Classsiﬁcation of Cellular Automata

Klaus Sutner

Carnegie Mellon University

Pittsburgh,PA 15213

Contents

Glossary 1

1 Deﬁnition 2

2 Introduction 3

3 Reversibility and Surjectivity 6

4 Deﬁnability and Computability 9

5 Computational Equivalence 15

6 Conclusion 18

References 20

Glossary

Cellular Automaton

For our purposes,a (one-dimensional) cellular automaton (CA) is given by a local map

ρ:Σ

w

→Σ where Σ is the underlying alphabet of the automaton and w is its width.As a

data structure,suitable as input to a decision algorithm,a CA can thus be speciﬁed by a

simple lookup table.We abuse notation and write ρ(x) for the result of applying the global

map of the CA to conﬁguration x ∈ Σ

Z

.

Wolfram Classes

Wolfram proposed a heuristic classiﬁcation of cellular automata based on observations of

typical behaviors.The classiﬁcation comprises four classes:evolution leads to trivial con-

ﬁgurations,to periodic conﬁgurations,evolution is chaotic,evolution leads to complicated,

persistent structures.

Undecidability

1

It was recognized by logicians and mathematicians in the ﬁrst half of the 20th century

that there is an abundance of well-deﬁned problems that cannot be solved by means of an

algorithm,a mechanical procedure that is guaranteed to terminate after ﬁnitely many steps

and produce the appropriate answer.The best known example of an undecidable problem

is Turing’s Halting Problem:there is no algorithm to determine whether a given Turing

Machine halts when run on an empty tape.

Semi-Decidability

A problem is said to be semi-decidable or computably enumerable if it admits an al-

gorithm that return “yes” after ﬁnitely many steps if this is indeed the correct answer.

Otherwise the algorithm never terminates.The Halting Problem is the standard example

for a semi-decidable problem.A problem is decidable if,and only if,the problem itself and

its negation are semi-decidable.

Universality

A computational device is universal it is capable of simulating any other computational

device.The existence of universal computers was another central insight of the early days

of computability theory and is closely related to undecidability.

Reversibility

A discrete dynamical system is reversible if the evolution of the system incurs no loss of

information:the state at time t can be recovered from the state at time t +1.For CAs this

means that the global map is injective.

Surjectivity

The global map of a CA is surjective if every conﬁguration appears as the image of

another.By contrast,a conﬁguration that fails to have a predecessor is often referred to as

a Garden-of-Eden.

Finite Conﬁgurations

One often considers CA with a special quiescent state:the homogeneous conﬁguration

where all cells are in the quiescent state is required to be ﬁxed point under the global map.

Inﬁnite conﬁgurations where all but ﬁnitely many cells are in the quiescent state are often

called ﬁnite conﬁgurations.This is somewhat of a misnomer;we prefer to speak about

conﬁgurations with ﬁnite support.

1 Deﬁnition

Cellular automata display a large variety of behaviors.This was recognized clearly when

extensive simulations of cellular automata,and in particular one-dimensional CA,became

computationally feasible around 1980.Surprisingly,even when one considers only elemen-

tary CA,which are constrained to a binary alphabet and local maps involving only nearest

neighbors,complicated behaviors are observed in some cases.In fact,it appears that most

behaviors observed in automata with more states and larger neighborhoods already have

qualitative analogues in the realm of elementary CA.Careful empirical studies lead Wol-

fram to suggest a phenomenological classiﬁcation of CA based on the long-term evolution of

conﬁgurations,see [75,78] and section 2.While Wolfram’s four classes clearly capture some

2

of the behavior of CA it turns out that any attempt at formalizing this taxonomy meets

with considerable diﬃculties.Even apparently simple questions about the behavior of CA

turn out to be algorithmically undecidable and it is highly challenging to provide a detailed

mathematical analysis of these systems.

2 Introduction

In the early 1980’s Wolfram published a collection of 20 open problems in the the theory

of CA,see [76].The ﬁrst problem on his list is “What overall classiﬁcation of cellular au-

tomata behavior can be given?” As Wolfram points out,experimental mathematics provides

a ﬁrst answer to this problem:one performs a large number of explicit simulations and ob-

serves the patterns associated with the long term evolution of a conﬁguration,see [74,78].

Wolfram proposed a classiﬁcation that is based on extensive simulations in particular of

one-dimensional cellular automata where the evolution of a conﬁguration can be visualized

naturally as a two-dimensional image.The classiﬁcation involves four classes that can be

described as follows:

• W1:Evolution leads to homogeneous ﬁxed points.

• W2:Evolution leads to periodic conﬁgurations.

• W3:Evolution leads to chaotic,aperiodic patterns.

• W4:Evolution produces persistent,complex patterns of localized structures.

Thus,Wolfram’s ﬁrst three classes follow closely concepts from continuous dynamics:

ﬁxed point attractors,periodic attractors and strange attractors,respectively.They corre-

spond roughly to systems with zero temporal and spatial entropy,zero temporal entropy but

positive spatial entropy,and positive temporal and spatial entropy,respectively.W4 is more

diﬃcult to associate with a continuous analogue except to say that transients are typically

very long.To understand this class it is preferable to consider CAas models of massively par-

allel computation rather than as particular discrete dynamical systems.It was conjectured

by Wolfram that W4 automata are capable of performing complicated computations and

may often be computationally universal.Four examples of elementary CA that are typical

of the four classes are shown in ﬁgure 1.Li and Packard [35,36] proposed a slightly modiﬁed

version of this hierarchy by reﬁning the low classes and in particular Wolfram’s W2.Much

like Wolfram’s classiﬁcation,the Li-Packard classiﬁcation is concerned with the asymptotic

behavior of the automaton,the structure and behavior of the limiting conﬁgurations.Here

is one version of the Li-Packard classiﬁcation,see [36].

• LP1:Evolution leads to homogeneous ﬁxed points.

• LP2:Evolution leads to non-homogeneous ﬁxed points,perhaps up a to a shift.

3

Figure 1:Typical examples of the behavior described by Wolfram’s classes among elementary

cellular automata.

4

• LP3:Evolution leads to ultimately periodic conﬁgurations.Regions with periodic

behavior are separated by domain walls,possibly up to a shift.

• LP4:Conﬁgurations produce locally chaotic behavior.Regions with chaotic behavior

are separated by domain walls,possibly up to a shift.

• LP5:Evolution leads to chaotic patterns that are spatially unbounded.

• LP6:Evolution is complex.Transients are long and lead to complicated space-time

patterns which may be non-monotonic in their behavior.

By contrast,a classiﬁcation closer to traditional dynamical systems theory was introduced

by K˚urka,see [29,30].The classiﬁcation rests on the notions of equicontinuity,sensitivity to

initial conditions and expansivity.Suppose x is a point in some metric space and f a map

on that space.Then f is equicontinuous at x if

∀ε > 0 ∃δ > 0 ∀y ∈ B

δ

(x),n ∈ N(d(f

n

(x),f

n

(y)) < ε)

where d(.,.) denotes a metric.Thus,all points in a suﬃciently small neighborhood of x

remain close to the iterates of x for the whole orbit.Global equicontinuity is a fairly strong

condition,it implies that the limit set of the automaton is reached after ﬁnitely many steps.

The map is sensitive (to initial conditions) if

∀x,ε > 0 ∃δ > 0 ∀y ∈ B

δ

(x) ∃n ∈ N(d(f

n

(x),f

n

(y)) ≥ ε)

Lastly,the map is positively expansive if

∃ε > 0 ∀x = y ∃n ∈ N(d(f

n

(x),f

n

(y)) ≥ ε)

K˚urka’s classiﬁcation then takes the following form.

• K1:All points are equicontinuous under the global map.

• K2:Some but not all points are equicontinuous under the global map.

• K3:The global map is sensitive but not positively expansive.

• K4:The global map is positively expansive.

This type of classiﬁcation is perfectly suited to the analysis of uncountable spaces such

as the Cantor space {0,1}

N

or the full shift space Σ

Z

which carry a natural metric structure.

For the most part we will not pursue the analysis of CA by topological and measure theoretic

means here and refer to [31] in this volume for a discussion of these methods.See section 4

for the connections between topology and computability.

Given the apparent complexity of observable CA behavior one might suspect that it is

diﬃcult to pinpoint the location of an arbitrary given CA in any particular classiﬁcation

scheme with any precision.This is in contrast to simple parameterizations of the space of

5

CA rules such as Langton’s λ parameter that are inherently easy to compute.Brieﬂy,the λ

value of a local map is the fraction of local conﬁgurations that map to a non-zero value,see

[32,36].Small λ values result in short transients leading to ﬁxed points or simple periodic

conﬁgurations.As λ increases the transients grow longer and the orbits become more and

more complex until,at last,the dynamics become chaotic.Informally,sweeping the λ value

from0 to 1 will produce CA in W1,then W2,then W4 and lastly in W3.The last transition

appears to be associated with a threshold phenomenon.It is unclear what the connection

between Langton’s λ-value and computational properties of a CA is,see [51,41].Other

numerical measures that appear to be loosely connected to classiﬁcations are the mean ﬁeld

parameters of Gutowitz [21,22],the Z-parameter by Wuensche [79],see also [49].It seems

doubtful that a structured taxonomy along the lines of Wolframor Li-Packard can be derived

from a simple numerical measure such as the λ value alone,or even from a combination of

several such values.However,they may be useful as empirical evidence for membership in a

particular class.

Classiﬁcation also becomes signiﬁcantly easier when one restricts one’s attention to a

limited class of CA such as additive CA,see [70].In this context,additive means that the

local rule of the automaton has the form ρ(x) =

i

c

i

x

i

where the coeﬃcients as well as the

states are modular numbers.Anumber of properties starting with injectivity and surjectivity

as well as topological properties such as equicontinuity and sensitivity can be expressed in

terms of simple arithmetic conditions on the rule coeﬃcients.For example,equicontinuity

is equivalent to all prime divisors of the modulus m dividing all coeﬃcients c

i

,i > 1,see

[38] and the references therein.It is also noteworthy that in the linear case methods tend to

carry over to arbitrary dimensions;in general there is a signiﬁcant step in complexity from

dimension one to dimension two.

No claim is made that the given classiﬁcations are complete;in fact,one should think of

them as prototypes rather than deﬁnitive taxonomies.For example,one might add the class

of nilpotent CA at the bottom.A CA is nilpotent if all conﬁgurations evolve to a particular

ﬁxed point after ﬁnitely many steps.Equivalently,by compactness,there is a bound n such

that all conﬁgurations evolve to the ﬁxed point in no more than n steps.Likewise,we could

add the class of intrinsically universal CA at the top.A CA is intrinsically universal if

it is capable of simulating all other CA of the same dimension in some reasonable sense.

For a fairly natural notion of simulation see [50].At any rate,considerable eﬀort is made

in the references to elaborate the characteristics of the various classes.For many concrete

CA visual inspection of the orbits of a suitable sample of conﬁgurations readily suggests

membership in one of the classes.

3 Reversibility and Surjectivity

A ﬁrst tentative step towards the classiﬁcation of a dynamical systems is to determine its

reversibility or lack thereof.Thus we are trying to determine whether the evolution of the

system is associated with loss of information,or whether it is possible to reconstruct the

state of the system at time t from its state at time t +1.In terms of the global map of the

6

system we have to decide injectivity.Closely related is the question whether the global map

is surjective,i.e.,whether there is no Garden-of-Eden:every conﬁguration has a predecessor

under the global map.As a consequence,the limit set of the automaton is the whole space.It

was shown of Hedlund that for CA the two notions are connected:every reversible CA is also

surjective,see [25,44].As a matter of fact,reversibility of the global map of a CA implies

openness of the global map,and openness implies surjectivity.The converse implications are

both false.By a well-known theorem by Hedlund [25] the global maps of CA are precisely

the continuous maps that commute with the shift.It follows from basic topology that the

inverse global map of a reversible CA is again the global map of a suitable CA.Hence,the

predecessor conﬁguration of a given conﬁguration can be reconstructed by another suitably

chosen CA.For results concerning reversibility on the limit set of the automaton see [67].

From the perspective of complexity the key result concerning reversible systems is the

work by Lecerf [33] and Bennett [7].They show that reversible Turing machines can compute

any partial recursive function,modulo a minor technical problem:In a reversible Turing

machine there is no loss of information;on the other hand even simple computable functions

are clearly irreversible in the sense that,say,the sum of two natural numbers does not

determine these numbers uniquely.To address this issue one has to adjust the notion of

computability slightly in the context of reversible computation:given a partial recursive

function f:N →N the function

f(x) = x,f(x) can be computed by a reversible Turing

machine where .,. is any eﬀective pairing function.If f itself happens to be injective

then there is no need for the coding device and f can be computed by a reversible Turing

machine directly.For example,we can compute the product of two primes reversibly.Morita

demonstrated that the same holds true for one-dimensional cellular automata [68,45,42,

28]:reversibility is no obstruction to computational universality.As a matter of fact,any

irreversible cellular automaton can be simulated by a reversible one,at least on conﬁgurations

with ﬁnite support.Thus one should expect reversible CA to exhibit fairly complicated

behavior in general.

For inﬁnite,one-dimensional CA it was shown by Amoroso and Patt [2] that reversibility

is decidable.Moreover,it is decidable if the the global map is surjective.An eﬃcient practical

algorithm using concepts of automata theory can be found in [61],see also [14,24,10].The

fast algorithm is based on interpreting a one-dimensional CA as deterministic transducer,

see [6,53] for background.The underlying semi-automaton of the transducer is a de Brujin

automaton B whose states are words in Σ

w−1

where Σ is the alphabet of the CA and w is its

width.The transitions are given by ax

c

−→xb where a,b,c ∈ Σ,x ∈ Σ

w−2

and c = ρ(axb),ρ

being the local map of the CA.Since B is strongly connected,the product automaton of B

will contain a strongly connected component C that contains the diagonal D,an isomorphic

copy of B.The global map of the CA is reversible if,and only if,C = D is the only

non-trivial component.It was shown by Hedlund [25] that surjectivity of the global map

is equivalent with local injectivity:the restriction of the map to conﬁgurations with ﬁnite

support must be injective.The latter property holds if,and only if,C = D and is thus easily

decidable.Automata theory does not readily generalize to words of dimensions higher than

one.Indeed,reversibility and surjectivity in dimensions higher than one are undecidable,see

7

Figure 2:A reversible automaton obtain by applying Fredkin’s construction to the irre-

versible elementary CA 90.

8

[27] and [28] in this volume for the rather intricate argument needed to establish this fact.

While the structure of reversible one-dimensional CA is well-understood,see [28,16],and

while there is an eﬃcient algorithm to check reversibility,few methods are known that allow

for the construction of interesting reversible CA.There is a noteworthy trick due to Fredkin

that exploits the reversibility of the Fibonacci equation X

n+1

= X

n

+X

n−1

.When addition

is interpreted as exclusive or this can be used to construct a second-order CA from any given

binary CA;the former can then be recoded as a ﬁrst-order CA over a 4-letter alphabet.For

example,for the open but irreversible elementary CA number 90 we obtain the CA shown

in ﬁgure 2.

Another interesting class of reversible one-dimensional CA,the so-called partitioned cel-

lular automata (PCA),is due to Morita and Harao,see [45,42,43].One can think of a PCA

as a cellular automaton whose cells are divided into multiple tracks;speciﬁcally Morita uses

an alphabet of the form Σ = Σ

1

× Σ

2

× Σ

3

.The conﬁgurations of the automaton can be

written as (X,Y,Z) where X ∈ Σ

1

Z

,Y ∈ Σ

2

Z

and Z ∈ Σ

3

Z

.Now consider the shearing

map σ deﬁned by σ(X,Y,Z) = (RS(X),Y,LS(Z)) where RS and LS denote the right and

left shift,respectively.Given any function f:Σ → Σ we can deﬁne a global map f ◦ σ

where f is assumed to be applied point-wise.Since the shearing map is bijective,the CA

will be reversible if,and only if,the map f is bijective.It is relatively easy to construct

bijections f that cause the CA to performparticular computational tasks,even when a direct

construction appears to be entirely intractable.

4 Deﬁnability and Computability

4.1 Formalizing Wolfram’s Classes

Wolfram’s classiﬁcation is an attempt to categorize the complexity of the CA by studying the

patterns observed during the long-term evolution of all conﬁgurations.The ﬁrst two classes

are relatively easy to observe,but it is diﬃcult to distinguish between the last two classes.In

particular W4 is closely related to the kind of behavior that would be expected in connection

with systems that are capable of performing complicated computations,including the ability

to perform universal computation;a property that is notoriously diﬃcult to check,see [58].

The focus on the full conﬁguration space rather than a signiﬁcant subset thereof corresponds

to the worst-case approach well-known in complexity theory and is somewhat inferior to an

average case analysis.Indeed,Baldwin and Shelah point out that a product construction

can be used to design a CA whose behavior is an amalgamation of the behavior of two given

CA,see [4,3].By combining CA in diﬀerent classes one obtains striking examples of the

weakness of the worst-case approach.A natural example of this mixed type of behavior

is elementary CA 184 which displays class II or class III behavior,depending on the initial

conﬁguration.Another basic example for this type of behavior is the well-studied elementary

CA 30,see section 6.

Still,for many CA a worst-case classiﬁcation seems to provide useful information about

the structural properties of the automaton.The ﬁrst attempt at formalizing Wolfram’s class

9

was made by Culik and Yu who proposed the following hierarchy,given here in cumulative

form,see [11]:

• CY1:All conﬁgurations evolve to a ﬁxed point.

• CY2:All conﬁgurations evolve to a periodic conﬁguration.

• CY3:The orbits of all conﬁgurations are decidable.

• CY4:No constraints.

The Culik-Yu classiﬁcation employs two rather diﬀerent methods.The ﬁrst two classes can

be deﬁned by a simple formula in a suitable logic whereas the third (and the fourth in

the disjoint version of the hierarchy) rely on notions of computability theory.As a general

framework for both approaches we consider discrete dynamical systems,structures of the

form A = C, where C ⊆ Σ

Z

is the space of conﬁgurations of the system and is the

“next conﬁguration” relation on C.We will only consider the deterministic case where for

each conﬁguration x there exists precisely one conﬁguration y such that x y.Hence we

are really dealing with algebras with one unary function,but iteration is slightly easier to

deal with in the relational setting.The structures most important in this context are the

ones arising from a CA.For any local map ρ we consider the structure A

ρ

= C, where

the next conﬁguration relation is determined by x ρ(x).

Using the standard language of ﬁrst order logic we can readily express properties of the

CA in terms of the system A

ρ

.For example,the system is reversible,respectively surjective,

if the following assertions are valid over A:

∀x,y,z (x z and y z implies x = y)

∀x∃y (y x)

As we have seen,both properties are easily decidable in the one-dimensional case.In fact,

one can express the basic predicate x y (as well as equality) in terms of ﬁnite state ma-

chines on inﬁnite words.These machines are deﬁned like ordinary ﬁnite state machines but

the acceptance condition requires that certain states are reached inﬁnitely and co-inﬁnitely

often,see [8,20].The emptiness problem for these automata is easily decidable using graph

theoretic algorithms.Since regular languages on inﬁnite words are closed under union,com-

plementation and projection,much like their ﬁnite counterparts,and all the corresponding

operations on automata are eﬀective,it follows that one can decide the validity of ﬁrst order

sentences over A

ρ

such as the two examples above:the model-checking problem for these

structures and ﬁrst order logic is decidable,see [37].For example,we can decide whether

there is a conﬁguration that has a certain number of predecessors.Alternatively,one can

translate these sentences into monadic second order logic of one successor,and use well-

known automata-based decision algorithms there directly,see [8].Similar methods can be

used to handle conﬁgurations with ﬁnite support,corresponding to weak monadic second

order logic.Since the complexity of the decision procedure is non-elementary one should not

expect to be able to handle complicated assertions.On the other hand,at least for weak

10

monadic second order logic practical implementations of the decision method exist,see [18].

There is no hope of generalizing this approach as the undecidability of,say,reversibility in

higher dimensions demonstrates.

Write x

t

→y if x evolves to y in exactly t steps,x

+

→y if x evolves to y in any positive

number of steps and x

∗

→y if x evolves to y in any number of steps.Note that

t

→is deﬁnable

for each ﬁxed t,but

∗

→ fails to be so deﬁnable in ﬁrst order logic.This is in analogy to the

undeﬁnability of path existence problems in the ﬁrst order theory of graphs,see [37].Hence

it is natural to extend our language so we can express iterations of the global map,either

by adding transitive closures or by moving to some limited system of higher order logic over

A

ρ

where

∗

→is deﬁnable,see [8].

Arguably the most basic decision problem associated with a system A that requires

iteration of the global map is the Reachability Problem:given two conﬁgurations x and y,

does the evolution of x lead to y?A closely related but diﬀerent question is the Conﬂuence

Problem:will two conﬁgurations x and y evolve to the same limit cycle?Conﬂuence is an

equivalence relation and allows for the decomposition of conﬁguration space into limit cycles

together with their basins of attraction.The Reachability and Conﬂuence Problem amount

to determining,given conﬁgurations x and y,whether

x

∗

→y,

∃z (x

∗

→z and y

∗

→z),

respectively.As another example,the ﬁrst two Culik-Yu class can be deﬁned like so:

∀x∃z (x

∗

→z and z z),

∀x∃z (x

∗

→z and z

+

→z).

It is not diﬃcult to give similar deﬁnitions for the lower Li-Packard classes if one extends

the language by a function symbol denoting the shift operator.

The third Culik-Yu class is somewhat more involved.By deﬁnition,a CA lies in the third

class if it admits a global decision algorithm to determine whether a given conﬁguration x

evolves to another given conﬁguration y in a ﬁnite number of steps.In other words,we are

looking for automata where the Reachability Problem is algorithmically solvable.While one

can agree that W4 roughly translates into undecidability and is thus properly situated in the

hierarchy,it is unclear howchaotic patterns in W3 relate to decidability.No method is known

to translate the apparent lack of tangible,persistent patterns in rules such as elementary

CA 30 into decision algorithms for Reachability.There is another,somewhat more technical

problem to overcome in formalizing classiﬁcations.Recall that the full conﬁguration space

is C = Σ

Z

.Intuitively,given x ∈ C we can eﬀectively determine the next conﬁguration

y = ρ(x).However,classical computability theory does not deal with inﬁnitary objects such

as arbitrary conﬁguration so a bit of care is needed here.The key insight is that we can

determine arbitrary ﬁnite segments of ρ(x) using only ﬁnite segments of x (and,of course,

the lookup table for the local map).There are several ways to model computability on

11

Σ

Z

based on this idea of ﬁnite approximations,we refer to [73] for a particularly appealing

model based on so-called type-2 Turing machines;the reference also contains many pointers

to the literature as well as a comparison between the diﬀerent approaches.It is easy to see

that for any CA the global map ρ as well as all its iterates ρ

t

are computable,the latter

uniformly in t.However,due to the ﬁnitary nature of all computations,equality is not

decidable in type-2 computability:the unequal operator U

0

(x,y) = 0 if x = y,U

0

(x,y)

undeﬁned otherwise,is computable and thus unequality is semi-decidable,but the stronger

U

0

(x,y) = 0 if x = y,U

0

(x,y) = 1,otherwise,is not computable.The last result is

perhaps somewhat counterintuitive,but it is inevitable if we strictly adhere to the ﬁnite

approximation principle.

In order to avoid problems of this kind it has become customary to consider certain sub-

spaces of the full conﬁguration space,in particular C

ﬁn

,the collection of conﬁgurations with

ﬁnite support,C

per

,the collection of spatially periodic conﬁgurations and C

ap

,the collection

of almost periodic conﬁgurations of the form...uuuwvvv...where u,v and w are all ﬁnite

words over the alphabet of the automaton.Thus,an almost periodic conﬁguration diﬀers

from a conﬁguration of the form

ω

uv

ω

in only ﬁnitely many places.Conﬁgurations with

ﬁnite support correspond to the special case where u = v = 0 is a special quiescent symbol

and spatially periodic conﬁgurations correspond to u = v,w = ε.The most general type of

conﬁguration that admits a ﬁnitary description is the class C

rec

of recursive conﬁgurations,

where the assignment of state to a cell is given by a computable function.

It is clear that all these subspaces are closed under the application of a global map.Except

for C

ﬁn

there are also closed under inverse maps in the following sense:given a conﬁguration

y in some subspace that has a predecessor x in C

all

there already exists a predecessor in the

same subspace,see [64,61].This is obvious except in the case of recursive conﬁgurations.

The reference also shows that the recursive predecessor cannot be computed eﬀectively from

the target conﬁguration.Thus,for computational purposes the dynamics of the cellular

automaton are best reﬂected in C

ap

:it includes all conﬁguration with ﬁnite support and we

can eﬀectively trace an orbit in both directions.It is not hard to see that C

ap

is the least

such class.Alas,it is standard procedure to avoid minor technical diﬃculties arising from

the inﬁnitely repeated spatial patterns and establish classiﬁcations over the subspace C

ﬁn

.

There is a arguably not much harm in this simpliﬁcation since C

ﬁn

is a dense subspace of C

all

and compactness can be used to lift properties from C

ﬁn

to the full conﬁguration space.

The Culik-Yu hierarchy is correspondingly deﬁned over C

ﬁn

,the class of all conﬁgurations

of ﬁnite support.In this setting,the ﬁrst three classes of this hierarchy are undecidable and

the fourth is undecidable in the disjunctive version:there is no algorithm to test whether a

CA admits undecidable orbits.As it turns out,the CA classes are complete in their natural

complexity classes within the arithmetical hierarchy [56,58].Checking membership in the

ﬁrst two classes comes down to performing an inﬁnite number of potentially unbounded

searches and can be described logically by a Π

2

expression,a formula of type ∀x∃y R(x,y)

where R is a decidable predicate.Indeed,CY1 and CY2 are both Π

2

-complete.Thus,

deciding whether all conﬁgurations on a CA evolve to a ﬁxed point is equivalent to the

classical problem of determining whether a semi-decidable set is inﬁnite.The third class is

12

even less amenable to algorithmic attack;one can show that CY3 is Σ

3

-complete,see [59].

Thus,deciding whether all orbits are decidable is as diﬃcult as determining whether any

given semi-decidable set is decidable.It is not diﬃcult to adjust these undecidability results

to similar classes such as the lower levels of the Li-Packard hierarchy that takes into account

spatial displacements of patterns.

4.2 Eﬀective Dynamical Systems and Universality

The key property of CA that is responsible for all these undecidability results is the fact

that CA are capable of performing arbitrary computations.This is unsurprising when one

deﬁnes computability in terms of Turing machines,the devices introduced by Turing in the

1930’s,see [69,52].Unlike the G¨odel-Herbrand approach using general recursive functions

or Church’s λ-calculus,Turing’s devices are naturally closely related to discrete dynamical

systems.For example,we can express an instantaneous description of a Turing machine as

a ﬁnite sequence

a

−l

a

−l+1

...a

−1

p a

1

a

2

...a

r

where the a

i

are tape symbols and p is a state of the machine,with the understanding that

the head is positioned at a

1

and that all unspeciﬁed tape cells contain the blank symbol.

Needless to say,these Turing machine conﬁgurations can also be construed as ﬁnite sup-

port conﬁgurations of a one-dimensional CA.It follows that a one-dimensional CA can be

used to simulate an arbitrary Turing machine,hence CA are computational universal:any

computable function whatsoever can already be computed by a CA.

Note,though,that the simulation is not entirely trivial.First,we have to rely on in-

put/output conventions.For example,we may insist that objects in the input domain,

typically tuples of natural numbers,are translated into a conﬁguration of the CA by a prim-

itive recursive coding function.Second,we need to adopt some convention that determines

when the desired output has occurred:we follow the evolution of the input conﬁguration

until some “halting” condition applies.Again,this condition must be primitive recursively

decidable though there is considerable leeway as to how the end of a computation should

be signaled by the CA.For example,we could insist that a particular cell reaches a special

state,that an arbitrary cell reaches a special state,that the conﬁguration be a ﬁxed point

and so forth.Lastly,if and when a halting conﬁguration is reached,we a apply a primitive

recursive decoding function to obtain the desired output.

Restricting the space to conﬁgurations that have ﬁnite support,that are spatially peri-

odic,and so forth,produces an eﬀective dynamical system:the conﬁgurations can be coded

as integers in some natural way,and the next conﬁguration relation is primitive recursive

in the sense that the corresponding relation on code numbers is so primitive recursive.A

classical example for an eﬀective dynamical system is given by selecting the instantaneous

descriptions of a Turing machine M as conﬁgurations,and one-step relation of the Turing

machine as the operation of C.Thus we obtain a system A

M

whose orbits represent the

computations of the Turing machine.Likewise,given the local map ρ of a CA we obtain a

system A

ρ

whose operation is the induced global map.While the full conﬁguration space C

all

13

violates the eﬀectiveness condition,any of the spaces C

per

,C

ﬁn

,C

ap

and C

rec

will give rise to an

eﬀective dynamical system.Closure properties as well as recent work on the universality of

elementary CA 110,see section 6,suggests that the class of almost periodic conﬁgurations,

also known as backgrounds or wallpapers,see [64,9],is perhaps the most natural setting.

Both C

ﬁn

and C

ap

provide a suitable setting for a CA that simulates a Turing machine:we

can interpret A

M

as a subspace of A

ρ

for some suitably constructed one-dimensional CA ρ;

the orbits of the subspace encode computations of the Turing machine.It follows from the

undecidability of the Halting Problem for Turing machines that the Reachability Problem

for these particular CA is undecidable.

Note,though,that orbits in A

M

may well be ﬁnite,so some care must be taken in setting

up the simulation.For example,one can translate halting conﬁgurations into ﬁxed points.

Another problem is caused by the worst-case nature of our classiﬁcation schemes:in Turing

machines and their associated systems A

M

it is only behavior on specially prepared initial

conﬁgurations that matters,whereas the behavior of a CAdepends on all conﬁgurations.The

behavior of a Turing machine on all instantaneous descriptions,rather than just the ones that

can occur during a legitimate computation on some actual input,was ﬁrst studied by Davis,

see [12,13],and also Hooper [26].Call a Turing machine stable if it halts on any instantaneous

description whatsoever.With some extra care one can then construct a CA that lies in the

ﬁrst Culik-Yu class,yet has the same computational power as the Turing machine.Davis

showed that every total recursive function can already be computed by a stable Turing

machine,so membership in CY1 is not an impediment to considerable computational power.

The argument rests on a particular decomposition of recursive functions.Alternatively,one

directly manipulate Turing machines to obtain a similar result,see [55,59].On the other

hand,unstable Turing machines yield a natural and coding-free deﬁnition of universality:a

Turing machine is Davis-universal if the set of all instantaneous description on which the

machine halts is Σ

1

-complete.

The mathematical theory of inﬁnite CAis arguably more elegant than the actually observ-

able ﬁnite case.As a consequence,classiﬁcations are typically concerned with CA operating

on inﬁnite grids,so that even a conﬁguration with ﬁnite support can carry arbitrarily much

information.If we restrict our attention to the space of conﬁgurations on a ﬁnite grid a more

ﬁne-grained analysis is required.For a ﬁnite grid of size n the conﬁguration space has the

form C

n

= [n] →Σ and is itself ﬁnite,hence any orbit is ultimately periodic and the Reach-

ability Problem is trivially decidable.However,in practice there is little diﬀerence between

the ﬁnite and inﬁnite case.First,computational complexity issues make it practically impos-

sible to analyze even systems of modest size.The Reachability Problem for ﬁnite CA,while

decidable,is PSPACE-complete even in the one-dimensional case.Computational hardness

appears in many other places.For example,if we try to determine whether a given conﬁgu-

ration on a ﬁnite grid is a Garden-of-Eden the problem turns out to be NLOG-complete in

dimension one and NP-complete in all higher dimensions,see [62].

Second,it stands to reason that the more interesting classiﬁcation problem in the ﬁnite

case takes the following parameterized form:given a local map together with boundary con-

ditions,determine the behavior of ρ on all ﬁnite grids.Under periodic boundary conditions

14

this comes down to the study of C

per

and it seems that there is little diﬀerence between this

and the ﬁxed boundary case.Since all orbits on a ﬁnite grid are ultimately periodic one

needs to apply a more ﬁne-grained classiﬁcation that takes into account transient lengths.

It is undecidable whether all conﬁgurations on all ﬁnite grids evolve to a ﬁxed point under

a given local map,see [60].Thus,there is no algorithm to determine whether

C

n

, |= ∀x∃z (x

∗

→z and z z)

for all grid sizes n.The transient lengths are trivially bounded by k

n

where k is the size

of the alphabet of the automaton.It is undecidable whether the transient lengths grow

according to some polynomial bound,even when the polynomial in question is constant.

Restrictions of the conﬁguration space are one way to obtain an eﬀective dynamical

system.Another is to interpret the approximation-based notion of computability on the

full space in terms of topology.It is well-known that computable maps C

all

→ C

all

are

continuous in the standard product topology.The clopen sets in this topology are the ﬁnite

unions of cylinder sets where a cylinder set is determined by the values of a conﬁguration

in ﬁnitely many places.By a celebrated result of Hedlund the global maps of a CA on

the full space are characterized by being continuous and shift-invariant.Perhaps somewhat

counter-intuitively,the decidable subsets of C

all

are quite weak,they consist precisely of the

clopen sets.Now consider a partition of C

all

into ﬁnitely many clopen sets C

0

,C

2

,...,

C

n−1

.Thus,it is decidable which block of the partition a given point in the space belongs

to.Moreover,Boolean operations on clopen sets as well as application of the global map

and the inverse global map are all computable.The partition aﬀords a natural projection

π:C

all

→ Σ

n

where Σ

n

= {0,1,...,n −1} and π(x) = i iﬀ x ∈ C

i

.Hence the projection

translates orbits in the full space C

all

into a class W of ω-words over Σ

n

,the symbolic orbits

of the system.The Cantor space Σ

Z

n

together with the shift describes all logically possible

orbits with respect to the given partition and W describes the symbolic orbits that actually

occur in the given CA.The shift operator corresponds to an application of the global map

of the CA.The ﬁnite factors of W provide information about possible ﬁnite traces of an

orbit when ﬁltered through the given partition.Whole orbits,again ﬁltered through the

partition,can be described by ω-words.To tackle the classiﬁcation of the CA in terms of

W it was suggested by Delvenne et al.,see [15],to refer to the CA as decidable if there

it is decidable whether W has non-empty intersection with a ω-regular language.Alas,

decidability in this sense is very diﬃcult,its complexity being Σ

1

1

-complete and thus outside

of the arithmetical hierarchy.Likewise it is suggested to call a CA universal if the problem

of deciding whether the cover of W,the collection of all ﬁnite factors,is Σ

1

-complete,in

analogy to Davis-universality.

5 Computational Equivalence

In recent work,Wolfram suggests a so-called Principle of Computational Equivalence,or

PCE for short,see [78,p.717].PCE states that most computational processes come in only

two ﬂavors:they are either of a very simple kind and avoid undecidability,or they represent a

15

universal computation and are therefore no less complicated than the Halting Problem.Thus,

Wolfram proposes a zero-one law:almost all computational systems,and thus in particular

all CA,are either as complicated as a universal Turing machine or are computationally

simple.As evidence for PCE Wolframadduces a very large collection of simulations of various

eﬀective dynamical systems such as Turing machines,register machines,tag systems,rewrite

systems,combinators,and cellular automata.It is pointed out in chapter 3 of [78],that in

all these classes of systems there are surprisingly small examples that exhibit exceedingly

complicated behavior–and presumably are capable of universal computation.Thus it is

conceivable that universality is a rather common property,a property that is indeed shared

by all systems that are not obviously simple.Of course,it is often very diﬃcult to give a

complete proof of the computational universality of a natural system,as opposed to carefully

constructed one,so it is not entirely clear how many of Wolfram’s examples are in fact

universal.As a case in point consider the universality proof of Conway’s Game of Life,or

the argument for elementary CA 110.If Wolfram’s PCE can be formally established in some

formit stands to reason that it will apply to all eﬀective dynamical systems and in particular

to CA.Hence,classiﬁcations of CA would be rather straightforward:at the top there would

be the class of universal CA,directly preceded by a class similar to the third Culik-Yu class,

plus a variety of subclasses along the lines of the lower Li-Packard classes.

The corresponding problem in classical computability theory was ﬁrst considered in the

1930’s by Post and is now known as Post’s Problem:is there a semi-decidable set that fails

to be decidable,yet is not as complicated as the Halting Set?In terms of Turing degrees the

problem thus is to construct a semi-decidable set A such that ∅ <

T

A <

T

∅

,or to rule out

the existence of any such set,see [52,58,34] for background on Turing degrees in general

and semi-decidable degrees in particular.Post’s Problem resisted all attempts at resolution

until Friedberg and Muchnik independently and almost simultaneously discovered a way to

construct a set of intermediate complexity,see [19,46].The construction is based on the idea

of a so-called priority argument and is signiﬁcantly more complicated than any construction

of semi-decidable sets previously known [58].Indeed,priority arguments have since become

the hallmark of computability theory and have even engendered some criticism as being so

very technical that,occasionally,the proofs seemto attract more attention than the theorems

being established,see [72].Be that as it may,it is striking how much more artiﬁcial and ad

hoc intermediate sets are,as compared to natural examples such as the theory of the reals

(decidable) or of Diophantine equations (equivalent to the Halting Problem).No natural

examples of intermediate semi-decidable sets are known to date.

Nonetheless,given an intermediate set A one can construct a one-dimensional CA whose

Reachability Problem has the same degree as A.This suggests a degree-based classiﬁcation:

given any computably enumerable degree d,deﬁne the class C

d

to consist of all CA whose

Reachability Problem has degree exactly d,see [63,65].The degree classiﬁcation is non-

trivial in the sense that every class is non-empty.Note that the ﬁrst three Culik-Yu classes are

all contained in C

0

whereas C

0

comprises all computationally universal CA.Unsurprisingly,

it is again undecidable whether a CA belongs to any particular class.At the bottom end

of the hierarchy it is Σ

3

-complete to determine membership in C

0

;at the top end it is

16

Σ

4

-complete to determine membership in C

0

.Thus,it is easier to determine decidability

than universality.In general,deciding membership in C

d

is Σ

d

3

-complete for any semi-

decidable degree d.Similar results hold for the analogous cumulative classes C

≤d

=

e≤d

C

e

.

Unlike the Culik-Yu classiﬁcation,the structure of the degree classiﬁcation between C

0

and C

0

is exceedingly complicated.For example,the proof of the Friedberg-Muchnik theo-

rem shows that there are incomparable semi-decidable degrees d

1

and d

2

.Hence there is are

CA whose orbits are undecidable but not as complicated as the Halting Problem.Indeed,

complete knowledge of the orbits of one of the two CA will not help in deciding membership

in the orbits of the other.Another surprising result in the theory of computably enumer-

able degrees is Sack’s Density Theorem,see [58]:between any two computably enumerable

degrees d

1

< d

2

there lies a third:d

1

< d < d

2

.Thus,between any two CA of strictly

increasing complexity there is an inﬁnite and dense hierarchy of other CA.The computably

enumerable degrees form a semi-lattice,so it is natural to try to understand the complexity

of the structure by analyzing its ﬁrst order theory.It is well-known that the Σ

1

-theory of

this semi-lattice is decidable.However,the reason for this decidability result lies in the fact

that any countable partial order can be embedded into the semi-lattice so that the relative

computational strength of cellular automata is indeed arbitrarily complicated.On the other

hand,the full theory of the semi-lattice of semi-decidable degrees is known to be highly un-

decidable,see [23];its degree is ∅

(ω)

.One might hope that restriction to reversible CA would

simplify the situation somewhat.Somewhat surprisingly it turns out that each class C

d

al-

ready contains an irreversible CA,see [66],so the same diﬃculties arise in the classiﬁcation

of reversible CA as in the classiﬁcation of ordinary CA.

While reachability is arguably the most basic relation between conﬁgurations,similar

diﬃculties also arise with conﬂuence.As a matter of fact,one can construct a CA whose

Reachability Problemhas complexity some arbitrarily chosen computably enumerable degree

d

1

while the Conﬂuence Problem for the same CA has degree d

2

,another arbitrarily chosen

computably enumerable degree.Thus,a classiﬁcation according to reachability is entirely

independent of a conﬂuence-based classiﬁcation.

How do these results relate to PCE?Wolfram would not accept any of the intermediate

classes of CAas a counterexample to PCE.The argument is that though intermediate degrees

exist,their construction is critically linked to universal computation.While the universal

computation is invisible when only the output of the system is observed,the associated

computational process includes the whole computation and is thus universal.As a case

in point,consider the standard Friedberg-Muchnik construction for an intermediate semi-

decidable set A.The construction actually builds two semi-decidable sets A and B that are

mutually incomparable with respect to Turing reducibility.Only A is output and B remains

hidden.However,even ignoring all the intricate technical details of the whole construction,if

we consider both A and B as output then the computation is indeed universal:the disjoint

union A ⊕ B is Σ

1

-complete,see [57].It remains to be seen if similar arguments can be

put forth in connection with priority-free constructions of intermediate degrees or if natural

examples of intermediate sets can be found.At any rate,by considering only the reachability

relation instead of a whole segment of the orbit we also achieve information-hiding,much as

17

Figure 3:A pseudo-random pattern generated by elementary CA 30.

in the classical Friedberg-Muchnik construction.

6 Conclusion

Classiﬁcation schemes of cellular automata based on the long-term evolution of pattern are

typically undecidable,even if the property in question can be expressed in a fairly week

system.While it is easy to construct examples of CA in particular classes it is usually very

diﬃcult to establish the position of a given CA in a particular classiﬁcation.An excellent

example for the diﬃculty of analyzing a given CA is Cook’s proof of the universality of

elementary CA number 110 whose local rule is given by ρ(x,y,z) = (

x∧y ∧z) ⊕y ⊕z where

⊕ denotes exclusive or,see [9,17].The argument shows that cyclic tag systems,which

are known to be complete,can be simulated by elementary CA 110 provided one allows an

almost periodic background.Recent work by Neary and Woods has shown that the whole

simulation can eﬀected with only a polynomial slow-down,see [47,48].This result suggests

that the appropriate setting for classiﬁcations is the space of almost periodic conﬁgurations

rather than ﬁnite ones.

In light of the successful analysis of elementary CA 110 it is tempting to ask about the

18

Figure 4:Interacting signals in Mazoyer’s optimal solution to the ﬁring squad problem.

classiﬁcation of elementary CA 30.Figure 3 shows a segment of the orbit of a one-point seed

conﬁguration under rule 30.It is striking how chaotic and apparently random the image is.

As a matter of fact,rule 30 has been used for many years as the default random number

generator in the commercial computer algebra systemMathematica,see [77].The underlying

local map is simply ρ(x,y,z) = x ⊕(y ∨ z).Alas,there appear to be no structures in the

evolution of conﬁgurations under rule 30 such as “moving particles” that might be exploited

in a universality argument along the lines of rule 110.On the other hand,it is unclear

how a decision procedure for reachability could be developed.This makes it tempting to

conjecture that rule 30 in C

ap

might be a member of one of the intermediate classes C

d

,

though at present there seems to be no way to either establish or refute this conjecture.

While undecidability results rule out the possibility of automatic classiﬁcation mecha-

nisms there is still ample room for the development of suﬃcient criteria for membership in

certain classes,see [79,1,71].For example,a proof of computational universality in a CA

that has not been artiﬁcially constructed to simulate some other device often rests on the

presence of “particles” or “gliders” that can be used to send “signals” between spatially

separated locations.Moreover,one has to be able to process these signals much in the way

of Boolean logic gates,to store state and so forth.A good example for complicate inter-

actions between signals are the various solutions to the ﬁring squad problem,albeit not in

the context of simulating arbitrary computations;see ﬁgure 4,[39].A more recent example

is Cook’s ingenious method of using natural gliders in elementary CA 110 to implement a

cyclic tag system in C

ap

,thereby establishing computational universality of rule 110,see [9].

Notable here is the fact that the automaton was ﬁxed from the start and the the appropriate

19

coding mechanisms had to be developed in a very constrained environment.This is in stark

contrast to other hardness arguments where the CA is carefully constructed to display the

desired behavior.Careful visual inspection of rule 110 orbits was a crucial component in

Cook’s proof,it is diﬃcult to imagine that the result could have been established in a purely

combinatorial or algebraic fashion.One can envision an interactive software system that

helps to tackle some algorithmically unsolvable classiﬁcation problems in special cases,much

as Baumslag’s Magnus project in group theory,see [5].

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