Permanent Errors May Contribute to Emergent

Behavior in One-Dimensional Cellular Automata

Ludek Zaloudek

Faculty of Information Technology

Brno University of Technology

Brno, Czech Republic

Email: izaloude@seznam.cz

Abstract—This paper describes the possibility of increasingthe complexity of behavior of one-dimensional cellular automatawith two states. The mechanism is based on simulating permanenterrors which may occur in hardware implementation of cellularautomata employed e.g. in Artificial Life. Complete exploration ofsimple 3-neighborhood is conducted and the change of behavioris illustrated in changes of Wolfram’s classification of saidautomata. Several 5-neighborhood examples of similar behaviorare provided to show the consistency of complexity-enhancingbehavior in different type of one-dimensional cellular automata.

Index Terms—Cellular automata, defects, emergence, Wolframclasses.

I. INTRODUCTION

Cellular automata (CA) have been used for a long time

for the purposes of Artificial Life and many other areas

such as chemical and behavioral simulation, random number

generation, art etc. During the decades, CA have been regarded

mostly as a model for computation in simulated environment.

With the advent of nanotechnology, we start to look at CA

from a different perspective that has been indicated from the

very beginning in von Neumann’s work on self-replicating

machines: hardware.

CA can be viewed not only as abstract computing machines

to be evaluated in our computers. With new technology un-

foreseen in the 50s, we can actually fabricate these massively

parallel machines in silicon and hopefully, we will be able to

do so in different media soon.

However, actual fabrication of CA-based computing systems

brings new challenges. The two most notable are (a) the

problem with imperfect synchronization in large cellular arrays

and (b) the presence of errors [7]. This paper does not

deal with problem (a) but shows some interesting discoveries

regarding the presence of errors, more precisely defects.

Important feature of CA which is linked with Artificial

Life is emergence. If we accept the assertion that ‘The

key concept of Artificial Life is emergent behavior,’ [4] we

should welcome any mechanism that can contribute to such

behavior. This paper shows that this can be achieved even

under the conditions, where the CA contains several permanent

errors (defects) and that such errors have not only destructive

influence as anticipated but in several cases can contribute

to increase of the complexity of patterns generated by CA.

This may perhaps enrich the viewpoint on Artificial Life being

computed on actual hardware which is prone to defects.

Behavior of CA may be classified by so-called Wolfram

Classes. These classes are used in this paper to more clearly

indicate the change of behavior of one-dimensional CA in the

presence of permanent errors.

The paper is organized as follows: Section II briefly de-

scribes the cellular automata and Section III introduces the

Wolfram classification scheme. Section IV outlines the ex-

periments made with one-dimensional CA and presents the

way the errors were generated. Section V presents selected

results from these extensive experiments. The paper ends with

discussion and conclusion.

II. CELLULAR AUTOMATA

A cellular automaton is a d–dimensional grid of cells, where

each cell is a finite automaton. The cells operate according to

their local transition functions (rules). Usually, the cells work

synchronously – a new state of every cell is calculated from its

previous state and the previous states of the cell’s ‘neighbors’

at each time step. By configuration of the cellular automaton

we mean the states of all the cells at a given moment. The

sequence of configurations, determined by the global transition

function, represents the computation of the cellular automaton.

In theory, the CA model operates with an infinite number of

cells. However, in the case of practical applications the number

of cells is finite. Then, it is necessary to define the boundary

conditions, i.e. the setting of the boundary cells. One of the

states is also usually used as quiescent or inactive state. By

convention, when a quiescent cell has an entirely quiescent

neighborhood, it will remain quiescent at the next time step.

Even a simple one-dimensional uniform CA, with only two

states and nearest neighbors neighborhood N = {−1, 0, 1}(only left and right neighbor cells together with the cell itself

are relevant for the local transition function), can exhibit very

complex behavior [8]. Each such CA is uniquely defined

by a mapping {0, 1}N → {0, 1}. Hence there are 28 such

cellular automata, each of which is uniquely specified by the

(transition) rule i (0 ≤ i < 256). The number of states is

usually denoted as k, the neighborhood radius is denoted as

r. Thus, the simple nearest neighbors neighborhood, two-state

CA can be specified as k = 2, r = 1.

57978-1-4577-1124-4/11/$26.00 c©2011 IEEE

In case of two-dimensional cellular automata, the neighbor-

hood usually comprises of five or nine cells. However, this

paper deals almost exclusively with one-dimensional CA and

thus more detailed description is unnecessary.

In order to specify the transition rules by simple codes, the

designation originating from Wolfram’s work [9] can be used

which denotes the number that comes from the new states’

values when the transition rule set is sorted by neighborhood

values in descending order. In case of k = 2, r = 1 CA, the

sequence goes from 111 to 000, in case of k = 2, r = 2 (five

cell neighborhood) from 11111 to 00000.

III. WOLFRAM CLASSIFICATION

One of the ways to determine the quality of a CA is

classification based on the behavior of the CA. The best known

classification scheme was developed by Wolfram [8].

All 1D CA can be divided into the 4 so-called Wolfram

Classes: homogenous fixed point (class I), periodic (class II),

chaotic (class III) and complex (class IV).

Class I is the simplest and it is the only one easily distin-

guishable of the four: The CA evolves into a configuration,

where all cells are in one of the possible states and no further

change is possible, i.e. in our case of two-state CA, either all

cells are in 0 or in 1.

Class II evolution leads to a state of separated simple stable

or periodic structures. In one of the more known attempts to

improve the Wolfram Classification, class II was separated into

three different classes [5]. In order to avoid confusion, these

classes can be labeled as IIa, IIa and IIc. CA in IIa evolve into

spatially inhomogenous fixed points or an uniform global shift

of fixed patterns. IIb represents periodic behavior or shifted

periodic behavior and IIc represents locally chaotic behavior.

Class III CA exhibit chaotic behavior where almost all initial

conditions lead to aperiodic chaotic patterns. Good example

of this is the well-known rule 30.

Class IV encompasses the most complex CA. It is cha-

racterized by long transient and complex space-time patterns

including both oscillating and propagating structures. CA in

this class balance on the edge between chaos and order which

makes them capable of unique computation. It has been shown

that class IV CA are capable of universal computation [9].

Generally, the borders between the classes are blurred, not

only because the original definition is vague. For example it

is hard and maybe even impossible to distinguish class III

and class IV using any statistical or numerical analysis for

sufficiently large CA. Several attempts were made [1], [2], [3],

[5], [6] to refine the classification. However, these methods

do not seem conclusive and as Wolfram himself stated [9],

it is not possible to predict class IV behavior in other way

than complete simulation. Moreover, mentioned methods do

not consider the introduction of defects, which applies also to

Langton’s λ parameter [4].

In light of this, CA evolutions from experiments with

permanent errors conducted for this paper are classified only

by author’s own cognitive processes, as it is not the purpose

of this paper to introduce a new method of CA classification.

Several examples will be shown in later sections so the reader

may consider the problem for himself.

IV. THE EXPERIMENTS

A. One-Dimensional Cellular Automata

This paper deals mainly with 1D CA. In order to show

two different rule sizes, k = 2, r = 1 and k = 2, r = 2automata were chosen. In the spirit of minimizing confusion,

the rules for 3-neighborhood will be denoted with index 3

and the 5-neighborhood will be denoted with index 5, e.g.

rule 303. Coding of rules corresponds with Wolfram’s scheme

mentioned in Section II. Also, 3-neighborhood rules will be

called ‘simple rules’ from this point on.

There are exactly 256 simple rules. All of them were

tested with permanent errors and compared with Wolfram’s

classification without errors which may be found in his book

[9]. These automata were 128 cells wide, used cyclic border

conditions and were seeded either with single black cell (state

1) at position 64 or with random initial conditions.

Seeding with single black cell may seem as an arbitrary

decision but that is not the case. As seen on some rules (e.g.

903, 1023, 1053), such seeding may cause interesting behavior

(Fig. 1). Such runs are listed as a special case in Section V.

The 5-neighborhood has more than 4 billion rules so it

practically impossible to assess all of them. Only a selection

from the 5-neghborhood range was used to prove the concept

in slightly different conditions.

Important feature of the experiments is the fact that these

were done only in simulation in special software. The goal of

this paper is not to investigate how the errors occur but how

the CA react to them. Moreover, neither the author nor his

department possess any hardware implementation of CA.

B. Generating Errors

Before describing, how the errors were generated, the type

of errors must be identified. In hardware systems, generally

two types of errors may occur: transient and permanent.

The first type is caused mostly by surrounding environmental

influences, e.g. radiation. Such errors last only limited amount

of time. Yet, even such temporal errors may cause faults. As

we strive to simulate CA from the viewpoint of hardware,

it is not possible to predict the exact behavior of transient

errors without such hardware, be it silicon-based circuit or

chemically synthesized future technology.

Focus of this paper is on permanent errors or in other words

– defects. Defects are commonly caused by imperfections in

the material or fabrication process, they may also be caused

during the lifetime of the computing system by physical

damage. Such errors may manifest as permanently dead or

alive (i.e. state 1 for 2-state CA) hardware elements. The

value of permanently damaged cell may also fluctuate in time.

However, such behavior is similar to transient errors.

Fluctuating cells also resemble experiments conducted by

Wolfram [9] on perturbation. These experiments were based

on randomly changing values of states independently on the

local transition function. The result was that many CA can

58 2011 Third World Congress on Nature and Biologically Inspired Computing

Fig. 1. 63 steps of rule 903 seeded with a) a single black cell, b) withrandom initial conditions. Note that at step 64, both versions stabilize to zeroacross all cells.

exhibit robustness in their behavior even in presence of several

perturbations (typically class III).

The defects in CA space in experiments for this paper were

generated randomly with uniform distribution. Several runs

were conducted for each rule with different initial conditions

(i.e. seeded by single black cell or randomly) and with

different random seed each run. More interesting cases were

tested more intensively and with higher number of CA steps

(thousands as opposed to hundreds).

Each defect manifests itself as a dead cell, i.e. there is

permanent 0 state in it. There is no need to test permanent

1 because the second half of simple rules is just a reversed

mirror image of first half (0 and 1 states’ roles are reversed).

The defects were generated with uniformly distributed pro-

Fig. 2. Rule 543 stabilizes into short periodic patterns after 49 steps withthis particular random initialization

bability of 1.5 %.

V. RESULTS

The automata in this section are often referenced by rule

code and Wolfram’s Class. All simple rule classes may be

found in [9]. There is though a problem with some assigned

classes which lies in their definition. Some rules show chaotic

behavior of class III but only for a limited number of steps.

This is caused by the original assumption that cellular space

is infinite. In this paper, cyclic boundary conditions have been

defined. Good example is rule 903 from Fig. 1, which puts

the automaton to a quiescent configuration after a number of

steps, falling into class I with the defined boundaries.

A. Destructive Effects

Most simple rules are class-wise robust and there is no

change in classification in the presence of errors. However,

if there is one, the most common effect of permanent errors

is a destructive one. Many CA just drop to a class of lower

complexity. Most of such rules belong to class II and the

behavior with errors drops to class I. This is the case for rules

(2, 10, 16, 24, 48, 52, 66, 80, 112, 130, 138, 152, 162, 170,

176, 184, 194, 208, 226 and 240)3.

Examples of 5-neighborhood rules are more scarce but they

can be found, e.g. (68, 76, 100, 104, 108, 5000, 28204)5 etc.

Degradation from class III to II includes (30, 86, 135, 149,

183 and 195)3. Rules in italics could be disputed as still being

in class III, however examination shows they are complex class

II with a long period of repetition.

There are also some special rules which fall in different

classes depending on the number of errors. In some cases,

these rules keep their old class but in other, they degrade.

This includes rules (41, 45, 137, 166, 180)3.

Degradation in class IV automata is not so easily recogniz-

able for some rules and it is arguable, whether there is actually

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Fig. 3. Rule 1693 from steps 4960 to 5000. The position of the defect isshown by the small arrow at the bottom.

Fig. 4. Wide gaps caused by defects cause very simple class II-like behaviorwith rule 2343. The defects are indicated by small arrows at the bottom.

any degradation at all, because after several hundred steps,

the behavior falls in class II with or without errors. Some of

these “class IV” rules fall into class II sooner, some later (see

Fig. 2). Example of rule 543, shows, that in most tested cases,

the class IV behavior lasted longer in the presence of defects

– in some cases more than 1500 steps. Somewhat rarer case

is degradation from IV to III, e.g. with rule 3265.

Nevertheless, there are some class IV rules which clearly

degrade into class I such as (106, 120)3 or class II such as

(110, 124)3 and e.g. 3705.

There are some surprisingly complex class IV rules which

show robustness even after 5000 steps with a defect (see Fig. 3)

and show no major degradation.

B. Complexity Enhancing Effects

The really interesting behavior with defects happens when

class I and II rules start to behave as one class higher CA.

Special case is false enhancement of complexity for some of

the highest code rules (249–255)3. This is again caused by the

definition of Wolfram Class I, which says that class I is a rule

which changes the cellular space into a constant homogenous

array of static cells. Because the defects are simulated as

constant 0, the cellular space is not homogenous so it falls

into class II.

It is arguable if actual increase of complexity by changing

class I behavior into class II can be seen with rules (234, 235,

238, 239 and 248)3. Illustration of such behavior is depicted

in Fig. 4.

TABLE ICHANGES IN WOLFRAM CLASSIFICATION FOR 256 SIMPLE RULES WITH

THE INTRODUCTION OF DEFECTS

Number of changes from class ..

.. to class I II III IV

I – 22 0 2II 5 – 7 2III 0 2 – 0IV 0 2 0 –

If we concur that change of class II to class III or class IV

is an increase in complexity, such behavior appears to emerge

with rules (97, 107, 109, 154 and 210)3. Some patterns formed

in later steps indicate this might be even class IV behavior.

For example, the behavior of clean rule 973 stabilizes into

shifting periodic patterns, falling into class II. Fig. 5 shows

rule 973 with one error, where class IV resembling behavior

is observed. The same applies for rule 1073.

Rule 1093 does not show tendencies to be class III or IV,

merely a very complex class II, therefore it does not count.

That changes with a special initialization by a single black

cell (see next subsection) which leads to class III.

Rule 1543 exhibits class III properties with defects, so

does rule 2103. Interesting thing about these rules is that

reported behavior appears only in the presence of 2 or more

defects and degrades significantly with the presence of more

than 5 defects. The degradation turns the CA into class II.

However segments separated by defects continue to oscillate

with various periods thus creating something like class III in

global view of the whole CA. The periods depend on the width

of the segment. Such segments were reported earlier and they

are called macrocells or membranes [6].

Interesting structures are formed by rule 2705. These appear

to be formed by random streams of 1s generated at the time

of the initialization colliding with defects and multiplying into

other streams thus creating complex behavior (Fig. 6).

Changes in CA behavior mentioned above are summarized

in Table. I.

C. Automata with Special Initial Condition

There is number of rules which behave very differently

when initialized by a single black cell. Most of these rules

exhibit what Wolfram calls “nested” behavior [9]. Typical

example is on Fig. 1a). Simple rules with nested behavior

are (18, 22, 26, 60, 82, 90, 102, 105, 126, 129, 146, 150, 153,

154, 161, 165, 167, 181, 182, 195, 210 and 218)3.

Of those listed rules, only (82 and 154)3 increase their class

from II to III.

There are also rules which do not exhibit nested behavior but

increase their class when initialized with a single black cell:

(73, 82)3 fall into class III, possibly IV only with 2 or more

defects, (107, 109)3 show random class III patterns with class

IV-like appearance (shortly propagating patterns), 2183 shows

clear class III behavior (in macrocells) with more than 1 defect.

(235, 239, 249–255)3 produce false complexity enhancement

described in previous subsection.

60 2011 Third World Congress on Nature and Biologically Inspired Computing

VI. DISCUSSION

Previous sections have shown how the presence of defects

in 1D CA increases the complexity of patterns generated by

such CA. That is clear by comparison with spatiotemporal

diagrams of same CA without the presence of defects.

What is not clear is the classification of such behavior. Since

Wolfram Classes are not so clearly defined, the classes of

such new behavior are at least arguable. That is not surprising

considering that some CA’s classes could be disputed even

without permanent errors under different boundary conditions.

Good example is the classification of rule 903, which is class

III according to Wolfram [9] but could be easily classified as

class I with cyclic boundary conditions because after varying

number of steps, the CA’s evolution stabilizes in a homogenous

state (Fig. 1).

These borderline cases seem to be prominent in experiments

yielding increased complexity described here. Identifying the

reason behind this could be an incentive for further study.

Reactions to defects in 1D CA with respect to complexity

can be divided into three basic cases: decrease of complexity,

no change and increase of complexity. Simple rules react

mostly with no change or by decrease of complexity. Decrease

may be either class-changing or just within one class which

impacts mostly class II, since it has actually several subclasses

with different level of complexity [5].

Reactions by complexity increase are often conditional. E.g.

it is necessary to have more than 2 or less than 5 defects. These

conditions vary with the width of the CA. Another condition is

the position of defects: As seen in Fig. 6, some structures may

pass ‘unharmed’ trough defects, some may trigger the creation

of new structures or be destroyed based on the position of

certain states relative to the defect. This is especially true for

5-neighborhood because the structure could cross one defect

gap since the neighborhood’s radius is 2.

Another point for discussion is that one may notice the

walls between macrocells described at e.g. rule 2103 function

as null boundary conditions. Wolfram stated that boundary

conditions do not have any significant effect [8]. In the exper-

iments described here, the walls created by defects prevent the

stabilization of some patterns which would otherwise happen

when cyclic boundary conditions apply. This prevention of

stabilization is based on a constant supply of 0 states into an

environment of stabilized shifting patterns which leads to their

destabilization and emergence of chaotic behavior. This may

be observed when employing null boundary conditions with

certain CA. However, the author believes, such behavior was

not particularly emphasized in previous works.

VII. CONCLUSION

It has been shown that defects (permanent errors) may cause

new emergent behavior in 1D cellular automata. The change is

manifested either as not significant, complexity decreasing or

complexity increasing. Rough estimation of Wolfram Classes

was used to demonstrate the shifts in complexity.

From 255 simple 3-neighborhood rules of 2-state CA, 33

rules were affected negatively in such a strong way that

they lost complexity by lowering their Wolfram Class. Not

included in this number are those rules which lost complexity

within their class. Surprisingly, 9 rules showed complexity

increase demonstrated by raised Wolfram Class. The increase

has been also presented by some examples of spatiotemporal

CA diagrams. Some examples have also been shown with 5-

neighborhood 2-state CA.

It has been shown that when initialized with a special

condition, i.e. a single black cell, some simple rules show an

increase in complexity in the presence of defects, even when

random initialization does not (2 cases – 733, 2183).

Only rough estimates of Wolfram Classes have been made.

There are various statistical and mathematical methods which

can more or less accurately determine the exact Wolfram

Classes. However, these methods are not quite ready for the

introduction of defects. Still, for this paper, classification was

not deemed necessary, because the interest was on complexity

of behavior, where classification served only as a support to

illustrate the major complexity shifts.

The author concludes, that from the viewpoint of imple-

menting Artificial Life or similar emergent phenomena in

hardware, supplying said hardware with defects (or ignoring

normally occurring ones) could provide a new mechanism for

emergence. Different perspective might also bring new ideas

into the field of cellular automata or Artificial Life because

errors generally are rarely considered in these areas.

Future work possibly includes refining the classification

(using some method modified for defects), finding more

examples in 5-neighborhood 2-state CA, increasing number

of states in our experiments and even expanding them beyond

one dimension. Different types of errors (e.g. transient) may

be relevant with some actual hardware implementation.

ACKNOWLEDGMENT

This work was partially supported by the grant Natural

Computing on Unconventional Platforms GP103/10/1517, the

FIT grant FIT-11-S-1 and the research plan Security-Oriented

Research in Information Technology, MSM0021630528.

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2011 Third World Congress on Nature and Biologically Inspired Computing 61

Fig. 5. Rule 973 with one defect (indicated by a small arrow at the bottom).Cutouts are up to 2000 steps.

Fig. 6. 167 steps of rule 2705. Note the two small arrows at the bottomindicating defects. At their locations, diagonal streams of double 1s collidingwith the defects merge existing streams, or new streams are created.

62 2011 Third World Congress on Nature and Biologically Inspired Computing