Simple networks on complex cellular automata: From de Bruijn diagrams to jump-graphs
Genaro J. Martínez, Andrew Adamatzky, Bo Chen, Fangyue Chen, Juan C.S.T. Mora
SSimple networks on complex cellular automata:From de Bruijn diagrams to jump-graphs
Genaro J. Mart´ınez , , Andrew Adamatzky ,Bo Chen , Fangyue Chen , Juan C.S.T. Mora
25 November 2017 ∗ Escuela Superior de C´omputo, Instituto Polit´ecnico Nacional, M´exico International Centre of Unconventional Computing, University of the Westof England, Bristol, United Kingdom School of Science, Hangzhou Dianzi University, Hangzhou, China ´Area Acad´emica de Ingenier´ıa, Universidad Aut´onoma del Estado deHidalgo, Pachuca, Hidalgo, M´exico Abstract
We overview networks which characterise dynamics in cellular automata.These networks are derived from one-dimensional cellular automaton rulesand global states of the automaton evolution: de Bruijn diagrams, sub-system diagrams, basins of attraction, and jump-graphs. These networksare used to understand properties of spatially-extended dynamical sys-tems: emergence of non-trivial patterns, self-organisation, reversibilityand chaos. Particular attention is paid to networks determined by travel-ling self-localisations, or gliders.
Cellular automata (CA) are arrays of finite state machines, or cells. Each celltakes a finite number of states. All cells update their states by the same rulesimultaneously. A cell updates its state depending on states of its immedi-ate neighbours. The CA originated from Stanislaw Ulam problem on a par-allel transformation of matrices, and further developed in a context of self-reproduction by John von Neumann [26]. CA are apparently simple systemsyet exhibiting sophisticated patterns of non-trivial behaviour. They are nowubiquitous tools in studying complex systems and non-linear dynamics in liter-ally all fields of science and engineering, see e.g. a visual guide of representativepower of CA in [4]. Rules space and global dynamics of CA are characterised bynetworks: the de Bruijn diagrams [20, 27], subsystem diagrams [9, 5], basins of ∗ Chapter published in the book
Swarm Dynamics as a Complex Network , Springer, (IvanZelinka and Guanrong Chen Eds.), chapter 12, pages 241-264, 2017. a r X i v : . [ n li n . C G ] F e b ttraction [29], and jump-graphs [32]. We analyse predictive power of the net-works using two CA rules which exhibit complex space-time dynamics: gliders,particles, waves, or localisations. The CA rules studied are the Rule 54 andthe Rule 110 . A cell in elementary CA (ECA) takes two states from Σ = { , } . A cell x i ,1 ≤ i ≤ n , updates its state by local function ϕ depending on its own state andstates of its two immediate neighbours: x t +1 i = ϕ ( x ti − , x ti , x ti +1 ) . There are 2 = 256 cell state transition rules. In 1983, Stephen Wolframestablished a classification ECA rules [30], based on space-time development ofautomata governed by the rules. The Wolfram classes are Class I : evolution to uniform behaviour;
Class II : evolution to periodic behaviour;
Class III : evolution to chaotic behaviour;
Class IV : evolution to complex behaviour.Wolfram’s classification is not the only one, there are 17 classifications ofCA rule space [13], however it is fully adequate and it became a classic ‘item’of CA theory.The class III includes rules 54 and 110 (decimal representation of a binaryrule-string). Rules in class III are called ‘complex’ because the automata gov-erned by these rules have longer transient periods and more sensitive to initialconditions than automata governed by rules form other classes. And, most im-portantly, they exhibit a wide range of travelling and stationary localisations,interactions that are the reason of ‘complexity’ of the rules’ behaviour. We study two ECA complex rules: Rules 54 and 110.The Rule 54 (‘54’ is a decimal representation of a binary string 00110110)automaton typically exhibits a rich dynamics of mobile localisations and out-comes of their collisions. Some of the mobile localisation collisions were used todemonstrate logical universality of the rule in [15]. The Rule 54 is representedby a function: ϕ R = (cid:26) , , , , , , . (1) Repository Rule 54 http://uncomp.uwe.ac.uk/genaro/Rule54.html Repository Rule 110 http://uncomp.uwe.ac.uk/genaro/Rule110.html Complex Cellular Automata Repository http://uncomp.uwe.ac.uk/genaro/Complex_CA_repository.html . ϕ R = (cid:26) , , , , , , . (2)Figure 2 shows a typical evolution of ECA Rule 110 from a random initialcondition. (a) (b) Figure 3: Mean field curves for (a) Rule 54 and (b) Rule 110.Complexity of CA rules can be estimated using polynomial approximationborrowed from the mean field theory [23]. Mean field theory techniques areefficient in discovering statistical properties of CA without analysing the wholeevolution space of individual rules. These techniques have been introducedin CA field by Howard Gutowitz in [10]. The approach assumes that cell-states do not correlate with each other in the local function ϕ . Thus we canstudy probabilities of states in a neighbourhood in terms of the probabilityof a single cell-state, and the probability of a neighbourhood would be theproduct of the probabilities of each cell in it. For one-dimensional CA with k -cell neighbourhood or radius r and k cell-states the probability is calculatedas follows: 5 t +1 = q r +1 − (cid:88) j =0 ϕ j ( X ) p vt (1 − p t ) n − v (3)where j of a neighbourhood state, q is a number of cell-states, X is a neighbour-hood x i − r , . . . , x i , . . . , x i + r , k is the number of cells in every neighbourhood, v indicates how often state ‘1’ occurs in X , n − v shows how often state ‘0’ occursin the neighbourhood X , p t is the probability of a cell being in state ‘1’, and q t is the probability of a cell being in state ‘0’; i.e., q = 1 − p . In our case of binarycell-states and three-cell neighbourhood the probability is p t +1 = (cid:88) j =0 ϕ j ( X ) p vt (1 − p t ) n − v In [17], Harold McIntosh proposed a classification based on curves derivedwith the mean field approximation. A complex rule has a curve tangential tothe identity, and an unstable fixed point that defines regions with unpredictablebehaviour:
Class IV: mean field curve horizontal plus diagonal tangency (no crossing theidentity, possibly complex dynamics) .Figure 3a shows a mean field curve for Rule 54 with a polynomial definedas: p t +1 = 3 p t q t + p t q t . (4)The origin value is a stable fixed point which guarantees the stable configu-ration in state zero. The maximum point p = 0 . p = 0 .
5. Complex dynamics in Rule 54 emerges on a periodicbackground with the same number of states zero and one, thus the stable fixedpoint well characterises the local function (see Eq. 1). Also, this fixed pointshows that a Rule 54 automaton starting from low or high densities of state 1cells, more likely will finish its evolution with the same ratio of states (Fig. 1).Figure 3b shows a mean field curve for Rule 110 with polynomial defined as: p t +1 = 2 p t q t + 3 p t q t . (5)The maximum point p = 0 . p = 0 . De Bruijn diagrams
For a one-dimensional CA of order ( k, r ) and a finite alphabet given Σ, its deBruijn diagram is a directed graph with k r vertexes and k r +1 edges calculatedas follows. Vertexes are labelled with elements of the alphabet of length 2 r ,i.e. neighbourhood states. An edge is directed from vertex i to vertex j , if andonly if, the 2 r − i are the same as 2 r − j forming a neighbourhood of 2 r + 1 states represented by i (cid:5) j . In this case, theedge connecting i to j is labelled by ϕ ( i (cid:5) j ) (the value of the neighbourhooddefined by the local function) [27].Thus the de Bruijn diagram is constructed as follow: M i,j = (cid:26) j = ki, ki + 1 , . . . , ki + k − k r )0 in other case (6)For ECA the module k r = 2 = 4 represents the number of vertexes inthe de Bruijn diagram and j takes values from k ∗ i = 2 i to ( k ∗ i ) + k − ∗ i ) + 2 − i + 1. The vertexes (indexes of M ) are labelled by fractions ofneighbourhoods beginning with 00, 01, 10 and 11, the overlap determines eachconnection (Fig. 4a). Paths in the de Bruijn diagram may represent chains,configurations, or classes of configurations in the evolution space. Also frag-ments of the diagram itself are useful in discovering periodic blocks of strings,pre-images, codes, and cycles [18].After the de Bruijn diagram is completed, we can calculate an extended deBruijn diagram. An extended de Bruijn diagram takes into account more signif-icant overlapping of neighbourhoods of length 2 r . We represent M (2) by indexes i = j = 2 r ∗ n , where n ∈ Z + . The de Bruijn diagram grows exponentially, order k r n , for each M ( n ) . We can calculate generic de Bruijn diagrams arranged in acircular pattern for r = 1 Fig. 4a (basic diagram), r = 2 Fig. 4b, r = 3 Fig. 4c, r = 4 Fig. 4d, r = 5 Fig. 4e, r = 6 Fig. 4f (constructing a circle).Generic diagrams calculate strings of different periods. These patterns arestructures without displacements. The complement diagrams calculates periodsplus displacements. In these diagrams we can find systematically any periodicstructure, including some mobile localisations.For extended de Bruijn diagrams we have shift registers to the right (+)or to the left ( − ). A mobile localisation can be identified as a cycle and thelocalisations interaction will be a connection with other cycles. Diagram (2 , x -displacements, y -generations), displays periodic strings moving two cells tothe right in two time steps, i.e., period of a mobile localisation. This way, wecan enumerate each string for every structure in this domain.The de Bruijn diagram than can calculate stationary localisations is of order M (4) R because localisations have period four without displacements. These pat-terns can be considered also as still life configurations. Figure 5 shows the fullde Bruijn diagram (0,4) used to calculate these stationary localisations. Thereare four main cycles, two largest cycles represent phases of each stationary lo-calisation plus its periodic background; and two smaller cycles characterising7 a) (b)(c) (d)(e) (f) Figure 4: Generic de Bruijn diagrams for ECA. Each generic diagram followsthe number of partial neighbourhoods k r for r = (a) 1, (b) 2, (c) 3, (d) 4, (e)5, and (f) 6. 8igure 5: De Bruijn diagram (0,4) calculating stationary localisations in Rule54. A snapshot for every cycle is showed below of every diagram. This way,patterns are defined as a code since its initial condition obtained from diagram.9igure 6: De Bruijn diagram (4,8) calculating non-stationary localisations inRule 110. The left evolution displays a fuse pattern produced by two mo-bile localisations colliding and both annihilated, the center evolution displaysa periodic pattern, and the right evolution displays a mobile localisation withdisplacement to the left. 10igure 7: De Bruijn diagram (2,10) calculating non-stationary localisations inRule 110. First, the original diagram is calculated with 1,048,576 vertices.Below, an evolution of non-stationary localisations beginning from the vertex652,687 (left) and a periodic background defining a small mosaic.11wo different periodic patterns in Rule 54 including the stable state representedwith a loop by vertex zero. Space-time configurations of ECA derived fromthese diagrams are illustrated on the left plate of Fig. 5. Position of each mobilelocalisation and periodic background follows arbitrarily routes into these cycles.Details on these regular expressions for Rule 54 are presented in [16].For Rule 110 we have calculated an extended de Bruijn diagram (4,8) thatdetermines non-stationary localisations. Figure 6 shows a diagram than ini-tially needs 65,536 vertexes. However, we can reduce the diagram just filteringcycles, this way we have a diagram of 145 vertexes and 153 links. This way,this diagram displays a stable state represented by loop zero and cycles definea phase of a mobile localisation, i.e., a string that determines how to input amobile localisation into its initial condition: left evolution shows mobile local-isations colliding constantly, right evolution shows mobile localisations movingto the left, and center evolution shows a periodic pattern. This characteristic isvery useful to control collisions of localisations in this automaton. Connectionsbetween cycles mean that you can connect several structures on a same phase.Figure 7 shows the de Bruijn diagram (2,10) than calculates non-stationary lo-calisations and periodic patterns coveting the evolution space of Rule 110. Thisdiagram surpass a million of vertexes and shows a snapshot of a particular ini-tial condition, coding several non-stationary localisations beginning in a diversenumber of phases, copies, and intervals. Details about these regular expressionsfor Rule 110 can be found in [22].De Bruijn diagrams contain all relevant information about of complex pat-terns emerging in cellular automata de Bruijn diagrams can proof exhaustivelythe number of periodic patterns that rule can yield. But they grown quickly,therefore not rarely a second algorithm must be implemented to extract usefulstrings from a diagram. Let us focus on presenting an analytical characterisation of symbolic dynamicsof mobile localisations in Rule 110. For each mobile localisation, a particularsubsystem can be found through enumeration and exhaustive analysis. Directedgraph theory and transition matrix provide are powerful tools for studying eachsub-shift of a finite type — such as a glider — which is topologically mixingand possesses the positive topological entropy on this subsystem. A positivetopological entropy implies the chaos in the sense of Li-Yorke [11, 28]. Devaney[8] and Li-Yorke types of chaos can be deduced from topological mixing. Herewe describe a non-stationary localisation, other types of mobile localisations canbe studied by analogy.Particularly, a non-stationary localisation with velocity to 1/5 in Rule 110 can be represented as, Λ A = { x ∈ S Z | x [ i − ,i +6] ∈ A , ∀ i ∈ Z } is a sub-shift ofa finite type, its determinative system A = { Gliders in Rule 110 http://uncomp.uwe.ac.uk/genaro/rule110/glidersRule110.html } . Let x [ i − ,i +6] denote a 14-bitstring ( x i − , x i − , x i − , . . . , x i +5 , x i +6 ) over S = { , } which is described by itsdecimal code expression, such as 9976 refers to the string (1,0,0,1,1,0,1,1,1,1,1,0,0,0), and 3569 refers to the string (0,0,1,1,0,1,1,1,1,1,0,0,0,1).A fundamental method for constructing finite shifts starts with a finite,directed graph and produces the collection of all bi-infinite walks (i.e., stringsof edges) on the graph. The Λ A can be described by a finite directed graph G A = {A , E } , where each vertex ( V ) is a string in A . Each edge e ∈ E ( G )starts at a string denoted by a = ( a , a , · · · , a ) ∈ A and terminates at thestring b = ( b , b , · · · , b ) ∈ V ( G ) if and only if a k = b k − , ≤ k ≤
13. One canrepresent each element of Λ A as a certain path on the graph G A . The entirebi-infinite walks on the graph constitute the closed invariant subsystem Λ A .The finite directed graph Λ A , is shown in Fig. 8. A finite path P = v → v →· · · → v m on a graph G is a finite string of vertexes v i from G . The length of P is | P | = m . A cycle is a path that starts and terminates at the same vertex.A graph G is irreducible if for each ordered pair of vertexes I and J there is apath in G starting at I and terminating at J . Each element of Λ A is a certainpath on the graph G A .Let (cid:98) S = { r , r , · · · , r , r } be a new symbolic set, where r i , i = 0 , · · · , A respectively. Then, one can construct a new sym-bolic space (cid:98) S Z on (cid:98) S , where B = { ( rr (cid:48) ) | r = ( b b · · · b ) , r (cid:48) = ( b (cid:48) b (cid:48) · · · b (cid:48) ) ∈ (cid:98) S, ∀ ≤ j ≤ s.t. b j = b (cid:48) j − } . The two-order sub-shift Λ B of σ is de-fined as Λ B = { r = ( · · · , r − , r ∗ , r , · · · ) ∈ (cid:98) S Z | r i ∈ (cid:98) S, ( r i , r i +1 ) ≺ B , ∀ i ∈ Z } .One can calculate the transition matrix Y of the sub-shift Λ B with Y ij = 1,if ( i, j ) ≺ Λ B ; otherwise Y ij = 0. We call the matrix Y positive if all of itsentries are non-negative; irreducible if ∀ i, j , there exist n such that Y nij > N , such that Y nij > , n > N, ∀ i, j . Then, Λ B istopologically mixing if and only if Y is irreducible and aperiodic. Thus, thetopologically conjugate relationship between (Λ A , σ ) and a two-order sub-shiftof finite type (Λ B , σ ) can be established. In addition, the transition matrix Y isrelatively large (the order of D is 167). Therefore, we only list the indexes ( i, j )of nonzero elements. Y = { (1 , , , , , , , , , , , , , , , , , v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v Figure 8: Graph representation for the subsystem Λ A of a non-stationary local-isation with velocity 1 /
5, it is the same mobile localisation calculated with a deBruijn diagram showed in Fig. 7. Where each vertex stands for the element of A by order, i.e., v = 9976, v = 3569, v = 7138, · · · , v = 14072, v = 11761.All bi-infinite walks on the graph constitute the closed invariant subsystem Λ A .1412 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , } .Denote the elements of Y n as Y ni,j ,1 ≤ i, j ≤ Y ni,j indicates the numberof the whole paths from i -th vertex to j -th vertex whose length is n . Thus, Y ni,i isthe number of all cycles of i -th vertex with the length n . As all Y ni,i , 1 ≤ i ≤ n = 39, it is easy to verify that each vertex possesses a particularcycle. Hence, the non-wandering set of subsystem Ω(Λ A ) = Λ A .The topological dynamics of Λ A is determined by the properties of its tran-sition matrix Y . The characteristic equation of Y is − λ − λ − λ + 5 λ + λ + 5 λ + 15 λ + 10 λ + λ − λ = 0. The spectral radius ρ ( Y )is the maximum positive real root λ ∗ of characteristic equation. We have ent (Λ A ) = log ( ρ ( Y )) = log (1 . . σ L on Λ A equals logρ ( Y ). The topological entropy ent (Λ A ) is posi-tive.The matrix ( Y + I ) n is positive for n ≥
47, where I is a 167 ×
167 identitymatrix. Y + I is aperiodic, and Y is irreducible. Λ A is topologically transitivebecause the transition matrix Y is irreducible.A two-order sub-shift of a finite type Λ A is topologically mixing if and onlyif its transition matrix is irreducible and aperiodic. Meanwhile, it is easy toverify that Y n is positive for n ≥ Y is irreducible andaperiodic. The Λ A is topologically mixing.In conclusion, the above discussions are summarised as the subsystem Λ A ischaotic in the sense of both Li-Yorke and Devaney.15 Basins of attraction
Basins of attraction or cycle diagrams calculate attractors in a dynamical sys-tem, as was extensively studied by Andrew Wuensche in CA and random Booleannetworks [29, 32, 20].Given a sequence of cells x i we define a configuration c of the system. Anevolution is represented by a sequence of configurations { c , c , c , . . . , c m − } given by the global mapping, Φ : Σ n → Σ n (7)and the global relation is given for the next function between configurations,Φ( c t ) → c t +1 . (8)An attractor is represented as a number of c i states, these states are con-nected in cycles in periods of global states.To obtain cycles for a given automaton we enumerate all the rings of thedesired length, and follow up their evolution. In doing so task, various short-cuts can be taken, such as generating the configurations in some order so thatonly a single cell changes state from one to the next, this way a number ofconfigurations can be compacted to avoid calculating again the same string.Still life configurations and small oscillators can be detected very quickly in thisway. Comparison of successive generations means that whenever the new gen-eration is smaller, it has been already examined and there is no need for furtherexploration [20].The number of global states c is defined by the length of the string n m . Thestructure of an attractor is given in three parts. Leaves represent Garden ofEden configurations for those global states, that means that these states haveno ancestors. Branches are configurations that have at least an ancestor andjust one successor. Height in branches determines the number of generationsnecessary before to reach the attractor. So, the attractor is the final state of astring of length n . Numbers labelling vertices’s represent the decimal value forthe string in study.Some properties are immediate. An attractor period one (loop) determinesan evolution dominated just for one state of the alphabet. Basins constructedby cycles for any n , they represent exactly reversible CA [19, 24], where you canmove back in the history of the system from any configuration, and thereforeany configuration has one ancestor.Generally a basin can recognise CA also with chaotic or complex behaviourusing prior results on attractors [29]. This way, Wuensche discovered that Wol-fram’s classes can be represented as a basin classification . Talking about com-plex rules, a basin will be defined as: Class IV: moderate transients, moderate-length periodic attractors, moderatein-degree, very moderate leaf density (possibly complex dynamics). k deriving just 6,326 basins. The attractoris defined for the next configurations: 0110111111110001001101111100010 → → → → → → • • a) (b) Figure 12: Patterns formation from two configurations of length 31 (leaves)in Rule 110 calculated in Fig. 11. (a) stationary localisations. (b) non-trivialmobile localisations colliding. A filter is selected to a better view of localisations.By calculating large attractors, we can discover landscapes of complexityin basins featured with non-symmetric, high, and dense ramifications: thesekind of ramifications are indicators of ‘unpredictable’ behaviour on most largeconfigurations. Frequently chaotic rules tend to have symmetric basins.
Basins of attraction can be constructed into a meta network, called the ‘jump-graph’ by Wuensche [32]. Jump-graphs determine the next level of CA com-plexity by showing a probability to jump to another attractor given a mutationon the same domain of strings.In a jump-graph the question is: What is the probability that a string w i could mutate to another string w j ? Which could induce a change of transitionto another attractor or itself (a loop, without change). This way, a configura-tion can remains in its attractor or jump to another attractor. We could findconnections that jump to other attractor and later back to the original attractorwith other mutation (a cycle between basins).Ψ(Φ( c i )) → Φ( c j ) (9)Let us consider a one-bit value mutation [32]. We have a configuration c i expressed as a string w i = a a . . . a n − , such than it can jump into otherconfiguration c j expressed as a string w j = b b . . . b n − . Hence a i can mutateto one b i , where each configuration c belongs at the same field of attractors Ψ.Also, if a i = b i , it represents a loop in the same basin.20igure 13: Jump-graph composed from a basin field with configurations of 20cells for ECA Rule 54. This meta-diagram consists of 428 basins yielding afield of 784,472 interconnections given by mutations of 1-bit value. Complexbehaviour transits with a high density of connections between basins in the fullgraph for this automaton. 21 ( ) Figure 14: A mutation between attractors characterised of a jump-graph inECA Rule 54 (Fig. 13) for strings of length 20. The first basin (label withnumber three in DDLab) has a period 18 with a mass of 6,866 configurations.The second basin (label with number five) has a period 86 with a mass of 9,946configurations. Both basins display dynamics with stationary localisations andsynchronisations of small mobile localisations colliding in ECA Rule 54. Basinsand evolutions are rotated 90 degrees. A filter is selected for a better view.22igure 13 shows a jump-graph for Rule 54 with configurations of length 20.This field is composed of 428 basins on a domain of k . Each node has at leastone mutation and therefore they all are connected (it is a graph strongly con-nected), determining 784,472 interconnections or mutations. For this automatonwe have no unreachable states, because all states are connected, consequently aglobal stability is achieved in this system. This meta-diagram is a local universecontaining basins with volume from 62 to 29,990 configurations and they haveperiods oscillating between 4 to 86 configurations. Non-trivial structures emergeduring these developments with dozens of mobile localisations colliding in longtransients; this implies a frequent change of basins into this jump-graph.Finally, Fig. 14 shows a mutation in detail, extracted from the full jump-graph calculated previously in Fig 13. There is a mutation between two attrac-tors and they determine a change of dynamics. This diagram displays a con-nection between two basis (the basin three and the basin five as enumerated inDDLab [32]). The mutation is expressed by the string x CA have very low entry fee — anyone with basic coding skills can start exper-imenting with CA — but very high exit fee — far from anyone can produceresults publishable in a reputable journal. This is because cell-start transitionrules are appealingly simple yet space-time behaviour might be shockingly diffi-cult to analyse and predict. There are few types of graphs/networks constructedover the cell-state transition rule spaces which might facilitate understanding ofemergence and dynamics of complexity. They are de Bruijn diagrams, subsys-tems graphs, basins of attraction, and jump-graphs. De Bruijn diagrams char-acterise interrelationships between strings representation of the functions, theyallows for some estimates of a distribution of states in a stable configuration.Subsystems graphs are determined by dynamics of travelling localisations, glid-ers, especially by interactions between mobile localisations. This is a promisingrepresentation especially when related to computational abilities, in a sense ofcollision-based computing, of CA rules supporting mobile localisations. Basinsof attractions give us a straightforward visualisation of complexity: a numberof cycles and heigh and bushiness of trees growing on them reflects sensitivityand ‘chaoticity’ of the cell-state transition rules. Jump-graphs might be seen ascharacterising rules’ tolerance to noise, mutations. There are two sides of thetolerance. Either a noise-tolerant system is very dull and therefore difficult tomove to another loci of a global state space, or the system is so sophisticatedthat it have an in-build ‘mechanism’ for dealing with mutation. Discrete Dynamics Lab . eferences [1] Adamatzky, A.: Computing in Nonlinear Media and Automata Collectives .Institute of Physics Publishing, Bristol and Philadelphia, Bristol (2001)[2] Adamatzky, A. (Ed.):
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