A guided tour of asynchronous cellular automata
AA guided tour of asynchronous cellular automata
Nazim Fatès [email protected]
Inria Nancy Grand-Est, LORIA UMR 7503F-54 600, Villers-lès-Nancy, FranceAugust 27, 2014
Abstract
Research on asynchronous cellular automata has received a great amountof attention these last years and has turned to a thriving field. We presenta state of the art that covers the various approaches that deal with asyn-chronism in cellular automata and closely related models.
Foreword : This article is the preprintof an article that is to appear in the
Journal of cellular automata . It is an extended verion of the invited paper thatappeared in the proceedings of Automata’13, 19th International Workshop onCellular Automata and Discrete Complex Systems, LNCS 8155, 2013, p. 15-30.
By their very simplicity, cellular automata are mathematical objects that occupya privileged situation in the study of complex systems. They are formed ofa regular arrangement of simple automata, the cells , which can hold a finitenumber of states. Cellular automata are as mosaics with tiles that autonomouslychange their colour: the cells are updated at discrete time steps and their newstate is calculated according to only a local information, usually limited to thestates of the neighbouring cells. These local laws of interaction may generateamazing behaviours at the global scale, even when they are simply expressed.Cellular automata were initially studied by von Neumann and Ulam to studythe properties of self-reproduction of living organisms with a simple “mechanical”tool [133]. Since then, they have been employed in various scientific domains.Their study can be divided into three main axes: (1) They are dynamical sys-tems where time, space and states are discrete. Their regular structure simplifiesthe mathematical definitions of the system but the exact or partial predictionof the trajectories of the system is often a highly challenging task. (2) Theyrepresent a model of spatially-extended, distributed and homogeneous comput-ing systems. As such, they represent an alternative to the classical computingframeworks that use sequential algorithms, variables, functions, etc. (3) Theyare employed to model the numerous complex systems seen in Nature. Re-searchers have been particularly interested in the properties of self-organisationor robustness they can display. 1 a r X i v : . [ n li n . C G ] A ug n important feature in the definition of cellular automata regards theirupdating: in their original definition, they are updated synchronously , that is,all the cells change their state at the same (discrete) time step. This globalupdate implies a strong simultaneity: cells need to gather simultaneously thestate of their neighbours, they need to process this information simultaneously,the transitions have to occur in a single time step.Making this hypothesis of perfect synchrony has many advantages, first ofall to simplify the description of the system. With a synchronous update, it isfor instance easy to build a Turing-universal system, to “program” the systemto obtain a given behaviour, to show that a given property is undecidable or tostudy under which restrictions this property becomes decidable, etc. (see thesurvey by Kari [60] for more details). Synchronous updates are also a convenienttool for modelling natural or artificial phenomena: there is no need to take intoaccount complex updating procedures as all the cells share the same time.In spite of these manifest advantages, there are reasons why the hypothesisof perfect synchrony needs to be questioned:(a) In the context of dynamical systems, the problem is to study how cellularautomaton “react” to perturbations of their updating. How can we interpret thepotential sensitivity of the system to changes of its definition? On the contrary,what can be said if the system “resists” to a change of its updating scheme?(b) In the context of parallel computing, we ask how to design a computingdevice that does not require a central clock. Various advantages can be expectedfrom the removal of a pace maker: increase of the speed of computations, econ-omy of energy, simplicity of design, etc. Beyond these potential gains, developingasynchronous massively parallel algorithms represents a research challenge byitself.(c) When cellular automata represent a model of a natural phenomenon, thequestion is to know what triggers the transitions of the cells’ state. How dowe represent this source of activity in the model? Answering is far from beingsimple and the argument that “there is no global clock in Nature to synchronisethe transitions” is somewhat incomplete. Indeed, it can be objected that a modelis a simplified representation of a phenomenon and does not need to faithfullyaccount for all the details of “reality”.All these questions raise rich problems and they are discussed in the worksthat we present in this survey. The field of asynchronous cellular automata hasattracted the interest of numerous authors and has evolved from a “marginal”to a “respected” topic during the last decade. The scientific production hasnow reached a level which makes it difficult to follow all the contributions thatappear. The purpose of this survey is thus to introduce the readers to thisquite diversified “landscape”, trying as much as possible to cover the various“sites” that it contains. As a “guided tour”, it does not claim to be an objectivedescription of the field: a guided tour is by definition a circuit that takes visitorsfrom place to place according to the arbitrary choices of the guide. It is thereforeimportant to bear in mind that the descriptions that will follow will be as briefas possible and should by no means prevent us from reading the texts mentioned themselves . Our hope is that readers that are unfamiliar with cellular automata2 ontinuousdiscrete Figure 1: Mapping from a continuous to a discrete time scale.will find landmarks for their orientation and those which are interested in aparticular topic will find useful references.
Our visit begins by considering the definitions of asynchronism. The etymol-ogy is clear: α-συν-χρόνος ( a-sun-chronos ) means not-same-time in Greek. Theword thus merely indicates that there are parts of the system that do not sharethe same time. As an illustration, we may figure out a choreography whereeach dancer has its own pace and its own sequence of movements: the choreog-raphy may be chaotic but the dancers may also succeed in forming a coherentperformance if some coordination is maintained between them.The privative nature of the definition of a-synchronism suggests that thereare many interpretations of the word. In fact, we are allowed to speak of asyn-chronism as soon as we break the framework of perfect updating. To date, thereis no agreement on how this word should defined. Moreover, it is frequent thatdifferent terms are used for naming the same updating scheme. The definitionsthat we present below are thus by no means “official”: we simply make the choiceto use in priority the terms that we have employed in our own research. In general, asynchronism is seen as an external and uncontrolled phenomenon,it is thus most often modelled as a stochastic process. The two main stochasticupdating schemes that have been employed are: • fully asynchronous updating: At each time step, the local rule is appliedto only one cell, chosen uniformly at random among the set of cells. • α -asynchronous updating: At each time step, each cell has a given proba-bility α to apply the rule and a probability − α to stay in the same state.The parameter α is called the synchrony rate . Note that the terms α - a synchronism and α -synchronism have been used and are bothrelevant: α can denote the name of the scheme and the synchrony rate. We use here the term α - a synchronism, as it is the form that was first proposed and which has been adopted byvarious authors such as Regnault, Correia, Worsch, Fukś, etc. the order in which cells are updated, (d) this order can beobtained by a sequential stochastic sampling on the set of cells (see Fig. 1 ande.g. Ref. [113] for a similar presentation).It can be remarked that this argument is physically relevant if the transitionsof the cellular automaton are “infinitely” short, that is, if the time to go fromone state to another can be neglected. This is surely a valid hypothesis forsome particular contexts (e.g. a radioactive disintegration) but this cannot beconsidered as the asynchronous updating model.In many cases, especially in biological systems, some synchrony between cellsneeds to be assumed. As this degree of synchrony is difficult to measure, theproblem is not so much about choosing the “right” model of updating but ratherto estimate the robustness of model, that is, if it will totally or partially resistthe perturbation of its updating scheme. In this context, the α -asynchronousmethod defines a system with a continuous variation from a perfect synchronism( α = 1 ) to the limiting case of full asynchronism ( α → ). Note however thatwhen looking at the asymptotic behaviour of a system, a discontinuity may existbetween the case α → and the fully asynchronous case. Indeed, the possibilitythat two neighbouring cells simultaneously update their state, be it as smallas wanted, may radically change the trajectory of a system. As an example,consider the minority rule in 2D with a von Neumann neighbourhood: witha fully asynchronous updating, the two uniform fixed points are not reachablefrom a non-uniform configuration [40] but this is not longer true if we allow asmall degree of synchrony.It should also be noted that fully asynchronous updating is defined with afinite set of cells. The passage to the limit for an infinite set of cells needs tobe done with a model that has a continuous time and the mathematical modelthat accurately describes a (stochastic) fully asynchronous updating is called an interacting particle system . (See e.g. Ref. [20] for examples where such systemsare used for solving the density classification problem in two dimensions.) The other question that is generally asked when defining asynchronism is toknow if the timing of the transitions should be defined with the use of a globalclock or with a clock that is proper to each cell. Schönfisch and de Roos call theformer step-driven methods and the latter time-driven methods [113]. It may bethought at first that “time-driven” methods are more adequate for making “re-alistic” simulations than “step-driven” methods. Indeed, it seems better to giveto the cells an explicit representation of time and to avoid to artificially sharea transition signal between all the cells. However, this idea needs to be exam-ined more closely. As remarked by various authors, this distinction is somewhat4rtificial as it is in general possible to build a “step-driven” method that emu-lates a “time-driven” method, and vice versa. For example, as discussed before,the random updates of a fully asynchronous scheme and the updates obtainedby independent clocks that use a continuous time are quasi-equivalent, up to arescaling. The α -asynchronous updating can also be defined from the point ofview of the cells by separating two updates by a random time which obeys ageometric law. (In other words, the probability that k time steps separate twoupdates of a cell is equal to: α (1 − α ) k − .)There are of course many other types of updating schemes where randomnessis involved. For instance, one may consider random sweeps where cells areupdated sequentially by following random permutations of the updating orders(this scheme is also called random order ) or fixed sweeps where the permutationorder is drawn at the beginning and kept fixed during the whole evolution ofthe system [113]. We will also present below how to define an asynchronismwhich results from an imperfect transmission of the state from the neighbours(see Sec. 3.4).Non-random updating schemes can also be considered: for instance, in the sequential ordered scheme, cells are updated sequentially following an order thatresults from their spatial arrangement (for example from left to right and fromtop to bottom); cells can also be updated depending on their parity at evenor odd time steps (see e.g. [96]). We refer to the work of Schönfisch and deRoos [113], Cornforth et al. [26], Bandini et al. [9] for the presentation of acollection of various deterministic or stochastic updating schemes.It is also necessary to distinguish the non-deterministic schemes from thestochastic ones. As in classical automata theory, non-determinism means thata given subset of cells may be updated and all the possibilities are considered,regardless of their “likeliness to appear”. The evolution of the system is thusrepresented by a set of configurations; this set evolves according to the outcomesof each transition that can be applied.The problem of the definition of asynchronism is thus completely open andwe end this section with the following questions: Questions 1
What taxonomy of the updating schemes can be issued? Whatare the guidelines that can drive modellers for choosing a particular updatingscheme? Under which restrictions (states, neighbourhoods, class of rules, etc.)can we establish equivalences between updating schemes?
Classifying “classical” cellular automata has been a central theme of researchand is far from being a closed question (for recent references, see e.g. thework by Schu¨le [115, 114] and the survey by Martínez [77]). It can then bethought at first that classifying asynchronous rules is a daunting task becauseof the additional complexity that is induced by the asynchronous updating. Infact, this is only partially true as in many cases the asynchrony may “break”5he complexity of a rule and render it more simple to study. In this section,we discuss the contributions that qualitatively or quantitatively estimate theeffects of asynchronism with numerical simulations.
In 1984, Ingerson and Buvel carried out a pioneering work where they couldshow that the behaviour of simple rules could be totally disrupted by simplemodifications of the updating [19]. Most importantly, they questioned to whichextent was the behaviour of a rule the consequence of the local rule and to whichextent it was due to the updating scheme.This question was re-examined by Bersini and Detours, who explored the dif-ference between the Game of Life and closely related asynchronous variants [14].Their main observation was the existence of a “stabilising effect” of asynchronousupdating. The experiments were made on small-size grids, no larger than 20by 20 cells. With such lattice sizes, they were able to observe that the fullyasynchronous Game of Life may “freeze” on some fixed-point patterns with alabyrinth-like aspect. However, more recent work has demonstrated that it wasnot possible to infer the large-size behaviour from these experiments and thatthe stabilising effect was intimately linked to finite-size effects of the numericalexperiments [15, 36].Schönfisch and de Roos gave a decisive impulse to the research on asynchro-nism by comparing various updating schemes and by exhibiting clear exampleswhere the schemes alter significantly the behaviour of a rule [113]. They gavea detailed analysis of the statistical properties of the schemes but their experi-ments were limited to some specific rules. The question thus remained open toknow how these observations could be generalised to a larger class of rules.On this basis, Fatès and Morvan examined how the 256 Elementary cellularautomata (ECA) reacted to α -asynchronism [41]. To estimate the changes ofbehaviour of the system quantitatively, the authors used an approximation ofthe asymptotic density, that is, the value of the density that would be reachedby an infinite-size system with an infinite simulation time. This parameterwas considered as a first means to detect changes in the behaviour: a strongvariation of the asymptotic density indicates that the system has undergone atransformation while an absence of variation does not necessarily imply that thesystem remained stable.The 256 rules were then classified into four qualitative sets according to theirresponses to the variation of the synchrony rate α : (a) continuous variation ofthe behaviour (e.g. ECA ), (b) discontinuity around α = 1 (e.g. ECA or ), (c) phase transition for a critical value α c < (e.g. ECA ), and (d) non-regular behaviour (e.g. ECA ). One of the surprising results of this studywas that no direct correspondence could be drawn between these new classesof robustness and the previously known classes of synchronous behaviour (e.g.,the informal Wolfram classes).Similar observations were made by Bandini et al., who tested the effects ofnumerous asynchronous schemes on one-dimensional binary rules where the local6unction depends only on two neighbours (also called “radius-1/2” rules) [9]. Blok and Bergersen were the first authors to identify the change that occurs inthe Game of Life when cells are updated with a given probability [15]. Theyused α -asynchronism to show the existence of a qualitative transition from a“static” behaviour, where the system would settle on fixed points, to a “living”behaviour, where the system evolves by forming labyrinth-like patterns that donot fixate. The change of behaviour is a second-order phase transition, thatis, the change of behaviour that separates the two qualitatively different phasesobey some well-known laws from statistical physics. In this case, the phasetransition was shown to belong to the directed percolation universality class,which means that at the critical point, the evolution of the order parameters(e.g. the density) obeys the same power laws as an oriented percolation processthat serves as a reference.Fatès identified that similar phenomena occurred in Elementary Cellular Au-tomata and that no less than nine rules also displayed phase transitions. It wasshown that the density follows a power-law decay for the critical synchrony rate,in good agreement with the behaviour expected from the directed percolationuniversality class [34]. A unique rule, namely ECA , was shown to belong toanother universality class, a fact that is explained by the symmetric role that isplayed by s and s in the transition rule.The phase transition occurring in the Game of life was also re-examinedby studying how this phenomenon was affected by perturbations of the topol-ogy [35, 36]. The main finding was that the critical value of the phase transitionstrongly depends on the regularity of the grid and that the qualitative change ofbehaviour becomes more difficult to observe as links between cells are removed.Concerning other two-dimensional rules, Regnault et al. carried out a pio-neering work by explaining in detail how the asynchronous minority rule dis-played various types of behaviour depending on the topology on which it is ap-plied [97, 99, 105]. A simple puzzling observation is that the minority rule willsettle out on a checkerboard or on a stripe-like pattern depending on whetherthe rule is defined with the von Neumann or the Moore neighbourhood. To ourknowledge, there is no mathematical explanation of this empirical observation.Remark that two different complementary views exist on phase transitions:the most common way of describing a phase transition is to establish that for an infinite system, a qualitative difference of behaviour occurs for an infinitesimalvariation of the control parameter. An alternative approach was adopted byRegnault who could prove that for a particular rule and a finite system, thetransition corresponds to a variation of the convergence time from a linear to apolynomial function of the system’s size [98].7 .3 Coalescence A curious phenomenon was remarked when comparing the evolutions of twodifferent initial conditions that were updated with the same local rule and the same sequence of updates: a rapid “coalescence” may occur, that is, the twosystems take the same state and then evolve with the same trajectory (as thesame sites are updated).This phenomenon is in some cases easy to understand, as when the coa-lescence occurs on an attractive fixed point, but it was also observed for anon-fixed-point region of the state space (as for ECA [41]). From a morepragmatic point of view, the following interpretation can be given: there areasynchronous systems whose evolution rapidly becomes governed by the randomnumber generator that dictates the updates, and not by the initial condition.Rouquier and Morvan studied systematically the coalescence phenomenonfor the 256 ECA [102, 104]. Their study revealed that it was possible to ob-serve that some ECA would always coalesce, while others would never coalesce,and that there existed some rules which displayed a phase transition betweena coalescing and non-coalescing behaviour. It is an open problem to explainanalytically the non-trivial cases of rapid coalescence. It is also interesting tocompare these results with those obtained by other methods of coupling (seee.g. [109, 103]). While the approaches of asynchronism studied so far are based on the dichotomyupdated versus not updated, Bouré et al. defined a model of asynchrony whichconsiders imperfect communications between neighbours [17, 16]. This approachis declined in two versions, called β - and γ -asynchronism, which respectivelyconsider stochastic failures of the communication of a state to the whole neigh-bourhood or to each neighbour independently.Among the various observations made with these two types of asynchronism,the most intriguing phenomenon is the disappearance of some, but not all, of thephase transitions that were obtained with the α -asynchronism. More precisely,ECA , and do not show any transition for β - and γ -synchronism. ECA gives an even more puzzling case as it does show a phase transition for α - and β -asynchronism but not for γ -asynchronism. It is an open problem tounderstand the origin of such radical differences of responses to the rate oftransmission failures.Experiments also displayed that in some cases, quantities can be conservedwhen using only a particular model of asynchronism (e.g., ECA has someparity conservation with β -asynchronism but not with α -asynchronism). Thisunderlines the necessity to continue to “invent” various perturbations of the clas-sical updating in order to gain insight on how cellular automata are dependenton their updating schemes. 8 .5 Other variants An interesting development on the work of asynchronism concerns how it mixeswith traditional noise, that is, on randomness imposed on the state of the cellsthat compose the automaton. An early reference that addresses this question isgiven by Gharavi and Anantharam, who revisited a well-known result of Toomand who considered delays in the cells’ communications [49]. We refer to thework of Kanada [59] on the 256 ECA rules, and to the work of Mamei et al. [75]for additional insights into this problem of mixing noise and asynchronism.More recently, Silva and Correia gave a detailed account on how some ECAcan react to asynchronism combined with noise [118]. Interestingly, they proposeto evaluate the robustness according to the difference patterns. This bringsthem to introduce a sampling compensation in order to cope with less frequentupdates.The case of asynchronous models simulated on a non-regular topology wastackled by Baetens et al., who examined an asynchronous updating with a non-regular topology generated with a Voronoi tessellation [8].To conclude this section, it seems that only a small part of the universe ofasynchronous cellular automata has been explored so far. This brings us to putan emphasis on the following questions:
Questions 2
What is a good protocol to numerically estimate the changes ofbehaviour induced by asynchrony? What are the relevant order parameters toquantify these changes? How common is it to observe discontinuities of be-haviour induced by a continuous change of the updating scheme?
We now turn our attention to the mathematical analysis of asynchronous cel-lular automata. It is important to remark that although this part is presentedseparated from the previous one, there is a joint movement of going from sim-ulations to analysis and back. (This co-development is not necessarily done bythe same authors of course.)
Agapie et al. conducted one of the first analytical studies of asynchronousrules using Markov chain theory [5]. They focused on several models of finitecellular automata with fully asynchronous updating. However, as far as wecould understand, their analysis was limited to a specific case where the localrule was stochastic, totalistic, symmetric with respect to an exchange of sand s, and with positive rates (the probability to reach each state is strictlypositive). It is worth noting that the number of borders of a configuration isa central parameter in their analysis and that this parameter is also found invarious other approaches. 9ne of the first analytical results of classification were given by Fatès etal. who analysed the doubly-quiescent ECA [42]. In this study, the 64 rulesconsidered are classified according to their worst expected convergence time toa fixed point. This time falls in the following classes: logarithmic time, lineartime, quadratic time, exponential time and non-converging rules . The visualinspection of the space-time diagrams of the rules of each class shows a goodcorrespondence between the visual “behaviour” and the class. In other words,the time of convergence to a fixed point is not an ad hoc parameter but doescapture a part of the “behaviour” of the stochastic rules.These results were later partially extended to the more difficult case of α -asynchronous updating by Regnault et al. [43], while Chassaing and Gerin exam-ined the continuous limit of the processes when the grid was made infinite [23].Fatès and Gerin also examined how to classify the two-dimensional totalisticrules with fully asynchronous updating [40]. They proposed a partial classifi-cation of 64 rules and an analysis of the convergence of some well-known rules.Among the interesting phenomena remarked, they exhibited a list of rules whichshowed an “erratic” behaviour: the question was to determine if these rules wereexhibiting a non-converging behaviour or a “metastable” behaviour, that is, if a(long) random sequence of updates could drive the system to a fixed point. Byadapting techniques from automatic planning, Hoffmann et al. could solve thisproblem for a specific rule and showed that it converged to a fixed point in (atmost) exponential time [55].Readers interested in the classification of rules with regard to their conver-gence time can refer to a recent synthesis note [38] and a recent work on thefast convergence of the ECA rules [39]. As mentioned above, for the α -asynchronous systems, the study of the asymp-totic density was mainly made with numerical simulations. By focusing theirefforts on eight simple ECA rules, Fukś and Skelton succeeded to give an ex-act calculation of this density [47]. They considered infinite systems where theinitial condition was generated by a Bernoulli measure and determined how theasymptotic density varies as a function of the initial density (that is, the param-eter of the Bernoulli measure). Such exact results are generally rather difficultto obtain and it is an open problem to extend them to a wider class of rules.Following this direction of research, Fukś and Fatès considered a develop-ment of Gutowitz’s “local structure theory”: contrary to a classical mean-fieldapproach where the state between neighbouring cells is assumed to be uncorre-lated, correlations of order 2 or larger are taken into account to try to predictthe asymptotic density of the system [46]. It was shown that this approachdoes detect the occurrence of phase transitions. The limit is that the positionof the critical synchrony rate remains difficult to find: for some rules, even ap- The classes are here given with a “rescaled time scale” where one step corresponds to asmany random updates as there are cells in the finite ring.
As mentioned above, the asynchronous updating of a system does not perturb itsfixed points. However, when the updating is stochastic cycles no longer exist andone needs to re-examine the meaning of reversibility. One such interpretationwas proposed by Das et al., who define reversibility as a possibility to returnto the initial condition in the case where the updating sequence (or “updatepattern”) could be set freely. They studied which are the Elementary CellularAutomata with null or periodic boundary conditions that allow to obtain sucha form of “cycles” [110, 27].Another point of view considered the case of fully asynchronous updating:as the evolution of the system is adequately described by a Markov chain, re-versibility is identified with the property of recurrence of this chain [116]. Aclassification of the ECA rules into three classes was then proposed based onthis tool: (a) The recurrent rules are those which make the system always returnto its initial condition. (b) The irreversible rules are those which contain initialconditions which are never returned to after a (random) time. Among this class,(c) the strongly irreversible rules are those which contain a state that is neverreturned to as soon as it is updated. It is an open problem to determine how toextend these results to a wider class of systems, in particular to deal with thecase of infinite-size systems.Wacker and Worsch also examined the question of reversibility of asyn-chronous cellular automata [134]. In their work, a rule is said to be reversibleif there is another rule whose state-transition graph is the “inverse” of the origi-nal. The novelty with respect to the synchronous case is that the out-degree ofthe nodes is no longer equal to one as a single configuration can lead to manyothers. Interestingly, the results presented on ECA are not far from those foundin Ref. [116] and it is an open question to determine which are the conditionsthat make the two points of view equivalent.
The study of the dynamical properties of cellular automata, such as injectivity,surjectivity, permutivity, etc., has been a central topic in the theoretical con-siderations of the field (see e.g. [60]). Manzoni examined how these propertiescould be re-defined and studied in the asynchronous updating context [76].This work was taken a step further by Dennunzio et al., who developed thenotion of m-asynchronous cellular automata in order to generalise the variousupdating methods used so far [30]. They provided a formal framework to de-scribe the updating probabilities on each cell, even in the case where the sizeof the system is infinite, and produced various theorems that allow to deal withthe non-deterministic nature of the updating.11or more details on this line of research, we refer to the recent survey byFormenti where more details and examples can be found [45].To synthesise, the contributions met in this section show the necessity toadapt the tools to the stochastic process theory for the specific case of cellularautomata. This brings us to ask:
Questions 3
What is the position of asynchronous cellular automata with re-spect to stochastic cellular automata? (a mere subset?) What are the analyticaltools that can ease the analysis of the Markovian systems obtained with randomupdates?
We now consider the contributions related to the computing abilities of asyn-chronous models and briefly describe the techniques that have been proposedto construct such (virtual or real) computing objects.
Nakamura was among the first authors to investigate how to compute with anasynchronous cellular automaton [81, 82]. He described several techniques toconstruct a universal rule and showed how to simulate a given q -state deter-ministic rule with an asynchronous rule that has the same neighbourhood andwhose state space is extended to q states (see also Lipton et al. [70], Tof-foli [126] and Nehaniv [84] for similar constructions). The construction relieson the idea that when a cell is updated, it then waits the neighbouring cells to“catch up” and makes the next transition only when all its neighbours are up todate. Additionally, it keeps its old state available for the neighbouring cells inorder for them to perform the “right” transitions. This construction was laterimproved by the use of only q + 2 q states by Lee, Peper et al. [67, 89].Peper et al. also proposed to consider the case where a cell can “activate”their neighbouring cells and showed that the cost in the number of states forthe simulation of q -state rule could be reduced to O ( q √ q ) states [90].Other discussions on the universality of asynchronous rules are found inthe study by Takada et al., in which many important arguments and usefulreferences can be found [124]. In particular, the authors present a result showingthe existence of an asynchronous, rotation-symmetric rule with 15 states andvon Neumann neighbourhood that has the property of universal constructionand computation.An alternative point of view was given by Golze who simulates an n -dimensionalsynchronous rule with an asynchronous rule defined on a space with n +1 dimen-sions [50]. This solution simplifies the problem as there is no longer the need tosave the previous and the current state in order to achieve correct computations.12nother advantage of having an additional dimension is to read one state of thesynchronous simulated system (guest) on the asynchronous simulating system(host): it simply corresponds to reading a line (or a hyperplane) of the host.This technique, called “global synchronisation”, is presented as a means to solvevarious problems, such as the Firing Squad Synchronisation Problem, whichwould not be solvable without this requirement. However, it can be noted thatthis technique can be interpreted as the “deployment” of Nakamura’s techniqueon an additional dimension. Reciprocally, one can also see Nakamura’s tech-nique as the “compressed” version of Golze’s solution, where only the necessaryinformation is retained.The case where asynchronous computations have to be made with stochastic and asynchronous components was tackled by Wang [135]. Unfortunately, thisauthor does not position his work with regard to the previous contributions(Nakamura, Golze) and it is difficult to see if this proposition significantly differsfrom the previous achievements.An original way to simulate a universal Turing machine with a fully asyn-chronous updating has also been proposed by Dennunzio et al. [29]. The authorsintroduce the notion of “scattered strict simulation” in which they tolerate thatonly a subset of cells is used to perform the simulation. They find that asyn-chrony induces a quadratic slowdown compared to the speed of the simulatedTuring machine. A key observation in the theory of asynchronous systems relies in the propertyof what we could call “non-overlapping influences”: if two cells c and c (cid:48) aresuch that the neighbourhood of c and c (cid:48) do not overlap (that is, have no cellin common), it does not matter whether c is updated before c (cid:48) , or c (cid:48) before c ,or both of them are updated at the same time. The study of this property hasgiven birth to various works that we now examine.Gács was one of the first authors to determine if the evolution of an asyn-chronous system could be independent of the order of updating [48]. He showedthat although this property was undecidable, there exists a sufficient conditionto verify this independence.This question was later re-examined by Mortveit, Macauley et al., who stud-ied in which cases repetitions of sequential updates on Elementary Cellular Au-tomata (ECA) could produce a set of periodic points that would be independentof the updating order [73, 74, 72]. This conducted the authors to present a list of104 ECA which display such an update independence. Their work also uses anoriginal representation of ECA that differs from the classical Wolfram code andthat could prove useful for future analysis of asynchronous systems. (Anothernotation is presented in Ref. [33, 42]).Order-independence was also a key point considered by Worsch, who ex-amined how to simulate an arbitrary rule by a universal asynchronous simu-lator [136]. He extended Golze’s results by tackling a large scope of updatingpolicies: purely asynchronous (no restriction on the set of cells to update), α -13synchronous, N-independent (where two neighbouring cells are never updatedat the same time), and non-deterministic fully asynchronous. He showed thatfor each such policy, there is a universal rule (the host) that can “simulate”,in a particular sense, any other guest. Worsch’s work raises many questions,in particular as to how to properly define the notion of simulation of an asyn-chronous rule by another. (See Ref. [7] for some reflexions made in the contextof stochastic cellular automata.)We also point out that Vielhaber has designed a formal framework in whichthe computations of functions on finite binary rings ( Z /n Z ) are made not bychanging the local rule but by a proper use of the order of updating on a fixed rule [132]. In particular, he showed that ECA with periodic boundaryconditions was a rule especially adapted for such a purpose. Interestingly, thistechnique could be generalised to make this particular rule Turing-universal inthe sense that the computation of an algorithm could be done only by settingup the proper sequence of updates. Among the early references that can be found on asynchronous cellular au-tomata, Priese wrote a note where he considers (two-dimensional) cellular au-tomata as a particular case of asynchronous rewriting systems (called Thue-systems) and widens the scope by considering also the case where more thanone cell may be re-written at a time (the overlapping problem) [93]. He uses hisconstruction to show how to build asynchronous circuits which are equivalentto asynchronous concurrent Petri nets.Following this path, Zielonka examined how asynchronous rules could beused to describe the situations of concurrency that arise in distributed sys-tems [25]. Pighizzini clarified the computing abilities of Zielonska’s models [92]and the problem of how to turn non-deterministic Büchi asynchronous cellularautomata into deterministic models was solved by Muscholl [80]. Droste gen-eralised to partially ordered multisets (pomsets) the original notion of Zielon-ska’s asynchronous mappings [31]; these questions were later re-investigated byKuske [32, 61, 62].With similar preoccupations, Hagiya et al. used formal methods from logicto verify the properties of some rewriting systems, showing the links betweentheir approach and (a)synchronous systems [53].
Another major field of research on asynchronous cellular automata was devel-oped by Peper, Lee and their collaborators. In their constructions, asynchronouscomputations are realised with particles that follow Brownian movements andwhich interact through special “gates” [4, 3, 66, 91]. These constructions re-sult in delay-insensitive circuits that are Turing-universal (see e.g. [64, 69] andreferences therein). 14ecently, Schneider and Worsch presented a 3-state rule that uses Mooreneighbourhood which can simulate any delay-insensitive circuit [112] and Lee etal. presented a generalisation of their work in the context of number-conservingcellular automata [68].To end this section, we propose to put an emphasis on the following ques-tions:
Questions 4
What is a good definition of the simulation of an (a)synchronoussystem by another (a)synchronous system? What are the techniques to simulatevarious asynchronous systems by other asynchronous systems? (e.g.: When canan α -asynchronous system with a given synchrony rate simulate another systemwith a different synchrony rate?) Asynchronous rules have been designed for specific goals such as finding newalgorithms, developing new types of computing devices, modelling various nat-ural or artificial complex systems, etc. In fact, giving a representative view ofthese contributions would necessitate a whole independent survey. The task isall the more difficult as often authors use asynchronous updating without evenmentioning it. For the sake of brevity, we will thus only give a few entry points,concentrating on the papers where the question of the updating is explicitlydiscussed.
As mentioned earlier, the hypothesis of perfect synchrony poses the problem ofthe realism of a model: How to interpret the behaviours that are only due to theupdating and not to the rule that governs the cells? Huberman and Glance gaveevidence of the existence of such “artifacts” and challenged the validity of thesimulations of spatially-extended models of the Prisoner’s dilemma: a changein the updating models brings out new conclusions, drastically opposed to whatwas known with the classical models [58]. This question was re-examined byNewth and Cornforth who showed that asynchronism could also lead to the ob-servation of new cooperative phenomena not seen in the synchronous setting [86].(See also Ref. [85] for a non-spatially-extended version of the problem.)Grilo and Correia also considered this problem but instead of restrictingtheir study to the fully asynchronous scheme, they employed α -asynchronousupdating to explore a wide range of degrees of synchrony [51]. Their studyrevealed that the changes induced by smooth variations of the synchrony ratemay brutally affect the level of cooperation in the system, a behaviour that isstrongly reminiscent of the second-order phase transition seen in binary systems(see Sec. 3.2). Saif and Gade also investigated this issue and found that therewas a first order transition between a regime with an all-defector state to amixed state [107]. All these works share in common the conclusion that some15reviously observed equilibrium states are artifacts of a synchronous updatingon a regular lattice.Ruxton and Saravia have discussed the importance of the ordering in thecontext of ecological modelling, studying a stochastic model of colonisation of anenvironment by a species [106]. They argue in favour of adapting the updatingscheme to the physical reality of the system that is modelled. The authorsalso emphasise the need to describe precisely the updating scheme that is usedto facilitate the reproducibility of the experiments. These arguments come tostrengthen the need for studying in detail the “emergence phenomena” that areseen in Ecology and question whether the predictions of the models can beobserved in “real-life” systems [100, 22]. In an approach close to the work of Turing on morphogenesis [130], Gunji usedasynchronous cellular automata to analyse the pattern formation mechanismsthat occur in molluscs [52]. Another interesting biological example is given byMessinger et al., who investigated the link between emergence of synchrony andthe simultaneous opening and closing stomatal arrays in plants [78].In Physics, we mention the work of Le Caër [63] and Radicchi et al. [94], whostudied how numerical simulations of cellular systems would be dependent onthe updating. In the latter work, the local rule is itself stochastic; the authorsemphasise the fact that neither totally synchronous nor totally asynchronousupdating is fully relevant for modelling natural systems.
Tomassini and Venzi [127], Capcarrere [21] and Nehaniv [83] have studied howasynchronous rules solve the density classification problem and the global syn-chronisation problem. Readers interested in this issue are referred to a studyby Vanneschi and Mauri, in which an enlightening discussion on these variouscontributions is found and where the authors present findings of robust andgeneric rules [131].Suzudo examined the use of genetic algorithms to find mass-conservative(also called number-conserving) asynchronous models that would generate non-trivial patterns [122, 123]. He classified these patterns into three categories:checkerboards, stripes and sand-like. In this work asynchronism is mainly usedto ensure that number of particles remain constant, but it is also a useful tech-nique for generating regular patterns out of randomness: this task is known tobe difficult in the synchronous setting (see e.g. [37]).Beigy and Meybodi investigated how asynchronous systems perform learningtasks and presented applications of their work for pattern generation and controlof cellular mobile networks [12].It is also worth mentioning that Lee et al. [65] and Huang et al. [57] designedmodels of self-reproduction that use asynchronous models (also called self-timedcellular automata). 16 .4 Other problems
On the simulation side, Overeinder and Sloot were among the first to exam-ine how to deal with the simulation of asynchronous automata on distributedsystems [88]. Bandman and other authors studied how to simulate chemicalsystems with asynchronous cellular automata [11, 117]. Hoseini et al. made animplementation of asynchronous rules with FPGAs [56]. They propose a partic-ular design of the FPGA in order to construct a “conformal computer”, that is,a computer made of physical cells “arrayed on large thin flexible substrates orsheets. Sheets may be cut, joined, bent, and stacked to conform to the physicaland computational needs of an application”.Original applications were considered by Bandini et al., who used asyn-chronous rules with memory for the design of an illumination facility [10] andby Minoofam et al., where asynchronism produce calligraphic patterns in theArabic Kufic style [79]. (Unfortunately, this paper lacks precision on the modelthat is used).As we have seen in this section, there is a broad range of domains whereasynchronous models have been employed and those which we cited above areonly a small part.
Questions 5
How can we develop a unified simulation environment to facilitatethe comparison of various updating schemes? Is there a method for identifyingthe artifacts that are due to a perfect synchronous updating? Are such effectsavoidable?
We end this guided tour on an opening on the use of asynchrony in the systemswhose structure is close to cellular automata. Again, this is such a wide topicthat we will indicate only a few entry points to the literature.
One first proposition to link the updating in multi-agent systems and cellularautomata was made by Cornforth et al., but the models they studied are in factstandard asynchronous cellular automata [26]. Spicher et al. considered thequestion of how to “translate” a multi-agent system with sequential updatinginto a synchronous cellular automaton [119]. So-called transactional cellular au-tomata were defined to model the movements of particles between neighbouringcells. One positive effect of using a synchronous cellular automaton is to removethe spurious effects that could be linked to a particular updating order. (Theauthors give the example of diffusion-limited aggregation.)17he link between large-scale multi-agent systems and asynchronous cellularautomata was also examined by Tošić [129]. This author argues that the struc-ture of cellular automata needs to be modified in several aspects, among whichit should be made asynchronous, in order to serve as a basis for modelling largegroups of interacting agents.An alternative approach to model (discrete) multi-agent systems was pro-posed by Chevrier and Fatès, who studied the dynamics of a simple multi-turmite systems, also known as multiple Langton’s ants. Their formalism, in-spired by cellular automata, captures the possibility to have synchronous in-teracting agents [24]. The difficulty relies in describing how to solve conflictsthat occur when two or more agents simultaneously want to modify the environ-ment. The solution relies on a framework invented by Ferber and Müller called influence-reaction [44]. Belgacem and Fatès later extended this work by con-sidering a wider range of updating procedures and discovered some phenomena(e.g., gliders) that resisted variations in the updating choices [13].Interesting observations were also made by Şamiloğlu et al. who analysedthe clustering effects in a group of self-propelled particles [108]. They modelasynchronism with the introduction of delays in the updating and observe thatthe coherence of the groups are strongly diminished as the bounds on the delaysare increased.
Lattice-Gas Cellular Automata (LGCA) can be seen as a “bridge” between cell-based updating and agent-based updating. Applying asynchrony in this contextis not a straightforward operation and a first proposition of an asynchronousLGCA was made by Bouré et al. [18]. In their model, movements of particlesare defined explicitly, like in multi-agents, but the updating is made cell by cell,like in classical cellular automata. Various responses to asynchrony are observeddepending on the patterns on which the system stabilises. In particular, strangepatterns such as checkerboards are shown to disappear where randomness in theupdating is added. It is an open problem to know if an infinitesimal amount ofasynchrony is sufficient to destroy this pattern.These first results show the need to explore various possibilities to define anasynchronous LGCA. In particular, it is interesting to look at a way to updateparticles independently.
The effect of asynchronous updating in genetic regulatory networks has alsobeen investigated by many authors. Aracena et al. introduce a labelled directedgraph that allows to determine to which extent deterministic update schedulesare equivalent [6]. Demongeot et al. [28], and Noual [87] examine the robustnessof the system under the variation of updating schemes and this perturbation is18oupled with various topological modifications of the network such as adding orremoving links in the graph or changing boundary conditions.The question of the effect of the updating in neural networks has been dis-cussed by Scherrer [111], Taouali et al. [125]. In particular, the latter authorsintroduce an interesting distinction between the use of (a)synchronous updatingat the modelling level and at the implementation level.In the context of “amorphous computing”, Stark discussed the computingabilities of a computing medium formed out of non-regularly placed cells whichobey asynchronous updating [121, 120]. This author suggests that asynchronyplays an enhancing role for the computing abilities of such systems.We mention that the differences between synchronous and asynchronousupdating were also investigated in coupled map lattices [71, 101, 1]. Similarly,the effects of the updating in the Asymmetric Exclusion Process (ASEP) havebeen studied by Rajewsky et al. [95]. Tomita et al. studied asynchronousgraph-rewriting systems and showed how to make such systems simulate theirsynchronous counterparts [128].To end this section, we wish to highlight the following questions:
Questions 6
What light can be shed by asynchronous cellular automata onother closely related models and vice versa? Can we transfer the techniquesused to analyse the simple asynchronous cellular systems to more complex mod-els? What is the interplay between the regular topology of cellular automata andthe regularity of their updating?
This guided tour allowed us to consider the various contributions that deal withthe question of asynchronism in cellular automata and closely related models.As we have seen, asynchrony is a privative property that does not in itself specifya system: there are plenty of ways to construct an asynchronous system and allof them are a priori valid.One of the main current challenges is to continue to explore this questionwith a joint work of mathematical analysis and numerical simulations. As wehave seen, analytical results have been more difficult to obtain than numericalones, but the situation is progressively changing as more techniques from theprobability theory are being developed for the specific case of cellular automata.We find it rather amazing that it is still an open question to determine theconvergence time of some simple binary rules [38, 39].It is also important to clarify the position of asynchronous cellular automatainto the wider field of stochastic cellular automata. Indeed, asynchrony is nota mere type of noise: recall for example that the addition of asynchrony to adeterministic model does not change its fixed points. However, many phenomenasuch as the existence of singularities or phase transitions can certainly find theirexplanations using the stochastic process theory and statistical physics.19s far as modelling is concerned, the main challenge would be to carry outan experimental work to validate some models of asynchronism or to dismisssome others for specific situations . As we mentioned earlier, the no-global-clockargument — “Nature does not possess a clock to synchronise the transitions.”— cannot be received directly and be taken alone as a valid objection to the useof synchronous models. Instead, we consider that studying a single updatingscheme is not sufficient and one should instead compare various possibilities tomodel a “natural computing” system.The principal observation from this guided tour is the existence of a greatvariety of approaches to asynchronism. This raises the question of what is timein the context of computer science and numerical simulations. The positivesciences define time as an object – identified with R , with Z , a collection ofcoordinates, etc. – but it may well be that time is not some “thing” that can bestudied “objectively”.Does this mean that time is subjective and that our models should reflect thissubjectivity? Such considerations would lead us out of the scientific method andwould therefore be dismissed as non rational. Can we then escape the dilemmaof “objective versus subjective time”? No simple answer can be given and forsure, time is one of the central problems of philosophy. It is certainly not acoincidence if one the most important philosophical contributions of the pastcentury bears as title: Sein und Zeit ( Being and Time ). Acknowledgements
May all the persons that have contributed to the elaboration of this text receiveour sincere expression of gratefulness. The author knows how much he owes tothem, including the organisers of Automata’13 and their precious support, andthe anonymous referees, whose constructive propositions were greatly appreci-ated. In echo to Heraclitus’ word that “the invisible harmony is greater thanthe apparent one” (DK B54), he hopes that they will pardon him for not beingnamed here. The author is aware of the limits of this text and he will be gratefulto all the corrections, indications, remarks, and suggestions that will be givento him.
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