesame behavior either averaged over time or averaged over space [14]. The pro-cess is stationary and homogeneous at mesoscopic scale. In other words, theirexists a smallest macro–cell C of size ξ × ξ, where ξ is the correlation length, asrepresentative (or statistical) volume element such that density d C in the macro–cell is close to the mean density averaged within the whole system [15, 16]. Thus d C ≈ = 0 .
375 that is, the exact ratio of ‘1’ filling ϕ R . It should be observed that a disordered sparse fractal pattern (DSF) mayevolve towards ergodicity (ERG) as in Fig. 5 but sometimes after a long, unpre-dictable time. We denote as “DSF” such evolution remaining in this state atleast within a time window of arbitrary length 10 . In the same way, sparse back-bone patterns (SBB) may evolve towards a dense backbone landscape (DBB).A simple way to check (not to prove) whether unstable evolutions becomeeventually ergodic or to get a more global overview upon evolution is to skipsome timesteps with skip time–lengths ∆ t . This transformation yields a pro-jective view upon the ( x, t )–landscape with angle arctan 1 / ∆ t. Various skippedscenarios of evolution from initial critical density d = 0 .
97 in a ring of 800 cellsand within a time window of length 10 · ∆ t are displayed in Fig. 7: • ( a ) Phase transitions ( a ) DSF → ERG with ∆ t = 32 from disordered8igure 5: Disordered sparse fractal (DSF) evolving from initial density d =0 .
93 towards fixed point d F after a long period: (0 − → (2000 − ↓ (3000 − ← (27000 − . DSF pattern appears as a long transient statebefore ergodicity. Compare with landscape in Fig. 3 derived from a regularexpression. 9igure 6: Rare periodic event occurring from initial density d = 0 .
94 with acomplete pattern moving leftward like a glider: (0 − → (200000 − . Compare with the patchwork of periodic patterns in Fig. 14. The probability ofoccurrence of such a pattern from a random initial distribution is about 10 − .sparse fractal to ergodicity – up to 32000 generations, transition occursafter about 8000 timesteps ( a ) SBB → DBB with ∆ t = 31 from sparse todense backbones – up to 31000 generations; transition occurs after about4000 timesteps. Note that apparent discrepancies between densities in ( a )and ( a ) before phase transition is no more than a side effect resulting fromeven or odd skip length parity. • ( b ) Stratified landscapes SBB with ∆ t = 64 ( b ) up to 64000 generationswith observable backbones... and sub–backbones evolving like a Cantordust ( b ) up to 4 · generations with perpetual phase transitions.Failing to prove the existence or not of ergodic evolution, this skipping approachnevertheless emphasizes several chaotic behavior with long–range correlations.As well as in statistical physics, renormalization methods [15] overcome theweakness of mean field approximations that may fail or, at least, produce insuf-ficient information. Mean field theory is a rough approximation which assumes independence between neighboring sites. On the contrary, other deterministicapproaches like de Bruijn diagrams assume dependence . They will be discussedthereafter. 10igure 7: Skipped scenarios in a ring of 800 cells with d = 0 .
97 within a timewindow of length 10 · ∆ t . ( a ) Phase transitions ( a ) with ∆ t = 32, DSF → ERGfrom disordered sparse fractal to ergodicity ( a ) with ∆ t = 31, SBB → DBBfrom sparse to dense backbones. ( b ) Stratified landscapes SBB with ∆ t = 64( b ) up to 64000 generations with observable backbones... and sub–backbonesevolving like a Cantor dust ( b ) up to 4 · generations with perpetual phasetransitions: a transition line separates a dense backbone (DBB) regime from asparse backbone (SBB) regime; evolution remains still unstable.11igure 8: Basin of attractors in ECA Rule 22 for rings of size 20. The number ofattractors are 108 with 12 non–equivalent types. Based in attractors characteri-zation, Rule 22 displays chaotic behavior with highly dense, not long transients,and several symmetric trees. 12igure 9: Discovering non–trivial patterns emerging in ECA Rule 22 display-ing a family of tilings of different sizes from a string of a basin of length 20(Fig. 8). A lot of these patterns can be reached with concatenation of the string00000001000100000000. 13 Attractors
Basins of attraction have been studied by Andy Wuensche in the framework ofECA and random Boolean networks [17, 18, 19]. A string of cell–states x ti isa configuration c . An evolution is represented by a sequence of configurations { c , c , c , . . . , c m − } , such that Φ : Σ n → Σ n , and the global transition can berepresented as Φ( c t ) → c t +1 . A number of all global states of c is determined bythe length of a string m n (where n is the length and m the number of symbols).The structure of an attractor (Fig. 8) is given in three parts. Leaves representGarden of Eden, i.e. unreachable in the evolution but only as initial globalstates (these states have no ancestors). Branches are configurations that haveat least one ancestor and just one successor. Height of branches determines anumber of generations to reach the attractor. An attractor is the final state ofa string of length n . Numbers labelling vertices represent the decimal values ofthe strings.Wuensche [17] proposed that Wolfram’s classes can be represented as a basinclassification . In this classification complex behavior is characterized by mod-erate transients, moderate–length periodic attractors, moderate in–degree andsmall density of leaves. This way, Fig. 9 displays a type of non–trivial behaviorlater of thousands of generations starting with a concatenation of one of thesestrings calculated by one attractor of length 20:00000001000100000000 → → → → → → → → → → → c i )) → Φ( c j ) [18]. A con-figuration c i expressed as a string w i = a a . . . a n − , such than it can jumpinto other configuration c j expressed as a string w j = b b . . . b n − . Hence a i can mutate to one b i , where each configuration c belongs at the same field ofattractors Ψ. Also, the mutation represents a loop in the same basin if a i = b i .Figure 10 shows a jump–graph from the basins of attraction of length 20 (Fig.8). This way, a chaotic system presents a high density of connectivity with allthe attractors in the jump–graph. In this section we will use the term state as an alias for global state c i , config-uration, or 1 d string. CA( n ) denotes an ECA Rule 22 with n cells and cyclicboundary. The aim of this section is to study • What is the length of the longest path until the zero-state (00 ...
0) isreached, and how does a related initial Garden of Eden state look like? • How many cycles exist, how long are they, and what is a representativestate for each cycle? Cycles that belong to the same class (cycles’ statesare equivalent under shift and mirroring) shall be listed only once as rep-resentative cycle , the kind of cycles we are interested in here. • Are there similar states (cyclically shifted) that appear periodically withina cycle? • What is the length of the longest path until a certain cycle is reached, andhow does a related initial state look like?The following terms and functions are used here: • path(A, C) is a sequence of states (from state A to state C ). • length(path) gives the number of states of a path or cycle. • prefix(C) is a path( A , B ) where B is direct predecessor of C . • maxprefix(C) is a prefix( C ) of maximum length. • α = length(maxprefix(0)) where (0) is the zero-state (00...0), the longestprefix(0). • cycle denotes a periodic attractor, a cyclic path. • k-cycle is a cycle of length k . • ω ( cycle ) gives the length of a cycle. • similar(S) is a state that can be derived from state S by cyclic shift andoptional mirroring. 16able 2: Longest path to the zero state. α : length normalized n max prefix(0) initial state3 2 0014 2 00015 6 010116 5 0101117 6 00011118 8 001010119 2 00010010110 7 001011110111 6 0010101010112 23 00100110011113 20 000100110011114 24 0001010110011115 32 00100101010011116 41 000000010101001117 53 0010101010111001118 8 00000101100001001119 17 000010101011101010120 18 00001000101010111101 • (cid:15) is called intra-cycle-period . • In some cycles similar states appear again after (cid:15) time-steps. • k/e-cycle is a k -cycle where e = (cid:15) ( k -cycle). We may call k/e -cycles strong if k = e , and weak if k > e . • cycle-prefix is a prefix( D ) where D belongs to a cycle. • λ ( cycle ) gives the length of the longest cycle-prefix. First method and results.
The CA( n ), n = 3 . . .
20, were simulated forall possible initial states and then analyzed by special programs and manualinspection. In order to avoid unnecessary simulations, similar initial states wereexcluded. Similar states are states which are equivalent under cyclic shift andmirroring. For instance the number of different cases to be simulated for n = 18is only 7685, which is significantly lower than 2 . It should be noted that thecycles for n up to 34 were already computed by McIntosh [35].Table 2 shows α (the length of the maxprefix (0)), and the related normalized initial state. A normalized state is a representative of all states which areequivalent under cyclic shift and mirroring. It is found by selecting the statewith the smallest binary number among all equivalents. E.g. normalize (100011)17able 3: Representative cycles for n = 3 . . . ω : cycle length, (cid:15) : intra-cycle-period, λ : length of longest cycle prefix. Not all leading zeroes are displayed. representative repetition initial state n ω cycle state (cid:15) within λ of longest(normalised) cycle cycle-prefix3, 5, 6 no cycle ω >
14 2 (0011)* 1 c t +1 =shl2( c t ) 0 no prefix7 7 (0001011)* 1 c t +1 =shl4( c t ) 0 no prefix2 (0011)* 1 c t +1 =shl2( c t ) 0 no prefix8 4 00000101 2 c t +2 =shl4( c t ) 2 001001116 00000011 3 c t +3 =shl4( c t ) 2 001011019 4 000000101 9 0001001114 0000001001 2 c t +2 =shl5( c t ) 7 001101010110 4 0000000101 5 00101100116 0000010011 3 c t +3 =shl3m( c t ) 1 00000100114 00000001001 7 0000010011111 5 00000001111 7 0010101001111 00001001111 1 c t +1 =shl7( c t ) 9 0001011001112 2 (0011)* 1 c t +1 =shl2( c t ) 0 no prefix= 3 × c t +2 =shl6( c t ) 6 0001001100115 000000001111 9 00101010110113 5 0000000001111 13 000010100110114 7 (0001011)* 1 c t +1 =shl3( c t ) 0 no prefix= 2 × c t +6 =shl9m( c t ) 4 0000010101001115 20 000000010000011 4 c t +4 =shl4( c t ) 20 0010101001011112 (0011)* 1 c t +1 =shl2( c t ) 0 no prefix16 4 (00000101)* 2 c t +2 =shl4( c t ) 2 (00100111)*= 2 × c t +3 =shl4( c t ) 2 (00101101)*= 4 × c t +6 =shl8( c t ) 29 000000100101011112 000000000100001 10 00101010101010114 000010100000101 2 0010010110010011117 12 000000000000101 21 0010101010110101126 000000000010011 13 c t +13 =shl15m( c t ) 26 001010100101100114 000100100000101 2 c t +2 =shl4( c t ) 2 001010101001011014 (000000101)* 45 0011100101010111118 4 000010100000101 5 00101010011001111= 2 × c t +6 =shl9( c t ) 35 0000000100001110112 000000000000101 55 0001001010011001118 000001101001011 9 c t +9 =shl9( c t ) 7 011110011001100114 000100100000101 21 000010111000101114 000101000000101 6 1010010101010101119 4 001001000000101 1 0011011100011011112 000000000001001 78 0000110001001011128 000001000011101 14 c t +14 =shl4m( c t ) 9 010011001100111012 (0011)* 1 c t +1 =shl2( c t ) 0 no prefix4 (0000001001)* 2 c t +2 =shl5( c t ) 44 001011000110100114 001000100000101 2 c t +2 =shl4( c t ) 5 001101100110010004 (0000000101)* 12 1101000111010111120 4 001001000001001 62 01111001010101111= 2 ×
10 4 001001000000101 10 00010000100001111= 4 × c t +3 =shl7m( c t ) 1 010110101001011116 (0000010011)* 3 c t +3 =shl7m( c t ) 8 000010101000101118 000000001000001 4 c t +4 =shl10( c t ) 42 0000001000111010112 000000000010001 6 c t +6 =shl10( c t ) 24 0010010110010001124 000000000100001 98 01010101110101101
18 000111, and normalize (110110100) = 001011011 (by mirror and shift). The α values are much smaller than 2 n , and not monotonically increasing with n .Table 3 shows the results obtained by analyzing all the simulations. Theoperator shlP ( c ) means shift c to the left by P positions, and shlP m ( c ) meansthat first the mirror operator is applied before shifting.We find always the trivial (00 .. → (00 ..
0) cycle of length 1. For even n there always exists the fixed point (01)* → (01)*, a lonely 1–cycle with no prefix.We will not further mention or pay special attention to these basic 1–cycles.For n = 4, and multiples of 4, we get the 2/1–cycle (0011)* ↔ (1100)*. Thetwo strings are similar under shift of two positions, so the inherent pattern isthe same.For n = 7, and multiples of 7, we get a 7–cycle. In order to characterize thiscycle, one representative is chosen, it is the one with the smallest normalizedvalue, i.e. (0001011)*.For n = 8, we get three cycles with length ω/(cid:15) = 2 / , / , / n = 10, there exists a 6/3–cycle. After every 3 time-steps, the samestring appears in mirrored form and shifted 3 positions to the left.For n = 12, the cycles of CA(4) form a subset (to be included if not detected)which is the 2/1-cycle (0011 0011 0011) ↔ (110 1100 1100).For n = 14, the cycles of CA(7) form a subset which is shown as 7/1–cycle.Fig. 8 shows all possible cycles where many of them are similar (equivalentunder shift and mirroring). From that figure we may anticipate a very complexattractor structure, but we should realize that the number of representativecycles in CA(20) is only 11.In general, because of the cyclic boundary, if k is a factor of n , then theCA( k ) cycles are a subset of the CA( n ) cycles. For example, the cycles ofCA( k = 4 , ,
10) form a subset of the CA( n = 20) cycles. However there is adifference: the strings of CA( n ) are cyclic repetitions of the CA( k ) strings, andthe original intra-cycle period (cid:15) may not appear in CA( n ). Second method and results.
For larger n , the first method cannot further beapplied due to extensive computational costs. Therefore, now only a relativelysmall random subset of all possible 2 n initial states is used in order to find asubset of all cycles and paths which are not necessarily the longest ones.10,0000 random initial states were generated for n = 25 , , , ...,
60. For5,000 of the states the probability 0.125 was used for each cell to generate a cellstate 1. For the other 5,000 of the initial states, at first a probability p between 0and 1 (in steps of 1/1000) was randomly selected. Then p was used for each cellto generate a cell state 1, otherwise 0. This technique of randomizing gave betterresults in experiments for Rule 22 compared to the usage of a fixed probabilityof 0.5. CAs were simulated and cycles and path lengths were computed andprocessed in a semi-automatic mode. In addition, a genetic algorithm was usedto find near-optimal α -values. The results are presented in Table 4. Note thatbecause of the statistical approach, the listed cycles are not complete and thetrue maximum path lengths could be longer, e.g. for CA(60) all cycles alreadyfound for the factors CA(4, 5, 10, 12, 15, 20, 30) have to be included.19able 4: Representative cycles for n = 25 . . . ω : cycle length, λ : length oflongest cycle prefix, ω + λ : length of longest path detected. The values wereobtained by simulation of 10,000 random initial states. ω ω + λ αn length of cycles longest path longestdetected ending in a cycle prefix(0)25 4 , , , , , ,
150 150 + 57 = 207 15230 1 , , , , , , , , , , , , , , , , , , , , , , , , ,
320 124 + 2551 = 2675 30345 4 , , , , , , , , , , , , ,
252 + 13956 = 14208 750252 , , , , , , , , , , , , , , , , , , , , , , We can summarize that for n ≤
60, the longest paths are much smaller than2 n − De Bruijn diagrams [22] were originally proposed in shift–register theory [23].For a 1 d CA( k, r ) the de Bruijn diagram is defined as a directed graph with k r vertices and k r +1 edges. Vertices are labelled with the elements of symbolsof length 2 r . An edge is directed from vertex i to vertex j , if and only if, the2 r − i are the same as the 2 r − j forminga neighborhood of 2 r + 1 states represented by i (cid:5) j . In this case, the edgeconnecting i to j is labelled with ϕ ( i (cid:5) j ) (the value of the neighborhood definedby the local function) [24], as shown in Fig. 11 for ECA(2,1) and for Rule 22 inFigs. 12–13. The connection matrix M corresponding to the de Bruijn diagram [25] is asfollows: M i,j = (cid:26) j ∈ { k i, k i + 1 , . . . , k i + k − k · r ) } k · r represents the number of vertices and j takes on values in20
31 2
Figure 11: Generic de Bruijn diagram for ECA (2,1). M R = . .. . . .. .
001 1 0 001
Figure 12: Connection matrix and de Bruijn diagram for ECA Rule 22. { k i, k i + 1 , . . . , k i + k − k · r ) } . Hence for ECA(2,1) M i,j = (cid:26) j ∈ { i, i + 1 (mod 4) } , ) (cid:5) ( , → , ) (cid:5) ( , → , ) (cid:5) ( , → , ) (cid:5) ( , → , ) (cid:5) ( , → , ) (cid:5) ( , → , ) (cid:5) ( , → , ) (cid:5) ( , → { , , , } corresponding tofour partial neighborhoods of two cells { , , , } , and eight edges repre-senting neighborhoods of size 2 r + 1. De Bruijn diagram for Rule 22 is derivedfrom the generic one (Fig. 11) and it is calculated in Fig. 12, where the edgesare labelled by the next state.Paths in the de Bruijn diagram may represent chains, configurations orclasses of configurations in the evolution space. Vertices are sequences of sym-bols in the set of states and the strings are sequences of vertices in the diagram.21
31 2
00 0 00
Figure 13: De Bruijn subdiagrams showing unreachable states.The edges represent overlapping of the sequences. Different intersection degreesevoke different de Bruijn diagrams (Fig. 13). Thus, the connection takes placebetween an initial symbol, the overlapping symbols and a terminal one. Forpractical reasons we can use colors, thus the color of an edge represents thenext state to which each neighborhood, as shown in Fig. 11, evolves.
An extended de Bruijn diagram [25, 6] takes into account wide overlapping ofneighborhoods. We represent M (2) R by indexes i = j = 2 r n , where n ∈ Z + , M (3) R and i = j = 3 r n , M (4) R and i = j = 4 r n , and so up to M ( m ) R with i = j = m r n ; consequently basic de Bruijn diagram is obtained when m = 1.The regular expressions derived from the de Bruijn diagram for Rule 22 (Tab. 5)can be linked to space–time dynamics phenomena exhibited by the rules. Theseincludes symmetric complex behavior, chaos, stable periodic behavior. Figure14 shows in detail every periodic pattern yielded from extended de Bruijn di-agrams. To read the diagram we use notation ( i, j ), where i is a displacement(left or right) and j is a number of generations. Thus the pattern in position(0 ,
0) (upper center) displays a periodic pattern without both displacement andperiod, the expression reproducing this pattern is (01) ∗ (de Bruijn subdiagramin Fig. 14 and Eq. 2 in Table 5). In this way, a lot of non-trivial patterns canbe extracted. Let us consider few examples. • (0 , → → → → → → → → → → → → → → → → → → → → → → → → → → → → n -111000101100010-(1100010) n , where n > • (4 , − , − , − , − , − ,
7) and ( − , • (10 , • (0 , , − , − , , , R ki,j = R k − i,j + R k − i,k ( R k − k,k ) ∗ R k − k,j (9)where i is the initial state and j the final state. Base case when k = 0 is thedirect path to every node. This way, by using the basic de Bruijn diagram inFig. 12, we have calculated the whole set of regular expressions, summarizedin Table 5. The first column shows an equation number, the second columnthe regular expression, and the third column the kind of behavior that emergeswhen we codify configurations by these regular expressions. This way, we couldevaluate these equations and explore an unlimited number of configurations. The question “does a complex CA contains a universal constructor?” is a classicproblem appearing in the CA literature since von Neumann works [27]. Aconfiguration of a universal constructor in the Game of Life CA is proposedby Goucher in 2010 [28]. In this context, our aim is to know if ECA Rule22 is able to construct any string. Previously this problem was studied byMcIntosh [6] who found that Rule 22 has a global injective relation and thereforeconfigurations without ancestors exist.We use a subset diagram to calculate
Garden of Eden configurations [6], i.e.,the configurations without ancestors [29]. A subset diagram has 2 k r vertices29able 5: Regular expressions derived in ECA Rule 22. The set of equations iscalculated using the recursive function R ki,j (Eq. 9) to recognise k paths betweennodes i to j in the de Bruijn diagram (Fig. 12).Eq expression evolution1 (0 + 1) ∗ stable state2 (01) ∗ stable periodic3 (001) ∗ stable state4 11(01) ∗
00 still life & symmetric complexbehavior5 (01) ∗ (0 + 1) stable periodic & symmetriccomplex behavior6 (000111) ∗ stable state7 (0 + 0(01) ∗ ∗ stable periodic, chaos & biggaps8 (((01) ∗ + 0)00 ∗ ∗ chaos, complex behavior9 (0 + 1) + 11(01) ∗ (0 + 1) complex behavior10 ((01) ∗ ∗ ∗ chaos, stable periodic, chaos &big gaps11 (11(01) ∗ ∗ ∗ chaos, still life & symmetriccomplex behavior12 (0 + 0(01) ∗ ∗ ∗ (0 + 1) stable state, stable periodic,chaos & big gaps13 (0 + 1(01) ∗ ∗ ∗ chaos, stable periodic, chaos &big gaps14 ((0 + 1) + 11(01) ∗
1) +(11(01) ∗ ∗ ∗ (0(01) ∗ (0 + 1)) –15 (0 ∗ ∗ )(10 ∗
10 + (10 ∗ +10 ∗ ∗ ) ∗ –with k states and r neighbors. If all the configurations of the certain lengthhave ancestors then all the configurations with extensions both to the left andto the right with the same equivalence must have ancestors. But if this is notthe case, then the vertices represent Garden of Eden configurations.We can define the subset diagram as the power set of 2 k r . Such that eachsubset S ∈ U S (where U S is a power set) and one symbol a ∈ Σ: α ( S, a ) = (cid:91) q i ∈ S ϕ ( q i , a ) . (10)Vertices of the subset diagram are formed by the combination of each subsetformed from the states of the de Bruijn diagram. Symbolic de Bruijn matrices M k,s or M s are characterized by k states and s number of states in the partialneighborhood. Thus, for Rule 22 we can obtain symbolic matrices, derived30igure 20: The simplified subset diagram for ECA Rule 22.from the de Bruijn sub–diagrams shown in Fig. 13. For any ECA we have foursequences of states in the Bruijn diagram enumerated as 0, 1, 2 and 3 (seeFig. 11).Union between subsets is represented by the state in which each sequenceevolves and is assigned to the states (subsets that form it) as governed by Eq. 10.Relations between subsets for Rule 22 are constructed in Table 6. Figure 20shows the full scalar subset diagram. Each class of edges defines a function onΣ or Σ . The subset diagram describes the union Σ ∪ Σ that by itself is notfunctional [25].We must distinguish four types of subsets, where it is possible to make atransition between its four unit classes. Also, we should observe that a residualof the de Bruijn diagram can be found in the subset diagram. This is becausea unit class is precisely defined by the nodes of the original diagram. At firstinstance, we can see some relations are more frequent than others. Also there arenodes without inputs, or nodes with most connections including loops. Mostimportant are cycles of different lengths. They are used to infer words, orsequences, that a CA could recognize. Thus, the subset diagram can be usedas a general machine to recognize the universe of words in which a CA couldevolve.By analyzing the full diagram we can derive a small subset diagram which isdeduced from the original diagram (de Bruijn diagram). This diagram includesonly vertexes with cycles, the universal and empty set and the subset withone element, yielding a new diagram that will be more practical for us. Thereduction gives yet a more small diagram to read quickly strings belonging toGarden of Eden configurations. The reduction is also useful to calculate the31able 6: Relations between states of the subset diagram in Rule 22. S label 0 1 φ { } { } { } { } { } { } { } { } { }
10 8 12 { }
12 9 6 { } { }
11 9 6 { }
13 9 6 { }
14 9 14 { }
15 9 14degree of Welch indices for reversible CAs [30]. The expressions that determineGarden of Eden configurations in ECA Rule 22 are listed below: • • • A fractal is constructed recursively from a self–replication of a pattern [31].Chaotic systems often bear properties of fractals.ECA Rule 22 produces a fractal pattern, known as Sierpi´nski triangle, start-ing from a single cell in state 1 (Fig. 1). Figure 21 shows a triangle constructedwith three small tiles derived in Rule 22, this triangle grows in power of twowith respect the number of cells. The main triangle (Fig. 21a) has three repli-cas in the next iteration (Fig. 21b), and following iteration produces nine basereplicas (Fig. 21c). The fractal dimension D can be calculated given the num-ber of replicates N and the scaling factor m [32]. The fractal dimension of thepatterns generated by Rule 22 is the following: D = log ( N ) log ( m ) = log (3) log (2) = 1 . . (11)32 a) (b)(c) Figure 21: Iterated function determines a fractal defining a Sierpi´nski trianglein ECA Rule 22 from a composition of three tiles starting with a 1.Also ECA Rule 22 displays non trivial behavior via fractals where theyemerge in different stages during the evolution. These fractals are combinedwith other fractals constructed from Rule 22 over thousands of generations.Using regular expressions we found two different periodic backgrounds emerg-ing in ECA Rule 22, as discussed in Sections 5, 6 and 7. Let us illustrate twofractals growing in intervals of other fractals with different composition of tiles.Figure 22a shows the initial state of fractals growing in a periodic backgroundwithout displacement conserving the same fractal dimension. Figure 22b showsthe same iterated function over thousands of generations. The same behavioris tested on a periodic background with displacement in Figs. 23ab. Composedfractals emerging in periodic backgrounds with or without displacement aredisjoint.
Rule 18 and Rule 22 are complex rules widely reported in ECA literature [33,34, 35, 11, 13]. Let us redefine them as follows: a cell takes state ‘1’ if exactlyone • (R18): of its neighbors is in state ‘1’: (100 , → • (R22): in its neighborhood is in state ‘1’: (100 , , → (cid:100) R = (cid:26) , ,
000 (12)33 a)(b)
Figure 22: Composition of non-trivial fractals emerging in ECA Rule 22 afterthousands of generations. The iterated function preserves its fractal dimension.These fractals evolve on a periodic background without displacement.34 a)(b)
Figure 23: Composition of non-trivial fractals emerging in ECA Rule 22 afterthousands of generations. The iterated function preserves its fractal dimension.These fractals evolve on a periodic background without displacement.35able 7: Mutation table from Rule 18 in the ECA subset ( b b b b b b b b ) =( b b b b ) . A rule R mutates into Rule R (cid:48) through bit b i ( R | b i (cid:32) R (cid:48) )with exactly a 1–bit change. Rule
111 110 101 100 011 010 001 000
Mutation | b (cid:32) | b (cid:32) | b (cid:32) | b (cid:32) | b (cid:32) | b (cid:32) | b (cid:32) | b (cid:32) | b (cid:32) | b (cid:32) | b (cid:32) | b (cid:32) | b (cid:32) | b (cid:32) | b (cid:32) that defines the sixteen rules displayed in Tab. 7. Referring to the genotypeparadigm in [36] with a rule defined by the sequence ( b b ...b b ) , a rule R mutates into Rule R (cid:48) through bit (or “ gene ”) b i , or ( R | b i (cid:32) R (cid:48) ) with exactly a1–bit change, that yields the mutation (or inheritance) tree in Fig. 24.This tree contains a set of rules with complex behavior inherited from Rule18 and where Sierpi´nski patterns often appear. Rules 18, 22, 122, 126 reveala similar behavior towards ergodic regime and unstable areas, as well as Rule90 but that displays a multilayered pattern, whereas Rule 30 and, by reflection,Rule 86 exhibit small striped patterns near the polygonal border. Rule 26 and,by reflection, Rule 82 exhibit large striped patterns at equilibrium, that mayevolve towards sparse backbones at low or high densities, or periodic patternssimilar to those in Fig. 6. Rule 94 displays a Sierpi´nski gasket, but evanescentand quickly entering a uniform polygon with vertical stripes. Finally it shouldbe emphasized that Rules 50, 54, 58 and by reflection 114, 62 and by reflection118, display patterns somewhat far from the sieve but nevertheless all of thementer the phase transition polygon .This tree does not contain the subset of rules associated either by conjugationor by conjugation-reflection. Beginning from Rule 129 –conjugate with Rule126– this subset would display a large collection of Sierpi´nski–like patterns butthis time in white on a black background. Rules 30, 54, 90, 126 are examined elsewhere [37, 38, 39, 40]. R to Rule R (cid:48) ( R | b i (cid:32) R (cid:48) ) bears exactly the 1–bit genetic change b i and d H ( R, R (cid:48) ) = 1 where d H is the Hamming distance.Even in the synchronous “1nCA” [41] with a minimal neighborhood of twocells –the cell itself and the either left or right adjacent cell alternating at eitherodd or even timesteps– Sierpi´nski patterns appears in Rules 6∆ and 9∆, namelyfor the only symmetric rules with an equal number of black and white cells, andfor the only rules fulfilling the maximal “sensitivity parameter”.Six rules (18, 22, 30, 90, 122, 126) evolving in their ergodic regime aredisplayed in Fig. 25. All patterns have the Hausdorff dimension log (3) ofEq. 11. They only differ from their average density d C of their mesoscopicminimal macrocell C . Referring again to the mean field curves in Sect. 2 now displayed for Rule18 and Rule 22 in Fig. 26 we observe that Rule 18 curve reaches its stable fixedpoint p t +1 = p t when crossing the identity at p t ≈ .
293 whereas for Rule 22the fixed point is got at p t ≈ .
423 whence the discrepancies between densitiesin ergodic regime, observable in Fig. 25. Moreover, Rule 18 curve shows a slope f (cid:48) R (0) = 2 whereas Rule 22 shows a slope f (cid:48) R (0) = 3 at the origin. Thatcomes from the fact that these rules induce from a single source their followingevolution: • (R18): 0 ∗ ∗ → ∗ (101)0 ∗ • (R22): 0 ∗ ∗ → ∗ (111)0 ∗ and that their density ratio at timesteps 2 p − p >
0) remains 2 / d = 0 .
50. Left to right, top to bottom: Rules 18, 22, 30, 90, 122, 126 in ergodicregime. All patterns have the Hausdorff dimension log (3) of Eq. 11.38igure 26: Rule 18 ( p t +1 = 2 p t q t ) and Rule 22 ( p t +1 = 3 p t q t ) with their meanfield curves.The Sierpi´nski gasket [42] appears in a wide variety of situations [43]. Thebinomial coefficients can be arranged to form Pascal’s triangle and Pascal’s tri-angle turns into Sierpi´nski gasket with coefficients modulo two. It may turninto something like a natural tree by some diffeomorphism. This tree is embed-dable into the 2 d diffusion graphs embedded into the triangulate lattice [44, 45]:its vertex dust forms the Sierpi´nski gasket patterns. Sierpi´nski gasket is oftenknown as a Banach fixed point from some contractive affine transformation intothree elements.But the most fascinating is the formation of patterns from random initialdistributions of pigmentations on certain varieties of seashells [33, 46]. This phe-nomenon can be explained from the Gierer–Meinhardt reaction-diffusion modelof the activator–inhibitor type, arising in various situations of pattern formationin morphogenesis [47, 48]. Despite the fact that Rule 22 is based on simple and determined interactions,the behavior generated by such system is visibly complex and seems to be non–deterministic. To test the kind of the rule’s behavior tests chaos 0–1 [49] (clas-sifier returning value near 1 if series is chaotic and near to 0 if deterministic)was used to determine whether behavior is chaotic/random or deterministic. Itis applied directly to the time series data and does not require phase space re-construction. These two kinds of behavior are significantly different. Physically,randomness has a stochastic nature, while deterministic chaos is generated byeven simple system that does not contain any source of randomness, as com-monly understood by public. If executed on a PC, then algorithms simulatingsuch behavior generate only pseudo-random/chaotic behavior and series gener-ated in such a way, are essential deterministic and periodic but with very longperiod. The period is usually long enough to simulate randomness/chaos.The term chaos covers a rather broad class of phenomena whose behaviormay seem erratic, chaotic at first glance. Till now chaos was observed in many39ystems (including evolutionary ones) and, in the last few years, it has beenalso used to replace pseudo-random number generators (PRGNs) in evolution-ary algorithms (EAs). Let us mention for example research papers like [50](a comprehensive overview of mutual intersection between EAs and chaos isdiscussed in this paper), one of the first use of chaos inside EAs [51], [52, 53]discussing the using of deterministic chaos inside particle swarm algorithm in-stead of PRGNs, [54, 55] investigating relations between chaos and randomnessor the latest one [56], and [57, 58, 59], using chaos with EAs in applications,amongst others.Another research joining deterministic chaos and pseudorandom numbergenerator has been done for example in [54]. The possibility of generation ofrandom or pseudorandom numbers by use of the ultra weak multidimensionalcoupling of 1-dimensional dynamical systems is discussed there. Another pa-per [60] deeply investigates a logistic map as a possible pseudo-random numbergenerator and it is compared with contemporary pseudo-random number gener-ators. A comparison of logistic map results is made with conventional methodsof generating pseudo-random numbers. The approach is used to determine thenumber, delay, and period of the orbits of the logistic map at varying degreesof precision (3 to 23 bits). Logistic map, we are using here, was also used in[61] like chaos-based true random number generator embedded in reconfigurableswitched-capacitor hardware. Another paper [62] proposed an algorithm of gen-erating a pseudorandom number generator, which is called (couple map latticebased on discrete chaotic iteration) and combine the couple map lattice andchaotic iteration. Authors also tested this algorithm in NIST 800-22 statisticaltest suits and is used in image encryption.In [63] authors exploit interesting properties of chaotic systems to design arandom bit generator, called CCCBG, in which two chaotic systems are cross-coupled with each other. For evaluation of the bit streams generated by theCCCBG, the four basic tests are performed: monobit test, serial test, auto-correlation, Poker test. Also the most stringent tests of randomness: the NISTsuite tests have been used. A new binary stream–cipher algorithm based ondual one-dimensional chaotic maps is proposed in [55] with statistic propertiesshowing that the sequence is of high randomness. Similar studies are also donein [51, 64, 65, 66]. For a long time various PRNGs were used inside evolutionaryalgorithms. During last few years deterministic chaos systems (DCHS) are usedinstead of PRNGs. As was demonstrated in [52, 53], very often the performanceof EAs using DCHS is better or fully comparable with EAs using PRNGs. Seefor example [52].The chaos test 0–1 [49] has been already and successfully used on varioustasks as for example on experimental data from a bipolar motor [67], behavior ofthe cutting process [68], real experimental time series of laser droplet generationprocess [69] and validated by applying to typical nonlinear dynamic systems,including fractional–order dynamic system [70] amongst others.The same approach was used here. Test 0-1 was used on different series inorder to verify and test the nature of Rule 22. Figure 27 visualizes results of ourexperiments. For evaluation 2000 Rule 22 behavior strings of length 2000 have40
500 1000 1500 2000 String No.0.9650.9700.9750.980Test 0 - (a) - (b) - (c)
500 1000 1500 2000 String No. - - - - - - (d) - (e) Figure 27: Test chaos 0-1 of (a) logistic equation, (b) Mersenne Twister gen-erator, (c) standard PRNG, (d) deterministic series (returned values are allaround 0, i.e. evaluated process is deterministic), (e) Rule 22. All results areconcentrated around 1. Chaos based on test chaos 0-1 was proved.41een used. The same was repeated for the Mersenne twister random numbergenerator [71] (MTPRNG) (Fig. 27(c)), chaos generated by logistic equation(LE) (Fig. 27(b)), and periodic series (PS): the sinus function generated fromthe randomly selected position (Fig. 27(d)). As clearly visible, test 0-1 hasclearly classified Rule 22, MTPRNG and LE as a non-deterministic series whileseries based on periodic pattern as deterministic. The random series were notdistinguished from chaotic ones. This was probably caused by insufficient length(2000 is likely not enough) of series, however, this was not a matter of thisexperiment.
In this section, we show that ECA Rule 22 with memory is strongly chaotic [72].Conventional CA are ahistoric (memoryless): i.e., the new state of a cell de-pends on the neighborhood configuration solely at the preceding time step of ϕ .Thus, CA with memory (CAM) can be considered as an extension of the stan-dard framework of CA where every cell x i is allowed to remember some periodof its previous evolution. A memory is based on the state and history of the sys-tem, thus we design a memory function φ , as follows: φ ( x t − τi , . . . , x t − i , x ti ) → s i ,such that τ < t determines the backwards degree of memory and each cell s i ∈ Σis a function of the series of states in cell x i up to time-step t − τ . To execute theevolution we apply the original rule as follows: ϕ ( . . . , s ti − , s ti , s ti +1 , . . . ) → x t +1 i .In CAM, while the mapping ϕ remains unaltered, a historic memory of pastiterations is retained by featuring each cell as a summary of its previous states;therefore cells canalize memory to the map ϕ . As an example, we can take thememory function φ as a majority memory : φ maj → s i , where in case of a tiegiven by Σ = Σ in φ , we shall take the last value x i . So φ maj representsthe classic majority function for three variables on cells ( x t − τi , . . . , x t − i , x ti ) anddefines a temporal ring before calculating the next global configuration c . Incase of a tie, it is allowed to break it in favor of zero if x τ − = 0, or to onewhether x τ − = 1.The representation of a ECAM is given as follows: φ CARm : τ (13)where CAR represents the decimal notation of a particular ECA rule and m the kind of memory given with a specific value of τ .Note that memory is as simple as any CA, and that the global behavior pro-duced by the local rule is rather unpredictable, it can lead to emergent propertiesand so can be classed as complex. Memory functions were developed and ex-tensively studied by Alonso-Sanz in [73]. Memory in ECA have been studied,showing its potential to produce complex behavior from chaotic systems andbeyond in [37, 72], and recently in [74] authors have included hybrid versions.Thus, we can conjecture that a memory function can produce complex behavior[72] as follows: φ ( ϕ chaos ) → complex. (14)42 a) (b)(c) (d) Figure 28: ECA Rule 22 with a memory function reveals complex behavior.(a) Evolution of the function φ R maj :3 . (b) The function φ R maj :4 (recentlyproven to be logically universal by simulating the Fredkin gate in [76, 77]). (c)The function φ R maj :7 (a glider gun was discovered in this rule [72]). (d) Thefunction φ R maj :8 (particles with long period).43ppstein [75] demonstrates that a CA class IV is a system where mobile self–localizations emerge. We can relate type of classes with memory functions inECA Rule 22 as: φ R maj :3 ⇒ chaos → chaos (15) φ R maj :4 ⇒ chaos → complexity (16) φ R maj :7 ⇒ chaos → complexity (17) φ R maj :8 ⇒ chaos → complexity (18) The state transition function ϕ ( x ti − r , . . . , x ti , . . . , x ti + r ) → x t +1 i can be re-writtenas Boolean formula with two xor operations: x ti + r = x ti − r ⊕ x ti ⊕ x ti + r . The xor gate is the most rare, most hard to find in natural non-linear systems, Booleangate. Let gates g and g discovered with occurrence frequencies f ( g ) and f ( g ),we say a gate g is easier to develop or evolve than a gate g : g (cid:66) g if f ( g ) >f ( g ). The hierarchies of gates obtained using evolutionary techniques in liquidcrystals [78], light–sensitive modification of Belousov–Zhabotinsky system [79],slime mould Physarum polycephalum [80] and protein molecules [81, 82]: • Gates in liquid crystals: { or , nor } (cid:66) and (cid:66) not (cid:66) nand (cid:66) xor • Gates in Belousov–Zhabotinsky medium: and (cid:66) nand (cid:66) xor • Gates in cellular automata [83]: or (cid:66) nor (cid:66) and (cid:66) nand (cid:66) xor • Gates in Physarum: and (cid:66) or (cid:66) nand (cid:66) nor (cid:66) xor (cid:66) xnor • Gates in protein molecules verotoxin and actin: and (cid:66) or (cid:66) and-not (cid:66) xor The xor gate is hard to find and space-time dynamics of Rule 22 automatais hard to predict. A strong link computational difficulty of a problem andits randomness was established by Yao [84]. His famous lemma, rephrased byImpagliazzo and Wigderson [85], can be seen in the framework of predictabilityof Rule 22 ECA behavior:Fix a non-uniform model of computation (with certain closure prop-erties) and a Boolean function f : { , } n → { , } . Assume thatany algorithm in the model of a certain complexity has a signifi-cant probability of failure when predicting f on a randomly choseninstance x . Then any algorithm (of a slightly smaller complexity)that tries to guess the xor f ( x ) ⊕ f ( x ) ⊕ . . . ⊕ f ( x k ) of k randominstances x , . . . , x k wont do significantly better than a random cointoss.Potential associations between dynamics in Rule 22 ECA and a role of xor functions in communication complexity [86, 87] could be explored in future.44 eferences [1] S. Wolfram, Statistical mechanics of cellular automata, Rev. Mod. Phys.55(3) (1983) 601–644.[2] S. 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