synchronous and asynchronous updating in cellular automata

BioSystems 51 (1999) 123–143

Synchronous and asynchronous updating in cellularautomata

Birgitt Schonfisch a,*, Andre de Roos b

a Biomathematik, Uni6ersitat Tubingen, Auf der Morgenstelle 10, 72076 Tubingen, Germanyb Institute for Systematics and Population Biology, Uni6ersity of Amsterdam, Kruislaan 320,

NL-1098 SM Amsterdam, The Netherlands

Received 6 October 1998; accepted 12 April 1999

Abstract

We analyze the properties of a synchronous and of various asynchronous methods to iterate cellular automata.Asynchronous methods in which the time variable is not explicitly defined, operate by specifying an updating orderof the cells. The statistical properties of this order have significant consequences for the dynamics and the patternsgenerated by the cellular automata. Stronger correlations between consecutive steps in the updating order result inmore, artificial structure in the patterns. Among these step–driven methods, using random choice with replacementto pick the next cell for updating, yields results that are least influenced by the updating method. We also analyse atime–driven method in which the state transitions of single cells are governed by a probability per unit time thatdetermines an exponential distribution of the waiting time until the next transition. The statistical properties of thismethod are completely independent of the size of the grid. Consecutive updating steps therefore show no correlationat all. The stationary states of a cellular automaton do not depend on whether a synchronous or asynchronousupdating method is used. Their basins of attraction might, however, be vastly different under synchronous andasynchronous iteration. Cyclic dynamics occur only with synchronous updating. © 1999 Elsevier Science Ireland Ltd.All rights reserved.

Keywords: Cellular automata; Synchronous; Asynchronous

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1. Introduction

In recent years cellular automata have fre-quently been used to model the dynamics of spa-tially extended biological systems. Examplesinclude a wide range of topics, such as prebiotic

evolution (Boerlijst and Hogeweg, 1991), the de-velopment of pigment patterns in mollusks(Gunji, 1990), growth of clonal plants (Inghe,1989), and the dynamics of interacting prey andpredators or hosts and parasites (Hassell et al.,1991). Cellular automata consist of a collection (a‘grid’) of cells that each adopts one of a finitenumber of states (‘elementary states’). Single cellschange in state following a rule (the ‘local rule’)

* Corresponding author. Tel.: +49-7071-2976843; fax: +49-7071-294322.

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B. Schonfisch, A. de Roos / BioSystems 51 (1999) 123–143124

that depends on the environment of the cell (the‘neighborhood’). The environment of a cell isusually taken to be a small number of neighboringcells. The dynamics of a cellular automaton isgenerated by repeatedly applying the local rule toall the cells on the grid. This iteration can becarried out in a number of different ways that wewill refer to as ‘updating methods’. With theclassical, synchronous or parallel updatingmethod all cells are evaluated and change state atthe same time. With asynchronous or sequentialupdating the cells are evaluated one after another.Asynchronous updating requires the specificationof an order in which the cells are considered. Anumber of asynchronous updating methods aredistinguished by different algorithms to obtainsuch an evaluation order.

If we use cellular automata or any other dy-namical system to model real (biological) systemswe demand a kind of ‘model stability’. This termrefers to the view that there are at least threekinds of stability levels: First stability of a specificdynamical system (stable stationary points,…).Second structural stability in the sense of Morse–Smale vector fields: the qualitative behavior of themodel doesn’t change if the parameters are varied,at least in a certain range. Finally on a third level‘model stability’—the qualitative results of themodel depend only on the basic assumptionsderived from the real biological system. They donot depend on the actual mathematical model weuse, whether we use for example cellular automataor partial differential equations. Recently espe-cially the updating methods in cellular automatacame into focus of discussion. Some of the deter-ministic cellular automata (with synchronous up-date) show very interesting patterns but thesevanish if any asynchronous updating is used. Forexample, in Huberman and Glance (1993) a spa-tially extended, cellular automata version of theprisoner’s dilemma is discussed. For this variantof the prisoner’s dilemma it is demonstrated thatsynchronous updating generates complicated pat-terns of defecting and cooperating individuals onthe grid. These patterns allow the two types ofindividuals to coexist indefinitely. With asyn-chronous updating these patterns are completelyabsent and in the long run the cellular automaton

reaches a state with only defecting individuals.Synchronous and asynchronous updating in thiscase lead to differences in both transient and longterm dynamics. These findings are confirmed insome studies while others found no qualitativedifferences in the dynamical behaviour. Exem-plary are the results of Ingerson and Buvel (1984)for one-dimensional Wolfram automata (Wol-fram, 1983). Every one of these 32 automata wassimulated with synchronous update and two dif-ferent asynchronous updating methods. For ruleswith ‘simple’ behavior, i.e. where finally all cellsare in state 0 or a stationary configuration evolves(Wolfram class 1 or 2), in most cases the finalpatterns are similar with synchronous and asyn-chronous updating. On the other hand complexpatterns generated by some rules with syn-chronous updating (Wolfram class 3 or 4) mostlydegenerate with each of the asynchronous updat-ing methods. This agrees with the findings ofGunji (1990) who also investigated Wolfram class4 rules showing ‘chaotic patterns’ with very largeperiods with synchronous updating. The periodsbecame small with asynchronous updating andthe space-time patterns look very different. Fur-thermore he shows that varying the asynchronousmethod produces different patterns (which agreewell with real mollusk shell patterns). Bersini andDetours (1994) conclude that the crucial factorfor the different behaviour of two prominent cel-lular automata, namely Conway’s Life game andthe immune network model, is the synchronousrespectively asynchronous updating. Le Caer(1997) gets diverse results: He simulates syn-chronous and one type of asynchronous updatingin cellular automata with two different totalisticrules. With one rule the qualitative behaviorchanges with the updating method. With the otherrule the changes are not significant. Divers resultsare also obtained in systems which are not cellularautomata but closely related models. Consideringa simple stochastic model where pairs of cells areevaluated Ruxton and Saravia (1998) observe dif-ferences in the quantitative results and variousspatial measurements when synchronous or asyn-chronous updating is used. Lumer and Nicolis(1994) found that ‘markedly different dynamicsarise when the standard synchronous model is

B. Schonfisch, A. de Roos / BioSystems 51 (1999) 123–143 125

made asynchronous’ when studying coupled maplattices (also with analytic methods). Contrastingare the findings of Rajewsky et al. (1998) on theasymmetric exclusion process with different up-dating methods. In this model particles jump toempty neighboring cells with a certain probability.The authors find that although there are differ-ences in quantitative measurements, ‘the updatedoes not change the fundamental properties of themodel’.

So far mostly simulation studies have beenperformed, analytic results have been obtainedonly on special models. In some cases the updateseems to have an qualitative effect on the dynam-ical behaviour of the model, while in other casesno difference could be shown. In spite of theimportance of the updating algorithm in somemodels it is only described ambiguous, ‘each cellresponds… in turn’ (Kier and Cheng, 1994) or‘for the sequential updating… we go through thelattice in a regular fashion’ (Le Caer, 1997).

In this paper we compare the properties ofdifferent updating methods for general cellularautomata. After defining cellular automata wewill discuss synchronous and different variants ofasynchronous iteration in detail. Asynchronousmethods are divided in ‘step-driven’ updatingmethods (characterized by the absence of an ex-plicit time variable) or ‘time-driven’ methods. Ascomparison criteria we focus on the number ofchanges of a particular cell and the number ofchanges in its environment. The expectation andvariance of these statistics are calculated within atotal number of n updating steps, for step–drivenupdating methods. For time–driven updatingmethods the expectation and variance of thesestatistics per unit time are evaluated. These resultsapply to a broad class of cellular automata. Weillustrate our results with patterns from simplecellular automata.

On the basis of the results obtained, two asyn-chronous methods seem most appropriate for sim-ulating real (biological) processes. These are astep–driven method where for each step in thesimulation a random cell of the grid is selected forupdating, and a time–driven method where theupdating of each single cell is governed by anexponentially distributed waiting time. The prop-

erties and differences of these two methods arecompared in more detail. Finally we discuss limitsets for synchronous and asynchronous dynamics,i.e. we compare stationary states, cycles and at-traction basins of the limit sets.

1.1. Cellular automata

To clearly work out the effects of differentupdating methods we will focus mainly on deter-ministic cellular automata. To be precise, wemostly consider cellular automata with a strictlydeterministic local function. With asynchronousmethods the algorithm to choose the next cell toupdate may, however, include stochastic elements.

A cellular automaton is defined by a grid G ofcells, a finite set E of elementary states (possiblestates of the cells), a neighborhood U and localfunction f0 which at updating assigns a new ele-mentary state to a cell for any given configurationof the neighborhood and its own state. A basicproperty we will need is translation invariance.Therefore the grid has to be ‘regular’ like forexample G=Zd, the d-dimensional line, square,cubic,… grid. The neighborhood and the localfunction must be ‘the same’ for all cells, i.e. wecan define both by the neighborhood of zerorespectively the local function acting on the neigh-borhood of zero. Strict translation invariance im-plies also that the grid is either infinite or a finitegrid closed to a torus but in principle the resultscan be applied to finite grids with ‘boundary’ cellsin fixed elementary states with an appropriatehandling of these cells and their neighbors.

Let A= (G, E, U, f0) be a cellular automaton.The state z of the automaton is a mapping z :G�E, z : x�(x), in which an elementary state isassigned to every cell x�G.

Every local function f0 uniquely defines a globalmapping from the state space into itself. Thesynchronous updating method can be describedby f : z�f(z) with:

f(z)= f0(z �U(x)) for all x�G.

With asynchronous updating the global functiondepends on the cell x to be evaluated next. Thenfx : z�fx(z) is given by:


fx(z)=! z(y)

f0(z �U(x))for y"xfor y=x

Asynchronous cellular automata are not cellularautomata in the classical (von Neumann) sense.

The results in Section 2 apply to any cellularautomaton A= (G, E, U, f0) with the followingproperties: The grid is finite with � G �=n cellsand we have translation invariance with regard tothe neighborhood, i.e. the neighborhood (not theactual neighboring cells) is the same for every cell.We do not actually need a square grid and theresults apply to any dimension. They can even becarried over to cellular automata with stochasticlocal rules (when f is a random variable) since weonly look at the updating statistics, not at whathappens then. Only in one case in Section 2 weneed more restrictions which is then indicated. InSection 3 after general considerations we considerthe example of an pure death process. Finally inSection 4 we restrict to cellular automata withdeterministic local functions. There we need trans-lation invariance of the local function (whichimplies the local function and consequently theneighborhood is the same for every cell), but thegrid may be infinite.

2. Asynchronous updating methods

In this section let A= (G, E, U, f0) be ancellular automaton with G finite, � G �=n andtranslation invariant neighborhood.

The term asynchronons cellular automata isused in a number of quite different meanings. Westart with a definition of the subject of this paper.In deterministic cellular automata with syn-chronous updating for a given initial configura-tion there is one possible sequence of the states ofstate space, i.e. there is exactly one possible trajec-tory. Using asynchronous updating the trajectorystarting from a given initial configuration in gen-eral depends on the evaluation order of the cells.We will now discuss several ways to define suchan evaluation sequence. We distinguish step-driven methods and time-driven methods. In step-driven methods an algorithm determines the orderof evaluation of the cells. The update of a singlecell is called a ‘single step’ and the iteration gives

a series of single steps. In step-driven methods wehave only this sequence of single steps while timeis not explicitly defined. In the time-driven meth-ods an algorithm assigns an explicit point in timeto every cell at which it will be evaluated next.The updating order results from these time points.In general the number of updates in a time inter-val of a given length will be different.

To compare the different updating methods wewill discuss the following statistics:1. The expected value E(X) and variance V(X) of

the number of single steps between two up-dates of the same cell x.

2. The expected value E(Z) and variance V(Z) ofthe number of single steps between an updateof cell x and the next update of a given celly"x cell in the neighborhood U(x) of x.

The above statistics are used to characterizestep–driven methods. In time-driven methods thestatistics are expressed in terms of the number ofupdates per unit time as opposed to the numberper n consecutive steps. Similarly, in time–drivenmethods we use the time interval between twoupdates instead of the number of single stepsbetween two updates that is used in step–drivenmethods. To compare the two classes of methodswe will use the expected number of updates persingle cell to relate the number of steps in step-driven methods to a unit of time in time–drivenmethods. The calculations are given in the ap-pendix and the results are summarized in Table 1.

To avoid confusion, we want to stress that withasynchronous updating we do not mean a processas described by Adamatzky (1994). He considerscellular automata with synchronous update where‘a cell calculates its next state depending on neigh-borhood configuration u(x)t, and leaves it withj(u(x)t) time steps, independently of states ofneighbors in the interval [t+1, t+j(u(x)t]% or asecond model where ‘cells calculates next state intime t, but takes new value at t+d(u(x)t)¦. Theseautomata model a delay. These models are alsocalled ‘asynchronous’ since every cell follows itsown cycle but are different (except some veryspecial cases) from the automata considered here.There are several applications which also use theterm asynchronous for models which are differentfrom the ones we consider here. For example if


some rules are applied every second, third,…iteration only (Biesiada, 1986).

2.1. Step-dri6en methods

For all step-driven methods we found that theexpected value E(X) of single steps between twoupdates of a cell is n, the variance V(X) howeveris not the same. The statistics of Z, the number ofsingle steps between updates of x and y, varies.

2.1.1. Fixed directional or line-by-line sweepThe simplest asynchronous updating method is

to put the cells of the grid in a predefined, fixedorder to form a sequence. One run through thegrid, i.e. n= � G � single steps, namely the evalua-tions of the cells x1,…,xn, is then called a sweep.Such an algorithm has been compared with othersin Rajewsky et al. (1998), Ruxton and Saravia(1998). In the context of numerical mathematics itis appears with the Gauss–Seidel algorithm. On atwo-dimensional grid we will use the examplewhere the cells are updated line by line. Generallysuch a sweep introduces a lot of additional struc-ture into the automaton. The number of singlesteps between two updates of a cell always equalsE(X)=n with variance V(X)=0. The generalformula for the expected number E(Z) of stepsbetween an update of x and a given neighbory"x an the variance V(Z) is given in the ap-pendix. However it can only be given explicitly forspecial grids and neighborhoods. In the Table 1

we give the formula for the von Neumann neigh-borhood on a l× l square grid.

2.1.2. Fixed random sweepTo avoid the systematics of a line–by–line

sweep one may built the sequence by choosing thefirst cell randomly out of the n cells, the secondout of the remaining n−1,… Each remaining cellhas the same probability to be chosen in a draw,i.e. this is a choice according to the uniformdistribution without replacement (compared inRuxton and Saravia (1998)). If we use the samesequence for all iterations, this will not be qualita-tively different from a directional choice. Theline–by–line sweep appears as a special case. InKitagawa (1974) this fixed random sweep is intro-duced as ‘deterministic cycle replication’ in thecontext of cell spaces.

The expected value E(X)=n and varianceV(X)=0 of the number of steps between twoupdates of a cell are the same as for the line-by-line sweep. Also the expected number of updatesbetween x and a neighbor cell is the same for bothmethods. The variance V(Z) however is smallerthan with the line-by-line sweep—at least with thevon Neumann neighborhood. With line-by-linesweep neighbors are updated either after few (1 orn) or after many (n−1 or n−n) single steps.Both gives large deviations to E(Z). With fixedrandom sweep all numbers of single steps between1 and n−1 occur with equal probability andtherefore the variance V(Z) is smaller.

Table 1Statistics of the different updating methods for a grid of n cells and arbitrary neighborhood

(1) a (2) b

V(Z)E(X) E(Z)V(X)

Line-by-line n 1/2n c0 1

2

�1−n n+

1

2n2� c

n 0Random fixed sweep 1/2n 1/12(n−2)nn 1/6(n2−1)Random new sweep 1/12(23/12n2−13/6n−13/12)1/12(7n+1)nUniform choice n(n−1) n n(n−1)1 1 1Exponential waiting times 1

a (1) X=number of single steps (time interval for exponential waiting time method) between two updates of the same cell.b (2) Z=number of single steps (time interval) between an update of x and an update of a given y�U(x), y"x.

The calculations are given in the Appendix.c Restricted to the von Neumann neighborhood on an l×l=n square grid.


2.1.3. Random new sweepAs an alternative to fixed sweeps we may build

a sequence like before by choosing the cells ac-cording to an uniform distribution without re-placement and make one sweep through the grid.Then we build a new sequence in the same wayand make one sweep, etc. (in Kitagawa (1974)refereed as ‘randomized cycle replication’, appliedin Seybold et al. (1997)). The fixed sweep is aspecial case where by chance, always the samesequence occurs.

The expected number of single steps betweentwo updates of the same cell is still E(X)=n buthere the variance is non–zero. This results fromthe fact that the positions of a cell within differentsweeps are different. If a cell y is updated beforex in the current sweep it may take more or fewersingle steps than with a fixed sweep until is up-dated in the following sweep. Therefore the vari-ance V(Z) is larger than with fixed sweeps. Sincethere are more cases where more single steps areneeded the expected value of single steps betweenupdates of x and a neighbor cell y"x, the ex-pected value E(Z) is also larger than with fixedsweep.

2.1.4. Uniform choiceThe asynchronous method near at hand is to

choose the cell to be evaluated randomly withuniform distribution on the grid, i.e. a randomchoice with replacement (Ingerson and Buvel,1984; Rajewsky et al., 1998). It can only be ap-plied on finite grids. This method is also used withMonte-Carlo Simulations on grids. Here, there isno sweep, if we make n= � G � asynchronous sin-gle steps then some cells may be evaluated twiceor more and some cells may have not been evalu-ated at all. To be exact, the probability that a cellis chosen k times in 6 updates is binomially dis-tributed with B6,1/n.

The expected number of single steps betweentwo updates of a cell again equals E(X)=n withvariance V(X)=n(n−1). The fact that the vari-ance V(X) is larger when using the uniform choicemethod than with the random new sweep showsthat if every cell is updated exactly once in everysweep, although not at the same position in thesweep, this leads to a lot more structure than with

a totally random choice. The distribution of Z isthe same as of X since here there is no differencewhether x is chosen again or y"x is chosen. Thismethod of choosing the next cell has no memorywhereas in the former methods the probability tochoose a certain cell depends on the cells alreadyevaluated. The expected value E(Z) is larger thanwith random new sweep but the variance V(Z) issmaller.

2.2. Time dri6en method

2.2.1. Exponential waiting timesThe methods presented above are motivated

heuristically. We proceed with the algorithm mostsatisfying from a theoretical point of view. Theidea is to simulate a process in continuous time(for example Durrett (1995), chapter 2). The up-dating of a cell is governed by a process indepen-dent of the other cells. Every cell has its own‘clock’ which rings when the cell is to be updated.The waiting times of the clock are exponentiallydistributed (with mean 1), i.e. the probability thatan event occurs at time t follows e− t. t is now areal number, t�R, tE0. To realize this poissonprocess we choose a number txi

according to theexponential distribution with mean one, for everycell of the grid. We evaluate the cell correspond-ing to the smallest of these numbers and choose anew t( xi

and set txi= txi

+ t( xi—this gives the next

time this cell is to be evaluated. Then we againlook for the smallest txi

and evaluate the corre-sponding cell.

This method can also be used for stochasticautomata. Suppose cells change from zero to onewith rate a. Then, the waiting times of the cellclocks are exponentially distributed (with meana), i.e. the probability that an event occurs at timet follows e−at. Such a process is usually called an(interacting) particle system (Durrett, 1995).

To compare this time driven method with thestep-driven we identify n single steps with one unitof time. The expected time between two updatesof the same cell is E(X)=1 with variance V(X)=1 both corresponding to n single steps. The distri-bution of Z, i.e. the time between an update of xand y� (x), y"x is again the same as of X.


2.3. Example patterns

We will illustrate the effects of different updat-ing methods with the two following cellular au-tomata. Let G be the two-dimensional squaregrid. The neighborhood is the von Neumannneighborhood with the five nearest neighborsU(0)={(1, 0), (0, −1), (0, 0), (0, 1), (−1, 0)}. Inthe first automaton the cells can be in state zero(white) or one (black), i.e. E={0, 1} and we havethe local function

f0(z �U(0))=ÍÃ

Ã

Á

Ä

1 for %x�U(0)

z(x)]1

0 otherwise

(1)

A cell becomes one if at least one neighbor is instate one, otherwise it becomes zero. In the sec-ond automaton we have E={0, 1, 2} and thelocal function

fo(z �U(0))=ÍÁ

Ä

2 for z(x)=0 and s]11 for z(x)=20 otherwise

(2)

with s= c{y :y�U(x) and z(y)=2}. A cell be-comes excited, i.e. state 2, if at least one neighboris excited (in state 2). In the next evaluation anexcited cell becomes immune, state 1 and with afurther evaluation it becomes resting, i.e. state 0,again. This is an special case of the Greenbergand Hastings (1978) or Threshold automata.

For the line-by-line sweep we see in Figs. 1, 2and 4 how the structure of the sweep dominatesthe patterns of the automata. The patterns of theexample automata generated with the fixed ran-dom sweep are quite different. Starting with oneblack cell in the middle we observe a growingdisc-like black area (Fig. 1). In the automaton (2)in some simulations after a few or even twosweeps there are only white (state 0) cells. Thishappens for example if the cell black (state 2) atthe beginning is updated before one of its neigh-bors. In most simulations a pattern of concentricbands of black, grey (state 1) and white cellsdevelops, giving the impression of spreading

waves (Fig. 2). In some runs we observe a patternlike in Fig. 5, the grey band is so large that noblack cell is inside the grey ring. In yet another setof runs the excitation travels only once across the(finite) grid. With random new sweep the patternsof the automaton (1) can not be distinguishedfrom the patterns with a fixed random sweep (Fig.1). As a result of the variance V(X) of X, with theautomaton (2) the concentric rings which hadappeared with fixed random sweep are broken upwhen in every sweep a new sequence is chosen. Insome cases also after few steps there are onlywhite cells left, but patterns like in Fig. 5 have notbeen observed—the fixed sweep is a special casebut it occurs very rarely. The patterns in Fig. 1with uniform choice show many white cells in theperiphery of the black spot and some black cellsfar away from the middle compared to the pat-terns with random new sweep. This was shown inSchonfisch (1997) where the propagation speedsof automaton (1) in different directions is consid-ered. With random new sweep the border betweenblack and white is sharper, and we note that usinguniform choice the propagation is slower com-pared to the sweep methods. The difference ismore evident with automaton (2). We can notrecognize broken waves like with random newsweep but get a pattern of scattered grey andblack cells. Both with automaton (1) and (2) thepatterns resulting from updates with uniformchoice and exponentially waiting times are notdistinguishable.

2.3.1. Further remarksThe results shown in Table 1 indicate that the

exponential waiting time method is very similar tothe uniform choice updating method. For theexponential waiting time method the number ofupdates of a single cell within a time interval t

Poisson distributed with parameter t. Hence, theprobability that a cell is updated at least oncewithin this interval equals P(W51)=1−P(W=0)=1−e−t. For t=1 we obtain 1−e−1. Forthe uniform choice updating method, the proba-bility that a cell is updated at least once in n singlesteps follows a binomial distribution, given byP(W51)=1−P(W=0)=1− (1−1/n)n. Tak-ing the limit n�� we find that limn�� 1− (1−


Fig. 1. Patterns resulting from the cellular automaton example (1) using different updating methods. A cell becomes black if at leastone von Neumann neighbor is black. From top-left to bottom-right: synchronous updating, line-by-line sweep, random fixed sweeprandom new sweep, uniform choice and exponential waiting time updating. The initial state consists of a single black cell in themiddle of the grid. The state of the automaton is shown after 15 updates (synchronous updating), 15 sweeps (all sweep methods),and 19n single-step updates (for uniform choice and exponential waiting time updating), respectively.

1/n)n=1−e−1. Therefore, for large grids (i.e. forn��) the exponential waiting time and the uni-form choice updating method yield the same limitfor the probability that a specific cell is updated atleast once per unit time or per n consecutive steps,

respectively. This applies also for other statisticsof the two updating methods since the underlyingbinomial distribution, characterizing the uniformchoice method, approaches the Poisson distribu-tion for n��. Hence, Eu(X)=1=E(X)e and the


variance with uniform choice Vu(x) approachesthe variance Ve(X) with exponential waiting timeslimn�� Vu(X)= limn�� 1−1/n=1=Ve(X). For

both methods the distribution of Z is the same asthe one of X. The convergence of the binomialdistribution to the Poisson is very fast. For exam-

Fig. 2. Patterns resulting from the cellular automaton example (2) using different updating methods. A white cell becomes black ifat least one von Neumann neighbor is black, one evaluation later it becomes grey and yet another evaluation later it turns back towhite (black=2, grey=1, white=0). From top-left to bottom-right: synchronous updating, line-by-line sweep, random fixed sweep,random new sweep, uniform choice and exponential waiting time updating. The initial state consists of a single black cell in themiddle of the grid. The state of the automaton is shown after 20 updates (synchronous updating), 20, 18, and 15 sweeps (line-by-line,random fixed, and random new sweep, respectively), and 35n single-step updates (for uniform choice and exponential waiting timeupdating).


Fig. 3. Comparison and identification of updating steps in the uniform choice method and event times in the exponential waitingtime method. The appropriate time scaling in the uniform choice method would involve stretching and squeezing the intervalsbetween two consecutive steps.

ple, already for n=100 simulation results fromboth methods are almost indistinguishable. Thisexplains also that the patterns of the automata inFigs. 1 and 2 look similar.

Instead of an exponential distribution of wait-ing times any other distribution can be used andsome have even been suggested in the literature.An asynchronous updating method with waitingtimes following a normal distribution is intro-duced by Hopfield in a neural network model(described in detail in Hoppensteadt (1986)). Aswith exponentially distributed waiting times, arandom time txi

is drawn for every cell on thegrid. A normal distribution N(1/W, s2) with‘mean updating rate’ W and variance s is used togenerate these random times. The cell with thesmallest of these numbers is updated first, while anew updating time txi

= txi+ t( xi

is generated byagain choosing a random time t( xi

from the normaldistribution N(1/W, s2). The process is repeatedby looking anew for the cell with the smallest txi

.Since the normal distribution will also give nega-tive values, some of the t( xi

will be negative. Thisproduces a ‘time jump’ or ‘time loop’, i.e. a newevaluation time is generated which lays before thecurrent time, maybe even before times at whichother (neighboring) cells were updated. Thesetime inconsistencies may be handled by eitherignoring negative values, i.e. cutting the distribu-tion at zero or by strictly following the algorithm,in which case the corresponding cell is updatednext (since it has the smallest value). Bothworkarounds are, however, not satisfying from atheoretical point of view. In the literature some

algorithms for asynchronous updating can befound which are often motivated by very practi-cal, i.e. programming, considerations. We willbriefly discuss some of them. Toffoli and Margolis(1987) proposed a method for automata withnearest neighbor interactions or, in two dimen-sions, with interactions within the von Neumannneighborhood. Using a checkerboard subdivisionof a two-dimensional square grid, all odd(or even) cells on the grid are updated firstfollowing either a systematic or a random se-quence. The remaining cells are evaluated subse-quently. Since two cells which are neighbors ofeach other can not change at the same time, itdoes not matter whether synchronous or (system-atic) asynchronous updating is used in thetwo ‘sub-steps’. In general this method can beapplied if the grid and neighborhood gives abipartite graph. Like the line–by–line sweep itintroduces however additional structure into theprocess.

Toffoli and Margolis (1987) remarked that asynchronous updating method approximates anasynchronous updating method if the probabilityP that something happens (with stochastic localrules) is very small. In every (synchronous) timesteps for every cell a random number r� [0, 1] ischosen. The cell is only updated if rBP. WithP�0 the probability that two neighboring cellschange in the same time step goes to zero. Con-sider for example a pure death process with syn-chronous updating. Let the probability for aliving cell to die be P=1/n. On the other handtake a death process with asynchronous updating-


according to the uniform choice method whereevery evaluated living cell dies with probabilityone. The two processes are similar, for examplethe probability for a living cell to die in onesynchronous time-step, respectively, one asynchronous single step is then 1/n. With syn-chronous updating two neighboring cells may beupdated at the same time. This never happenswith asynchronous updating. Usually in cellularautomata the evolution of a cell depends on thestates of its neighbors. Therefore in general syn-chronous and asynchronous processes will be dif-ferent. We see however that a process withsynchronous updating with n large, a small neigh-borhood and P small approximates an corre-sponding asynchronous process.

Mixtures of parallel and serial iterations havealso been described in the literature (Sipper et al.,1997). For block sequential iterations Goles andMartınez (1990) (or serie-parallel Robert (1995))an ordered partition g1…gp of the grid G is cho-sen. Within each partition iterations take placesynchronously, while the evaluation order of theorder of the blocks is asynchronous. In the check-erboard example introduced by Toffoli and Mar-golis (1987) we have two blocks g1, g2 one beingthe ‘white’ and one being the ‘black’ cells. Syn-chronous iteration is the special case with onlyblock g1=G and asynchronous iteration is thecase g1…gn with one cell in each block.

Finally we give an example of a so called eventdriven process (Overeinder et al., 1992). In asimplified form, this process is initialized by gen-erating for every cell a time ti at which it will beupdated for the first time. Subsequently, a fixedtime t elapses before it is updated again. The cellsnow act as independent automata, each followingits own internal clock, while the local functioncouples the cells together. After the elapse of thelargest initial time among the set of ti values thismethod is, however, equivalent to a fixed randomsweep. These event driven processes are general-ized by drawing a new time interval ti for everycell when it is updated according to a givendistribution. If this distribution depends on thecells itself, different groups of cells can be updatedwith different frequencies. Moreover the distribu-tion may even depend on the state of the cell.

3. Uniform choice versus exponential waiting timemethod

Using exponential waiting times seems the mostsatisfying updating method from a theoreticalpoint of view. With this method transitions ofsingle cells (possibly representing biological indi-viduals) between different states are formulated interms of probabilities per unit time. Table 1 showsthat it is the only method for which the grid sizen does not affect the updating statistics. In con-trast to step–driven methods, the dynamics of thecellular automaton are therefore independent ofthe (statistical) properties of the updating method.The synchronous updating method and all asyn-chronous sweep methods are likely to induce dy-namical artifacts, given their rather differentstatistical properties (see Table 1). When n singleupdating steps in step–driven methods are takento be equivalent to one unit of time in time–driven methods, the variance in the number ofsteps between two consecutive updates of thesame cells is always smaller with asynchronoussweep methods. Also, the expected number ofsteps between an update of a cell and an updateof one of its neighbours and its variance areusually smaller than with the exponential waitingtime method. As a result, the patterns that aregenerated with synchronous updating or asyn-chronous sweep methods contain more structurethan those generated with the exponential waitingtime method (cf. Huberman and Glance, 1993).The asynchronous, uniform choice method seemsidentical to the exponential waiting time methodfor sufficiently large grids. In this section, theproperties of these two updating methods will becompared in more detail.

Given a cellular automaton A= (G, U, E, f0),both the uniform choice updating method andexponential waiting time updating determine aparticular stochastic process. Any trajectory orrealization of these processes is a sequence S=fxm

fxm−1…fx 2

fx 1, xi�G, which maps the initial state

z0 into z=S(z0). In Bandt et al. (1998) it is shownthat the two stochastic processes determined bythe uniform choice and exponential waiting timemethod, respectively, are identical in the sensethat for every trajectory S of the uniform choice


process there exists a corresponding trajectory inthe exponential waiting time process and that thesetwo trajectories have an equal likelihood ofoccurrence. As a consequence, the probability dis-tribution of possible states z after M updates fromany particular initial state z0 is identical for boththe uniform choice and the exponential waitingtime process. Both processes hence have the samestationary states and limit sets.

The single major difference between the uniformchoice and exponential waiting time updatingmethod is the definition of the variable time, whichis explicitly defined in the exponential waiting timemethod but not in the uniform choice method. Withthe exponential waiting time method the updatestake place at times t1, t2,… with ti�R+. With theuniform choice method the attachment of events totimes is arbitrary. An appropriate scaling of timein the uniform choice method would involvestretching and squeezing the intervals between twoevents, matching the events in this process with thecorresponding events in the exponential waitingtime process (Fig. 3). More formally, this scalinginvolves the introduction of an additional Poissonprocess into the uniform choice method, generatingthe time-value at which the updates of cells in theuniform choice method take place. The details ofthis time-scaling procedure depend on the proper-ties of the cellular automaton, especially if the latteris stochastic.

In practice, however, time in the uniform choicemethod is almost always defined by taking n singleupdating steps as one unit of time. This particulartime scaling implies that exactly n single updatesoccur per unit of time when the uniform choiceupdating method is used. In contrast, if eventsoccur with rate 1 when the exponential waiting timemethod is used, the number of single updates per

unit of time is a stochastic variable, following aPoisson distribution with parameter 1. Hence, bothits expectation and variance equal unity. Below wewill illustrate that this scaling of time has conse-quences at least for quantitative comparisons be-tween the uniform choice updating method and theexponential waiting time method when measuresthat involve the variable time or rates are consid-ered.

3.1. Comparing time measurements

To illustrate the differences between the uniformchoice and exponential waiting time method we willinvestigate the behavior of one of the most simplecellular automata, representing a pure death pro-cess of a population of N individuals, populatinga l× l=n grid G at time t=0. Therefore, the cellsare either in state zero (representing no individualpresent) or one (occupied by an individual), i.e.E={0, 1}. The local function equals:

f0(z �U(0))=0

Note that this example does not require thespecification of a neighborhood U, because the celldynamics are completely independent (However,the cellular automaton is technically identical toexample (1) when s\4). Furthermore, the grid Gis only functional when using the uniform choicemethod for the selection of sites to be updated. Forthe exponential waiting time method the grid isirrelevant.

When the exponential waiting time updatingmethod is used, the total time T. (p) that elapsesuntil the death of the pth individual is subdividedinto time intervals Ti (i=0, …,p−1), in which thepopulation consists of N− i individuals, respec-tively. These intervals Ti are independently dis-tributed, following an exponential distribution:

Fig. 4. Patterns resulting from the cellular automaton example (2) for four consecutive line-by-line sweeps. A white cell becomesblack if at least one von Neumann neighbor is black, one evaluation later it becomes grey and yet another evaluation later it turnsback to white (black=2, grey=1, white=0).


h(Ti=t)= (N− i )e− (N− i )t, t]0

The expected value and variance of T. (p) aretherefore given by:

E(T. (p))=E� %

p−1

i=0

Ti

�= %

p−1

i=0

E(Ti)

= %p−1

i=0

1(N− i)

(3)

V(T. (p))=V� %

p−1

i=0

Ti

�= %

p−1

i=0

V(Ti)

= %p−1

i=0

1(N− i)2 (4)

Using the uniform choice updating method, thenumber of updating steps U0 (p) that are carriedout until the death of the pth individual, is at leastequal to p, but can be larger if an unoccupied cellis selected for updating in a specific step. U0 (p)hence consists of p steps in which an occupied cellis updated resulting in the death of an individual,plus sequences of single updating steps Si (i=0,…,p−1), during which the population remainsstable at N− i individuals, respectively. Thelengths of these sequences Si are independentlydistributed, following a geometric distribution:

P(Ui=u)=(N− i)

n�

1−(N− i)

n�u

u=0, 1,…,�

The expected value and variance of U0 (p) aretherefore given by:

E(U0 (p))=E�

p+ %p−1

i=0

Ui�

=p+ %p−1

i=0

E(Ui)

=n %p−1

i=0

1(N− i)

V(U0 (p))=V�

p+ %p−1

i=0

Ui�

= %p−1

i=0

V(Ui)

=n2 %p−1

i=0

1(N− i)2−n %

p−1

i=0

1(N− i)

Defining n single updating steps as a unit oftime, the expected value and variance of the timeT0 (p)=U0 (p)/n that elapses until the death of the

pth individual, equals

E(T0 (p))= %p−1

i=0

1(N− i)

(5)

V(T0 (p))= %p−1

i=0

1(N− i)2−

1n

%p−1

i=0

1(N− i)

(6)

Comparing Eqs. (3) and (5), it can be concludedthat the expected time until the death of the pthindividual is the same for both the exponentialwaiting time and the uniform choice updatingmethod. The variance for the uniform choicemethod (Eq. (6)) is, however, a factor

ÃÃ

Ã

Á

Ä

1−1n

%p−1

i=0

(N− i)−1

%p−1

i=0

(N− i)−2

ÃÃ

Ã

Â

Å

smaller than the variance for the exponential wait-ing time method (Eq. (4)). The variances V(T. (p))and V(T0 (p)) will, therefore, be substantially dif-ferent in case the variance to mean ratio is in theorder of 1/n, i.e. when the variance is small withrespect to the expected value T. (p) (or T0 (p)).

3.2. Comparing rate measurements

With the exponential waiting method, all indi-viduals are totally independent and have a proba-bility to survive a time period of length t equal toexp(−t). The number of individuals S. (T) surviv-ing t units of time, given there are initially Nindividuals present, follows a binomial distribu-tion with parameters N and exp(−t):

P(S. (t)=m)=�N

m�

(e−t)m(1−e−t)N−m

The average value and variance of S. (t) equal:

E(S. (t))=Ne−t

V(S. (t))=Ne−t(1−e−t)

For the uniform choice updating method, theprobability Pm(u) that after u single updatingsteps m of the initial N individuals have survived,follows a recurrence relation:

Pm(u)=ÍÃ

Ã

Á

Ä

�1−

m

n

�Pm(u−1)�

1−m

n

�Pm(u−1)+

m+1

nPm+1(u−1)

if m=N

otherwise


with initial conditions

Pm(0)=!1

0if m=Notherwise

The expected number of surviving individuals,E(S0 (u)), after u single updating steps, equals thesolution of a recurrence relation, which can bederived from the recurrence relation for Pm(u):

E(S0 (u))=�

1−1n�

E(S0 (u−1)), E(S0 (0))=N

The solution of this equation is:

E(S0 (u))=N�

1−1n�u

.

To compute the variance V(S0 (u)) we define

W(u)= %N

i=0

i2Pi(u).

Just like E(S0 (u)), a recurrence relation can bederived for the dynamics of W(u):

W(u)=�

1−2n�

W(u−1)+1n

E(S0 (u−1)),

W(0)=N2

which yields the solution

W(u)=N2�1−2n�u

−N��

1−2n�u

−�

1−1n�un

The variance V(S0 (u)) is thus given by

V(S0 (u))=N2��1−2n�u

−�

1−1n�2un

−N��

1−2n�u

−�

1−1n�un

To assess the differences in total death rates,the expectation and variance of the number ofsurviving individuals after one time unit for theexponential waiting time method:

E(S0 (1))=Ne−1 (7)

and

V(S0 (1))=Ne−1(1−e−1), (8)

Fig. 5. Example pattern resulting from the cellular automatonexample (2) using the random fixed sweep method. A whitecell becomes black if at least one von Neumann neighbor isblack, one evaluation later it becomes grey and yet anotherevaluation later it turns back to white (black=2, grey=1,white=0). The initial state consists of a single black cell in themiddle of the grid. The state of the automaton is shown after20 sweeps.

respectively, have to be compared with theanalogous quantities after n single updating stepsin the uniform choice method:

E(S0 (1))=N�

1−1n�n

(9)

and

V(S0 (n))=N2��1−2n�n

−�

1−1n�2nn

−N��

1−2n�n

−�

1−1n�nn

(10)

Obviously, limn�� E(S0 (n))=E(S0 (1)), so thatfor sufficiently large grids the expectations ap-proach the same value. Comparing Eqs. (7) and(8) shows that the ratio of variance and mean forthe exponential waiting time method is indepen-dent of the initial number of individuals N andthe grid size n and approximately equal to 0.63.Numerical comparisons of Eqs. (8) and (10) (seeFig. 6) shows that the variances are only com-parable for a combination of small values of Nand large grid sizes n. The differences in varianceare largest for initial numbers of individuals Nclose to the grid size n. In absolute terms the


differences increase with increasing grid sizes n, inrelative terms the ratio V(S0 (n))=V(S0 (1)) appearsto approach a limit value of approximately 0.4 forN close to n.

In summary, for the pure death process ana-lyzed here it can be concluded that the exponen-tial waiting time updating method will yieldquantitative estimates of times and rates that aremore variable than when the uniform choice up-dating method is used. For time measurements,this variability of the exponential waiting timemethod is only substantially larger in cases wherethe variance is small with respect to the expectedvalue of the time variable. For rate measurements,the variability of the exponential waiting timemethod will always be substantially larger in caseswhere the grid G exhibits high levels of occupa-tion. The variance in the number of survivingindividuals is in this case not negligible with re-spect to the mean value.

4. Stationary states, cycles and reachable states

In this section we will focus on the differencesin the long-term dynamics of cellular automata

using synchronous and asynchronous updatingmethods. The results apply to any automatonA= (G, E, U, f0) with deterministic local functionand translation invariance with regard to thislocal function. Here, we do not focus on onesimulation run or mean values from different runsbut consider all possible asynchronous evalua-tions from a given initial configuration. This viewis stimulated by the ideas of theoretical computerscience (see for example the fundamental workCori et al. (1993)). We already noted that startingfrom a given initial configuration with syn-chronous updating we get one trajectory, i.e. onesuccession of states. Asynchronous updating witha fixed sweep will also lead to one trajectory. Incontrast, with random choices the trajectories willdiffer between different realizations of the pro-cess. Let R(z0) be the set of all states which canbe reached from a given state z0, i.e. for everystate z�R(z0) there is a sequence of evaluationsS= fxm

fxm−1…fx 2

fx 1such that S(z0)=z. One can

imagine the set of reachable states as a tree. Sinceseveral sequences may result in the same statesome ‘branches’ of the tree will join. If we use arandom method to determine the sequence ofevaluations different realizations will generally

Fig. 6. Variance in the number of surviving individuals after 1 unit of time with the exponential waiting time method (solid line)and n single updating steps with the uniform choice method (dashed line). Left: as a function of the grid size n; the initial numberof individuals N equals the grid size n. Right: as a function of the initial number of individuals N for a grid size n=10 000.


follow different branches. With a fixed sweep asingle branch is selected. Fig. 8 gives an example.

In the following discussion we will restrictourselves to asynchronous updating sequences inwhich every cell is e6aluated e6er and e6er again,sequences which leaves out certain cells seem notto make much sense. We start with the stationarystates.

Def: Let A= (G, U, E, f0) be a cellular automa-ton with synchronous updating f:z�f(z). A statez is called stationary state if

f(z)= z.

Def: Let A= (G, U, E, f0) be a cellular automa-ton with asynchronous dynamic fx:z�fx(z). Astate z is called stationary state if for all x�G

fx(z)= z.

Lemma. States which are stationary under syn-chronous updating are stationary under asyn-chronous updating and vice versa.

Proof. Let z be stationary under f and not station-ary under fx. The latter implies that there is atleast one x such that f0(z �U(x))" z(x). As a conse-quence, also f(z)" z which is a contradiction.

Let z be stationary under fx and not stationaryunder f. Because there must exist at least one xsuch that f0(z �U(x))" z(x), z is not stationary un-der fx either.

Remark: If we would have a sequence in whichfor at least one cell there exists a time T, such thatthe cell is never evaluated for all t\T we wouldnot need to require fx(z(x))= z(x) for that cell.Then states stationary with this asynchronous se-quence need not be stationary with synchronousupdate. Also we need a deterministic local func-tion since otherwise we would have to considerstationary distributions of states instead of simplestationary states.

Although the set of stationary states is the samefor both synchronous and asynchronous updatingmethods their basins of attraction may be totallydifferent. This implies that starting from a giveninitial configuration, synchronous and asyn-

chronous updating may lead to totally differentstationary states. In general, different stationarystates may even be reached with different se-quences of a particular asynchronous updatingmethod, while obviously only one can be reachedwith synchronous updating. The following exam-ple shows that this may even be a state that is notat all reachable with any asynchronous updatingmethod.

Consider a one-dimensional nearest-neighborautomaton A= (G=Z, E={0, 1}, U={−1, 0, +1}, f0) with the following rule (Wolf-ram rule 76) 111, 000, 001, 100, 101�0 and010, 011, 110�1, i.e. the middle one of three onesmaps to zero, everything else stays the same.Starting from the initial state z=…011110… withsynchronous updating the stationary state…010010… is immediately reached. This state isalso stationary under any asynchronous updating,although there exists no asynchronous sequenceleading to it from the given initial configuration z.Asynchronous updating will lead to one of thestationary states …010110… or …011010… whichare also stationary under synchronous dynamics,but can not be reached with it from the giveninitial configuration z. Generally the stationarystate reached from an initial state z by syn-chronous updating may not be an element of theset of reachable stationary states E(z)¦R(z) withasynchronous updating.

With synchronous updating some cellular au-tomata may exhibit cycles as long-term dynamics,which is only possible with asynchronous updat-ing if a fixed sweep is used. Consider for examplethe automaton A= (G=Z, E={0, 1}, U={−1, 0, +1}, f0) where f0 maps 111�0 and 101�1otherwise f(z(x))=z(x) (Wolfram rule 108).Starting from the configuration …000111000…with synchronous updating a cycle of period 2results. With a line-by-line or random fixed sweepthe same cycle is obtained, but with a period of2×n single steps. With a random new sweep wewill likewise swap between the states...000111000... and ...000101000..., although thenumber of single steps between the two states willnot be constant. Even though this example seemsvery artificial, it can be concluded that cycles aregenerally not likely to be observed with asyn-chronous updating methods.


Fig. 7. An example of cyclic dynamics resulting from thecellular automaton example (3) using synchronous updating. Acell becomes black if at least three von Neumann neighborsare black. Otherwise it turns white.

updating. Fig. 8 shows the states reachable byasynchronous updating, consisting of five station-ary states. With line-by-line sweep only one ofthem can be reached (a six block of black cells atthe top and a four block at the bottom). Withrandom fixed sweep, random new sweep, uniformchoice and exponential waiting times in every runonly one stationary state will be reached, butdifferent ones in different realizations.

5. Discussion

We have compared several methods of updatingfor cellular automata. With asynchronous updat-ing, algorithms involving fixed sweeps usually in-troduce a lot of additional, unintended structureinto the dynamics and patterns exhibited by thecellular automaton. The latter can be reduced bychoosing the cell to be updated next according tosome random process. The line-by-line methodinduces most structure, because the choice of the‘next’ cell strongly depends on the choice of itspredecessor. The fixed sweep and new randomsweep methods induce less structure, because thecorrelation in the updating order of the cells is

Goles and Martınez (1990) could prove for acertain class of cellular automata that both sta-tionary states and cycles can be obtained withsynchronous updating while with asynchronousupdating only stationary states are reachable. InFig. 7 we show as an example automaton similarto automaton (1) only here the cell needs threeneighbors in state one to become state one, too,otherwise it goes to state zero. This automatonbelongs to the class considered by Goles andMartınez. From a particular initial configurationa cycle of period two is reached with synchronous

Fig. 8. The stationary states of the cellular automaton example (3) that can be reached from a specific initial configuration usingasynchronous updating. A cell becomes black if at least three von Neumann neighbors are black. Otherwise it turns white.


smaller. The uniform choice method involves arandom choice with replacement for the selectionof the next cell to be updated. This minimizes anycorrelation in the updating order of the cells.From all step-driven methods, the uniform choicemethod most closely resembles the exponentialwaiting time method. We argue that the lattermethod is most satisfying from a theoretical andform biological point of view as its justified by aderivation from continuous time processes andintroduces least undesired structure.

The uniform choice and exponential waitingtime method are identical if we neglect time mea-sures and consider only the sequences of states ofthe automaton. The latter implies that the qualita-tive features of both processes (stationary states,limit sets, probability distributions of states) arethe same. The two methods only differ in quanti-tative measurements of time periods and rates.Our analysis suggests that especially the estimatesof rates will exhibit more variability when theexponential waiting time method is used. Forspecific biological applications, this variabilitymight be important. Furthermore, if time is anexplicit part of the underlying model, for examplewhen studying the dynamics of an age-structuredpopulation in a spatially explicit context, thequantitative differences in variability between thetwo methods might actually translate into qualita-tive differences in the observed dynamics. In suchcases the exponential waiting time method seemsmost appropriate to iterate the cellularautomaton.

Synchronous and asynchronous updating maylead to qualitatively very different dynamics. Eventhough the stationary states of a cellular automa-ton are identical whether synchronous or asyn-chronous updating is used, which of these arereached from a given initial state with syn-chronous and asynchronous updating may be dif-ferent. Moreover, the cycles or stationary states ofsynchronous iteration may be not reachable at allby any asynchronous sequence. Especially cyclesand highly symmetric or fractal patterns producedby synchronous updating are in a kind artificial.These phenomena seem to be more the result ofthe updating method than of the underlying, bio-logical processes since they are destroyed by the

smallest perturbation. Asynchronous updatingwith random sequences seem to produce indeedpatterns which are more robust to methodologicaldetails.

Synchronous updating may of course bejustified if there is an external zeitgeber or trigger.In other applications it might be sufficient toconsider the process between two synchronousupdates as a black box and only assess the out-come of the underlying, continuous time processat specific times. The specifics of the applicationwill determine whether this phenomenological ap-proach is appropriate or not.

Following this view the difference between syn-chronous and asynchronous update is a questionof how we look at the (real) process. If we observeonly in large time intervals we will see that allcells have been updated (at least) once in one timestep, implying synchrony. If we refine the timescale such that in every time interval at the mostone event will happen, then we find asynchrony. Itis true that in the real world everything evolves inparallel, i.e. synchronously. But at most points intime at most places nothing happens. In this senseasynchronous updating is an discretization andapproximation of (real) continuous time.

The comparisons of different updates in theliterature and in our study show that in somecases different updating methods lead to not onlyquantitatively but also qualitatively different dy-namical behavior of the models. But these differ-ences are not expressed in all cases. Especially ifstrong stochastic components like a stochasticlocal function are involved, then automata withdifferent updating methods may show at leastqualitatively similar behavior. This is not surpris-ing since stochasticity may act the same wayunregarding whether it is introduced via asyn-chronous update or via stochastic local functions.Furthermore it is plausible that for every asyn-chronous automaton one can find a correspond-ing synchronous automaton with (perhaps verycomplicated) stochastic local function. The effectsof different updating rules propagate most instrictly deterministic cellular automata, in au-tomata with stochastic components they may notbe obvious. In summary it will depend on theactual cellular automaton how strong the influ-ence of different updating methods will be.


Appendix A

A.1. Line-by-line sweep

(1) Number of single steps between two updates ofsame cell: Since every cell always appears at thesame position in every sweep: E(X)=n, V(X)=0.

(2) Number of single steps between an update ofcell x and an update of a gi6en y�U(x), y"x: Thiscase depends on the actual neighborhood. For ageneral formula we need the ‘distance’ (in the senseof single steps between updates) between two cells.Let h= c{y�U(x), y"x} be the number ofneighbors excluding the cell itself. Number thecells of the grid from top left to right bottom bynatural numbers ci=1,…,n. Let dab be the ‘dis-tance’ between two cells with numbers ca and cb

with

da,b=!cb−ca

n− (cb−ca)if cb\ca

if cb5ca

We can calculate E(Z) and V(Z) for an arbitrarycell since we have translation invariance of theneighborhood. Then:

E(Z)=1h

%y�U(x),y"x

dxy

V(Z)=1h

%y�U(x),y"x

(dxy−E(Z))2

Consider for example the von Neumann neighbor-hood and a grid of l× l=n cells. We have h=4.Name a cell and its neighbors in the following way:

da,b=!cb−ca

n− (cb−ca)if cb\ca

if cb5ca

The cells are evaluated in the order: c, d, x, e, f.Start with cell x. The number of single stepsbetween x and cell c is dxc=n−n, likewisedxd=n−1, dxe=1 and dxy=n. Hence,

E(Z)=14

(n−n+n−1+1+n)=12

n

V(Z)=14

[(n−n−E(Z))2(n−1−E(Z))2

+ (1−E(Z))2+ (n−E(Z))2]

=12�

1−n n+12

n2�For the other asynchronous updating methods

the probability that a cell is chosen to be updateddoes not depend on its grid coordinates. Thereforethe following results do not depend on the neigh-borhood and actually we do not even have to havea square grid. We only need that the number ofcells of the grid is n and the translation invarianceof the neighborhood.

A.2. Random fixed sweep

(1) Number of single steps between two updates ofsame cell: Since every cell always appears at thesame position in every sweep: E(X)=n, V(X)=0.

(2) Number of single steps between an update ofcell x and an update of a gi6en y�U(x), y"x:Suppose x is just updated. Then, y is updatedwith certainty in the next (n−1) single steps.With the random fixed sweep method the proba-bility to be at a certain position in the sweep is thesame for every neighbor y of x therefore we don’taverage over all neighbors. Also, the probabilitythat a given cell occurs at a specific position in thesweep is the same for all positions. For n]2 weget

E(Z)=1

n−1%

n−1

i=1

i=12

n

V(Z)=1

n−1%

n−1

i=1

(i−E(Z))2=112

n(n−2)

A.3. Random new sweep

(1) Number of single steps between two updatesof same cell: Consider two successive sweeps. Leti be the position of cell x in the first sweep and jits position in the second sweep.

E(X)=1n2 %

n

i=1

%n

j=1

(n− i+ j)=n


V(X)=1n2 %

n

i=1

%n

j=1

(n− i+ j−E(X))2=16

(n2−1)

(2) Number of single steps between an update ofcell x and an update of a gi6en y�U(x), y"x:Consider a single sweep. Let i be the position ofcell x in this sweep. With probability (n−1)/(n−1) the update of cell y will occur after the updateof cell x in the current sweep and with probability(i−1)/(n−1) it will have occurred before thecurrent update of x. In the first case the numberof single steps between the two updates is justj− i, where j is the position of cell y in the sweep(iB j5n). In the second case, the next update ofcell y will only occur in the following (new ran-dom) sweep. Here it may occur at any positionj=1,…,n in the sweep. The number of single stepsbetween the two updates in this case equals n−i+ j. Hence for n]2

E(Z)=1n

%n

i=1

!(n− i)(n−1)

%n

j= i+1

( j− i)1

n− i

+(i−1)n−1

%n

j=1

(n− i+ j)1n"

=1

12(7n+1)

V(Z)=1n

%n

i=1

!(n−i)(n−1)

%n

j= i+1

(( j−i)−E(Z))2 1n−i

+(i−1)(n−1)

%n

j=1

((n− i+ j)−E(Z))2 1n"

=112�23

12n2−

136

n−1312�

A.4. Uniform choice

The probability that a cell x is chosen forupdating in a particular step equals 1/n. Theprobability that a cell x is chosen k times in 6single steps is binomially distributed, i.e.

P(x is chosen k times)=�6

k� �1

n�k�

1−1n�6−k

(1) Number of single steps between two updatesof same cell:

E(X)= %�

i=1

i1n�

1−1n�i−1

=n

V(X)= %�

i=1

[i−E(X)]21n�

1−1n�i−1

=n(n−1)

(2) Number of single steps between an update ofcell x and an update of a gi6en y�U(x), y"x:Since the uniform choice method representschoice with replacement, the distribution of Z isidentical to the distribution of X. Hence, E(Z)=n, V(Z)=n(n−1).

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