Noise and Intermediate Asynchronism in Cellular

Automata with Sampling Compensation

Fernando Silva1, Luís Correia

1,

1 LabMAg, Dept. Informatics, Faculty of Sciences,

University of Lisbon, Portugal

{fsilva, luis.correia}@di.fc.ul.pt

Abstract. Cellular Automata (CAs) are a class of discrete dynamical systems

that are widely used to model complex systems in which the dynamics is

specified locally at cell scale. In its classic definition, CAs performs with

perfect synchronism. However, this does not stand for what happens at a

microscopic level for physical and biological systems. Recent research has

studied the CAs behavioral consequences of using intermediate levels of

asynchronism, where only a fraction of the cells is updated at each time step. In

this work we examine, in addition to intermediate asynchronism, the impact of

different levels of noise, a perturbation that causes a cell to randomly change its

state when it is updated. To conclude, we explore an observation mechanism in

which sampling normalizes the updating differences introduced by

asynchronism. Results show that this method reduces CAs behavior into

two classes, chaotic and fixed-point. Explanations for observed behaviors are

proposed.

Keywords: Asynchronism, Noise, Cellular Automata

1 Introduction

Cellular Automata are a common approach to model the dynamics of real-world

(biological, physical) systems, such as pedestrian movement [3], the interaction

between tumors and immune system [4], prey and predators, host and parasites [2],

and even the development of pigment patterns in mollusks [1]. In its classic

definition, CA consists of a regular lattice of cells where each adopts one of a finite

number of states. Single cells update their state according to a rule, the update rule,

which depends on the cell environment, i.e. its neighborhood. The evolution of a CA

is performed by iteratively applying the update rule to the cells on the lattice.

The most common update method in CAs contemplates perfect synchronism. This

means that, at each time step, all cells update their state simultaneously. However, this

practice has been widely questioned since it does not reflect what happens at a

microscopic level for physical and biological systems. In these systems, perfectly

timed elements are absent even when a strong and regular synchronization component

exists. For example, it is known that tropical fireflies can routinely synchronize their

flashes among large groups, a phenomenon explained as phase synchronization [5].

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Also, the referred update method presupposes the existence of a global

synchronization mechanism that can coordinate the cells‟ instantaneous update.

Given that, our motivation for this work consists on establishing a first step in the

direction of modeling real-world complex systems by studying the effects of different

levels of noise and intermediate asynchronism update methods on the evolution of

cellular automata. We use a one-dimensional, nearest neighbor, two-state CA and

vary two main parameters on the system‟s dynamics, the cells update and noise

probabilities, which respectively control the CA‟s intermediate asynchronism and

noise levels

The next section includes a description of related work. In Section 3, we provide a

formal definition of the used CAs models and methods. After that, in Section 4, we

present results of CA behavior when using different synchrony rates. By changing this

rate, it is possible to use quite different approaches to model the dynamics of complex

systems. For a CA made by N cells, the rate can vary from 1 (perfect synchronism)

down to 1/N (sequential intermediate asynchronism, in which, only one cell is

updated at each time step). This protocol allows not only to study the CA behavior at

extreme cases, synchronous and sequential asynchronous updates, but also to evaluate

the behavior manifested in between, where most of real-world systems‟ dynamics

seem to lie. In Section 5, we study the impact of the above mentioned spatial

perturbations on the evolution of CAs and show that intermediate asynchronism can

be seen as a form of perturbation through time. In Section 6, we explore an

observation mechanism in which sampling normalizes the updating differences

introduced by asynchronism. Contrary to what one might expect, this approach does

not exhibit system behavior closer to synchrony but instead reveals that asynchronism

can reduce the CAs behavior into two classes, chaotic and fixed-point. We end this

section by providing explanations for the observed behaviors. Finally, in Section 7,

conclusions are drawn and future work is presented.

2 Related Work

It has been shown that different CA update methods alter the system‟s behavior.

These can be divided into two classes, synchronous and asynchronous. Asynchronous

update methods may be purely sequential, ranging all the way from completely

random orders to sequential left-to-right sweeps. There is also intermediate

asynchronism, the one approached in this work, where only a fraction of the cells is

updated at each time step. Asynchronous methods can strongly affect the exhibited

behavior when comparing to the one revealed by the use of synchronous ones.

In [6], the authors performed the first qualitative study of the impact of

asynchronism. Also using a one-dimensional, nearest neighbor CA model, the authors

have compared the properties of a synchronous and two asynchronous models. The

two latter consisted on a random iteration scheme in which, at each time step, the cells

iterate randomly, one at a time, and an independent clock scheme with cells also

iterating one at a time, each having its own definite period. The main focus of the

work was to understand and estimate how much of the CA‟s remarkable behavior

comes from synchronous modeling and how much is intrinsic to the iteration process.

Their research showed that some of the properties exhibited when using the

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synchronous update method, namely some apparent self-organization, are

artificialities introduced by the global clock phenomenon.

In 1994, Bersini and Detours studied an asynchronous version of two-dimension

cellular automata, the Game of Life (GL], a model that belongs to Wolfram‟s

empirical class IV (complex behavior) [7]. By adding to this model some of the

properties associated with immune network model and an asynchronous update

method, they observed that asynchrony was the key factor for inducing stability in

game of life type of simulations, transforming its behavior into a typical class I CA,

fixed-point, and thus removing its dynamic complex characteristics. Although the

results revealed that variants of the GL with IMN-like thresholds were much faster to

freeze than purely asynchronous GL, they were of great importance since raised the

question if either all members of the class IV CA could lose their dynamical richness

by increasing the level of asynchrony. Also, it was another demonstration that

asynchronism really does alter CA‟s behavior in some very interesting ways.

In [8], Fatès and Morvan made an exhaustive study of the one-dimension CA rules

space. The aim was to explore the robustness to asynchronism for cellular automata

and, by using intermediate asynchrony, trying to understand to which extent the CA

behavior depends on the synchrony of the transitions. The most significant point of

interest work was to determine if results shown by Bersini and Detours applied to

most of class IV CA or were just specific to the models presented. In reaction to this

subject of great deal, it was demonstrated that there is no universal answer to the

question of knowing what part of the interesting behavior of classical CA is due to

synchronism and that each modeling problem should be studied with a specific

approach. Additionally, the followed methodology allowed to define robustness

classes according to the type of changes observed when asynchronism is added in the

update rule and suggested that “a CA model might be robust enough to produce the

same output when evolved with perturbations”. As for what our work is concerned,

we continue this line of thought and examine not only the consequences of

intermediate asynchronism in CA dynamical characteristics but also compare it with

the effects of perturbing the configuration, i.e., how the system reacts to localized

spatial changes, or noise.

3 Formal Definitions

In this section, we formally define the notions of asynchronous cellular automata

(ACA) and asynchronous cellular automata with noise (ACAN).

3.1 Asynchronous Cellular Automata

An Asynchronous Cellular Automata (ACA) can be defined as a 5-tuple

),,,,( mfGQL , as follows:

A cell is the most elementary component of a CA. It is basically a variable that

takes values in Q, the set of possible states.

The set of all cells is called lattice, is denoted by L and can be in any finite number

of dimensions.

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The neighborhood of a cell, Neigh(c), is a function that associates c to a set of cells

located nearby. This neighborhood is constant, i.e., does not change through time.

f: QN → Q is the local transition rule and defines the conditions in which each cell

updates its state, taking into account the state of the cells that constitute its

neighborhood.

m: N → P(L) is the update method and defines, at each time t, the set of cells from

the lattice to which the local transition rule will be applied and the subsequent

updating order.

The structure is homogenous, i.e., every cell c has the same type of neighborhood

and the same local transition rule.

The distinction made between synchronous and asynchronous cellular automata is

due to the kind of updating discipline used. This is said to be synchronous if

Ltmt )(: and all cells perform this operation simultaneously. On the other

hand, asynchronous methods exists when ),()(: tLPtmt , i.e., at each time step

only a subset P(L, t) of the cells is updated (intermediate asynchronism) or all cells

are updated but in a specific order and thus the process is not simultaneous. Update

methods can be time-driven, in which time is explicitly defined, or step-driven

(discrete time) [10]. We focus our work on the latter because in this purpose of this

work both models are equivalent, it is the updating order that matters. Also, from all

existing step-driven methods, we restrict our study to asynchronous stochastic

dynamics, as in [8]. This method which we denoted by mα, is defined by, at each time

step, considering all cells of L and assigning an equal probability α that each cell is in

m(t), thus satisfying a fair sampling condition. The parameter α is called the

synchrony rate and takes values of the interval ]0, 1]. Within this context, one can

view a synchronous CA as a special case of an ACA for which α = 1.

3.2 Asynchronous Cellular Automata with Noise

The act of assigning a state to each cell in L is called a configuration. In order to

study the impact of localized changes on the system‟s orbits, i.e., the set of following

configurations, we have extended the previously presented model. We added a

parameter ɸ that allowed controlling the level of functional perturbation to the cell

state, when it is updated. Analogously to the synchrony rate this parameter ϵ ]0, 1]

and is called noise rate.

The parameter ɸ determines the probability of a cell to error the calculation of the

new state when it is updated. Since we use two-state CA (see 3.3), these perturbations

can only occur in two ways, from 0 to 1 or vice-versa.

3.3 Elementary Cellular Automata

In this paper, we restrict our study to one-dimensional cellular automata, in

particular to the elementary case. This model refers to the simplest class of CAs. Each

cell has two possible values, Q = {0, 1}, and local transition rules depend only on

nearest neighbors‟ states. This means the evolution of elementary cellular automata

(ECA) can be represented in a space-time diagram where time is represented

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vertically and configurations are represented horizontally (Fig. 1). This diagram

allows to easily examining the system‟s orbits and thus the CA exhibited behavior.

As referred in [11], every ECA f can be described by a code, according to the

employed local transition rule:

( ) ( ) ( ) ( ) . (1)

An ECA having the code R = W (f) is denoted by ECA R.

Fig. 1. Space-time diagram of ECA 22 made by 300 cells and evolving for 200 generations.

Time goes from top to bottom.

3.4 Measurements

As previously mentioned, the purpose of this work is to compare the effects of

different types of perturbations, either temporal (asynchronism) or spatial (noise). To

gain some insight on how these aspects alter the CA behavior, there was the need to

go further than the simple visual analysis of the orbits. In order to do this, we

developed a correlation factor based on Hamming Distance measure which we

denoted by ρ* and whose purpose was to measure the correlation coefficient between

two different CA configurations, c1 e c2. Given the Hamming distance (Dh) between

c1 e c2, [13], and their cardinality N:

⁄ (2)

Since ρ* can take values of the interval [-1, 1], different qualitative correlation levels

were established (Table 1).

When comparing two models, we also analyze the relation between the CAs

cardinality, i.e. number of cells, and the required generations for models to reach a

certain correlation level. These data provide a mechanism to directly relate the three

involved variables: the perturbation level, the number of cells in each CA and how the

first affect the latter through time, in a short and long term perspective.

Table 1. Qualitative levels for the Hamming correlation.

Absolute value Correlation Level

[0.9, 1] Very High

[0.7, 0.9[ High

[0.5, 0.7[ Moderate

[0.3, 0.5[ Low

[0, 0.3[ Non-existent

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4 Study on the Effects of Different Synchrony Rates

In this section, we analyze the exhibited CA behavior for different synchrony rates.

In particular, ECAs are divided according to Wolfram‟s empirical classes. This

division has the advantage of facilitating the search for CA‟s common behavioral

responses to intermediate asynchronism. For each class, we examine how the

dynamical system evolves and characterize it based on the resulting orbits. In

particular, we show that general class I and II CAs are less susceptible to

asynchronism and that class III and IV are disrupted in a relatively small number of

generations.

The experimental results were obtained by testing the behavior of several rules,

initial random conditions (the assignment of the initial state to a cell is an

equiprobable operation), lattice size from LS = [500, 5000] (500) and synchrony rate

[1/LS, 1.0].

4.1 Class I and II

Class I revealed the higher level of robustness to intermediate asynchronism. This

class is called a fixed-point and is characterized by a very simple behavior, in which

almost all initial conditions lead to exactly the same uniform state. The demonstrated

stability concedes the system a very robust mechanism to deal with intermediate

asynchronism.

Figure 2 shows the evolution of ECA 160 for different synchrony rates. Results

show that lowering the synchrony rate for class I CA delays the convergence to the

stable state. In particular, this ECA is α-invariant approximately down to a synchrony

rate of 0.5.

Fig. 2. Evolution of a 500 cell ECA 160 with random initial conditions. From top to bottom,

synchrony rate of α = 1 and α = 0.4.

In class II CAs, the exhibited behavior is characterized by the periodic repetition of

a certain set of simple structures that appear and either remain the same forever or

repeat every few time steps. We now examine the behavior of two class II systems,

ECA 232, an ECA version of the “Majority Vote Rule”, in which the most common

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state in the neighborhood is the one the cell adopts, and of ECA 58. In a synchronous

version, ECA 232 has a short period and ECA 58 presents a long period.

Using ρ* as the measure unit, we iteratively evolved and compared two models, a

synchronous and asynchronous with a certain value of α, both starting from the same

randomly generated initial configuration. Notice that when comparing the evolution

of two or more models, either quantitatively or qualitatively, they need to be

submitted to the same set of initial conditions in order for the comparison to be

accurate. In this case, both systems are stopped when a non-existent correlation level

or a threshold of 50000 generations is reached. To avoid artifacts introduced by a

certain initial configuration and inaccurate data, we performed 30 runs for each test

case. For a more appropriated comparison, it was also measured the degenerative

generations (DG), i.e., the number of generations elapsed since the first perturbation

appeared until a non-existent correlation threshold (ρ* < 0.3) was reached.

Table 2. Results for a 500 cell ECA 58 performing with high levels of synchrony

α Degenerative Generations Standard Deviation for DG

0.99999 32707.67 10156.8

0.9999 4063 1378.714

0.999 376.33 165.56

0.99 63.33 15.37

0.9 11.33 0.57

Experimental results show that although belonging to a periodic class, ECA 58 can

only tolerate a certain level of asynchrony. These results are arguably due to the long

period it presents and to the fact that when there is a fault in a cell, it is contained and

only spreads in a much localized way, requiring a very high number of generations to

be completely uncorrelated to the synchronous version. Decreasing the synchrony rate

allows for more faults and consequently, for more localized perturbations to appear.

As for ECA 232, it reached the upper threshold maintaining a high or very high

correlation with the synchronous model, even varying the synchrony rate from 1

down to a sequential asynchronous update (Fig. 3). The manifested behavior has to do

with the fact that the most common state in the neighborhood is the one a cell adopts.

This property attracts the CA to a certain set of configurations, in which the effects of

different synchrony rates are overcome by the system‟s evolution.

Fig. 3. A 500 cells ECA 232‟s correlation values when reaching the upper threshold.

0

0,5

1

1

0,99866…

0,94666…

0,89066…

0,88133…

0,87733…

0,80133…

Synchrony Rate

ρ* at generation 50000

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4.2 Class III and IV

In this section, we approach ECA classes for which a minimal introduction of

asynchronism results in massive behavioral changes and very fast disassociation with

the synchronous model. Collected data show both classes have high susceptibility to

asynchronism.

Fig. 4. The behavioral consequences of asynchronism for three 500 cell ECA 22. From left to

right, α = 1, α = 0.9999, α = 0.5.

In class III and IV CAs, as can be seen in Fig. 4, introducing asynchronism makes

the structures, an apparent of form of self-organization, to dissolve. These results

corroborate those presented in [6], where these structures are characterized as artifacts

of the synchronization of the clocks phenomenon, i.e., perfect synchronism. The

difference in the orbits of the asynchronous dynamical system makes it to rapidly

achieve a stage of non-correlation with the synchronous one. Experimental data

(Table 3) have shown that even with high levels of synchrony, the non-correlation

happens extremely rapidly and that lowering this rate fastens the process.

Table 3. Degenerative generations„ count for several 500 cell ECA – Rule (Class).

α ECA 22 (III) ECA 90 (III) ECA 54 (IV) ECA 110 (IV)

0.99999 435.5 496.7 894.8 1430.23

0.9999 255.5 348.1 438.3 552.53

0.999 101.53 141.77 151.73 159.2

0.99 33.13 39.37 34.13 43.97

0.9 8.23 9.73 7.37 10.6

5 Introducing Noise

In this section we shift the methodology used so far and focus our study on the

impact of errors in the updating function of cells. First, an analysis is performed on

the effects of a single localized perturbation. After that, the system is subjected to

several noise rates ɸ. Finally, we compare the results with those obtained in the

previous sections.

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5.1 Sensitivity to Local Perturbations

This zone consists of a preliminary study on how different classes deal with

localized changes in their configuration. The task is performed by introducing a

random state-change in one of the cells in the initial state. The random change is

justified by the fact that a perturbation can occur in both ways, either changing the

cell state from 0 to 1 or the opposite. The purpose of this activity is to measure how it

affects the system, i.e., determining if as with intermediate asynchronism, it

propagates and recursively affects the neighborhood, is contained or if it eventually

dies out.

Determining how the CA reacts can be done by using ρ*. If the perturbation dies

out or is contained, we‟ll have a constant correlation value between a system with the

imposed error and the one without it. On the other end, if the perturbation spreads to

the system, after a certain threshold both models will no longer be correlated.

Although with a different purpose and a quantitative measure, what is done here is

similar to what has been performed in [12], where the sensitivity to initial conditions

is tested.

The results are rather different for each class. In class I, the spatial perturbation

always dies out and the same uniform final state is reached. In class II, the

perturbation remains contained to a small portion of the space and thus does not

spread to the rest of the system. However, as with intermediate asynchronism, classes

III and IV are also more susceptible to this type of changes. Figure 5 shows the

evolution of a class III (complex) and class IV (chaotic) CAs. In class III (Fig. 5 left),

the perturbation spreads at a constant rate, recursively affecting the entire system.

This means that for a CA of cardinality N, it will be necessary approximately N

generations for the entire system to get affected. In class IV, the perturbations also

spread but in a much more sporadic way (Fig. 5 right).

Fig. 5. From left to right, the propagation of a perturbation in the initial state for a 500 cell

ECA 30 (chaotic) and ECA 54 (complex), respectively.

5.2 The Impact of Different Noise Rates

As previously mentioned, class I and II systems can forget or contain the lag

introduced by intermediate asynchronism and noise. Given that, we focus the rest of

this work on the impact of different noise rates on the evolution of class III and IV

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CAs. In this zone we examine the responses of systems in both these classes to noise,

based on a noise rate ɸ that determines the probability of a cell changing its state

when it is updated. Notice that intermediate asynchronism restricted the set of cells to

which the update method was to be applied. With this noise rate, all cells are updated

but after this operation they may change their state due to the above mentioned rate.

We use the ρ* measure to obtain the correlation level between a noisy and a noise

free system. In order to study only the effects of this parameter, a synchronous update

method is used. Results show that by introducing only a minimal noise rate, the

system quickly languishes and loses its noise free behavior.

Table 4. Degenerative generations for 500 cell ECAs with various noise rates – Rule (Class).

ɸ ECA 22 (III) ECA 90 (III) ECA 54 (IV) ECA 110 (IV)

0.00001 385.33 487 1099 1261

0.0001 159.33 296.67 524.67 408

0.001 72.67 98 108 97

0.01 24.33 27.33 29 26.33

0.1 5 6.33 5.67 6.67

As previously stated for intermediate asynchronism, class IV systems exhibit a

greater value for the degenerative generations‟ count, which means that they tend to

resist more to noise than class III. The latter systems are more susceptible to this kind

of perturbation. The comparison of both experimental data related to intermediate

asynchronism and noise (Table 3 and 4) reveal that systems subjected to both kinds of

perturbations languish at the same pace, when in comparison with a synchronous

system. These behavioral consequences tend to suggest that although being

perturbations of different kinds, the impact they have on the systems evolution is, in

practice, the same. This is arguably due to the way each class of system handles

information.

In class I, perturbations that occur are forgotten and the system quickly evolves

into the uniform final state. In class II, partial information about previous stages of the

system is retained and thus perturbations remain in localized spaces and are never

communicated to the other parts of the system. As referred in [12], class III “show

long range of communication”, justifying the fact that any change introduced into the

system, either a cell that is not updated or a state-change of a cell will spread and be

communicated even to most distant parts of the system. Since the local transition rule

is based on the state of the nearest neighbors, perturbations recursively affect the

nearest neighbors until it reaches the entire system. As for class IV, they present a

very interesting behavior. Being a somewhat intermediate between class II and III, the

mentioned above long-range of communication is possible in principle and it does

occur but in a sporadic way, which leads to a greater resistance to perturbations than

in class III systems.

In this context, asynchronism can be seen as a form of noise through time,

inhibiting the cells from updating their state and thus creating perturbations that,

depending on the system‟s class, may affect the overall behavior. This means that

inducing a localized perturbation in the configuration will affect the system in the

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same way intermediate asynchronism does when the updating process is not supposed

to maintain the cell‟s state.

6 Sampling Compensation of Asynchronism

Intermediate asynchronism is based on a sampling scheme in which every cell has

the same probability of being selected for update. However, this mechanism implies

that, at each time step, only a fraction of the cells is updated. In this section, we

explore an observation mechanism in which sampling normalizes the updating

differences introduced by asynchronism. This method, denoted as compensatory

sampling asynchronism, can be defined as follows.

For a synchrony rate α, at each time step, 1/α updates are performed. For example,

if α = 0.1, 10 intermediate asynchronism steps are executed, the resulting

configuration is considered as the system‟s next stage and is the one presented in

space-time diagram. This method has the advantage of, from a theoretical point of

view, allowing an average of one update per cell at each time step and thus

normalizing the visualization differences introduced by asynchronism in the updating.

An a priori intuitive analysis may indicate that results produced by this update

method, either visual (orbits) or analytical (at a correlation level), should be

somewhat close to the ones produced by a synchronous scheme, although with some

lag due to its asynchronous basis. To test this hypothesis, we perform an analysis of

the behavioral differences induced by this method when compared to intermediate

asynchronism and perfect synchronism. Both visual and analytical data are examined.

Table 5. Compensatory sampling vs. Intermediate asynchronism. Degenerative generations for

several 500 cells ECAs – Rule (Class).

α ECA 30 (III) ECA 150 (III) ECA 54 (IV) ECA 110 (IV)

0.99999 420.73 299.13 738.67 1177.4

0.9999 217.17 182.93 326.2 367.43

0.999 69.07 67.2 94.93 102.9

0.99 21.97 18.63 20.27 24.03

0.9 4.6 4.4 4.27 5.07

Table 6. Compensatory sampling vs. Perfect synchronism. Degenerative generations for

several 500 cells ECAs – Rule (Class).

α ECA 30 (III) ECA 150 (III) ECA 54 (IV) ECA 110 (IV)

0.99999 493.53 343 885.33 1859.77

0.9999 273.5 242.33 441.77 533.13

0.999 104.07 91.9 140.23 145.7

0.99 28.03 26.83 28.8 36.07

0.9 5.9 5.23 6.03 6.5

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Results show that, although not a significant difference in some cases,

compensatory model performs closer to perfect synchronism that intermediate

asynchronism (Table 5 and 6) since it is required a higher number of generations for

the system to reach a non-existent correlation level. This means that contrary to what

one might expect, compensatory asynchronism is not too far from the intermediate

levels. In order to further understand this phenomenon, it is necessary to examine the

system‟s evolution and, consequently, its orbits.

Fig. 6. From left to right, the evolution of the same initial state for a 500 cell ECA 150

(chaotic) using respectively the synchronous, intermediate asynchronism and compensatory

observation mechanism. α = 0.6.

Examining the exhibited behavior, intermediate asynchronism (Fig. 6 center)

completely flattens the structures that usually appear in the synchronous update

method (Fig. 6 left). In the compensatory scheme (Fig. 6 right) these structures appear

but present a lesser degree of deformation. This difference is due to the latter

normalizing the sampling differences introduced by asynchronism and thus exhibiting

a somewhat intermediate behavior between this one and perfect synchronism.

Fig. 7. Behavior of a 500 cell ECA 54 (complex). From left to right, synchronous update and

compensatory observation (α = 0.6).

One important consequence of the compensatory scheme is that below a certain

threshold of asynchrony, in which sampling requires for a greater normalization level,

this update method reduces the demonstrated behavior into two categories, namely

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some class II systems are attracted to a fixed-point, after a disordered stage, and class

IV lose their combination of order and randomness and starts to exhibit a completely

chaotic behavior (Figs. 7 and 8, respectively).

Fig. 8. Behavior of a 640 cell ECA 58 (periodic). From left to right, compensatory mechanism

with α = 1, α = 0.8, α = 0.6 and α = 0.1, respectively.

7 Conclusions

In this paper we analyzed the behavior of one-dimensional binary Cellular

Automata with a neighborhood of radius one, under asynchronous update and with

noise in the updating function. Results indicate that asynchronism may be considered

as a kind of noise and that inducing a localized perturbation in the configuration will

affect the system in the same way intermediate asynchronism does when the updating

process is not supposed to maintain the cell‟s state. This has important implications

when modeling natural systems, which do not have perfect synchronism. By using

synchronous CAs we introduce an artificiality that may produce results seldom found

in nature.

We have also introduced a sampling compensation to observe CAs under

asynchronous updates. Commonly CA states are displayed for each updating, which is

not fair for asynchronous updates. In case of systems where cells all have similar

updating periods, a synchronous updating should be approximately equivalent to N

asynchronous updates of individual cells. The sampling compensation shows that

asynchronous updates are more similar to the synchronous, although differences are

still noticeable due to the modification of the updating.

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In the future we intend to study modifications to the updating function that may

turn it into a compliant model, where different updates are self-adjusted. As

mentioned above, there are natural systems where local updating disturbances do not

propagate in a way to modify the global behavior of the system. We also will extend

this study to two-dimensional cellular automata and to more complex models, with

more than two states and larger neighborhoods.

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