Classification of Complex Systems Based on Transients
CClassification of Complex Systems Based on Transients
Barbora Hudcova , and Tomas Mikolov Charles University, Prague Czech Institute of Informatics, Robotics and Cybernetics, CTU, Prague
Abstract
In order to develop systems capable of modeling artificial life,we need to identify, which systems can produce complex be-havior. We present a novel classification method applicable toany class of deterministic discrete space and time dynamicalsystems. The method distinguishes between different asymp-totic behaviors of a systems average computation time be-fore entering a loop. When applied to elementary cellular au-tomata, we obtain classification results, which correlate verywell with Wolfram’s manual classification. Further, we useit to classify 2D cellular automata to show that our techniquecan easily be applied to more complex models of computa-tion. We believe this classification method can help to de-velop systems, in which complex structures emerge.
Introduction
There are many approaches to searching for systems capableof open-ended evolution. One option is to carefully design amodel and observe its dynamics. Iconic examples were de-signed by Ray (1991), Ofria and Wilke (2004), or Soros andStanley (2014). However, as we lack any formal definitionof open-endedness or complexity, there is no formal methodof proving the system is indeed ”interesting”. Conversely,lacking definitions of such key terms, it seems extremelydifficult to design such systems systematically.Approaching the problem of searching for open-endedness bottom up, we define a classification of determin-istic discrete dynamical systems based on their asymptoticcomputation time with increasing space size. This methodgives a surprisingly clear classification of the toy class of el-ementary cellular automata, which seems to correspond wellto Wolfram’s established yet informal four types of cellularautomata dynamics. Subsequently, we use the classificationto discover two-dimensional automata with emergent behav-ior. This demonstrates that the transient classification can beused to navigate us towards regions of interesting systems.Even though we are far from giving sufficient conditions forcomplexity, we hope this method helps us understand, whichformally defined properties correlate with it.
Introducing Cellular Automata
Informally, a cellular automaton (CA) can be perceived asa k -dimensional grid consisting of identical finite state au-tomata. They are all updated synchronously in discrete timesteps based on an identical update function depending onlyon the states of automata in their local neighborhood. A for-mal definition can be found in Kari (2005).CAs were first studied as models of self replicating struc-tures (Neumann and Burks (1966), Langton (1984), Reggiaet al. (1993)). Subsequently, they were examined as dynam-ical systems (Hedlund (1969), Vichniac (1984), Gutowitzet al. (1987), Crutchfield and Young (1989)), or as mod-els of computation (Toffoli (1977), Mitchell (1998)). Beingso simple to simulate, yet capable of complex behavior andemergent phenomena (Crutchfield and Hanson (1993), Han-son (2009)), CAs provide a convenient tool to examine thekey, yet undefined notions of complexity and emergence. Basic Notions
We study the simple class of elementary cellular automata (ECAs), which are one-dimensional CAs with two states { , } and neighborhood of size 3. We examine the caseof a finite cyclic grid of size n ∈ N and denote each ECA bythe tuple ( { , } n , F ) where F : { , } n → { , } n is theglobal update rule.We identify each local rule f determining an ECA withthe Wolfram number of the ECA defined as: f (0 , , f (0 , , f (0 , , . . . +2 f (1 , , . We will refer to each ECA as a ”rule k ” where k is the cor-responding Wolfram number of its underlying local rule.Given an ECA ( { , } n , F ) we define the trajectory of aconfiguration u ∈ { , } n as the sequence ( u, F ( u ) , F ( u ) , . . . ) . The space-time diagram of such simulation is obtained byplotting the configurations as a horizontal rows of black andwhite squares (corresponding to states 1 and 0) with verticalaxis determining the time, which is progressing downwards. a r X i v : . [ n li n . C G ] A ug s the set of all grid configurations { , } n is finite, ev-ery trajectory ( u , u = F ( u ) , u = F ( u ) , . . . ) even-tually becomes periodic. We call the preperiod of this se-quence the transient of initial configuration u and denote itslength by t u . More formally, we define t u to be the small-est positive integer i , for which there exist j ∈ N , j > i ,such that F i ( u ) = F j ( u ) . The periodic part of the se-quence is called an attractor . The phase space of an ECA ( { , } n , F ) is a graph with vertices V = { , } n and edges E = { ( u, F ( u )) , u ∈ { , } n } . Such a graph is composedof components each containing one attractor and multipletransient paths leading to the attractor. The phase spacecompletely characterizes the dynamics of the system, it ishowever infeasible to describe it for large n .We note that properties of CA phase spaces were exam-ined among others by Wuensche and Lesser (2001). Pre-cisely for this purpose, a software was designed by Wuen-sche (2016). Cellular Automata Classifications
Even though there are many interesting definitions of com-plexity (Chaitin (1966), Bennett (1988), McShea (1996)),none of them seems to be perfectly suitable for studyingcomplex systems. A crucial result helping us understand thenotion of complexity in the context of dynamical systemswould be a suitable classification of CAs, which would nav-igate us toward a region of CAs with complex behavior. Anideal classification would be based on a rigorously definedand easily measurable property.In this section we describe three qualitatively differentclassifications of ECAs and subsequently, we will compareour results to them.
Wolfram’s Classification
The most intuitive and simple approach to examining the dy-namics of CAs is to observe their space-time diagrams. Thismethod was particularly proclaimed by Wolfram (2002).Therein, he established an informal classification of CA dy-namics based on such diagrams. He distinguishes the fol-lowing classes, which are shown in Figure 1.Class 1 . . . quickly resolves to a homogenous stateClass 2 . . . exhibits simple periodic behaviorClass 3 . . . exhibits chaotic or random behaviorClass 4 . . . produces localized structures that interactwith each other in complicated waysThe main issue is that we have no formal method of classi-fying CAs in this way. Moreover, the behavior of some CAscan vary with different initial configurations. An examplebeing rule 126, which oscillates between Class 2 and Class3 behavior, as shown in Figure 2. The transient classificationwe present in this paper deals with both these issues.
Figure 1: Space-time diagrams of rules from each Wolfram’sclass. Class 1 rule 32 is on top left, Class 2 rule 108 on topright. Both are simulated for 40 time steps on a grid of size50. At the bottom row we have Class 4 rule 110 on the leftand Class 3 rule 30 on the right. The two are simulated for200 steps on a grid of size 250.
Figure 2: On the left, rule 126 is simulated with an initialcondition consisting of a single 1 bit padded with 0’s. Onthe right, the same rule is simulated with a random initialconfiguration.
Zenil’s Classification
Zenil (2009) studied the compression size of the space-timediagrams of each ECA simulated for a fixed amount ofsteps. Using a clustering technique, he obtained two classesroughly distinguishing between Wolfram’s simple classes 1and 2 and complex classes 3 and 4. We show our reproduc-tion of Zenil’s results in Figure 3.His method nicely formalizes Wolfram’s observations ofthe space-time diagrams. However, the results depend on thechoice of initial conditions as well as the grid size, data rep-resentation, and the compression algorithm. We conductedmultiple experiments presented in Figure 4, which show thatZenil’s results are very sensitive to the choice of such param-eters.In vast CA spaces, where it is not feasible to examine ev-ery CA and mark it into one of Wolfram’s classes by hand,it would not be clear how the parameter values should be
50 100 150 200 250ECA Rules0200040006000800010000 C o m p r e ss e d s i z e i n b y t e s Figure 3: Reproduction of Zenil’s results (Zenil (2009)).The purple cluster corresponds to the interesting Class 3 and4 rules, the yellow cluster to the rest. C o m p r e ss e d s i z e i n b y t e s C o m p r e ss e d s i z e i n b y t e s Figure 4: Graphs representing the results of Zenil’s methodwhen different parameter values were used. They demon-strate how sensitive the results are. On the left, the ECAswere simulated for longer time, which caused complex rules110, 124, 137, and 193 to no longer belong to the ”inter-esting” purple cluster. On the right, the ECAs were sim-ulated from a fixed, randomly chosen initial condition. Insuch case, we obtain entirely different clusters.chosen. Moreover, the data representation causes the exten-sion of this method to more general dynamical systems to beproblematic, as for example using gzip to compress space-time diagrams of a 2D cellular automaton is suboptimal.
Wuensche’s Z-parameter
In Wuensche and Lesser (2001), Wuensche chose an in-teresting approach by studying the ECA’s behaviour whenreversing the simulations and computing the preimages ofeach configuration. He introduces the Z-parameter, whichrepresents the probability that a partial preimage can beuniquely prolonged by one symbol and suggests that Class4 CAs typically occur at Z ≈ . . However, no clear clas-sification is formed. The crucial advantage is that the Z-parameter depends only on the CA’s local rule and can becomputed effectively. It is, however, questionable whetherstudying the local rule only could describe overall dynamicsof a system sufficiently well. Classification Based on Transients
The classification we present is based on the asymptoticgrowth of the average transient length with increasing gridsize. We will refer to it as the Transient Classification. For a given CA and a grid size n , we randomly sample initialconfigurations u ∈ { , } n and estimate the average tran-sient length µ n = n (cid:80) u ∈{ , } n t u . Using regression, weestimate the asymptotic growth of the sequence ( µ n ) ∞ n =1 .Below, we motivate the study of this property.In non-classical models of compuation (Stepney (2012)),the process of traversing CA’s transients can be perceived asthe process of self-organization, in which information canbe aggregated in an irreversible manner. The attractors arethen viewed as memory storage units, from which the infor-mation about the output can be extracted. This is exploredin Kaneko (1985). Measuring the average transient growththen corresponds to the average computation time of the CA.CAs with bounded transient lengths can only perform trivialcomputation. On the other hand, CAs with exponential tran-sient growth can be interpreted as inefficient computationmodels.In the context of artificial evolution, we can view the lo-cal rule of a CA as the physical rule of the system whereasthe initial configuration as the particular ”setting of the uni-verse”, which is then subject to evolution. If we are inter-ested in finding CAs capable of complex behavior automati-cally, it would be beneficial for us if such behavior occurredon average, rather than having to select the initial configura-tions carefully from some narrow region. The probability tofind such special initial configurations would be extremelylow as the overall number of configurations grows exponen-tially with increasing grid size. This motivates our study ofthe growth of average transient lengths rather than the max-imum transient lengths.We note that transients of CAs have been examined, as inWuensche and Lesser (2001) or Saclay and Gutowitz (1994).However, we are not aware of an attempt to compare theasymptotic growth of transients for different ECAs. Transient Classification of ECAs
We consider all 256 ECAs up to equivalence classes ob-tained by changing the role of ”left” and ”right” neighbor,the role of 0 and 1 state, or both. It can be easily shownthat automata in the same equivalence class have isomor-phic phase spaces for any grid size. Thus, they performthe same computation. This yields 88 effectively differentECAs, each being a representative with the minimum Wol-fram number from its corresponding equivalence class. Inthis section, we present the classification of the 88 uniqueECAs based on their asymptotic transient growth. First, wedescribe the details of the classification process.
Data Sampling and Regression Fits
Suppose we have an ECA operating on a large grid of size n . In such case, computing the average transient length µ n is infeasible. Therefore, we randomly sample initial con-figurations u , u , . . . , u m and estimate µ n by m (cid:80) mi =1 t u i .t remains to estimate the number of samples m so that theerror | m (cid:80) mi =1 t u i − µ n | is reasonably small.More formally, we fix n ∈ N and let ( C n , P n ) be a dis-crete probability space where C n = { , } n is the set of all n -bit configurations and P n is a uniform distribution. Let X : C n → N be a random variable, which sends each u to its transient length t u . This gives rise to a probabilitydistribution of transient lengths on N with mean E ( X ) andvariance var ( X ) . It can be easily shown that E ( X ) = µ n .Our goal is to obtain a good estimate of E ( X ) by the MonteCarlo method (Owen (2013)).Let ( X , X , . . . , X m ) be a random sample of iid randomvariables, X i d = X for all i . Let µ ( m ) n = m (cid:80) mi =1 X i bethe sample mean and σ ( m ) n = (cid:113) m − (cid:80) mi =1 ( X i − µ ( m ) n ) the sample standard deviation. As var ( X ) < ∞ , we haveby the Central limit theorem the convergence to a normaldistribution, and the interval (cid:16) µ ( m ) n − u − α σ ( m ) n √ m , µ ( m ) n + u − α σ ( m ) n √ m (cid:17) where u β is the β quantile of the normalized normal dis-tribution, covers µ n for m large with probability approxi-mately − α . We will take α = 0 . . Hence, with proba-bility approximately | µ n − µ ( m ) n | < u . σ ( m ) n √ m . From the nature of our data, both the values E ( x ) = µ n and var ( X ) tend to grow with increasing grid size. There-fore, to employ a general method of estimating the numberof samples, we normalize the error by the sample mean andconsider | µ n − µ ( m ) n | µ ( m ) n . Therefore for m sufficiently large suchthat u . σ ( m ) n √ mµ ( m ) n < (cid:15) (1)we have that µ ( m ) n differs from µ n by at most (cid:15) · withprobability approximately .In practice, we put (cid:15) = 0 . and produce the observationsin batches of size 20 until condition (1) is met. For eachECA we obtained a dataset of the form (˜ µ n ) n max n = n min where ˜ µ n is the estimate of the average transient length on the gridof size n . We typically put n min = 20 and n max = 200 .We examined different regression fits of the dataset to es-timate the asymptotic growth of ˜ µ n . This included estimat-ing the fit to constant, logarithmic, linear, polynomial, andexponential functions. We picked the best fit with respect tothe R score. Surprisingly, we found a very good fit with R > for most ECAs. Results
We obtained a surprisingly clear classification of all the 88unique ECAs with four major classes corresponding to thebounded, logarithmic, linear, and exponential growth of av-erage transients. Below, we give a more detailed descriptionof each class.
Bounded Class: . ). The average tran-sient lengths were bounded by a constant independent of thegrid size. This suggests that the long term dynamics of suchautomata can be predicted efficiently.
50 100 150 200grid size1.751.801.851.901.952.00 a v e r a g e t r a n s i e n t l e n g t h Rule 36
Figure 5: Bounded Class rule 36. The average transient plotis on the left, the space-time diagram on the right.
Log Class: . ). The largest ECA classexhibits logarithmic average transient growth. The event oftwo cells at large distance ”communicating” is improbablefor this class.
50 100 150 200grid size3.54.04.55.05.56.06.57.0 a v e r a g e t r a n s i e n t l e n g t h Rule 28
Figure 6: Log Class rule 28. The average transient plot is onthe left, the space-time diagram on the right.
Lin Class: . ). On average, informationcan be aggregated from cells at arbitrary distance. Thisclass contains automata whose space-time diagrams resem-ble some sort of computation. This is supported by the factthat this class contains two rules known to have a nontrivialcomputational capacity: rule 184, which computes the ma-jority of black and white cells, and rule 110, which is theonly ECA proven to be Turing complete (Cook (2004)).e note that rules in this class are not necessarily complexas the interesting behavior seems to correlate with the slopeof the linear growth. Most of the Class Lin rules had onlya very gradual incline. In fact, the only two rules with suchslope greater than 1, rules 110 and 62, seem to be the oneswith the most interesting space-time diagrams.
50 100 150 200grid size50100150200 a v e r a g e t r a n s i e n t l e n g t h Rule 62
Figure 7: Lin Class rule 62. The average transient plot is onthe left, the space-time diagram on the right.We are aware that average transients of rules in Lin Classmight turn out to grow logarithmically or exponentiallygiven enough data samples. This could explain why the be-havior of ECAs in Lin Class depends on the slope of the tran-sient growth. More formally, the class could be interpretedas consisting of rules, which might have a logarithmic or ex-ponential growth, but this could not be decided given only alimited amount of data points. However, given such limiteddata, the best fit for such rules is to a linear function.
Exp Class: . ). This class has a strikingcorrespondence to automata with chaotic behavior. Visu-ally, there seem to be no persistent patterns in the configura-tions. Not only the transients, but also the attractor lengthsare significantly larger than for other rules. This class con-tains rules 45, 30 and 106 whose transients grow the fastest,as well as rules 54, 73, and 22. a v e r a g e t r a n s i e n t l e n g t h Rule 45
Figure 8: Exp Class rule 45. The average transient plot is onthe left, the space-time diagram on the right.
Affine Class: . ). This class contains rules60, 90, 105, and 150 whose local rules are affine boolean functions. Such automata can be studied algebraically andpredicted efficiently. It was shown in Martin et al. (1984)that the transient lengths of rule 90 depend on the largestpower of 2, which divides the grid size. Therefore, the mea-sured data did not fit any of the functions above but formeda rather specific pattern.
20 25 30 35 40 45grid size0.02.55.07.510.012.515.0 a v e r a g e t r a n s i e n t l e n g t h Rule 90
Figure 9: Affine Class rule 90. The average transient plot ison the left, the space-time diagram on the right.
Fractal Class: . ). This class containsrules 18, 122, 126, and 146, which are sensitive to initialconditions. Their evolution either produces a fractal struc-ture resembling a Sierpinski triangle or a space-time dia-gram with no apparent structures. We could say such rulesoscillate between easily predictable behavior and chaotic be-havior. Their average transients and periods grow quite fast,which makes it difficult to gather data for larger grid sizes.
20 30 40 50 60 70grid size0100200300400500 a v e r a g e t r a n s i e n t l e n g t h Rule 126
Figure 10: Fractal Class rule 126. The average transient plotis on the left, the space-time diagram on the right.
Discussion
We note that we have also tried to measure the asymptoticgrowth of the average attractor size a u , u ∈ { , } n as wellas the average rho value defined as ρ u = t u + a u . Thishowever produced data points, which could not be fitted tosimple functions well. This is due to the fact that many au-tomata have attractors consisting of a configuration, whichis shifted by one bit to the left, resp. right, at every timestep. The size of such an attractor then depends on the great-est common divisor of the size of the period of the attractornd the grid size, and this causes such oscillations. We con-clude that such phase-space properties are not suitable forthis classification method.Below, we compare our results to other classifications de-scribed earlier. Exhaustive comparison for each ECA is pre-sented in Table 1. Classification ComparisonECA Transient Wolfram Zenil Wuensche
Classification ComparisonECA Transient Wolfram Zenil Wuensche
56 log 2 1 or 2 0.7557 lin 2 1 or 2 0.7558 log 2 1 or 2 0.7560 affine 2 1 or 2 162 lin 2 1 or 2 0.7572 bounded 1 1 or 2 0.573 exp 3/4 3 0.7574 log 2 1 or 2 0.7576 bounded 2 1 or 2 0.62577 log 2 1 or 2 0.578 log 2 1 or 2 0.7590 affine 2 1 or 2 194 log 2 1 or 2 0.75104 log 1 1 or 2 0.75105 affine 2 1 or 2 1106 exp 3 1 or 2 1108 bounded 1 1 or 2 0.75110 lin 4 4 0.75122 fractal 2/3 1 or 2 0.75126 fractal 2/3 1 or 2 0.5128 log 1 1 or 2 0.25130 log 2 1 or 2 0.5132 log 2 1 or 2 0.5134 log 2 1 or 2 0.75136 log 1 1 or 2 0.5138 bounded 2 1 or 2 0.75140 log 2 1 or 2 0.625142 lin 2 1 or 2 0.5146 fractal 2/3 1 or 2 0.75150 affine 2 1 or 2 1152 log 2 1 or 2 0.75154 bounded 2/3 1 or 2 1156 log 2 1 or 2 0.75160 log 1 1 or 2 0.5162 log 2 1 or 2 0.75164 log 2 1 or 2 0.75168 log 1 1 or 2 0.75170 bounded 2 1 or 2 1172 log 2 1 or 2 0.75178 log 2 1 or 2 0.5184 lin 2 1 or 2 0.5200 bounded 1 1 or 2 0.625204 bounded 2 1 or 2 1232 log 1 1 or 2 0.5Table 1: Comparing classifications of the 88 unique ECAs.
Wolfram’s Classification - Discussion
The significanceof our results stems precisely from the fact that the TransientClassification corresponds to Wolfram’s so well. As it is notclear for many rules, which Wolfram class they belong to,the main advantage is that we provide a formal criterion,pon which this could be decided.In particular, rules in Classes Bounded and Log corre-spond to rules in either Class 1 or 2. Class Exp correspondsto the chaotic Class 3 and Class Lin contains Class 4 to-gether with some Class 2 rules. We mention an interestingdiscrepancy: rule 54, which is possibly considered by Wol-fram to be Turing complete, belongs to the Class Exp. Thismight suggest that computations performed by this rule canbe on average quite inefficient.
Zenil’s Classification - Discussion
Zenil’s Classificationof ECAs offers a great formalization of Wolfram’s andseems to roughly correspond to it. Compared to the Tran-sient Classification, it is however less fine grained. More-over, it contains some arbitrary parameters such as the datarepresentation and compression algorithm used. In addition,it uses a clustering technique, which requires data of multi-ple automata to be mutually compared in order to give riseto different classes. In contrast, the Transient Class can bedetermined for a single automaton without any context.
Wuensche’s Z-parameter - Discussion
Wuensche sug-gests that complex behavior occurs around Z = 0 . , whichagrees with the fact that Lin Class rules with steep slope(rule 110, 62, and 25) have precisely this Z value. How-ever, the Z = 0 . is in fact quite frequent. This suggeststhat thanks to its simplicity, the Z parameter can be usedto narrow down a vast space of CA rules when searchingfor complexity. However, more refined methods have to besubsequently applied to find concrete CAs with interestingbehavior. Transient Classification of 2D CAs
So far we have examined the toy model of ECAs. The trueusefulness of the classification would stem from its applica-tion to more complex CAs where it could be used to discoverautomata with interesting behavior.We therefore applied it on a subset of two-dimensionalCAs with a × neighborhood and 3 states to see whether2D automata would still exhibit such clear transient growths.We consider the 2D CAs to operate on a finite square gridof size n × n . We consider the topology of the grid to be thatof a torus in order for each cell to have a uniform neighbor-hood. In such scenario, the definition of transients is analo-gous to the one-dimensional case.To reduce the vast automaton space, we only consideredsuch automata whose local rules are invariant to all the sym-metries of a square. As there are still such symmetrical2D CAs, we randomly sampled 10 000 of them.We estimated the average transient length analogously tothe 1D case and measured the asymptotic growth with re-spect to n - the size of the side of the square grid. This is motivated by the fact that in a n × n grid the greatest dis-tance between two cells depends linearly on n rather thanquadratically.We were able to classify . of 10 000 sampled au-tomata with a time bound of 40 seconds for the computa-tion of one transient length value on a single CPU. We es-timate that most CAs are unclassified due to such compu-tation resources restriction or due to rather strict conditionswe imposed on a good regression fit. We obtained the samemajor classes - the Log, Lin, and Exp Class. However, inthis case, the Exp Class seems to dominate the rule space.Another interesting difference is that a new class was ob-served - the polynomial class - which contains rules whosetransients grow approximately quadratically. Moreover, ourresults suggest that the occurence of bounded class CAs in2D is much scarcer as we found no such CAs in our sample.Classification of 2D 3-state CAs (10 000 samples)Transient Class Percentage of CAsBounded Class 0%Log Class 18.21%Lin Class 1.17%Poly Class 1.03%Exp Class 72.62%Unclassified 6.97%Table 2: Classification of 10 000 randomly sampled sym-metric 2D 3-state CAs.We observed the space-time diagrams of randomly sam-pled automata from each class to infer its typical behavior.On average, the Log Class automata quickly enter attrac-tors of small size. Lin Class exhibit emergence of variouslocal structures. For automata with more gradual incline,such structures seem to die out quite fast. However, au-tomata with steeper slopes exhibit complex interactions ofsuch structures. The Poly class automata with steep slopeseem to produce spatially separated regions of chaotic be-havior against a static background. In the case of more grad-ual slopes, some local structures emerge. Finally, the ExpClass seems to be evolving chaotically with no apparent lo-cal structures. We present various examples of CA evolutiondynamics in the form of GIF animations here .These observations suggest that the region of Lin Classwith steep slope and Poly class with more gradual inclineseems to contain a non-trivial ratio of automata with com-plex behavior. In this sense, the Transient Classification canassist us to automatically search for complex automata simi-larly to the method designed by Cisneros et al. (2019), whereinteresting novel automata were discovered by measuringgrowth of structured complexity using a data compressionapproach. http://bit.ly/transient_classification ransients Classification of Other Well KnownCAs We were interested whether some well-known complex au-tomata from larger CA spaces would conform to the tran-sient classification as well. Surprisingly, the result is posi-tive.
Game of Life
As the left plot in Figure 11 suggests, theTuring complete Game of Life (Gardener (1970)) seems tofit the Lin Class. This is confirmed by the linear regressionfit with R ≈ . .
20 40 60 80 100grid size0.00.51.01.52.0 a v e r a g e t r a n s i e n t l e n g t h Game of Life
Figure 11: Game of Life. The average transient growth plotis on the left. On the right, we show a space-time diagram attime t = 200 started from a random initial configuration. Genetically Evolved Majority CA
Mitchell et al. (2000)studied how genetic algorithms can evolve CAs capable ofglobal coordination. The authors were able to find a 1D CAdenoted as φ par with two states and radius r = 3 , whichis quite successful at computing the majority task with theoutput required to be of the form of a homogenous state ofeither all 0’s or all 1’s.
50 100 150 200grid size20406080100120 a v e r a g e t r a n s i e n t l e n g t h CA par Figure 12: Cellular automaton φ par . The average transientgrowth plot is on the left. On the right, we show a space-timediagram simulated from a random initial configuration.This CA seems to belong to the Lin Class, which is con-firmed by the linear regression fit with R ≈ . . Totalistic 1D 3-state CA
A totalistic CA is any CA whoselocal rule depends only on the number of cells in each stateand not on their particular position. Wolfram studied variousCA classes, one of them being the totalistic 1D CAs withradius r = 1 and 3 states S = { , , } .Wolfram (2002) presents a list of possibly complex CAsfrom this class. We applied the Transient Classification tosuch CAs and learned that most of them were classified aslogarithmic. This agrees with our space-time diagram obser-vations that the local structures in such CAs ”die out” quitequickly. Nonetheless, some of the CAs were classified aslinear. An example of such a CA is in Figure 13 where thelinear regression fit has R ≈ . . a v e r a g e t r a n s i e n t l e n g t h Figure 13: Totalistic cellular automaton with code .The average transient growth plot is on the left. On the right,we show a space-time diagram of the evolution from a ran-dom initial configuration.
Conclusion
We presented a classification method based on the asymp-totic growth of average computation time, which is applica-ble to any deterministic discrete space and time dynamicalsystem. We did present a good correspondence between theTransient and Wolfram’s classification in the case of ECAs.We also did show that the transient classification works intwo dimensions and used it to discover CAs capable of emer-gent phenomena. By demonstrating that famous CAs suchas Game of Life or rule 110 belong to the Lin Class, we be-lieve that the linear transient growth navigates us toward aregion of complex and interesting CAs.Our future work includes publishing an open-source li-brary with the classification techniques. We also plan tocompare the classification results for 2D CAs with variousneighborhoods and number of states to compare, which oneshave the largest ratio of the Lin and Poly Class automata. Wewould also like to discover, which other types of discrete dy-namical systems can this method be applied to.
Acknowledgements
We would like to thank Jiri Tuma for all his help and supportas well as Jaromir Antoch, Ondrej Tybl, and Hugo Cisnerosfor numerous inspiring discussions. eferences
Bennett, C. H. (1988). Logical depth and physical complexity.
TheUniversal Turing Machine – a Half-Century Survey , pages227–257.Chaitin, G. J. (1966). On the length of programs for computingfinite binary sequences.
J. ACM , 13(4):547569.Cisneros, H., Sivic, J., and Mikolov, T. (2019). Evolving structuresin complex systems.
Proceedings of the 2019 IEEE Sympo-sium Series on Computational Intelligence , pages 230–238.Cook, M. (2004). Universality in elementary cellular automata.
Complex Systems , 15.Crutchfield, J. and Young, K. (1989). Inferring statistical complex-ity.
Physical review letters , 63:105–108.Crutchfield, J. P. and Hanson, J. E. (1993). Turbulent pattern basesfor cellular automata.
Physica D: Nonlinear Phenomena ,69(3):279 – 301.Gardener, M. (1970). The fantastic combinations of John Conwaysnew solitaire game life by Martin Gardner.
Scientific Ameri-can , 223:120–123.Gutowitz, H. A., Victor, J. D., and Knight, B. W. (1987). Localstructure theory for cellular automata.
Physica D: NonlinearPhenomena , 28(1):18 – 48.Hanson, J. E. (2009). Emergent Phenomena in Cellular Automata.
Meyers R. (eds) Encyclopedia of Complexity and Systems Sci-ence .Hedlund, G. A. (1969). Endomorphisms and automorphisms ofthe shift dynamical system.
Mathematical systems theory ,3:320–375.Kaneko, K. (1985). Complexity in basin structures and informationprocessing by the transition among attractors.
Theory andApplications of Cellular Automata , pages 367–399.Kari, J. (2005). Theory of cellular automata: A survey.
TheoreticalComputer Science , 334(1):3 – 33.Langton, C. G. (1984). Self-reproduction in cellular automata.
Physica D: Nonlinear Phenomena , 10(1):135 – 144.Martin, O., Odlyzko, A., and Wolfram, S. (1984). Algebraic prop-erties of cellular automata.
Communications in MathematicalPhysics , 93.McShea, D. (1996). Perspective: Metazoan complexity and evolu-tion: Is there a trend?
Evolution , 50.Mitchell, M. (1998). Computation in cellular automata: A selectedreview.
NonStandard Computation , pages 95–140.Mitchell, M., Crutchfield, J., and Das, R. (2000). Evolving cel-lular automata with genetic algorithms: A review of recentwork.
First Int. Conf. on Evolutionary Computation and ItsApplications , 1.Neumann, J. V. and Burks, A. W. (1966).
Theory of Self-Reproducing Automata . University of Illinois Press, Urbana,USA. Ofria, C. and Wilke, C. (2004). Avida: A software platform for re-search in computational evolutionary biology.
Artificial Life ,10(2):191–229.Owen, A. B. (2013).
Monte Carlo theory, methods and examples .Ray, T. S. (1991). An approach to the synthesis of life.
ArtificialLife II, Santa Fe Institute Studies in the Sciences of Complex-ity , XI:371408.Reggia, J., Armentrout, S., Chou, H., and Peng, Y. (1993). Simplesystems that exhibit self-directed replication.
Science (NewYork, N.Y.) , 259:1282–7.Saclay, C. and Gutowitz, H. (1994). Transients, cycles, and com-plexity in cellular automata.
Physical Review A , 44.Soros, L. and Stanley, K. (2014). Identifying necessary conditionsfor open-ended evolution through the artificial life world ofchromaria.
Artificial Life Conference Proceedings , (26):793–800.Stepney, S. (2012).
Nonclassical Computation — A DynamicalSystems Perspective . Springer Berlin Heidelberg, Berlin, Hei-delberg.Toffoli, T. (1977). Computation and construction universality ofreversible cellular automata.
Journal of Computer and SystemSciences , 15(2):213 – 231.Vichniac, G. Y. (1984). Simulating physics with cellular automata.
Physica D: Nonlinear Phenomena , 10(1):96 – 116.Wolfram, S. (2002).
A New Kind of Science . Wolfram Media,Champaign, USA.Wuensche, A. (2016).
Exploring discrete dynamics - Second Edi-tion. The DDLab manual . Luniver Press.Wuensche, A. and Lesser, M. (2001). The global dynamics ofcelullar automata: An atlas of basin of attraction fields ofone-dimensional cellular automata.
J. Artificial Societies andSocial Simulation , 4.Zenil, H. (2009). Compression-based investigation of the dynam-ical properties of cellular automata and other systems.