A Full Computation-relevant Topological Dynamics Classification of Elementary Cellular Automata
AA Full Computation-relevant Topological Dynamics Classification ofElementary Cellular Automata
Martin Sch¨ule a) and Ruedi Stoop Institute of Neuroinformatics, ETH and University of Zurich, 8057 Zurich, Switzerland
Cellular automata are both computational and dynamical systems. We give a complete classification of thedynamic behaviour of elementary cellular automata (ECA) in terms of fundamental dynamic system notionssuch as sensitivity and chaoticity. The “complex” ECA emerge to be sensitive, but not chaotic and noteventually weakly periodic. Based on this classification, we conjecture that elementary cellular automatacapable of carrying out complex computations, such as needed for Turing-universality, are at the “edge ofchaos”.
In the rich classical history of the theory ofcomputation, models of computation were typi-cally compared to the Turing machine concept,which allows us to characterize their computa-tional power in great detail.
If, however, onewould like to ascribe “computational” capacityto processes and systems observed in nature, oneis naturally pushed toward using dynamical sys-tems notions as the natural framework, leavingthe problem open of how to fit this approach into,or how to link this approach with, Turing com-putation. A paradigmatic class of systems thatcomprise in a generic manner both computationaland dynamical system aspects are the cellular au-tomata (CA). While CAs are defined as a classof discrete dynamical systems, they also serve asa mathematical model of massively parallel com-putation, a paradigm often observed when “na-ture computes”. Remarkably, already very sim-ple rules make CAs computationally universal,i.e. capable of carrying out arbitrary computa-tional tasks. By clarifying the dynamical systemproperties of the most popular and best-studiedsubclass of CA, the so-called elementary cellu-lar automata (ECA), we will contribute here to amore profound understanding of CA as both com-putational and dynamical systems. We will fullyclassify the dynamic behavior of ECA using ex-clusively topological dynamics attributes such assensitivity and chaos. Based on this classification,we will finally conjecture that the computation-ally most complex and biologically relevant ECAare those located at the “edge of chaos”.
I. INTRODUCTION
By definition, CA are discrete dynamical systems act-ing in a discrete space-time. The state of a CA is specifiedby the states of the individual cells of the CA, i.e. by the a) [email protected] values taken from a finite set of states associated withthe sites of a regular, uniform, infinite lattice. The stateof a CA then evolves in discrete time steps according to arule acting synchronously on the states in a finite neigh-bourhood of each cell. Despite the simplicity of theserules, CA can exhibit strikingly complex dynamical be-haviour. A well-known example of a CA with intricatedynamics is the so-called Game of Life . CA have alsobeen extensively applied as models for a wide variety ofphysical and biological processes.Obtaining a dynamical system classification of ECAis part of the long-standing problem in CA theoryto characterise the “complexity” seen inherent in CAbehaviour. In a series of influential papers, Wolframstudied the dynamical system and statistical propertiesof CA and devised a classification scheme.
Accordingto this scheme, CA behaviour can be divided into thefollowing classes:(W1) almost all initial configurations lead to the samefixed point configuration,(W2) almost all initial configurations lead to a periodicconfiguration,(W3) almost all initial configurations lead to randomlooking behaviour,(W4) localized structures with complex behaviouremerge.Wolfram’s classification attempt was largely based onsimulations of ECA. Since his pioneering work manymore classification schemes have been proposed, e.g. byLi et al. or Culik et al. It is however still an openproblem of CA theory to obtain a completely satisfying,formal classification of CA behaviour.In this paper, we will put forward a complete topo-logical dynamics classification of ECA. Our approach isbased on the symbolic dynamics treatment of CA ini-tiated by the seminal paper of Hedlund. The topolog-ical dynamics approach allows to use the fundamentalnotions of dynamics system theory such as sensitivity,chaos, etc. More specifically, the classification is basedon a scheme, introduced by Gilman and modified byKurka , which proposes four classes: EquicontinuousCA, CA with some equicontinuous points, sensitive butnot positively expansive CA and positively expansive CA.Each one-dimensional CA belongs to exactly one class,but class membership is generally not decidable. We a r X i v : . [ n li n . C G ] F e b determine for every ECA, as far as we know for thefirst time, to which class it belongs. We also (re-)derivefurther properties such as surjectivity and chaoticity ofECA. Taken together this gives a fairly complete pictureof the dynamical system properties of ECA.The paper is organised as follows. In Sect. II, we in-troduce one-dimensional CA and ECA formally. In Sect.III, we give basic notations and definitions of the topolog-ical dynamics approach to CA. In Sect. IV, we introducea scheme that allows to express ECA rules algebraically.This will prove helpful in Secs. V and VI, where wewill classify ECA in the topologically dynamics sense ofKurka. In Sect. VII, we discuss our results. II. DEFINITION OF ELEMENTARY CELLULARAUTOMATA
We start with the definitions of the basic concepts un-derlying the theory of one-dimensional CA. The config-uration of a one-dimensional CA is given by the double-infinite sequence x = ( x i ) i ∈ Z with x i ∈ S being elementsof the finite set of states S = { , , ... } . The configurationspace X is the set of all sequences x , i.e., X = S Z . TheCA map F , simply called the CA F , is a map F : X → X where the local function is the map f : S r +1 → S , r ≥ F ( x ) i = f ( x i − r , ..., x i , ..., x i + r ). The integer r iscalled the radius of the CA. The iteration of the map F acting on an initial configuration x generates the orbit x, F ( x ) , F ( x ) , ... of x . The orbits of all configurations x are a discrete space-time dynamical system also referredto as CA F . Instances of the system can be visualised inso-called space-time patterns .A spatially periodic configuration is a configurationwhich is invariant under translation in space, that is, x is periodic if there is q > σ q ( x ) = x where σ : X → X is the shift map σ ( x ) i = x i +1 . A temporallyperiodic or simply periodic configuration x for some CA F is given if F n ( x ) = x for some n >
0. If F ( x ) = x , x iscalled a fixed point . A configuration x is called eventuallyperiodic , if it evolves into a temporally periodic configu-ration, i.e. if F k + n ( x ) = F k ( x ) for some k ≥ n > x , the correspondingCA is called eventually periodic .An elementary cellular automaton (ECA) is an one-dimensional CA with two states and “nearest neighbour-hood coupling”, that is, S = { , } and r = 1. Thereare then 256 different possible local functions f : S → S with F ( x ) i = f ( x i − , x i , x i +1 ). Local functions are alsocalled rules and usually given in form of a rule table . Anexample is:111 110 101 100 011 010 001 0000 1 1 0 1 1 1 0Every ECA rule is, following Wolfram , referred to by thesequence of the values of the local function, as given in therule table, written as a decimal number. In the example above one speaks of ECA rule 110, because 01101110written as a decimal number equals 110. III. TOPOLOGICAL AND SYMBOLIC DYNAMICSDEFINITIONS AND CONCEPTS
The framework we use to study the dynamical proper-ties of ECA is given by the symbolic dynamics approachthat views the state space S Z of one-dimensional CA asthe Cantor space of symbolic sequences. The topology ofthe Cantor space is induced by the metric d C ( x, y ) = + ∞ (cid:88) i = −∞ δ ( x i , y i )2 | i | , where δ ( x i , y i ) is the discrete metric δ ( x i , y i ) = (cid:40) , x i (cid:54) = y i , x i = y i .Under this metric the configuration space S Z is compact,perfect and totally disconnected, i.e., a Cantor space. From now on, the configuration space S Z endowed withthis metric will be referred to as X . The ECA functions F are continuous in X , hence ( X, F ) is a (discrete) dy-namical system.Now we introduce some key concepts of the topologi-cal dynamics treatment of CA. A configuration x is an equicontinuity point of CA F , if ∀ (cid:15) > , ∃ δ > , ∀ y ∈ X : d ( x, y ) < δ, ∀ n ≥ d ( F n ( x ) , F n ( y )) < (cid:15). (1)If all configurations x ∈ X are equicontinuity points thenthe CA is called equicontinuous . If there is at least oneequicontinuity point, the CA is almost equicontinuous .A CA is sensitive (to initial conditions), if ∃ (cid:15) > , ∀ x ∈ X, ∀ δ > , ∃ y ∈ X : d ( x, y ) < δ, ∃ n ≥ d ( F n ( x ) , F n ( y )) ≥ (cid:15). (2)A CA is positively expansive , if ∃ (cid:15) > , ∀ x (cid:54) = y ∈ X, ∃ n ≥ d ( F n ( x ) , F n ( y )) ≥ (cid:15). (3)Positively expansive CA are sensitive .If a configuration is an equicontinuity point, its orbitremains arbitrarily close to the orbits of all sufficientlyclose configurations. If a CA is sensitive, there exists forevery configuration at least one configuration arbitrarilyclose to it such that the orbits of the two configurationswill eventually be separated by some constant. Positiveexpansivity is a stronger form of sensitivity: the orbits ofall configurations that differ in some cell will eventuallybe separated by some constant. The long term behaviourof a sensitive CA can thus only be predicted if the initialconfiguration is known precisely.With these concepts, CA as dynamical systems canbe classified according to a classification introducedby Gilman and modified by Kurka . Every one-dimensional CA falls exactly in one of the followingclasses :(K1) equicontinuous CA,(K2) almost equicontinuous but not equicontinuous CA,(K3) sensitive but not positively expansive CA,(K4) positively expansive CA.The typical emergent dynamics of the different classesare illustrated by the space-time patterns of Figure 1. (a) Rule 108 (b)
Rule 73 (c)
Rule 110 (d)
Rule 90
FIG. 1.
Examples of space-time patterns that illustrate the dynamicbehaviour of the classes (K1)-(K4): Equicontinuous ECA rule 108 (a),almost equicontinuous but not equicontinuous ECA rule 73 (b), sensi-tive but not positively expansive ECA rule 110 (c), positively expansiveECA rule 90 (d). Finite arrays of 200 (rule 73 and rule 108) and 400cells, respectively (rule 110 and rule 90), with periodic boundary con-ditions are used; black dots code state 1, white dots state 0. Time runsfrom top to bottom.
It has been shown that for one-dimensional CA it isnot decidable whether a given CA belongs to class (K1),(K2) or (K3) ∪ (K4), whereas it is still open whether theclass (K4) is decidable. We will show that it can bedetermined to which class an ECA belongs.
IV. ALGEBRAIC EXPRESSIONS OF ELEMENTARYCELLULAR AUTOMATA RULES
Here, we devise an algebraic expression scheme forECA. The main idea is to derive in a consistent way alge-braic expressions for the local ECA rules from a Booleanfunction form of ECA rules. The algebraic expressionsof ECA rules are of use in Secs. V-VI. Algebraic ex-pressions of specific ECA rules have been derived ear-lier, usually for additive ECA rules. For example, rule90 is usually given as F ( x ) i = x i − + x i +1 mod 2. Other approaches, e.g. by Chua , do not yield the samesimple polynomial forms as obtained below. The ap-proach taken here was introduced earlier by the presentauthors , where, to the best of our knowledge, for thefirst time simple, algebraic expressions were given for allECA rules. Note that Betel and Flocchini used a simi-lar approach in their study on the relationship betweenBoolean and “fuzzy” cellular automata. The rule tables which define the ECA rules can beregarded as truth tables familiar from propositionallogic. Any ECA rule hence corresponds to a
Booleanfunction , which can always be expressed as a disjunc-tive normal form (DNF) (or a conjunctive normal formrespectively). The DNF of a Boolean function is a dis-junction of clauses, where a clause is a conjunction ofBoolean variables. Any ECA rule can thus be expressedas (cid:95) m (cid:94) j = − ( ¬ ) X mi + j (4)where X i + j are Boolean variables associated with thestates of the cells in the neighbourhood of an ECA. Forexample the DNF expression of ECA rule 110 reads to:( X i − ∧ X i ∧¬ X i +1 ) ∨ ( X i − ∧¬ X i ∧ X i +1 ) ∨ ( ¬ X i − ∧ X i ∧ X i +1 ) ∨ ( ¬ X i − ∧ X i ∧ ¬ X i +1 ) ∨ ( ¬ X i − ∧ ¬ X i ∧ X i +1 ).The representation of ECA rules in DNF is well knownand has e.g. been studied by Wolfram. We may express now the Boolean operations ( ∧ , ∨ , ¬ )arithmetically as x ∧ y = xy (5) x ∨ y = x + y − xy ¬ x = 1 − x. We found it convenient to express the Boolean operationsin this way, instead of using the more common modulo-2operations. This replacement takes the Boolean algebra( A, ∧ , ∨ , ¬ , , A = { , } , into a Booleanring ( R, + , − , · , , R = { , } and theusual arithmetical operations.Replacing the Boolean operations in the DNF expres-sions of ECA rules with their arithmetic counterpartsyields, for all ECA, Boolean polynomials of the form α + α x i − + α x i + α x i +1 + α x i − x i + α x i x i +1 + α x i − x i +1 + α x i − x i x i +1 , (6) with x i ∈ { , } and α j ∈ Z . ECA rules are completelydetermined by the appropriate set of coefficients α j inexpression (6).As examples we list here a few algebraic expressionsof some interesting ECA rules.Rule 30: F ( x ) i = x i − + x i + x i +1 − x i − x i − x i x i +1 − x i − x i +1 + 2 x i − x i x i +1 Rule 90: F ( x ) i = x i − + x i +1 − x i − x i +1 Rule 108: F ( x ) i = x i + x i − x i +1 − x i − x i x i +1 Rule 110: F ( x ) i = x i + x i +1 − x i x i +1 − x i − x i x i +1 Rule 184: F ( x ) i = x i − − x i − x i + x i x i +1 Rule 232: F ( x ) i = x i − x i + x i x i +1 + x i − x i +1 − x i − x i x i +1 Note how simple, for example, the algebraic expressionof the “complex” ECA rule 110 is!It is well-known that the ECA rule space can be parti-tioned into 88 equivalence classes, because ECA rules areequivalent under the symmetry operations of exchangingleft/right and 0 / f ( x ) i = f ( x i − , x i , x i +1 ) these symmetry operationsare given by T left/right ( f ( x ) i ) = f ( x i +1 , x i , x i − ) and T / ( f ( x ) i ) = 1 − f (1 − x i − , − x i , − x i +1 ).For example, for ECA rule 110 the equivalent rules are:Rule 110: F ( x ) i = x i + x i +1 − x i x i +1 − x i − x i x i +1 Rule 137: F ( x ) i = 1 − x i − − x i − x i +1 + x i − x i +2 x i x i +1 + x i − x i +1 − x i − x i x i +1 Rule 124: F ( x ) i = x i − + x i − x i − x i − x i − x i x i +1 Rule 193: F ( x ) i = 1 − x i − − x i − x i +1 + 2 x i − x i + x i x i +1 + x i − x i +1 − x i − x i x i +1 From now on we will use the lowest decimal ECA rulenumber present within the group to refer to the wholegroup. For example, referring to ECA rule 110 impliesin this way the four rules { , , , } .Note that the approach developed here can be ex-tended in various ways, for example to one-dimensionalCA with state space { , } with larger neighbourhood, orto two-dimensional CA with state space { , } , etc. V. CLASSIFICATION OF ELEMENTARY CELLULARAUTOMATA
We will now classify ECA from their topological dy-namics properties, that is, according to the scheme in-troduced by Gilman and modified by Kurka .First, we need some more symbolic dynamics defini-tions and notions. A word u is a finite symbolic sequence u = u ...u l − , with u i ∈ S , where S is a finite alphabet , e.g. in the case of ECA the state set { , } . The length of u is denoted by l = | u | . The set of words of S of length l isdenoted by S l , the set of all words of S with l > S + .The cylinder set [ u ] of u consists of all points x ∈ S Z with leading part u , i.e. [ u ] = { x ∈ S Z : x [0 ,l ) = u } .A word u ∈ S + with | u | ≥ m, m >
0, is m-blocking for a one-dimensional CA F , if there exists an offset q ∈ [0 , | u | − m ] such that ∀ x, y ∈ [ u ] , ∀ n ≥ , F n ( x ) [ q,q + m ) = F n ( y ) [ q,q + m ) .For an illustration of the mathematical definition see Fig-ure 2. FIG. 2.
A word u of length | u | = l is said to be blocking , if it has aninterior of size m , located from position q , that remains unaffectedby the states of the cells left and right to the word u , at all times. One-dimensional CA, and therefore ECA, are eithersensitive or almost equicontinuous. The latter propertyis equivalent to having a blocking word:
Proposition 1 (Kurka ) . For any one-dimensional CA F with radius r > the following conditions are equiva-lent.(1) F is not sensitive.(2) F has an r-blocking word.(3) F is almost equicontinuous. If a configuration x contains a m -blocking word u , thenthe sequence x [ q,q + m ) , i.e. the states of the cells in thesegment [ q, q + m ), are at all times independent of theinitial states outside of the blocking word u . Hence, thefollowing corollary holds: Corollary 2.
For any one-dimensional CA F with ra-dius r > the following conditions are equivalent.(1) F has a m-blocking word with m ≥ r . (2) F has a word u ∈ S + with | u | ≥ m, m > andan offset q ∈ [0 , | u | − m ] such that ∀ x ∈ [ u ] thesequence x [ q,q + m ) is eventually temporally periodic.Proof. (1) ⇒ (2): Denote the sequence x [ q,q + m ) of ablocking word u that is at all times independent of the ini-tial states outside of u by v . The configuration x = ( u ) ∞ is spatially periodic and hence eventually temporally pe-riodic. Because the sequence v is independent of thestates of the cells outside of u , the sequence v is alsoeventually temporally periodic.(2) ⇒ (1): The condition (2) says that for all x ∈ [ u ] there is t ≥ p > F t + p ( x ) [ q,q + m ) = F t ( y ) [ q,q + m ) . Thus, for all x, y ∈ [ u ] and all n ≥ F n ( x ) [ q,q + m ) = F n ( y ) [ q,q + m ) must be indepen-dent of the initial states outside of u , hence the word u is m -blocking.We will now systematically search for blocking words.We know by proposition 1 that whenever a blocking wordcan be found, the corresponding ECA is almost equicon-tinuous. By corollary 2, we know that this corresponds tofinding a word u that contains a sequence that is eventu-ally temporally periodic, independent of the initial statesoutside of u . As it turns out, we can thereby effectivelydetermine all almost equicontinuous ECA, because anyalmost equicontinuous ECA corresponds to a blockingword u for which the length l = | u | is bounded. Proposition 3.
Each almost equicontinuous ECA hasat least one blocking word of length l ≤ .Proof. In the following, we look for blocking words, start-ing with the smallest possible length l = 1 and then suc-cessively for words of greater length (for a visualisationof the definition of a blocking word see again Figure 2).If a blocking word can be found, one or several almostequicontinuous ECA rules will satisfy the blocking condi-tions. The ECA rules are specified by a rule table whichwe denote by ( t , t , t , t , t , t , t , t ). For example,ECA rule 110 is given by the table (0 , , , , , , , , , , , , , , t ) refers to the two ECA rules 110 and111. If a blocking word can be found, we put the ECArule table admitted by the blocking conditions in a list. Ablocking word u and the admitted rule table is denoted by t ( u, p ) = ( t , t , t , t , t , t , t , t ), where p is the periodwith which the eventually periodic sequence in the word u (i.e. the sequence x [ q,q + m ) referred to in corollary 2) isrepeated. For example, t (00 ,
1) = ( t , t , t , , t , t , , p = 1 that cor-responds to 2 = 32 ECA rules, as denoted by the ruletable. If a newly found blocking word admits ECA rulesgenerated by a rule table obtained by a blocking wordalready in the list (hence of smaller length), the wordand the rule table admitted by it is not listed. We alsodo not list blocking words, and the rule tables admittedby them, if they correspond to ECA rules equivalent to ECA rules admitted by a blocking word already in thelist.Let us further assume the following notation: The vari-able c i always denotes the states of cells i of a blockingword u that are at all times independent of the initialstates of the cells outside of the blocking word u . Thevariable x i on the other hand denotes the states of cells i that are in principle influenceable by the initial statesof the cells outside of u . The state x i of such a cell i isleft undetermined, i.e. the value can either be 0 or 1. Ifit is known for configurations x, y ∈ [ u ] that the states x i and y i of some cell i differ, we write ¯ x i . For example,the “scenario” ¯ x − c ¯ x x − c x refers to two configurations x, y ∈ [ u ] that share the blocking word u = c of length l = 1 that is repeated with period p = 1. At the bound-aries of the blocking word u , here at the cells i = − i = 1, we can assume that the configurations x and y dif-fer, which is denoted by ¯ x − i and ¯ x i , whereas in the nexttime step this may not necessarily be the case anymore(at the cells i = − i = 1).The proof has two parts. In part A, we determineall blocking words of length l ≤
4. In part B, we showthat for any blocking word u of length l > l ≤ l = 1, (b) l = 2, (c) l = 3 and (d) l = 4, where l , as said, denotes the lengthof a blocking word u .(a) With l = 1, the following scenarios are possible:(1) ¯ x − c ¯ x x − c x , (2) ¯ x − c ¯ x c c x , (3) ¯ x − c ¯ x x − c c , (4)¯ x − c ¯ x c c c , (5) ¯ x − c ¯ x x − c (cid:48) x , (6) ¯ x − c ¯ x c (cid:48)− c (cid:48) x ,(7) ¯ x − c ¯ x x − c (cid:48) c (cid:48) and (8) ¯ x − c ¯ x c (cid:48)− c (cid:48) c (cid:48) , where at least forone i , c (cid:48) i (cid:54) = c i . Note that there are further scenariospossible that however do not yield further valid rule ta-bles and are not listed here. Scenario (1) yields therule table t (1 ,
1) = (1 , , t , t , , , t , t ) (and the ta-ble t (0 ,
1) = ( t , t , , , t , t , , x − c ¯ x x − c (cid:48) x c (cid:48) and ¯ x − c ¯ x x − c (cid:48) x c , where c (cid:48) (cid:54) = c . The firstcase does not lead to new ECA rules. The second caseyields the rule table t (1 ,
2) = (0 , , , , , , , t (1 ,
2) = (0 , , , , , , , l = 2, we deal with essentially the same sce-narios as in case (a). For example, in analogy to thescenario (1) ¯ x − c ¯ x x − c x of case (a), we have the sce-nario ¯ x − c c ¯ x x − c c x . However, for reasons of space,we cannot list all possible scenarios and from now ononly list the scenarios that yield blocking words thatadmit rule tables not yet obtained. These are: (1)¯ x − c c ¯ x x − c c x and (2) ¯ x − c c ¯ x x − c (cid:48) c (cid:48) x , where c (cid:48) i (cid:54) = c i .Scenarios (1) and (2) yield, as can easily be checked,the following blocking words and rule tables: t (00 ,
1) =( t , t , t , , t , t , , t (01 ,
1) = ( t , t , , t , , , , t ), t (10 ,
1) = ( t , , , , t , , t , t ) and t (00 ,
2) =(0 , , t , , , t , , t (10 ,
2) = ( t , , , , , , , t ).(c) For l = 3, the scenarios that yield rule ta-bles not listed above are: (1) ¯ x − c c c ¯ x x − c c c x and (2) ¯ x − c c c ¯ x x − ¯ x c (cid:48) ¯ x x c c c , where c (cid:48) (cid:54) = c . Sce-nario (1) yields the blocking words and rule ta-bles t (010 ,
1) = ( t , t , , , t , , , t ) and t (101 ,
1) =( t , , , t , , , t , t ). Scenario (2) yields t (000 ,
2) =(0 , , t , , , , , x − c c c ¯ x x − ¯ x c ¯ x x c c c does not yield new rule tables.(d) For l = 4, the only scenario that leads to ablocking word corresponding to a rule table not yetlisted is ¯ x − c c c c ¯ x x − c c c c x , yielding the rule table t (0110 ,
1) = ( t , , , , , t , , t ). Note again that e.g.the scenario ¯ x − c c c c ¯ x x − ¯ x c (cid:48) c (cid:48) ¯ x x c c c c , where at least forone i c (cid:48) i (cid:54) = c i , does not yield new rule tables.With this we conclude Part A. Let us list the blockingwords and the rule tables admitted by them that we havefound so far: t (0 ,
1) = ( t , t , , , t , t , , t (1 ,
2) = (0 , , , , , , , t (1 ,
2) = (0 , , , , , , , t (00 ,
1) = ( t , t , t , , t , t , , t (01 ,
1) = ( t , t , , t , , , , t ) t (10 ,
1) = ( t , , , , t , , t , t ) t (00 ,
2) = (0 , , t , , , t , , t (01 ,
2) = ( t , , , , , , , t ) t (010 ,
1) = ( t , t , , , t , , , t ) t (101 ,
1) = ( t , , , t , , , t , t ) t (000 ,
2) = (0 , , t , , , , , t (0110 ,
1) = ( t , , , , , t , , t )Part B: In the general case, i.e. for l >
4, we canconclude in analogy to the cases already considered, i.e.the cases with l ≤
4, that the following scenarios couldpossibly lead to new blocking words:(1) (cid:18) ¯ x − c c ... c l − c l − ¯ x l x − c c ... c l − c l − x l (cid:19) , (2) (cid:18) ¯ x − c c ... c l − c l − ¯ x l c − c c ... c l − c l − c l (cid:19) ,(3) ¯ x − c c ... ... c l − c l − ¯ x l ... ¯ x q − c q ... c q + m − ¯ x q + m x q − c q ... c q + m − x q + m ,(4) ¯ x − c c ... ... c l − c l − ¯ x l ... ¯ x q − c q ... c q + m − ¯ x q + m c q − c q ... c q + m − c q + m , (5) (cid:18) ¯ x − c c ... c l − c l − ¯ x l x − c (cid:48) c (cid:48) ... c (cid:48) l − c (cid:48) l − x l (cid:19) ,(6) (cid:18) ¯ x − c c ... c l − c l − ¯ x l c (cid:48)− c (cid:48) c (cid:48) ... c (cid:48) l − c (cid:48) l − c (cid:48) l (cid:19) , (7) ¯ x − c c ... ... c l − c l − ¯ x l ... ¯ x q − c q ... c q + m − ¯ x q + m x q − c (cid:48) q ... c (cid:48) q + m − x q + m ,(8) ¯ x − c c ... ... c l − c l − ¯ x l ... ¯ x q − c q c q +1 ... c q + m − c q + m − ¯ x q + m ¯ x q c (cid:48) q +1 ... c (cid:48) q + m − ¯ x q + m − c q c q +1 ... c q + m − c q + m − , with m ≥ i , c (cid:48) i (cid:54) = c i .Case (1) yields blocking words already listed, becausefor l > u are entailed in the conditions to obtain ablocking word u with l ≤
4. The same reasoning appliesto cases (2), (3) and (4). The basic reason that such areduction is possible is due to the fact that the conditionsto be satisfied in order to obtain a blocking word dependon the values of the boundary cells, here the values ¯ x − and ¯ x l (respectively the values ¯ x q − and ¯ x q + m in cases(3) and (4)), but not on the values of the cells to the left(of i = −
1) and right (of i = l ) of the boundary cells, ascan be checked with the scenarios treated in Part A.Let us then look closer at case (5). We will showthat if there is a blocking word c c ...c l − c l − , the wordis repeated with period p = 2, because if the wordis blocking, the word at the next time step (in case(5)) must be ¯ c ¯ c ... ¯ c l − ¯ c l − . The bar signifies that thestate c i of the cell i must change, i.e. ¯ c i = (1 − c i ).Without loss of generality, we can consider only the2 boundary conditions for blocking words at successivetime steps. That is, given the word c c ...c l − c l − , weconsider at the next time-step all the (2 −
2) possiblecases: c c ...c l − ¯ c l − , c c ... ¯ c l − c l − , etc., excluding thetwo cases c c ...c l − c l − and ¯ c ¯ c ... ¯ c l − ¯ c l − . It sufficesto consider the case c c ...c l − ¯ c l − . The other cases canbe dealt with analogously. The temporal evolution of theECA generates in this case the following scheme:¯ x − c c ... c l − c l − ¯ x l x − c c ... c l − ¯ c l − x l x − c c ... ¯ c l − c l − x l ...x − c ¯ c ... c l − l − c l − l − x l x − ¯ c c l − ... c l − l − c l − l − x l The superscript denotes the time-step n . The third, fifthand sixth line are due to the fact that if the state of e.g.the cell l − n = 2 did not change, onewould obtain a blocking word of shorter length ( l − possible values for the boundary statesof the initial word c c ...c l − c l − it can be shown that theabove scheme cannot be satisfied. Thus, any initial word c c ...c l − c l − evolves in the next time step into eitherthe word c c ...c l − c l − or the word ¯ c ¯ c ... ¯ c l − ¯ c l − . Inthe first case, a blocking word of period p = 1 is found,in the second case, i.e. for p = 2, one can find a blockingword of length l = 2, as can easily be shown.The case (6) can be reduced to the case already treatedunder (a (8)) in Part A, the case (7) to the case (5) andthe case (8) again to the case treated under (c (2)) (orthe example in (d) respectively) in Part A.With this we conclude our analysis. In Part A, wehave identified all blocking words of length l ≤
4. For l ≥
2, we omitted, for reasons of space, the presentationof the cases that do not lead to blocking words or toblocking words already identified. In Part B, we haveconcluded from the cases for l ≤ l >
4. These general scenarios could then be reducedto the scenarios obtained for l ≤
4. One case (case (5))required a separate treatment and was analysed by meansof an example.To arrive at a complete list of blocking words for l ≤ l >
4, greatcare and efforts have been invested. We have tested thecompleteness of the list also by extensively sampling thespace of initial configurations for ECA, which yielded noadditional blocking words. One may also check the cor-rectness and completeness of the cases investigated inour analysis by hand with the help of a computer, run-ning a program that follows the lines of the proof above.Alternatively, to demonstrate the impossibility of addi-tional blocking words, the systems of equations generatedfrom the conditions for blocking words and the algebraic expressions of ECA rules could be used, systematicallyevaluated for each single case.The proof of proposition 3 allows to give, for ECA, astronger version of proposition 1. Let us call a word u of length l invariant for an ECA F , if for all x ∈ [ u ] ,there is a p > F p ( x ) [0 ,l ) = x [0 ,l ) . Corollary 4.
An ECA F is almost equicontinuous if andonly if F has an invariant word.Proof. See the proof of proposition 3.Proposition 3 (or corollary 4 respectively) allows usto determine for each ECA rule whether it is almostequicontinuous or not. It is almost equicontinous if thereis an associated blocking word on the list composed ofthe invariant words of shortest length. Below we providethis list together with the corresponding almost equicon-tinuous ECA rules.
Corollary 5.
Invariant words of period p = 1 and cor-responding ECA rules: : 0, 4, 8, 12, 72, 76, 128, 132, 136, 140, 200, 204. : 32, 36, 40, 44, 104, 108, 160, 164, 168, 172, 232. : 13, 28, 29, 77, 156. : 78. : 5. : 94. : 73.Invariant words of period p = 2 and corresponding ECArules: : 1. : 51. : 19, 23. : 50, 178. : 33. Conversely, we now also know the sensitive ECA rules.
Proposition 6.
The following rules are sensitive:2, 3, 6, 7, 9, 10, 11, 14, 15, 18, 22, 24, 25, 26, 27, 30,34, 35, 37, 38, 41, 42, 43, 45, 46, 54, 56, 57, 58, 60, 62,74, 90, 105, 106, 110, 122, 126, 130, 134, 138, 142, 146,150, 152, 154, 162, 170, 184.Proof.
Follows from Proposition 1, 3 and corollary 5.The class of sensitive ECA is large, because in the Can-tor space left- or right-shifting rules are sensitive. We willlater return to this point.From the almost equicontinuous ECA rules, we canfurther specify the equicontinuous ones. We use the fol-lowing lemma.
Lemma 7 (Kurka ) . A one-dimensional almostequicontinuous CA F is equicontinuous if and only if:(1) There exists a preperiod m ≥ and a period p > ,such that F m + p = F m . It is almost equicontinuous but not equicontinuous if andonly if:(2) There is at least one point x ∈ X for which thealmost equicontinuous CA F is not equicontinuous. Proposition 8.
The following rules are equicontinuous:0, 1, 4, 5, 8, 12, 19, 29, 36, 51, 72, 76, 108, 200, 204.Proof.
The proof is by showing that condition (1) ofLemma 7 holds. We only give an example for a specificECA rule.Rule 72 is equicontinuous with preperiod m = 2 andperiod p = 1, because, by using the algebraic expressionfor the local function, we obtain F ( x ) i = x i − x i + x i x i +1 − x i − x i x i +1 F ( x ) i = x i − x i − x i − x i − x i + x i x i +1 − x i − x i x i +1 + x i − x i − x i x i +1 − x i x i +1 x i +2 + x i − x i x i +1 x i +2 F ( x ) i = x i − x i − x i − x i − x i + x i x i +1 − x i − x i x i +1 + x i − x i − x i x i +1 − x i x i +1 x i +2 + x i − x i x i +1 x i +2 .Hence, F ( x ) i = F ( x ) i , ∀ i ∈ Z . Thus, F = F . Proposition 9.
The following rules are almost equicon-tinuous but not equicontinuous:13, 23, 28, 32, 33, 40, 44, 50, 73, 77, 78, 94, 104, 128,132, 136, 140, 156, 160, 164, 168, 172, 178, 232.Proof.
The proof is by showing that condition (2) ofLemma 7 holds. We only give an example for a specificECA rule.ECA rule 104 is almost equicontinuous but notequicontinuous, because (10) ∞ is not an equicontinuouspoint.Assume the configuration x = (10) ∞ and an integer q > ∀ y ∈ X, ( x [ − q,q ] = y [ − q,q ] ) ⇒ ( d ( x, y ) < − q ) . Assume that y differs from x at cells ( − q −
1) and ( q + 1),that is y − q − = 1 − x − q − and y q +1 = 1 − x q +1 . Then,as can easily be shown by using the algebraic expressionof ECA rule 104, d ( F n ( x ) , F n ( y )) > − ( q − n ) for all n ≤ q . Hence, ECA 104 is not equicontinuous atthe point x = (10) ∞ .From the sensitive ECA, we can distinguish further the positively expansive ECA.First, we need the definition of permutivity for ECA. An ECA F is left-permutive if ( ∀ u ∈ S ) , ( ∀ b ∈ S ) , ( ∃ ! a ∈ S ): f ( au ) = b . It is right-permutive if ( ∀ u ∈ S ) , ( ∀ b ∈ S ) , ( ∃ ! a ∈ S ): f ( ua ) = b . The ECA F is permutive if itis either left-permutive or right-permutive.We will use the following lemma: Lemma 10 (Kurka ) . A one-dimensional CA F is pos-itively expansive if the following condition holds.(1) The CA is both left- and right-permutive. A one-dimensional sensitive CA F is not positively ex-pansive if and only if the following condition holds.(2) There is no (cid:15) > such that for all x (cid:54) = y ∈ X thereis n ≥ with d ( F n ( x ) , F n ( y )) ≥ (cid:15) . Proposition 11.
The following ECA rules are sensitivebut not positively expansive:2, 3, 6, 7, 9, 10, 11, 14, 15, 18, 22, 24, 25, 26, 27, 30,34, 35, 37, 38, 41, 42, 43, 45, 46, 54, 56, 57, 58, 60,62, 74, 106, 110, 122, 126, 130, 134, 138, 142, 146, 152,154, 162, 170, 184.Proof.
The proof is by showing that condition (2) ofLemma 10 holds. We only provide the example of a spe-cific ECA rule.ECA rule 110 is sensitive but not positively expansive.Assume the expansivity constant (cid:15) = 2 − m , then ∀ x (cid:54) = y ∈ X ⇒ ∃ n ≥ , F n ( x ) [ − m,m ] (cid:54) = F n ( y ) [ − m,m ] (7)must hold. Assume the configuration x =(00110111110001) ∞ and an integer q > q > m . Then, for a configuration y ∈ X that differsfrom x at the cells 14 q, q + 1 , q + 2, (7) does nothold. Proposition 12.
The following ECA rules are positivelyexpansive:90, 105, 150.Proof.
For ECA rules 90, 105 and 150 condition (1) ofLemma 10 holds.For ECA left- and right-permutivity is equivalent topositive expansivity.
Proposition 13.
ECA are positively expansive if andonly if they are both left- and right-permutive.Proof.
Follows from Proposition 6, 11 and 12.Note that Proposition 13 does not hold generally forone-dimensional CA. We summarize the findings of this section in Table I,which shows all ECA rules according to whether theyhave the property of equicontinuity, almost equicontinu-ity, sensitivity or positively expansivity.
VI. CLASSIFICATION OF SENSITIVE ELEMENTARYCELLULAR AUTOMATA
Since the beginning of CA research, the classificationof the degree of “complexity” seen in CA behaviour hasbeen a main research focus. It is intuitively clear that thesensitivity property is a source of the apparent “complex-ity” of ECA behaviour. Among the sensitive ECA rules,we find, however, rules that show in their space-timedynamics “travelling waves” patterns (Fig. 3). These
TABLE I. Topological dynamics classification of ECA rules.almost equicontinuous sensitiveequicontinuous positively expansive0, 1, 4, 5, 8, 13, 23, 28, 32, 2, 3, 6, 7, 9, 90, 105, 15012, 19, 29, 36, 33, 40, 44, 50, 73, 10, 11, 14, 15,51, 72, 76, 108, 77, 78, 94, 104, 18, 22, 24, 25,200, 204 128, 132, 136, 140, 26, 27, 30, 34,156, 160, 164, 168, 35, 37, 38, 41,172, 178, 232 42, 43, 45, 46,54, 56, 57, 58,60, 62, 74, 106,110, 122, 126, 130,134, 138, 142, 146,152, 154, 162, 170, 184 non-complex shift-dynamics patterns are from eventuallyweakly periodic ECA defined as follows.A configuration x is called weakly periodic , if there is q ∈ Z and p > F p σ q ( x ) = x . We definea configuration x as eventually weakly periodic if thereis q ∈ Z and n, p > F n + p σ q ( x ) = F n ( x ).We call an ECA eventually weakly periodic , if the ECAis not eventually periodic, but for all configurations x eventually weakly periodic. (a) Rule 170 (b)
Rule 90
FIG. 3.
Space-time patterns of two chaotic ECA rules (rule 170 (a)and rule 90 (b)). The eventually weakly periodic ECA rule 170 simplyshifts the values of cells and exhibits a “travelling wave”. Finite arraysof 200 cells with periodic boundary conditions were used; black dotscode state 1, white dots state 0. Time runs from top to bottom.
Proposition 14.
The following sensitive ECA rules areeventually weakly periodic:2, 3, 10, 15, 24, 34, 38, 42, 46, 138, 170.Proof.
The general proof follows the argument exhibitedfor a specific ECA rule as follows.Employing the algebraic expression for ECA rule 10,it can easily be shown that F σ − ( x ) = F ( x ) for allconfigurations x .Hence, ECA rule 10 is eventually weakly periodic with n = 1 , p = 1 and q = − n or p that prevents calculating the forward orbits aseasily as in the proof of Proposition 14.Surprisingly, some of the eventually weakly periodicECA are also chaotic (but not positively expansive),while others are sensitive, but not chaotic. For this state-ment, we adhere to the standard definition of (topolog-ical) chaos given by Devaney . A map F : X → X ischaotic, if F is sensitive, transitive and if the set of peri-odic points of F is dense in X . The class of chaotic ECAhas already been determined by Cattaneo et al. ; for thesake of completeness, we rederive the result below.First, we shall study the surjectivity property sharedby some ECA maps F . For sensitive ECA F , surjectivityis already sufficient to establish the transitivity of F andthe density of periodic points in X under F , so that thechaoticity of F is implied.A CA is surjective if and only if it has no Garden-of-Eden configurations, that is configurations which have nopre-image. A necessary (but not sufficient) condition forsurjectivity is that the local rule is balanced. For ECArules this means that the local rule table contains 4 zerosand 4 ones. Further, any permutive CA is surjective. Proposition 15.
The following ECA rules are surjec-tive:15, 30, 45, 51, 60, 90, 105, 106, 150, 154, 170, 204.Proof.
Apart from rule 51 and rule 204, the above listedrules are permutive, hence surjective. Rule 51 and rule204 are surjective, because they are, trivially, bijective.For the ECA rules that are not listed, but satisfy thebalance condition, it can be shown that they possessGarden-of-Eden configurations. For example, ECA rule184 satisfies the balance condition, nevertheless it is notsurjective, because any configuration containing pattern(1100) is a Garden-of-Eden as can easily be shown.Next, we show that for ECA transitivity is equivalentto permutivity. An one-dimensional CA F is transitive iffor any nonempty open sets, U, V ⊆ X there exists n > F n ( U ) ∩ V (cid:54) = ∅ . Proposition 16.
A ECA is transitive if and only if it ispermutive. Proof.
Transitivity of one-dimensional CA implies its sur-jectivity and sensitivity . From Proposition 6 and 15and the definition of permutivity, we gain that ECA thatare surjective and sensitive are permutive.Conversely, permutive ECA are surjective . From thesurjective ECA that are sensitive (the surjective ECArules 51, 204 are not sensitive, hence not transitive), thepositively expansive ECA are permutive and transitive .The ECA rules 15 and 170 are also permutive andtransitive. Rule 106 is permutive and has been showntransitive . Proofs of the transitivity of the remainingpermutive and sensitive rules 30, 45, 60, 154 can be sim-ilarly constructed. Corollary 17.
A ECA map is transitive if and only ifit is surjective and sensitive.Proof.
Follows from Proposition 6, 15 and 16.Next, we show that for ECA surjectivity implies thatthe set of periodic points of F is dense in X . Proposition 18.
Surjective ECA have a dense set ofperiodic points in X .Proof. Surjective ECA are either almost equicontinuousor sensitive. Almost equicontinuous one-dimensional CAthat are surjective have a dense set of periodic points .The sensitive ECA that are surjective are permutiveand permutive one-dimensional CA are known to havea dense set of periodic points (through the property ofclosingness ).While for general one-dimensional CA it is still animportant open question whether surjectivity implies adense set of periodic points, for ECA, transitivity or per-mutivity implies chaos. Corollary 19.
The following ECA rules are chaotic inthe sense of Devaney:15, 30, 45, 60, 90, 105, 106, 150, 154, 170.
The distinction between the chaotic and non-chaoticECA is not necessarily seen in the space-time patterns.The eventually weakly periodic ECA that are chaotic andthe eventually weakly periodic ECA that are sensitive butnot chaotic both show similar “travelling wave” patterns.The difference between the chaotic ECA and the sensitivebut not chaotic ECA is not in the space-time patternsthey generate, but in how they react to perturbations.While the eventually weakly periodic ECA show toosimple behaviour to be called “complex”, chaotic ECAare in a sense “too complex”: their mixing properties dono allow for the memory capacities apparently neededfor “complex” behaviour. In the following final sectionwe will expand on this observation. Figure 4 summarisesthe results of our analysis.
VII. DISCUSSION
The results of this paper show that one can classify thedynamic behaviour of every elementary cellular automa-ton (ECA) in terms of the standard notions of dynami-cal system theory, that is, according to the classificationproposed by Gilman and Kurka . We also determinedwhich ECA are chaotic in the sense of Devaney, rederiv-ing a result by Cattaneo et al. This gives a fairly com-plete picture of the dynamical system properties of ECAin the standard topology, as summarised in Fig. 4. Thetopological dynamics approach to CA thus delivers a rel-evant and coherent account of the dynamical behaviourof ECA.In the light of our results, the class of “complex” ECAcan be characterised as those ECA that are sensitive,but not surjective, and not eventually weakly periodic.This class corresponds well to what one would intuitivelyregard as “complex”, given the space-time patterns ofECA. In particular, the ECA rules of Wolfram’s class(W4) seem to fall into this class.Among the ECA rules, a few deserve special interestfrom a computational point of view. The most promi-nent example is ECA rule 110 which has been shownto be computationally universal. Based on our resultswe conjecture that sensitivity is a necessary condition ofcomputational universality. In contrast, Wolfram con-jectured that, for example, ECA rule 73, which is notsensitive, may be computationally universal. This dif-ference is due to the fact that our results hold generallyfor ECA without any restrictions on the initial condi-tions, whereas Wolfram considers specific sets of initialconfigurations on which the rule acts. On such a re-stricted set of configurations, ECA rule 73 might indeedbe sensitive.If a CA is sensitive, then its dynamics defies numericalcomputation for practical purposes, because a finite pre-cision computation of an orbit may result in a completelydifferent orbit than the real orbit. Hence, while sensitiv-ity seems inherent to the in principle computationallymost powerful rules, as e.g. rule 110, their limited ro-bustness to small changes in the initial conditions mayimpair their practical usage in a physical or biologicalsystem: Even a single bit-flip in the input of a sensitiveECA may completely change the computed output.Among the many questions left open, a natural ex-tension of our study would consist in giving a completecharacterisations in the topological dynamics sense formore general CA than ECA. Examples by Cattaneo etal. , however, show that the approach taken here to es-tablish chaoticity can already fail in slightly more gen-eral settings. In the general case, long-term propertiesof CA and hence classification schemes based on theseproperties are typically undecidable. It would thereforebe useful to pinpoint where exactly undecidability enters.Establishing a verifiable notion of computational uni-versality in the Turing-machine sense in terms of nec-essary and sufficient conditions related to the dynamic1 FIG. 4.
Classification diagram for the elementary cellular automata (ECA). The chaotic ECA are inside the double-framed box. Theclass of the sensitive and eventually weakly periodic ECA is not complete. behaviour of the underlying system would greatly ad-vance our understanding of the relation between compu-tational and dynamic properties of physical and biologi-cal systems. Part of the problem to clarify this relation isthat there is no unanimous accepted definition of compu-tational universality for computational systems such asCA (see e.g. the discussion by Ollinger and Delvenne etal. Delvenne et al. also prove necessary conditions fora symbolic system to be universal, according to their def-inition of universality, and demonstrate the existence ofa universal and chaotic system on the Cantor space.). Todifferent definitions of universality, there might thus cor-respond different topological dynamics properties. De-spite this fact, we conjecture that for ECA sensitivity andnon-surjectivity are necessary conditions of universality.This conjecture is in accordance with the intuitive ideathat systems at the “edge of chaos”, i.e. systems withneither too simple nor chaotic dynamical behaviour, arethe computationally relevant systems for biology. Suchintermittent systems have, moreover, been characterisedas having the largest complexity in the sense that theirbehaviour is the hardest to predict. If computation ismeasured as a reduction of complexity , the intermit-tent systems may then be said to provide the complexityneeded for efficient computations.The extension of the results and observationsfrom ECA to general one-dimensional CA or higher-dimensional CA is thus not without problems. Being much more tractable, ECA provide an important bench-mark to test ideas on universality, the “edge of chaos”hypothesis and, generally, on how “computation” occursin nature. ACKNOWLEDGMENTS
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