Revisiting the Edge of Chaos
Evolving Cellular Automata to Perform
Computations
Melanie Mitchell Peter T Hrab er and James P Crutch eld
Santa Fe Institute Working Pap er
Submitted to Complex Systems
Abstract
We present results from an exp eriment similar to one p erformed by Packard in
which a genetic algorithm is used to evolve cellular automata CA to p erform a particular
computational task Packard examined the frequency of evolved CA rules as a function of
Langton s parameter and interpreted the results of his exp eriment as giving evidence
for the following two hyp otheses CA rules able to p erform complex computations are
most likely to b e found near critical values which have b een claimed to correlate with
a phase transition b etween ordered and chaotic b ehavioral regimes for CA When CA
rules are evolved to p erform a complex computation evolution will tend to select rules with
values close to the critical values Our exp eriment pro duced very di erent results and
we suggest that the interpretation of the original results is not correct We also review and
discuss issues related to dynamical b ehavior classes and computation in CA
The main constructive results of our study are identifying the emergence and comp etition
of computational strategies and analyzing the central role of symmetries in an evolutionary
system In particular we demonstrate how symmetry breaking can imp ede the evolution
toward higher computational capability
1
Santa Fe Institute Old Pecos Trail Suite A Santa Fe New Mexico U S A
Email mm santafe edu pth santafe edu
2
Physics Department University of California Berkeley CA U S A
Email chaos go jira b erkeley edu
Intro duction
The notion of computation at the edge of chaos has gained considerable attention in the
study of complex systems and arti cial life e g This notion is related
to the broad question What is the relation b etween a computational system s ability for
complex information pro cessing and other measures of the system s b ehavior In particular
do es the ability for nontrivial computation require a system s dynamical b ehavior to b e near
a transition to chaos There has also b een considerable attention given to the notion of
the edge of chaos in the context of evolution In particular it has b een hyp othesized that
when biological systems must p erform complex computation in order to survive the pro cess
of evolution under natural selection tends to select such systems near a phase transition from
ordered to chaotic b ehavior
This pap er describ es a re examination of one study that addressed these questions in
the context of cellular automata The results of the original study were interpreted
as evidence that an evolutionary pro cess in which cellular automata rules are selected to
p erform a nontrivial computation preferentially selected rules near the transition to chaos
We show that this conclusion is neither supp orted by our exp erimental results nor consistent
with basic mathematical prop erties of the computation b eing evolved In the pro cess of
this demonstration we review and clarify notions relating to terms such as computation
dynamical b ehavior and edge of chaos in the context of cellular automata
Cellular Automata and Dynamics
Cellular automata CA are discrete spatially extended dynamical systems that have b een
studied extensively as mo dels of physical pro cesses and as computational devices
In its simplest form a CA consists of a spatial lattice of cel ls each of which at time
t can b e in one of K states We denote the lattice size or numb er of cells as L A CA has a
single xed rule used to up date each cell the rule maps from the states in a neighb orho o d
of cells e g the states of a cell and its nearest neighb ors to a single state which is the
up date value for the cell in question The lattice starts out with some initial con guration
of lo cal states and at each time step the states of all cells in the lattice are synchronously
up dated In the following we will use the term state to refer to the value of a single
cell e g or and con guration to mean the pattern of states over the entire lattice
The CA we will discuss in this pap er are all one dimensional with two p ossible states p er
cell and In a one dimensional CA the neighb orho o d of a cell includes the cell itself
and some numb er of neighb ors on either side of the cell The numb er of neighb ors on either
side of the center cell is referred to as the CA s radius r All of the simulations will b e of CA
with spatially p erio dic b oundary conditions i e the one dimensional lattice is viewed as a
circle with the right neighb or of the rightmost cell b eing the leftmost cell and vice versa
The equations of motion for a CA are often expressed in the form of a rule table This is
a lo ok up table listing each of the neighb orho o d patterns and the state to which the central
cell in that neighb orho o d is mapp ed For example Figure displays one p ossible rule table
for an elementary one dimensional two state CA with radius r The left hand column
gives the p ossible neighb orho o d con gurations and the states in the right hand column
Figure Rule table for the elementary binary state nearest
3
neighb or CA with rule numb er The neighb orho o d
patterns are given in the left hand column For each of these the
right hand column gives the output bit the center cell s value at
the next time step
are referred to as the output bits of the rule table To run the CA this lo ok up table is
applied to each neighb orho o d in the current lattice con guration resp ecting the choice of
b oundary conditions to pro duce the con guration at the next time step
A common metho d for examining the b ehavior of a two state one dimensional CA is to
display its space time diagram a two dimensional picture that vertically strings together the
one dimensional CA lattice con gurations at each successive time step with white squares
corresp onding to cells in state and black squares corresp onding to cells in state Two
such space time diagrams are displayed in Figure These show the actions of the Gacs
Kurdyumov Levin GKL binary state CA on two random initial con gurations of di erent
densities of s In b oth cases over time the CA relaxes to a xed pattern in one
case all s and in the other case all s These patterns are in fact xed p oints of the
GKL CA That is once reached further applications of the CA do not change the pattern
The GKL CA will b e discussed further b elow
CA are of interest as mo dels of physical pro cesses b ecause like many physical systems
they consist of a large numb er of simple comp onents cells which are mo di ed only by
lo cal interactions but which acting together can pro duce global complex b ehavior Like
the class of dissipative dynamical systems even the class of elementary one dimensional CA
exhibit the full sp ectrum of dynamical b ehavior from xed p oints as seen in Figure to
limit cycles p erio dic b ehavior to unpredictable chaotic b ehavior Wolfram considered
a coarse classi cation of CA b ehavior in terms of these categories He prop osed the following
four classes with the intention of capturing all p ossible CA b ehavior
Class All initial con gurations relax after a transient p erio d to the same xed
con guration e g all s
Class All initial con gurations relax after a transient p erio d to some xed p oint or
some temp orally p erio dic cycle of con gurations but which one dep ends on the initial
con guration
Class Some initial con gurations relax after a transient p erio d to chaotic b ehavior (a) (b) 0 0
t t
148 148
0 i 148 0 i 148
Figure Two space time diagrams for the binary state Gacs
Kurdyumov Levin CA L sites are shown evolving with time
increasing down the page from two di erent initial con gurations over
time steps In a the initial con guration has a density of s of
approximately in b a density of approximately Notice
that by the last time step the CA has converged to a xed pattern of
a all s and b all s In this way the CA has classi ed the initial
con gurations according to their density
The term chaotic here and in the rest of this pap er refers to apparently unpredictable
space time b ehavior
Class Some initial con gurations result in complex lo calized structures sometimes
long lived
Wolfram do es not state the requirements for memb ership in Class any more precisely than
is given ab ove Thus unlike the categories derived from dynamical systems theory Class
is not rigorously de ned
L
It should b e p ointed out that on nite lattices there is only a nite numb er of
p ossible con gurations so all rules ultimately lead to p erio dic b ehavior Class refers not
L
to this typ e of p erio dic b ehavior but rather to cycles with p erio ds much shorter than
Cellular Automata and Computation
CA are also of interest as computational devices b oth as theoretical to ols and as practical
highly e cient parallel machines
Computation in the context of CA has several p ossible meanings The most common
meaning is that the CA do es some useful computational task Here the rule is interpreted
as the program the initial con guration is interpreted as the input and the CA runs
for some sp eci ed numb er of time steps or until it reaches some goal pattern p ossibly a
xed p oint pattern The nal pattern is interpreted as the output An example of this is
using CA to p erform image pro cessing tasks
A second meaning of computation in CA is for a CA given certain sp ecial initial con
gurations to b e capable of universal computation That is the CA can given the right
initial con guration simulate a programmable computer complete with logical gates timing
devices and so on Conway s Game of Life is such a CA one construction for univer
sal computation in the Game of Life is given in Similar constructions have b een made
for one dimensional CA Wolfram sp eculated that all Class rules have the capacity
for universal computation However given the informality of the de nition of Class
not to mention the di culty of proving that a given rule is or is not capable of universal
computation this hyp othesis is imp ossible to verify
A third meaning of computation in CA involves interpreting the b ehavior of a given CA on
an ensemble of initial con gurations as a kind of intrinsic computation Here computation
is not interpreted as the p erformance of a useful transformation of the input to pro duce
the output Rather it is measured in terms of generic structural computational elements
such as memory information pro duction information transfer logical op erations and so on
It is imp ortant to emphasize that the measurement of such intrinsic computational elements
do es not rely on a semantics of utility as do the preceding computation typ es That is
these elements can b e detected and quanti ed without reference to any sp eci c useful
computation p erformed by the CA such as enhancing edges in an image or computing
the digits of This notion of intrinsic computation is central to the work of Crutch eld
Hanson and Young
Generally CA have b oth the capacity for all kinds of dynamical b ehaviors and the ca
pacity for all kinds of computational b ehaviors For these reasons in addition to the compu
tational ease of simulating them CA have b een considered a go o d class of mo dels to use in
studying how dynamical b ehavior and computational ability are related Similar questions
have also b een addressed in the context of other dynamical systems including continuous
state dynamical systems such as iterated maps and di erential equations Bo olean
networks and recurrent neural networks Here we will con ne our discussion to CA
With this background we can now rephrase the broad questions presented in Section
in the context of CA
What prop erties must a CA have for nontrivial computation
In particular do es a capacity for nontrivial computation in any of the three senses
describ ed ab ove require a CA to b e near a transition from ordered to chaotic b ehavior
When CA rules are evolved to p erform a nontrivial computation will evolution tend
to select rules near such a transition to chaos
Structure of CA Rule Space
Over the last decade there have b een a numb er of studies addressing the rst question ab ove
Here we fo cus on Langton s empirical investigations of the second question in terms of the
structure of the space of CA rules The relationship of the rst two questions to the
third evolving CA will b e describ ed subsequently
One of the ma jor di culties in understanding the structure of the space of CA rules
and its relation to computational capability is its discrete nature In contrast to the well
develop ed theory of bifurcations for continuous state dynamical systems there app ears to
b e little or no geometry in CA space and there is no notion of smo othly changing one CA to
get another nearby in b ehavior In an attempt to emulate this however Langton de ned
a parameter that varies incrementally as single output bits are turned on or o in a given
rule table For a given CA rule table is computed as follows For a K state CA one state
3
q is chosen arbitrarily to b e quiescent The of a given CA rule is then the fraction of
non quiescent output states in the rule table For a binary state CA if is chosen to b e the
quiescent state then is simply the fraction of output bits in the rule table Typically
there are many CA rules with a given value For a binary CA the numb er is strongly
p eaked at due to the combinatorial dep endence on the radius r and the numb er of
states K It is also symmetric ab out due to the symmetry of exchanging s and
s Generally as is increased from to K the CA move from having the most
homogeneous rule tables to having the most heterogeneous
Langton p erformed a range of Monte Carlo samples of two dimensional CA in an attempt
to characterize their average b ehavior as a function of The notion of average b e
havior was intended to capture the most likely b ehavior observed with a randomly chosen
initial con guration for CA randomly selected in a xed subspace The observation was
that as is incremented from to K the average b ehavior of rules passes through
the following regimes
xed p oint p erio dic complex chaotic
That is according to Figure in for example the average b ehavior at low is for a
rule to relax to a xed p oint after a relatively short transient phase As is increased rules
tend to relax to p erio dic patterns again after a relatively short transient phase As reaches
a critical value rules tend to have longer and longer transient phases Additionally
c
the b ehavior in this regime exhibits long lived complex non p erio dic but non random
patterns As is increased further the average transient length decreases and rules tend
to relax to apparently random space time patterns The actual value of dep ends on r K
c
and the actual path of CA found as is incremented
These four b ehavioral regimes roughly corresp ond to Wolfram s four classes Langton s
claim is that as is increased from to K the classes are passed through in the
order He notes that as is increased one observes a phase transition b etween
highly ordered and highly disordered dynamics analogous to the phase transition b etween
the solid and uid states of matter p
According to Langton as is increased from K to the four regimes o ccur in the
reverse order sub ject to some constraints for K For two state CA since b ehavior
is necessarily symmetric ab out there are two values of at which the complex
c
regime o ccurs
3
In all states ob eyed a strong quiescence requirement For any state s f K g the
neighb orho o d consisting entirely of state s must map to s γ
0.0 λ 1.0
Figure A graph of the average di erence pattern spreading rate
of a large numb er of randomly chosen r K CA as a function
of Adapted from with p ermission of the author No vertical
scale was provided there
How is determined Following standard practice Langton used various statistics such
c
as single site entropy two site mutual information and transient length to classify CA b e
havior The additional step was to correlate b ehavior with via these statistics Langton s
Monte Carlo samples showed there was some correlation b etween the statistics and But
the averaged statistics did not reveal a sharp transition in average b ehavior a basic prop
erty of a phase transition in which macroscopic highly averaged quantities do make marked
changes We note that Wo otters and Langton gave evidence that in the limit of an increasing
numb er of states the transition region narrows The main result indicates that in one
class of two dimensional in nite state sto chastic cellular automata there is a sharp transition
in single site entropy at
c
The existence of a critical and the dep endence of the critical region s width on r and
K is less clear for nite state CA Nonetheless Packard empirically determined rough values
of for r K CA by lo oking at the di erence pattern spreading rate as a function
c
of The spreading rate is a measure of unpredictability in spatio temp oral patterns
and so is one p ossible measure of chaotic b ehavior It is analogous to but not the
same as the Lyapunov exp onent for continuous state dynamical systems In the case of CA
it indicates the average propagation sp eed of information through space time though not
the rate of pro duction of lo cal information
At each a large numb er of rules was sampled and for each CA was estimated The
average over the selected CA was taken as the average spreading rate at the given The
results are repro duced in Figure As can b e seen at low and high s vanishes at
intermediate it is maximal and in the critical regions centered ab out and
it rises or falls gradually
While not shown in Figure for most values s variance is high The same is true
for single site entropy and two site mutual information as a function of That is the
b ehavior of any particular rule at a given might b e very di erent from the average b ehavior
at that value Thus the interpretations of these averages is somewhat problematic This
recounting of the b ehavioral structure of CA rule space as parameterized by is based on
statistics taken from Langton s and Packard s Monte Carlo simulations Various problems
in correlating with b ehavior will b e discussed in Section A detailed analysis of some of
these problems can b e found in Other work on investigating the structure of CA rule
space is rep orted in
The claim in is that predicts dynamical b ehavior well only when the space of rules
is large enough Apparently is not intended to b e a go o d b ehavioral predictor for the
space of elementary CA rules r K and p ossibly r K rules as well
CA Rule Space and Computation
Langton hyp othesizes that a CA s computational capability is related to its average dynam
ical b ehavior which is claimed to predict In particular he hyp othesizes that CA
capable of p erforming nontrivial computation including universal computation are most
likely to b e found in the vicinity of phase transitions b etween order and chaos that is near
values The hyp othesis relies on a basic observation of computation theory that any form
c
of computation requires memory information storage and communication information
transmission and interaction b etween stored and transmitted information Ab ove and b e
yond these prop erties though universal computation requires memory and communication
over arbitrary distances in time and space Thus complex computation requires signi cantly
long transients and space time correlation lengths in the case of universal computation
arbitrarily long transients and correlations are required Langton s claim is that these phe
nomena are most likely to b e seen near values near phase transitions b etween order
c
and chaos This intuition is b ehind Langton s notion of computation at the edge of chaos
4
for CA
Evolving CA
The empirical studies describ ed ab ove addressed only the relationship b etween and the
dynamical b ehavior of CA as revealed by several statistics Those studies did not correlate
or b ehavior with an indep endent measure of computation Packard addressed this
issue by using a genetic algorithm GA to evolve CA rules to p erform a particular
computation This exp eriment was meant to test two hyp otheses CA rules able to
p erform complex computations are most likely to b e found near values and When
c
CA rules are evolved to p erform a complex computation evolution will tend to select rules
near values
c
The Computational Task and an Example CA
The original exp eriment consisted of evolving two state s f g one dimensional CA
with r That is the neighb orho o d of a cell consists of itself and its three neighb ors
4
This should b e contrasted with the analysis of computation at the onset of chaos in and in
particular with the discussion of the structure of CA space there
on each side The computational task for the CA is to decide whether or not the initial
con guration contains more than half s If the initial con guration contains more than
half s the desired b ehavior is for the CA after some numb er of time steps to relax to
a xed p oint pattern of all s If the initial con guration contains less than half s the
desired b ehavior is for the CA after some numb er of time steps to relax to a xed p oint
pattern of all s If the initial con guration contains exactly half s then the desired
b ehavior is unde ned This can b e avoided in practice by requiring the CA lattice to b e of
o dd length Thus the desired CA has only two invariant patterns either all s or all s In
the following we will denote the density of s in a lattice con guration by the density of
s in the con guration at time t by t and the threshold density for classi cation by
c
Do es the classi cation task count as a nontrivial computation for a small
c
radius r L CA Though this term was not rigorously de ned in or one p ossible
de nition might b e any computation for which the memory requirement increases with L
i e any computation which corresp onds to the recognition of a non regular language and
in which information must b e transmitted over signi cant space time distances on the order
of L Under this de nition the classi cation task can b e thought of as a nontrivial
c
computation for a small radius CA The e ective minimum amount of memory is prop ortional
to l og L since the equivalent of a counter register is required to track the excess of s in a
serial scan of the initial pattern And since the s can b e distributed throughout the lattice
information transfer over long space time distances must o ccur This is supp orted in a CA
by the non lo cal interactions among many di erent neighb orho o ds after some p erio d of time
Packard cited a K r rule constructed by Gacs Kurdyumov and Levin
which purp ortedly p erforms this task The Gacs Kurdyumov Levin GKL CA is de ned by
the following rule
If s t then s t ma jority s t s t s t
i i i i 1 i 3
If s t then s t ma jority s t s t s t
i i i i+1 i+3
where s t is the state of site i at time t
i
In words this rule says that for each neighb orho o d of seven adjacent cells if the state
of the central cell is then its new state is decided by a ma jority vote among itself its left
neighb or and the cell two cells to the left away Likewise if the state of the central cell is
then its new state is decided by a ma jority vote among itself its right neighb or and the
cell two cells to the right away
Figure gives space time diagrams for the action of the GKL rule on an initial con gu
ration with and on an initial con guration with It can b e seen that although
c c
the CA eventually converges to a xed p oint there is a transient phase during which a spa
tial and temp oral transfer of information ab out lo cal neighb orho o ds takes place and this
lo cal information interacts with other lo cal information to pro duce the desired nal state
Very crudely the GKL CA successively classi es lo cal densities with the lo cality range
increasing with time In regions where there is some ambiguity a signal is propagated
This is seen either as a checkerb oard pattern propagated in b oth spatial directions or as
a vertical white to black b oundary These signals indicate that the classi cation is to b e
made at a larger scale Note that b oth signals lo cally have the result is that the
c
signal patterns can propagate since the density of patterns with is not increased or
c
decreased under the rule In a simple sense this is the CA s strategy for p erforming the
computational task
It has b een claimed that the GKL CA p erforms the task but actually this is
c
true only to an approximation The GKL rule was invented not for the purp ose of p erforming
any particular computational task but rather as part of studies of reliable computation and
phase transitions in one spatial dimension The goal in the former for example was to
nd a CA whose b ehavior is robust to small errors in the rule s up date of the con guration
It has b een proved that the GKL rule has only two attracting patterns either all s or
all s Attracting patterns here are those invariant patterns which when p erturb ed
a small amount return to the same pattern It turns out that the basins of attraction
for the all and all patterns are not precisely the initial con gurations with or
5
resp ectively On nite lattices the GKL rule do es classify most initial con gurations
according to this criterion but on a signi cant numb er the incorrect attractor is reached
One set of exp erimental measures of the GKL CA s classi cation p erformance is displayed
in Figure To make this plot we ran the GKL CA on randomly generated initial
con gurations at each of densities The fraction of correct classi cations
was then plotted at each The rule was run either until a xed p oint was reached or for a
maximum numb er of time steps equal to L This was done for CA with three di erent
lattice sizes L f g
Note that approximately of the initial con gurations with were misclassi ed
c
All the incorrect classi cations are made for initial con gurations with In fact the
c
worst p erformances o ccur at Interestingly although the error region narrows with
c
increasing lattice size the p erformance at decreases when the lattice size is increased
c
from to
The GKL rule table has not Since it app ears to p erform a computational
c
task of some complexity at a minimum it is a deviation from the edge of chaos hyp othesis
for CA computation The GKL rule s puts it right at the center of the chaotic
region in Figure This may b e puzzling since clearly the GKL rule do es not pro duce chaotic
b ehavior during either its transient or asymptotic ep o chs far from it in fact However the
parameter was intended to correlate with average b ehavior of CA rules at a given
value Recall that in Figure represent an average over a large numb er of randomly
chosen CA rules and while not shown in that plot for most values the variance in is
high Thus it can b e claimed that the b ehavior of any particular rule at its value might
b e very di erent from the average b ehavior at that value
More to the p oint though we exp ect a value close to for a rule that p erforms well on
the task This is largely b ecause the task is symmetric with resp ect to the exchange
c
of s and s Supp ose for example a rule that carries out the task has
c
This implies that there are more neighb orho o ds in the rule table that map to output bit
than to output bit This in turn means that there will b e some initial con gurations
with on which the action of the rule will decrease the numb er of s And this is the
c
5
The terms attractor and basin of attraction are b eing used here in the sense of and This
di ers substantially from the notion used in for example There attractor refers to any invariant or
time p erio dic pattern and basin of attraction means that set of nite lattice con gurations relaxing to it 1
0.75
0.5 149 cell lattice 599 cell lattice 0.25 999 cell lattice fraction of correct classifications 0 0 0.25 0.5 0.75 1
ρ
Figure Exp erimental p erformance of the GKL rule as a function
of for the task Performance plots are given for three
c
lattice sizes L the size of the lattice used in the GA runs
and
opp osite of the desired action However if the rule acts to decrease the numb er of s on
an initial con guration with it risks pro ducing an intermediate con guration with
c
which then would lead under the original assumption that the rule carries out the
c
task correctly to a xed p oint of all s misclassifying the initial con guration A similar
argument holds in the other direction if the rule s value is greater than This informal
argument shows that a rule with will misclassify certain initial con gurations
Generally the further away the rule is from the more such initial con gurations
there will b e Such rules may p erform fairly well classifying most initial con gurations
correctly However we exp ect any rule that p erforms reasonably well on this task in the
sense of b eing close to the GKL CA s average p erformance shown in Figure to have a
value close to
This analysis p oints to a problem with using this task as an evolutionary goal in order
to study the relationship among evolution computation and As was shown in Figure
for r K CA the values o ccur at roughly and and one hyp othesis
c
that was to b e tested by the original exp eriment is that the GA will tend to select rules
close to these values But for the classi cation tasks the range of values required for
c
go o d p erformance is simply a function of the task and sp eci cally of For example the
c
underlying exchange symmetry of the task implies that if a CA exists to do
c
the task at an acceptable p erformance level then it has Even though this do es not
directly invalidate the adaptation hyp othesis or claims ab out s correlation with average
b ehavior it presents problems with using classi cation tasks as a way to gain evidence
ab out a generic relation b etween and computational capability
The Original Exp eriment
Packard used a GA to evolve CA rules to p erform the task His GA started out
c
with a randomly generated initial p opulation of CA rules Each rule was represented as a bit
string containing the output bits of the rule table That is the bit at p osition in the string
is the state to which the neighb orho o d is mapp ed the bit at p osition in the string
is the state to which the neighb orho o d is mapp ed and so on The initial p opulation
was randomly generated but it was constrained to b e uniformly distributed across values
b etween and
A given rule in the p opulation was evaluated for ability to p erform the classi cation task
by cho osing an initial con guration at random running the CA on that initial con guration
for some sp eci ed numb er of time steps and at the nal time step measuring the fraction
of cells in the lattice that have the correct state For initial con gurations with the
c
correct nal state for each cell is and for initial con gurations with the correct
c
nal state for each cell is For example if the CA were run on an initial con guration with
and at the nal time step the lattice contained s the CA s score on that initial
c
6
con guration would b e The tness of a rule was simply the rule s average score over a
large numb er of initial con gurations For each rule in the p opulation Packard generated a
large set of initial con gurations that were uniformly distributed across values from to
Packard s GA worked as follows At each generation
The tness of each rule in the p opulation is calculated
The p opulation is ranked by tness
Some fraction of the lowest tness rules are removed
The removed rules are replaced by new rules formed by crossover and mutation from
the remaining rules
Crossover b etween two strings involves randomly selecting a p osition in the strings at random
and exchanging parts of the strings b efore and after that p osition Mutation involves ipping
one or more bits in a string with some low probability
A diversity enforcement scheme was also used to prevent the p opulation from converging
to o early and losing diversity If a rule is formed that is to o close in Hamming distance
i e the numb er of matching bits to existing rules in the p opulation its tness is decreased
The results from Packard s exp eriment are displayed in Figure The two histograms
display the observed frequency of rules in the GA p opulation as a function of with rules
merged from a numb er of di erent runs The top graph gives this data for the initial
6
A slight variation on this metho d was used in Instead of measuring the fraction of correct states in
the nal lattice the GA measured the fraction of correct states over con gurations from a small numb er n
of nal time steps This prevented the GA from evolving rules that were temp orally p erio dic viz those
with patterns that alternated b etween all s and all s Such rules obtained higher than average tness at
early generations by often landing at the correct phase of the oscillation for a given initial con guration
That is on the next time step the classi cation would have b een incorrect In our exp eriments we used a
slightly di erent metho d to address this problem This is explained in subsection Figure from The b ottom gure is Figure repro duced here for reference p opulations CA γ
0.0 Prob(rule) Prob(rule) for with p ermission of the author the Results of c CA from at generations classi cation the original λ and exp erimen task The resp ectiv t top on GA t w ely o ev gures olution v Adapted ersus 1.0 are of
generation As can b e seen the rules are uniformly distributed over values The middle
graph gives the same data for the nal generation in this case after the GA has run for
generations The rules now cluster around the two regions as can b e seen by comparison
c
with the di erence pattern spreading rate plot reprinted here at the b ottom of the gure
Note that each individual run pro duced rules at one or the other p eak in the middle graph
so when the runs were merged together b oth p eaks app ear Packard interpreted these
results as evidence for the hyp othesis that when an ability for complex computation is
required evolution tends to select rules near the transition to chaos He argues like Langton
that this result intuitively makes sense b ecause rules near the transition to chaos have the
capability to selectively communicate information with complex structures in space time
thus enabling computation p
New Exp eriments
As the rst step in a study of how well these general conclusions hold up we carried out a set
of exp eriments similar to that just describ ed We were unable to obtain some of the exact
details of the original exp eriment s parameters such as the exact p opulation size for the
GA the mutation rate and so on As a result we used what we felt were reasonable values
for these various parameters We carried out a numb er of parameter sensitivity tests which
indicated that varying the parameters within small b ounds did not change our qualitative
results
Details of Our Exp eriments
In our exp eriments as in the original the CA rules in the p opulation all have r and
2r +1
K Thus the bit strings representing the rules are of length and the
128
size of the search space is huge the numb er of p ossible CA rules is The tests
for each CA rule are carried out on lattices of length L with p erio dic b oundary
conditions The p opulation size is which was roughly the p opulation size used in the
original exp eriment The initial p opulation is generated at random but constrained to
b e uniformly distributed among di erent values A rule s tness is estimated by running
the rule on randomly generated initial con gurations that are uniformly distributed over
Exactly half the initial con gurations have and exactly half have
c
7
c
We allow each rule to run for a maximum numb er M of iterations where a new M
is selected for each rule from a Poisson distribution with mean This is the measured
maximum amount of time for the GKL CA to reach an invariant pattern over a large numb er
7
It was necessary to have this exact symmetry in the initial con gurations at each generation to avoid
early biases in the of selected rules If say of the initial con gurations have and of initial
c
con gurations have high rules would obtain slightly higher tness than low rules since high
c
rules will map most initial con gurations to all s A rule with say would in this case classify
of the initial con gurations correctly whereas a rule with would classify only correctly But such
slight di erences in tness have a large e ect in the initial generation when all rules have tness close to
since the GA selects the best rules even if they are only very slightly b etter than the worst rules
This biases the representative rules in the early p opulation And this bias can p ersist well into the later
generations
8
of initial con gurations on lattice size A rule s tness is its average score the fraction
of cell states correct at the last iteration over the initial con gurations We term this
tness function proportional tness to contrast with a second tness function performance
tness which will b e describ ed b elow A new set of initial con gurations is generated
every generation At each generation all the rules in the p opulation are tested on this set
Notice that this tness function is sto chastic the tness of a given rule may vary from
generation to generation dep ending on the set of initial con gurations used in testing it
Our GA is similar to Packard s In our GA the fraction of new strings in the next
generation the generation gap is That is once the p opulation is ordered according
to tness the top half of the p opulation the set of elite strings is copied without mo di
cation into the next generation For GA practitioners more familiar with nonoverlapping
generations this may sound like a small generation gap However since testing a rule on
training cases do es not necessarily provide a very reliable gauge of what the tness
would b e over a larger set of training cases our selected gap is a go o d way of making a rst
cut and allowing rules that survive to b e tested over more initial con gurations Since a
new set of initial con gurations is pro duced every generation rules that are copied without
mo di cation are always retested on this new set If a rule p erforms well and thus survives
over a large numb er of generations then it is likely to b e a genuinely b etter rule than those
that are not selected since it has b een tested with a large set of initial con gurations An
alternative metho d would b e to test every rule in every generation on a much larger set of
initial con gurations but given the amount of compute time involved that metho d seems
unnecessarily wasteful Much to o much e ort for example would go into testing very weak
rules which can safely b e weeded out early using our metho d
The remaining half of the p opulation for each new generation is created by crossover
9
and mutation from the previous generation s p opulation Fifty pairs of parent rules are
chosen at random with replacement from the entire previous p opulation For each pair a
single crossover p oint is selected at random and two o spring are created by exchanging the
subparts of each parent b efore and after the crossover p oint The two o spring then undergo
mutation A mutation consists of ipping a randomly chosen bit in the string The numb er
of mutations for a given string is chosen from a Poisson distribution with a mean of this
is equivalent to a p er bit mutation rate of Again to GA practitioners this may seem
to b e a high mutation rate but one must take into account that at every generation half
the p opulation is b eing copied without mo di cation
Results of Prop ortional Fitness Exp eriment
8
It may not b e necessary to allow the maximum numb er of iterations M to vary In some early tests with
smaller sets of xed initial con gurations though we found the same problem Packard rep orted that
if M is xed then p erio d rules evolve that alternate b etween all s and all s These rules adapted to
the small set of initial con gurations and the xed M by landing at the correct pattern for a given initial
con guration at time step M only to move to the opp osite pattern and so wrong classi cation at time step
M These rules did very p o orly when tested on a di erent set of initial con gurations evidence for
over tting
9
This metho d of pro ducing the non elite strings di ers from that in where the non elite strings were
formed from crossover and mutation among the elite strings only rather than from the entire p opulation We
observed no statistically signi cant di erences in our tests using the latter mechanism other than a mo dest
di erence in time scale 1
best fitness 0.75 mean fitness
(a) 0.5
fraction of total rules 0.25
0 0 0.25 0.5 0.75 1 λ
1
best fitness 0.75 mean fitness
(b) 0.5
fraction of total rules 0.25
0 0 0.25 0.5 0.75 1
λ
Figure Results from our exp eriment with prop ortional tness The
top histogram a plots as a function of the frequencies of rules
merged from the initial generations of runs The b ottom histogram
b plots the frequencies of rules merged from the nal generations
generation of these runs Following the x axis is divided
into bins of length each The rules with are included
in the rightmost bin In each histogram the b est cross and mean
circle tnesses are plotted for each bin The y axis interval for
tnesses is also
We p erformed di erent runs of the GA with the parameters describ ed ab ove each with
a di erent random numb er seed On each run the GA was iterated for generations We
found that running the GA for longer than this up to generations did not result in
improved tness The results of this set of runs are displayed in Figure Figure a is
a histogram of the frequency of rules in the initial p opulations as a function of merging
together the rules from all initial p opulations thus the total numb er of rules represented
in this histogram is The bins in this histogram are the same ones that were used by
Packard each of width Packard s highest bin contained only rules with that
is rules that consist of all s We have merged this bin with the immediately lower bin
As was said earlier the initial p opulation consists of randomly generated rules uniformly
spread over the values b etween and Also plotted are the mean and b est tness
values for each bin These are all around which is exp ected for a set of randomly
generated rules under this tness function The b est tnesses are slightly higher in the very
low and very high bins This is b ecause rules with output bits that are almost all s or
s correctly classify all low density or all high density initial con gurations In addition
these CA obtain small partial credit on some high density low density initial con gurations
Such rules thus have tness sightly higher than
Figure b shows the histogram for the nal generation merging together rules
from the nal generations of all runs Again the mean and b est tness values for each
bin are plotted
In the nal generation the mean tnesses in each bin are all around The exceptions
are the central bin with a mean tness of and the leftmost bin with a mean tness of
The leftmost bin contains only ve rules each at right next to the the bin s
upp er limit The standard deviations of tness for each bin not shown in the gure are
all approximately except the leftmost bin which has a standard deviation of The
b est tnesses for each bin are all b etween and except the leftmost bin which has
a b est tness of Under this tness function the GKL rule has tness the GA
never found a rule with tness ab ove
As was mentioned ab ove the tness function is sto chastic a given rule might b e assigned
a di erent tness each time the tness function is evaluated The standard deviation under
the present tness scheme on a given rule is approximately This indicates that the
di erences among the b est tnesses plotted in the histogram are not signi cant except for
that in the leftmost bin
The lower mean tness in the central bin is due to the fact that the rules in that bin
largely come from non elite rules generated by crossover and mutation in the nal generation
This is a combinatorial e ect the density of CA rules as a function of is very highly p eaked
ab out as already noted We will return to this combinatorial drift e ect shortly
Many of the rules in the middle bin have not yet undergone selection and thus tend to have
lower tnesses than rules that have b een selected in the elite This e ect disapp ears in
Figure which includes only the elite rules at generation for the runs As can b e
seen the di erence in mean tness disapp ears and the height of the central bin is decreased
by half
The results presented in Figure b are strikingly di erent from the results of the original
exp eriment In the nal generation histogram in Figure most of the rules clustered around 1
best fitness 0.75 mean fitness
0.5
fraction of total rules 0.25
0 0 0.25 0.5 0.75 1
λ
Figure Histogram including only the elite rules from the nal gen
erations of the runs cf Figure b with the prop ortional tness
function
either or Here though there are no rules in these regions Rather the
c
rules cluster much closer with a ratio of variances of b etween the two distributions to
Recall this clustering is what we exp ect from the basic exchange symmetry of
the task
c
One rough similarity is the presence of two p eaks centered around a dip at a
phenomenon which we will explain shortly and which is a key to understanding how the GA
is working But there are signi cant di erences even within this similarity In the original
exp eriments the p eaks are in bins centered ab out and In Figure b
though the p eaks are very close to b eing centered in the neighb oring bins those
with and Thus the ratio of original to current p eak spread is roughly a
factor of Additionally in the nal generation histogram of Figure the two highest bin
p opulations are roughly ve times as high as the central bin whereas in the nal generation
histogram of Figure b the two highest bins are roughly three times as high as the central
bin Finally the nal generation histogram in Figure shows the presence of rules in every
bin but in the new nal generation histogram there are only rules in six of the central bins
Similar to the original exp eriment we found that on any given run the p opulation was
clustered ab out one or the other p eak but not b oth Thus in the histograms that merge
all runs two p eaks app ear This is illustrated in Figure which displays histograms from
the nal generation of two individual runs In one of these runs the p opulation clustered to
the left of the central bin in the other run it clustered to the right of the center The fact
that di erent runs result in di erent clustering lo cations is why we p erformed many runs
and merged the results rather than p erforming a single run with a much larger p opulation
The latter metho d might have yielded only one p eak Said a di erent way indep endent of
the p opulation size a given run will b e driven by and the p opulation organized around the
t individuals that app ear earliest Thus examining an ensemble of individual runs reveals
more details of the evolutionary dynamics
The asymmetry in the heights of the two p eaks in Figure b results from a small
statistical asymmetry in the results of the runs There were out of runs in which
the rules clustered at the lower bin and out of runs in which the rules clustered at
the higher bin This di erence is not signi cant but explains the small asymmetry in the
p eaks heights
We extended of the runs to generations and found that not only do the tnesses
not increase further but the basic shap e of the histogram do es not change signi cantly
E ects of Drift
The results of our exp eriments suggest that for the task an evolutionary pro cess
c
mo deled by a genetic algorithm tends to select rules with This is what we exp ect
from the theoretical discussion given ab ove concerning this task and its symmetries We
will delay until the next section a discussion of the curious feature near viz the
dip surrounded by two p eaks Instead here we fo cus on the larger scale clustering in that
region
To understand this clustering we need to understand the degree to which the selection of
rules close to is due to an intrinsic selection pressure and the degree to which it is
due to drift By drift we refer to the force that derives from the combinatorial asp ects of
CA space as explored by random selection genetic drift along with the e ects of crossover
and mutation The intrinsic e ects of random selection with crossover and mutation are to
move the p opulation irresp ective of any selection pressure to This is illustrated by
the histogram mosaic in Figure These histograms show the frequencies of the rules in the
p opulation as a function of every generations from runs on which selection according
to tness was turned o That is on these runs the tness of the rules in the p opulation
was never calculated and at each generation the selection of the elite group of strings was
p erformed at random Everything else ab out the runs remains the same as b efore Since
there is no selection drift is the only force at work here As can b e seen under the e ects
of random selection crossover and mutation by generation the p opulation has largely
drifted to the region of and this clustering b ecomes increasingly pronounced as the
run continues
This drift to is related to the combinatorics of the space of bit strings For
N
2
binary binary CA rules with neighb orho o d size N r the space consists of all
N
strings of length Denoting the subspace of CA with a xed and N as CA N we
see that the size of the subspace is binomially distributed with resp ect to
N
jCA N j
N
N
The distribution is symmetric in and tightly p eaked ab out with variance
Thus the vast ma jority of rules is found at The steepness of the binomial distribu
tion near its maximum gives an indication of the magnitude of the drift force Note that
the last histogram in Figure gives the GA s rough approximation of this distribution
Drift is thus a p owerful force moving the p opulation to cluster around For
comparison Figure gives the rule frequency versus histograms for the runs of our 1
0.75
0.5
fraction of total rules 0.25
0 0 0.25 0.5 0.75 1 λ 1
0.75
0.5
fraction of total rules 0.25
0 0 0.25 0.5 0.75 1
λ
Figure Histograms from the nal generations of two individual runs
of the GA employing prop ortional tness Each run had a p opulation
of rules The nal distribution of rules in each of the runs we
p erformed resembled one or the other of these two histograms 0.6 0 5 10 15 20 25 30
P
0
0.6 35 40 45 50 55 60 65
P
0
0.6 70 75 80 85 90 95 100
P
0
0 λ 1 0 λ 1 0 λ 1 0 λ 1 0 λ 1 0 λ 1 0 λ 1
Figure No selection mosaic Rule frequency versus histograms
given every ve generations for p opulations evolved under the genetic
algorithm with no selection that is the tness function was not cal
culated As b efore each histogram is merged from runs each run
had a p opulation of rules The generation numb er is given in the
upp er left corner of each histogram
0.6 0 5 10 15 20 25 30
P
0
0.6 35 40 45 50 55 60 65
P
0
0.6 70 75 80 85 90 95 100
P
0
0 λ 1 0 λ 1 0 λ 1 0 λ 1 0 λ 1 0 λ 1 0 λ 1
Figure Proportional tness mosaic Rule frequency versus his
tograms given every ve generations merged from the GA runs
with prop ortional tness Each run had a p opulation of rules
prop ortional tness exp eriment every ve generations The last histogram in this gure is
the same one that was displayed in Figure b Figure gives the merged data from the
entire p opulation of each run every ve generations A similar mosaic plotting only the elite
strings at each generation lo oks qualitatively similar
Figure lo oks very similar to Figure up to generation The main di erence in
generations is that Figure indicates a more rapid p eaking ab out = The in
creased sp eed of movement to the center over that seen in Figure is presumably due to the
additional evolutionary pressure of prop ortional tness At generation something new
app ears The p eak in the center has b egun to shrink signi cantly and the two surrounding
bins are b eginning to rival it in magnitude By generation the right of center bin has
exceeded the central bin and by generation the histogram has develop ed two p eaks sur
rounding a dip in the center The dip b ecomes increasingly pronounced as the run continues
but stabilizes by generation or so
The di erences b etween Figure and Figure over all generations shows that the
p opulation s structure in each generation is not entirely due to drift Indeed after generation
the distinctive features of the p opulation indicates new qualitatively di erent and unique
prop erties due to the selection mechanism The two p eaks represent a symmetry breaking
in the evolutionary pro cess the rules in each individual run initially are clustered around
= but move to one side or the other of the central bin by around generation The
causes of this symmetry breaking will b e discussed in the next subsection
Evolutionary Mechanisms Symmetry Breaking and the Dip at =
At this p oint we move away from questions related to the original exp eriment and instead
concentrate on the mechanisms involved in pro ducing our results Two ma jor questions need
to b e answered Why in the nal generation are there signi cantly fewer rules in the central
bin than in the two surrounding bins And what causes the symmetry breaking that b egins
near generation seen in Figure
In the briefest terms the answer obtained by detailed analysis of the GA runs is
the following The course of CA evolution under our GA roughly falls into four strategy
ep o chs Each ep o ch is asso ciated with an innovation discovered by the GA for solving the
problem Though the absolute time at which these innovations app ear in each run varies
somewhat each run basically passes through each of these four ep o chs in succession The
ep o chs are shown in Figure which plots the b est tness the mean tness of the elite
strings and the mean tness of the p opulation versus generation for one typical run of the
GA The b eginnings of ep o chs through are p ointed out on the b est tness plot Ep o ch
b egins at generation
Ep o ch Randomly generated rules
The rst ep o ch starts at generation when the b est tness in the initial generation is
approximately and the values are uniformly distributed b etween and No rule
is much tter than any other rule though as was seen in Figure a rules with very low
and very high tend to have slightly higher tness The strategy here if it can b e called
K
o
epoch
epoch
Fitness
epoch
b est tness
elite mean tness
p opulation mean tness
Generation
Figure Best tness elite mean tness and p opulation mean tness
versus generation for one typical run The b eginnings of ep o chs
are p ointed out on the b est tness plot Ep o ch b egins at generation
this at all derives from only the most elementary asp ect of the task Rules either sp ecialize
for > con gurations by mapping high density neighb orho o ds in the CA rule table to
c
or sp ecialize for < con gurations by mapping low density neighb orho o ds to
c
Ep o ch Discovery of two halves of the rule table
The second ep o ch b egins when a rule is discovered in which most neighb orho o d patterns in
the rule table that have < map to and most neighb orho o d patterns in the rule table
c
that have > map to This is roughly correlated with the left and right halves of the
c
rule table namely neighb orho o ds to and to resp ectively
Such a strategy is presumably easy for the GA to discover due to single p oint crossover s ten
dency to preserve contiguous sections of the rule table It di ers from the accidental strategy
of ep o ch in that there is now an organization to the rule table output bits are roughly
asso ciated with densities of neighb orho o d patterns It is the rst signi cant attempt at dis
tinguishing initial con gurations with more s than s and vice versa Under our tness
function the tness of such rules is approximately b etween and which is signi cantly
higher than the tness of the initial random rules This innovation typically o ccurs b etween
generations and in the run displayed in Figure it o ccurred in generation and can
b e seen as the steep rise in the b est tness plot at that generation All such rules tend to
have close to There are many p ossible variations on these rules with similar tness so
such rules all close to = b egin to dominate in the p opulation This along with the
natural tendency for the p opulation to drift to = is the cause of the clustering around
= seen by generation in Figure For the next several generations the p opulation
tends to explore small variations on this broad strategy This can b e seen in Figure as
the leveling o in the b est tness plot b etween generations and (a) (b) 0 0
t t
148 148
0 i 148 0 i 148
Figure Space time diagrams of one ep o ch rule with
that increases su ciently large blo cks of adjacent or nearly adjacent
s Both diagrams have L and are iterated for time steps
the time displayed here is shorter than the actual time allotted under
the GA In a and In b and
Thus in a the classi cation is incorrect but partial
credit is given in b it is correct
Ep o ch Growing blo cks of s or s
The next ep o ch b egins when the GA discovers one of two new strategies The rst strategy is
to increase the size of a su ciently large blo ck of adjacent or nearly adjacent s the second
strategy is to increase the size of a su ciently large blo ck of adjacent or nearly adjacent s
Examples of these two strategies are illustrated in Figures and These gures give
space time diagrams from two rules that marked the b eginning of this ep o ch in two di erent
runs of the GA Figure illustrates the action of a rule discovered at generation of one
run This rule has : which means that the rule maps most neighb orho o ds to Its
strategy is to map initial con gurations to mostly s the con gurations it pro duces have
< unless the initial con guration contains a su ciently large blo ck of s in which
c
case it increases the size of that blo ck The left space time diagram Figure a shows how
the rule evolves an initial con guration with < to a nal lattice with mostly s This
c
pro duces a fairly go o d score The right space time diagram Figure b shows how the
rule evolves an initial con guration with > The initial con guration contains a few
c
su ciently large blo cks of adjacent or nearly adjacent s and the size of these blo cks is
quickly increased to yield a nal lattice with all s for a p erfect score The tness of this
rule at generation was
Figure illustrates the action of a second rule discovered at generation in another
run This rule has : which means that the rule maps most neighb orho o ds to Its
strategy is the inverse of the previous rule It maps initial con gurations to mostly s unless
the initial con guration contains a su ciently large blo ck of s in which case it increases the (a) (b) 0 0
t t
148 148
0 i 148 0 i 148
Figure Space time diagrams of one ep o ch rule with
that increases su ciently large blo cks of adjacent or nearly adjacent
s In a the initial con guration with maps to a correct
classi cation pattern of all s In b the initial con guration with
is not correctly classi ed but partial credit
is given
size of that blo ck The left space time diagram a illustrates this for an initial con guration
with < here a su ciently large blo ck of s app ears in the initial con guration and is
c
increased in size yielding a p erfect score The right space time diagram b shows the action
of the same rule on an initial con guration with > Most neighb orho o ds are mapp ed
c
to so the nal con guration contains mostly s yielding a fairly high score The tness
of this rule at generation was
The general idea b ehind these two strategies is to rely on statistical uctuations in the
initial con gurations An initial con guration with > is likely to contain a su ciently
c
large blo ck of adjacent or nearly adjacent s The rule then increases this region s size to
yield the correct classi cation Similarly this holds for the CA in Figure with resp ect to
blo cks of s in initial con gurations with < In short these strategies are assuming
c
that the presence of a su ciently large blo ck of s or s is a go o d predictor of
Similar strategies were discovered in every run They typically emerge by generation
A given strategy either increased blo cks of s or blo cks of s but not b oth These
strategies result in a signi cant jump in tness typical tnesses for the rst instances of
such strategies range from to This jump in tness can b e seen in the run of
Figure at approximately generation and is marked as the b eginning of ep o ch This
is the rst ep o ch in which a substantial increase in tness is asso ciated with a symmetry
breaking in the p opulation which will b e explained b elow
The rst instances of ep o ch strategies typically have a numb er of problems As can
b e seen in Figures and the rules often rely on partial credit to achieve fairly high
tness on structurally incorrect classi cation They typically do not get p erfect scores on (a) (b) 0 0
t t
148 148 0 i 148 0 i 148 (c) 0
t
148
0 i 148
Figure Space time diagrams illustrating three typ es of classi ca
tion errors committed by ep o ch rules a growing a blo ck of s in
a sea of < b growing a blo ck of s for an initial con guration
c
with > to o slowly the correct xed p oint of all s do es not o ccur
c
until iteration and c generating a blo ck of s from a sea of
< and growing it so that > the incorrect xed p oint of all
c c
s o ccurs at iteration The initial con guration densities are a
: b : and c : (a) (b) 0 0
t t
148 148
0 i 148 0 i 148
Figure Space time diagrams of one ep o ch rule with :
that increases su ciently large blo cks of adjacent or nearly adjacent
s In a : in b : Both initial con gurations
are correctly classi ed
many initial con gurations The rules also often make mistakes in classi cation Three
common typ es of classi cation errors are illustrated in Figure Figure a illustrates a
rule increasing a to o small blo ck of s and thus misclassifying an initial con guration with
< Figure b illustrates a rule that do es not increase blo cks of s fast enough on an
c
initial con guration with > leaving many incorrect bits in the nal lattice Figure c
c
illustrates the creation of a blo ck of s that did not app ear in an initial con guration with
< ultimately leading to a misclassi cation The rules that pro duced these diagrams
c
come from ep o ch in various GA runs
The increase in tness seen in Figure b etween generation and or so is due to
further re nements of the basic strategies that correct these problems to some extent
Ep o ch Reaching and staying at a maximal tness
In most runs the b est tness is typically at its maximum value of to by generation
or so In Figure this o ccurs at approximately generation and is marked as the
b eginning of ep o ch The b est tness do es not increase signi cantly after this the GA
simply nds a numb er of variations of the b est strategies that all have roughly the same
tness When we extended of the runs to generations we did not see any signi cant
increase in the b est tness
The actions of the b est rules from generation of two separate runs are shown in Fig
ures and The leftmost space time diagrams in each gure are for initial con gurations
with < and the rightmost diagrams are for initial con gurations with > The
c c
rule illustrated in Figure has : its strategy is to map initial con gurations to s
unless there is a su ciently large blo ck of adjacent or nearly adjacent s which if present (a) (b) 0 0
t t
148 148
0 i 148 0 i 148
Figure Space time diagrams of one ep o ch rule with :
that increases su ciently large blo cks of adjacent or nearly adjacent
s In a : in b : Both initial con gurations
are correctly classi ed
is increased The rule shown in Figure has : and has the opp osite strategy Each
of these rules has tness They are b etter tuned versions of the rules in Figures
and
Symmetry breaking in ep o ch
Notice that the values of the rules that have b een describ ed are in the bins centered around
and rather than In fact it seems to b e much easier for the GA to discover
versions of the successful strategies close to : and : than to discover them
close to = though some instances of the latter rules were found Why is this One
reason is that rules with high or low work well by specializing The rules with low
map most neighb orho o ds to s and then increase su ciently large blo cks of s when they
app ear Rules with high sp ecialize in the opp osite direction A rule at = cannot
easily sp ecialize in this way Another reason is that a successful rule that grows su ciently
large blo cks of say s must avoid creating a su ciently large blo ck of s from an initial
con guration with less than half s Doing so will lead it to increase the blo ck of s and
pro duce an incorrect answer as was seen in Figure An easy way for a rule to avoid
creating a su ciently large blo ck of s is to have a low This ensures that low density
initial con gurations will quickly map to all s as was seen in Figure Likewise if a rule
increases su ciently large blo cks of s it is safer for the rule to have a high value so it will
avoid creating su ciently large blo cks of s where none existed A rule close to = will
not have this safety margin and may b e more likely to inadvertently create a blo ck of s or
s that will lead it to a wrong answer A nal element that contributes to the di culty of
nding go o d rules with = is the combinatorially large numb er of rules there In e ect
the search space is much larger which makes the global search more di cult Lo cally ab out
a given adequate rule at = there are many more rules close in Hamming distance and
thus reachable via mutation that are not markedly b etter
Once the more successful versions of the ep o ch strategies are discovered in ep o ch
their variants spread in the p opulation and the most successful rules have on the low or
high side of = This explains the shift from the clustering around = as seen
in generations in Figure to a two p eaked distribution that b ecomes clear around
generation The rules in each run cluster around one or the other p eak sp ecializing
in one or the other way We b elieve this typ e of symmetry breaking is a key mechanism
that determines much of the p opulation dynamics and the GA s success or lack thereof in
optimization
How do es this analysis of the symmetry breaking jib e with the argument given earlier
that the b est rules for the = task must b e close to = None of the rules found
c
by the GA had a tness as high as the tness of the GKL rule whose is exactly
That is the evolved rules make signi cantly more classi cation errors than the GKL rule
To obtain the tness of the GKL rule a numb er of careful balances in the rule table must b e
achieved This is evidently very hard for the GA to do esp ecially in light of the symmetries
in the task and their sub optimal breaking by the GA
Performance of the Evolved Rules
Recall that the prop ortional tness of a rule is the fraction of correct cell states at the nal
time step averaged over initial con gurations This tness gives a rule partial credit
for getting some nal cell states correct However the actual task is to relax to either
all s or all s dep ending on the initial con guration In order to measure how well the
evolved rules actually p erform the task we de ne the performance of a rule to b e the fraction
of times the rule correctly classi es initial con gurations averaged over a large numb er of
initial con gurations Here credit is given only if the initial con guration relaxes to exactly
the correct xed p oint after some numb er of time steps We measured the p erformance of
each of the elite rules in the nal generations of the runs by testing it on randomly
generated initial con gurations that were uniformly distributed in the interval
letting the rule iterate on each initial condition for time steps Figure displays the
mean p erformance diamonds and b est p erformance squares in each bin This gure
shows that while the mean p erformances in each bin are much lower than the mean tnesses
for the elite rules shown in Figure the b est p erformance in each bin is roughly the same as
the b est tness in that bin In some cases the b est p erformance in a bin is slightly higher
than the b est tness shown in Figure This is b ecause di erent sets of initial conditions
were used to calculate tness and p erformance This di erence can pro duce small variations
in the tness or p erformance values The b est p erformance we measured was Under
this measure the p erformance of the GKL rule is Thus the GA never discovered
a rule that p erformed as well as the GKL rule even up to generations In addition
when we measure the p erformance of the ttest evolved rules on larger lattice sizes their
p erformances decrease signi cantly while that of the GKL rule remains roughly the same 1
best performance 0.75 mean performance
0.5
fraction of total rules 0.25
0 0 0.25 0.5 0.75 1
λ
Figure Performances of the rules evolved with the prop ortional
tness function Only the elite rules from generation are included
in this histogram The mean p erformance in each bin op en dia
monds and the b est p erformance in each bin black squares is plot
ted
Using Performance as the Fitness Criterion
Can the GA evolve b etter p erforming rules on this task To test this we carried out an
additional exp eriment in which p erformance as de ned in the previous section is the tness
criterion As b efore at each generation each rule is tested on initial con gurations that
are uniformly distributed over density values However in this exp eriment a rule s tness
is the fraction of initial con gurations that are correctly classi ed An initial con guration
is considered to b e incorrectly classi ed if any bits in the nal lattice are incorrect Aside
from this mo di ed tness function everything ab out the GA remained the same as in the
prop ortional tness exp eriments We p erformed runs of the GA for generations each
The results are given in Figure which gives a histogram plotting the frequencies of the
elite rules from generation of all runs as a function of As can b e seen the shap e
of the histogram again has two p eaks centered around a dip at = This shap e results
from the same symmetry breaking e ect that o ccurred in the prop ortional tness case these
runs also evolved essentially the same strategies as the ep o ch strategies describ ed earlier
The b est and mean p erformances here are comparable to the b est p erformances in the
prop ortional tness case the b est p erformances found here are The p erformance as
a function of for one of the b est rules is plotted in Figure for lattice sizes of
the lattice size used for testing the rules in the GA runs and We used the same
pro cedure to make these plots as was describ ed earlier for Figure As can b e seen the
p erformance according to this measure is signi cantly worse than that of the GKL rule cf
Figure esp ecially on larger lattice sizes The worst p erformances for all lattice sizes are
centered close to the rule s value of 1
best performance 0.75 mean performance
0.5
fraction of total rules 0.25
0 0 0.25 0.5 0.75 1
λ
Figure Results from our exp eriment with p erformance as the
tness criterion The histogram plots the frequencies of elite rules
merged from the nal generations generation of runs in which
the p erformance tness function was used
1
0.75
0.5 149 cell lattice 599 cell lattice 0.25 999 cell lattice fraction of correct classifications 0 0 0.25 0.5 0.75 1
ρ
Figure Performance of one of the b est rules evolved using p erfor
mance tness plotted as a function of Performance plots are
given for three lattice sizes the size of the lattice used in the
GA runs and This rule has : 1
best fitness 0.75 mean fitness
0.5
fraction of total rules 0.25
0 0 0.25 0.5 0.75 1
λ
Figure Results from our exp eriment in which a diversity
enforcement mechanism was added to the GA The histogram plots
the frequencies of rules merged from the entire p opulation at gener
ation of runs in which our diversity enforcement scheme was
used
Adding A Diversity Enforcement Mechanism
The description given ab ove of the four ep o chs in the GA s search explains the results of
our exp eriment but it do es not explain the di erence b etween our results and those of the
original exp eriment rep orted in One di erence b etween our GA and the original was
the inclusion in the original of a diversity enforcement scheme that p enalized newly formed
rules that were to o similar in Hamming distance to existing rules in the p opulation To
test the e ect of this scheme on our results in one set of exp eriments we included a similar
scheme In our scheme every time a new string is created through crossover and mutation
the average Hamming distance b etween the new string and the elite strings the strings
that are copied unchanged is measured If this average distance is less than of the
string length here bits then the new string is not allowed in the new p opulation New
strings continue to b e created through crossover and mutation until new strings have met
this diversity criterion We note that many other diversity enforcement schemes have b een
develop ed in the GA literature e g crowding
The results of this exp eriment are given in Figure The histogram in that gure
represents the merged rules from the entire p opulation at generation of runs of the
GA using the prop ortional tness function and our diversity enforcement scheme As can
b e seen the histogram in this gure is very similar to that in Figure b The only ma jor
di erence is the signi cantly lower mean tness in the middle and leftmost bins which results
from the increased requirement for diversity in the nal non elite p opulation We conclude
that the use of this diversity enforcement scheme was not resp onsible for the di erence
b etween the results from and our results
Di erences Between Our Results and the Original Exp eriment
As was seen in Figure b our results are strikingly di erent from those rep orted in
These exp erimental results along with the theoretical argument that the most successful
rules for this task should have close to lead us to conclude that the interpretation of
the original results as giving evidence for the hyp otheses concerning evolution computation
and is not correct However we do not know what accounted for the di erences b etween
our results and those obtained in the original exp eriment We sp eculate that the di erences
are due to additional mechanisms in the GA used in the original exp eriment that were not
rep orted in
Although the results were very di erent there is one qualitative similarity the rule
frequency versus histograms in b oth cases contained two p eaks separated by a dip in the
center As already noted in our histogram the two p eaks were closer to by a
factor of but it is p ossible that the original results were due to a mechanism similar to
i the ep o ch sensitivity to initial con guration and p opulation asymmetry ab out
or ii the symmetry breaking we observed in ep o ch as describ ed ab ove Perhaps these
were combined with some additional force in the original GA that kept rules far away from
Unfortunately the b est and mean tnesses for the bins were not rep orted for
the original exp eriment As a consequence we do not know whether or not the p eaks in
the original histogram contained high tness rules or even if they contained rules that were
more t than rules in other bins Our results and the basic symmetry in the problem suggest
otherwise
General Discussion
What We Have Shown
The results rep orted in this pap er have demonstrated that the results from Packard s original
exp eriment do not hold up under our exp eriments We conclude that the original exp eriment
do es not give rm evidence for the hyp otheses it was meant to test rst that rules capable of
p erforming complex computation are most likely to b e found close to values and second
c
that when CA rules are evolved by a GA to p erform a nontrivial computation evolution will
tend to select rules close to values
c
As we argued theoretically and as our exp erimental results suggest the most successful
rules for p erforming a given classi cation task will b e close to a particular value of that
dep ends on the particular of the task Thus for this class of computational tasks the
c
values asso ciated with an edge of chaos are not correlated with the ability of rules to
c
p erform the task
In the remainder of this section we step back from these particular exp eriments and
discuss in more general terms the ideas that motivated these studies
Dynamical Behavior and Computation
As was noted earlier Langton presented evidence that given certain caveats regarding the
radius r and numb er of states K there is some correlation b etween and the b ehavior
of an average CA on an average initial con guration Behavior was characterized
in terms of quantities such as single site entropy two site mutual information di erence
pattern spreading rate and average transient length The correlation is quite go o d for very
low and very high values which predict xed p oint or short p erio d b ehavior However
for intermediate values there is a large degree of variation in b ehavior Moreover there
is no precise correlation b etween these values and the lo cation of a b ehavioral phase
transition other than that describ ed by Wo otters and Langton in the limit of in nite K
The remarks ab ove and all the exp erimental results in are concerned with the re
lationship b etween and the dynamical b ehavior of CA They do not directly address the
relationship b etween and computational capability of CA The basic hyp othesis was that
correlates with computational capability in that rules capable of complex and in particular
universal computation must b e or at least are most likely to b e found near values As
c
10
far as CA are concerned the hyp othesis was based on the intuition that complex compu
tation cannot b e supp orted in the short p erio d or chaotic regimes b ecause the phenomena
that apparently o ccur only in the complex non p erio dic non chaotic regimes such as
long transients and long space time correlation are necessary to supp ort complex computa
tion There has thus far b een no exp erimental evidence correlating with an indep endent
measure of computation Packard s exp eriments were intended to address this issue since
they involved an indep endent measure of computation p erformance on a particular complex
computational task but as we have shown these exp eriments do not provide evidence for
the hyp othesis linking values with computational ability In order to test this hyp othesis
c
more general measures of computation need to b e used
The argument that complex computation cannot o ccur in the short p erio d or chaotic
regimes may seem intuitively correct but there is actually a theoretical framework and
strong exp erimental evidence to the contrary Hanson and Crutch eld have devel
op ed a metho d for ltering out chaotic domains in the space time diagram of a CA some
times revealing particles that have the non p erio dic non chaotic prop erties of structures
in Wolfram s Class CA That is with the appropriate lter applied complex structures
can b e uncovered in a space time diagram that to the human eye and to the statistics used
in app ears to b e completely random As an extreme example it is conceivable that
such lters could b e applied to a seemingly chaotic CA and reveal that the CA is actually
implementing a universal computer with glider guns implementing AND OR and NOT
gates etc Hanson and Crutch eld s results strikingly illustrate that apparent complexity
of b ehavior and apparent computational capability can dep end on the implicit lter
imp osed by one s chosen statistics
10
In the context of continuous state dynamical systems it has b een shown that there is a direct relationship
b etween intrinsic computational capability of a pro cess and the degree of randomness of that pro cess at the
phase transition from order to chaos Computational capability was quanti ed with the statistical complexity
a measure of the amount of memory of a pro cess and via the detection of an emb edded computational
mechanism equivalent to a stack automaton
What Kind of Computation in CA Do We Care Ab out
In the section ab ove the phrases complex computation and computational capability
were used somewhat lo osely As was discussed in Section there are at least three di erent
interpretations of the notion of computation in CA The notion of a CA b eing able to
p erform a complex computation such as the task where the CA p erforms the
c
same computation on all initial con gurations is very di erent from the notion of a CA
b eing capable under some sp ecial set of initial con gurations of simulating a universal
computer Langton s sp eculations regarding the relationship b etween dynamical b ehavior
and computational capability seemed to b e more concerned with the latter than the former
though the implication is that the capability to sustain long transients long correlation
lengths and so on are necessary for b oth notions of computation
If computationally capable is taken to mean capable under some initial con gura
tion s of universal computation then one might ask why this is a particularly imp ortant
prop erty of CA on which to fo cus In CA were used merely as a vehicle to study
the relationship b etween phase transitions and computation with an emphasis on universal
computation But for those who want to use CA as scienti c mo dels or as practical compu
tational to ols a fo cus on the capacity for universal computation may b e misguided If a CA
is b eing used as a mo del of a natural pro cess e g turbulence then it is currently of limited
interest to know whether or not the pro cess is in principle capable of universal computation
if universal computation will arise only under some sp ecially engineered initial con guration
that the natural pro cess is extremely unlikely to ever encounter Instead if one wants to
understand emergent computation in natural phenomena as mo deled by CA then one should
try to understand what computation the CA intrinsically do es rather than what it
is in principle capable of doing only under some very sp ecial initial con gurations Thus
understanding the conditions under which a capacity for universal computation is p ossible
will not b e of much value in understanding the natural systems mo deled by CA
This general p oint is neither new nor deep Analogous arguments have b een put forth
in the context of neural networks for example While many constructions have b een made
of universal computation in neural networks e g some psychologists e g have
argued that this has little to do with understanding how brains or minds work in the natural
world
Similarly if one wants to use a CA as a parallel computer for solving a real problem
such as face recognition it would b e very ine cient if not practically imp ossible to solve
the problem by say programming Conway s Game of Life CA to b e a universal computer
that simulates the action of the desired face recognizer Thus understanding the conditions
under which universal computation is p ossible in CA is not of much practical value either
In addition it is not clear that anything like a drive toward universal computational
capabilities is an imp ortant force in the evolution of biological organisms It seems likely
that substantially less computationally capable prop erties play a more frequent and robust
role Thus asking under what the conditions evolution will create entities including CA
capable of universal computation may not b e of great imp ortance in understanding natural
evolutionary mechanisms
In short it is mathematically imp ortant to know that some CA are in principle capable
of universal computation But we argue that this is by no means the most scienti cally
interesting prop erty of CA More to the p oint this prop erty do es not help scientists much
in understanding the emergence of complexity in nature or in harnessing the computational
capabilities of CA to solve real problems
Conclusion
The main purp ose of this study was to examine and clarify the evidence for various hyp othe
ses related to evolution dynamics computation and cellular automata We hop e this study
has shed some new and constructive light on these issues As a result of our study we have
identi ed a numb er of evolutionary mechanisms such as the role of combinatorial drift and
the role of symmetry and the imp ediments to emerging computational strategies caused by
symmetry breaking For example we have found that the breaking of the goal task s sym
metries in the early generations can b e an imp ediment to further optimization of individuals
in the p opulation The symmetry breaking results is a kind of sub optimal sp eciation in the
p opulation that is stable or at least meta stable over long times The symmetry breaking
e ects we describ ed here may b e similar to symmetry breaking phenomena such as bilateral
symmetry and handedness that emerge in biological evolution It is our goal to develop a
more rigorous framework for understanding these mechanisms in the context of evolving CA
We b elieve that a deep understanding of these mechanisms in this relatively simple context
can yield insights for understanding evolutionary pro cesses in general and for successfully
applying evolutionary computation metho ds to complex problems
Though our exp eriments did not repro duce the results rep orted in we b elieve that
the original conception of using GAs to evolve computation in CA is an imp ortant idea
Aside from its p otential for studying various theoretical issues it also has a p otential prac
tical side that could b e signi cant As was mentioned earlier CA are increasingly b eing
studied as a class of e cient parallel computers the main b ottleneck in applying CA more
widely to parallel computation is programming in general it is very di cult to program CA
to p erform complex tasks Our results suggest that the GA has promise as a metho d for ac
complishing such programming automatically In order to further test the GA s e ectiveness
as compared with other search metho ds we p erformed an additional exp eriment comparing
the p erformance of our GA on the task with the p erformance of a simple steep est
c
ascent hill climbing metho d We found that the GA signi cantly outp erformed hill climbing
reaching much higher tnesses for an equivalent numb er of tness evaluations This gives
some evidence for the relative e ectiveness of GAs as compared with simple gradient ascent
metho ds for programming CA Koza has also evolved CA rules using a very di erent
typ e of representation scheme it is a topic of substantial practical interest to study the
relationship of representation and GA success on such tasks
Acknowlegments
This research was supp orted by the Santa Fe Institute under the Adaptive Computation and
External Faculty Programs and by the University of California Berkeley under contract
AFOSR Thanks to Doyne Farmer Jim Hanson Erica Jen Chris Langton Wentian
Li Cris Mo ore and Norman Packard for many helpful discussions and suggestions concerning
this pro ject Thanks also to Emily Dickinson and Terry Jones for technical advice
References
E Berlekamp J H Conway and R Guy Winning ways for your mathematical plays Aca
demic Press New York NY
J P Crutch eld and J E Hanson Turbulent pattern bases for cellular automata Technical
Rep ort SFI Santa Fe Institute
J P Crutch eld and K Young Inferring statistical complexity Physical Review Letters
J P Crutch eld and K Young Computation at the onset of chaos In W H Zurek edi
tor Complexity Entropy and the Physics of Information pages Addison Wesley
Redwo o d City CA
P Gonzaga de S a and C Maes The Gacs Kurdyumov Levin automaton revisited Journal of
Statistical Physics
D Farmer T To oli and S Wolfram editors Cel lular Automata Proceedings of an Inter
disciplinary Workshop North Holland Amsterdam
P Gacs G L Kurdyumov and L A Levin One dimensional uniform arrays that wash out
nite islands Probl Peredachi Inform
D E Goldb erg Genetic Algorithms in Search Optimization and Machine Learning Addison
Wesley Reading MA
J Guckenheimer and P Holmes Nonlinear Oscil lations Dynamical Systems and Bifurcations
of Vector Fields Springer Verlag New York
H A Gutowitz editor Cel lular Automata MIT Press Cambridge MA
J E Hanson and J P Crutch eld The attractor basin p ortrait of a cellular automaton
Journal of Statistical Physics
J H Holland Adaptation in Natural and Arti cial Systems The University of Michigan Press
Ann Arb or MI
S A Kau man Requirements for evolvability in complex systems Orderly dynamics and
frozen comp onents Physica D
S A Kau man and S Johnson Co evolution to the edge of chaos Coupled tness landscap es
p oised states and co evolutionary avalanches In C G Langton G Taylor J Doyne Farmer
and S Rasmussen editors Arti cial Life II pages Redwo o d City CA Addison
Wesley
J R Koza Genetic programming On the programming of computers by means of natural
selection MIT Press Cambridge MA
C G Langton Computation at the edge of chaos Phase transitions and emergent computa
tion Physica D
W Li Non lo cal cellular automata In L Nadel and D Stein editors Lectures in
Complex Systems pages Addison Wesley Redwo o d City CA
W Li and N H Packard The structure of the elementary cellular automata rule space
Complex Systems
W Li N H Packard and C G Langton Transition phenomena in cellular automata rule
space Physica D
K Lindgren and M G Nordahl Universal computation in a simple one dimensional cellular
automaton Complex Systems
N Packard Complexity of growing patterns in cellular automata In J Demongeot E Goles
and M Tchuente editors Dynamical Behavior of Automata Theory and Applications Aca
demic Press New York
N H Packard Personal communication
N H Packard Adaptation toward the edge of chaos In J A S Kelso A J Mandell and
M F Shlesinger editors Dynamic Patterns in Complex Systems pages Singap ore
World Scienti c
J B Pollack The induction of dynamical recognizers Machine Learning
K Preston and M Du Modern Cel lular Automata Plenum New York
A Rosenfeld Parallel image pro cessing using cellular arrays Computer
D E Rumelhart and J L McClelland PDP mo dels and general issues in cognitive science
In D E Rumelhart and J L McClelland editors Paral lel Distributed Processing volume
Cambridge MA MIT Press
H Siegelman and E D Sontag Neural networks are universal computing devices Technical
Rep ort SYCON Rutgers Center for Systems and Control Rutgers University New
Brunswick NJ
T To oli and N Margolus Cel lular Automata Machines A new environment for modeling
MIT Press Cambridge MA
S Wolfram Universality and complexity in cellular automata Physica D
S Wolfram editor Theory and applications of cel lular automata World Scienti c Singap ore
W K Wo otters and C G Langton Is there a sharp phase transition for deterministic cellular
automata Physica D
A Wuensche and M Lesser The global dynamics of cel lular automata Santa Fe Institute
Studies in the Sciences of Complexity Addison Wesley Reading MA