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Physica A 379 (2007) 465–470

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Long-range memory elementary 1D cellular automata:Dynamics and nonextensivity

Thimo Rohlfa,�, Constantino Tsallisb

aSanta Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USAbCentro Brasileiro de Pesquisas Fisicas, Xavier Sigaud 150, 22290-180 Rio de Janeiro-RJ, Brazil

Received 11 February 2007

Available online 2 March 2007

Abstract

We numerically study the dynamics of elementary 1D cellular automata (CA), where the binary state siðtÞ 2 f0; 1g of acell i does not only depend on the states in its local neighborhood at time t� 1, but also on the memory of its own past

states siðt� 2Þ; siðt� 3Þ; . . . ; siðt� tÞ; . . . . We assume that the weight of this memory decays proportionally to t�a, withaX0 (the limit a!1 corresponds to the usual CA). Since the memory function is summable for a41 and nonsummable

for 0pap1, we expect pronounced changes of the dynamical behavior near a ¼ 1. This is precisely what our simulations

exhibit, particularly for the time evolution of the Hamming distance H of initially close trajectories. We typically expect the

asymptotic behavior HðtÞ / t1=ð1�qÞ, where q is the entropic index associated with nonextensive statistical mechanics. In all

cases, the function qðaÞ exhibits a sensible change at a ’ 1. We focus on the class II rules 61, 99 and 111. For rule 61, q ¼ 0

for 0papac ’ 1:3, and qo0 for a4ac, whereas the opposite behavior is found for rule 111. For rule 99, the effect of the

long-range memory on the spread of damage is quite dramatic. These facts point at a rich dynamics intimately linked to the

interplay of local lookup rules and the range of the memory. Finite size scaling studies varying system size N indicate that

the range of the power-law regime for HðtÞ typically diverges / Nz with 0pzp1.

r 2006 Elsevier B.V. All rights reserved.

Keywords: Cellular automata; Nonextensive statistical mechanics; Long-range memory

Cellular automata (CA) have a long tradition as models of complex, emergent space time dynamics encodedin simple, local rules. Originally introduced by von Neumann as a theoretical framework for self-replication [1],it was realized that they could serve as models for much broader classes of phenomena, including evenuniversal computation (e.g. in Conway’s ‘‘game of life’’ [2]). In the context of statistical mechanics, detailedstudies on the simplest class of CA, elementary 1D CA with k ¼ 2 states and 3-cell neighborhood, were carriedout in the 1980s. Wolfram [3,4] developed a qualitative classification scheme of the 22

3

¼ 256 elementary CArules (only 88 fundamentally inequivalent) that distinguished four different ‘complexity classes’ of theirdynamics (class I: fixed-point attractors, class II: space–time periodic limit cycles, class III: space–time chaos,class IV: ‘complex’ dynamics). It was shown that rule 110, a member of class IV, is a universal computer in theTuring sense [5]. Many attempts where undertaken to obtain a more quantitative characterization of CA

e front matter r 2006 Elsevier B.V. All rights reserved.

ysa.2007.02.015

ing author.

ess: rohlf@santafe.edu (T. Rohlf).

ARTICLE IN PRESST. Rohlf, C. Tsallis / Physica A 379 (2007) 465–470466

dynamics, e.g. mean field models [6,3], local structure theory [7], quantification of pre-images [8] and relatingcertain class III rules to low-dimensional deterministic chaos [9]. While certain classes of CA, for example,lattice gas automata, have proved to provide perfectly relevant models for, e.g., the statistical mechanicaltheory of fluids [10], in most other cases (and, in particular, for elementary CA) the relation to statisticalmechanics is still obscure (the same happens, in fact, for similar models such as coupled maps: see [11], amongmany other examples). Fractal patterns and strong spatial and temporal correlations observed in CAdynamics for certain rules, as well as the absence of a Hamiltonian, seem to suggest a natural connection tononextensive statistical mechanics and the related concept of q-entropies (with qa1) [12–14], rather than thetraditional Boltzmann–Gibbs (BG) framework (i.e., the limiting case q ¼ 1). Whereas systems in the BG classare characterized by strong sensitivity on initial conditions with trajectories typically diverging exponentially

with time (hence strong mixing in phase space, and ergodicity), systems with qa1 typically show a weakerdependence on initial conditions, e.g. trajectories diverging asymptotically as a power of time. Therefore, BGsystems may becharacterized as ‘memoryless systems’, whereas the systems studied in nonextensive statisticalmechanics often exhibit long-range memory.

In this paper, we first define a new model by explicitly introducing long-range memory. In our model, thebinary state siðtÞ 2 f0; 1g of a cell i does not only depend on the states in its local neighborhood at time t� 1,but also on the memory of its own past states siðt� 2Þ; siðt� 3Þ; . . . ; siðt� tÞ; . . . . We assume that the weightof this memory decays proportionally to t�a, with aX0. The case a!1 corresponds to the usual CA, wherethe states at time t are determined solely by the neighbor states at time t� 1. This case is compared to thegeneralized update scheme in Fig. 1. We then study its sensitivity to initial conditions. More precisely, wedetermine the a-dependence of the entropic index q (sometimes noted qsen in the literature, where sen stands forsensitivity). By analogy with what occurs in simple dissipative maps at the edge of chaos [15], in vanishingLyapunov exponent conservative maps [16], and in self-organized critical models [17], one expects thesensitivity to the initial conditions (the Hamming distance in the present case) to be asymptoticallyproportional to t1=ð1�qÞ with qo1. While for a few elementary CA, e.g. rule 110, long-range memory seems toemerge as a result of purely local update rules, the way this happens is poorly understood. Our approach, thatcontains the conventional CA scheme as a limit case, aims at providing a generalized framework to addresssuch questions, and also relating them to nonextensive statistical mechanical concepts.

Model: Consider a 1D CA, consisting of N cells in a line, with periodic boundary conditions. The state spaceof the system is defined by ~s 2 f1; 0gN . Each cell i is updated in parallel according to the following update rule:

siðtÞ ¼ f ½si�1ðt� 1Þ;YðXiðtÞ � 1=2Þ;siþ1ðt� 1Þ�, (1)

where XiðtÞ is defined by

XiðtÞ ¼ limT!1

XT

t¼1

siðt� tÞta

XT

t¼1

1

ta

,, (2)

i−1 i i+1

t−1

t

Ξi(t)

i−1 i+1i

Fig. 1. Left panel: in conventional CA, the state of cell i at time t depends only on the states in its local neighborhood at time t� 1. Right

panel: in a-CA, the state of cell i at time t depends on the states of its neighbors at time t� 1 and the (infinite) memory XiðtÞ of its own past

states (grey shading indicates power-law decay of the weighting of memory states).

ARTICLE IN PRESST. Rohlf, C. Tsallis / Physica A 379 (2007) 465–470 467

with a 2 ½0;1Þ. YðxÞ is the Heaviside step function, i.e., due to the normalization of X 2 ½0; 1�, it returns either0 or 1, depending on XiðtÞ �

12 being smaller or larger than zero, respectively. f is one of the well-known 256

elementary local update rules for neighborhood 3, binary CA. However, XiðtÞ introduces long-range memory,decaying with a power a of (discrete) system time. Notice that for a!1, this maps on the conventional 1DCA without memory. In practice (i.e., simulations), T does not go to infinity, but rather is fixed to some largefinite value (T up to 960). Let us briefly discuss the spirit of this approach. While it would be perfectlyreasonable to choose any other functional form for memory decay (e.g. an exponential), for our purposes, theone chosen in Eq. (2) appears to be the most sensible: it becomes, for arbitrary configurations and T !1,nonsummable for 0pap1; hence, we expect a transition from a short-range memory phase to a long-rangememory phase if we approach the critical value ac from above. Consequently, it is the simplest functional formfor which we may expect a qualitative change in dynamics for a finite value of the control parameter a, whilekeeping the update rule f constant.

Results: We have carried out extensive numerical simulations for many of the 256 elementary CA rules. Itturns out that class I and class III CA usually are very insensitive to long-range memory; however, quitepronounced changes in system dynamics are found for several class II and class IV rules. Here, we focus ourdiscussion on three rules that were classified as class II, namely rule 61, 111 and 99. In the conventionalupdating scheme, all three rules behave very similarly, i.e., their dynamics converges to space–time periodic,checker-board like patterns traveling over the lattice (compare left panels of Fig. 2). Near ac � 1, however,profound changes in the dynamics are observed: due to increased sensitivity towards initial conditions, we findcomplex, fractal spatio-temporal patterns (perturbations) traveling on a regular background. Whereas forsmall N these perturbations tend to die out after a finite number of updates, for large N they appear to be(asymptotically) stable. Similar striking changes are found in the time evolution of difference patterns, whencompared to the conventional update scheme (Fig. 3).

Let us now systematically study sensitivity towards initial conditions. We measured the time evolution ofthe Hamming distance HðtÞ of initially close trajectories for different CA rules and different values of thememory parameter a. Typically, we find asymptotic scaling proportional to a power-law with an exponent gthat is a function of a:

HðtÞ / tgðaÞ for t4t0, (3)

where t0 is small, e.g. in the order of 10 for rule 61 if the initial configurations differ in one (randomlychosen) bit. Fig. 4 compares HðtÞ of rules 61 and 111, for different values of a. In all four cases shown, we find

Fig. 2. Space–time plots starting from random initial configurations of conventional CA, i.e., a!1 (left panels), and a ¼ 1:2 for rule 61,a ¼ 1 for rule 99 (right panels). States si ¼ 0 are shown yellow, si ¼ 1 in red.

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Fig. 3. Difference patterns for CA with initial configurations differing in one randomly chosen bit. Cells with different states in both

configurations at time t are shown in red. Right panels: a ¼ 1:2 (a ¼ 1) for rule 61 (rule 99).

α =3.5

t

H (

t)

100001000100101

1000

100

10

1

α = 1.15

t

H (

t)

100001000100101

1000

100

10

1

α = 0.9

t

H (

t)

100001000100101

1000

100

10

1

α = 0.75

t

H (

t)

100001000100101

1000

100

10

1

Fig. 4. Time dependence of the Hamming distance HðtÞ for rule 111 (filled curves) and rule 61 (dashed curves) for typical values of a and

N ¼ 1000, ensemble averages over 200 different initial conditions. All curves approximately follow straight lines in the log–log plots for

two decades or more, indicating power-law behavior. Memory size T ¼ 320.

T. Rohlf, C. Tsallis / Physica A 379 (2007) 465–470468

power-law scaling for two decades or more, however, the functional dependence of the slope gðaÞ is obviouslyvery different for both rules. This is also evident in Fig. 5: whereas for rule 61, g is very close to unity for allaoa61c � 1:4 and shows a steep, step-like transition to g � 1

2for a4a61c , rule 111 shows an (nearly) opposite

behavior—for a4a111c , one has g � 1, for a below the critical point, a smooth transition to gð0Þ � 0 is found.

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Rule 61

Rule 111

q(α)

43.532.521.510.50

10

-1-2-3-4-5-6-7-8-9

Rule 61

Rule 111

γ(α)

43.532.521.510.50

1.2

1

0.8

0.6

0.4

0.2

α α

Fig. 5. The a-dependence of g (left) and q (right) for rules 61 and 111. Memory size T ¼ 320.

N = 250N = 500

N = 1000N = 2000N = 4000

α = 4

t

H (

t)

100001000100101

100

10

1N = 250N = 500

N = 1000N = 2000N = 4000

α = 1

t/N

H (

t/N

)/N

1010.10.010.001

1

0.1

0.01

0.001

1e-04

Fig. 6. Time dependence of the Hamming distance for rule 61 at different sizes N. Left: t and H are rescaled by N. The collapse of the

curves indicates that the range of the power-law regime for HðtÞ diverges with system size N (straight dashed line with g ¼ 1 shown for eye

guidance). Right: For aX1:4 curves collapse without rescaling (straight dashed line has slope g ¼ 12). Memory size T ¼ 320.

T. Rohlf, C. Tsallis / Physica A 379 (2007) 465–470 469

The exact value of lima!0 gðaÞ is, however, hard to measure in simulations due to finite size effects of thelimited memory size. We studied finite size scaling for memory sizes between T ¼ 32 and 1600; our findings,that are also robust with regard to different types of memory initialization, indeed suggest thatlima!0 limT!1 gða;TÞ ¼ 0.

The sensitivity to the initial conditions for weakly chaotic systems (i.e., with vanishing maximal Lyapunovexponent) is typically given [15,16], for say a one-dimensional case, by

xðtÞ ¼ ½1þ ð1� qÞlqt�1=ð1�qÞ, (4)

where lq40 and qo1. For tb1, we have xðtÞ�t1=ð1�qÞ. By comparison with Eq. (3), we obtain the relation

g ¼ 1=ð1� qÞ 3 q ¼ 1� 1=g. (5)

The functional behavior of qðaÞ for rules 61 and 111 is plotted in Fig. 5 (right panel). For both rules, we findstrong deviations from q ¼ 1 (i.e., the exponential divergence typical of classical BG statistics): in the case ofrule 61, one finds q � 0 for aoa61c and q � �1 for larger values of a; for rule 111, q is strongly negative belowa111c (with a probable divergence qðaÞ ! �1 for a! 0), and zero for larger a.

Last, let us look at the critical exponents zðaÞ that describe the divergence of the range of the power-lawregime with system size / Nz. For rule 111, one finds z ¼ 1 independent from a, whereas for rule 61, one findszðaÞ � 1 for aoa61c and zðaÞ � 0 above a61c (Fig. 6). This seems to suggest that the transition at ac is second-order like for rule 111 and first-order like for rule 61, which is also confirmed by the step-like discontinuity ingðaÞ in the latter case and the smooth descent of this quantity below ac in the former case (Fig. 5, right panel).

Discussion: We demonstrated that long-range memory with weights decaying / t�a leads to completelyunexpected, exciting new dynamical phenomena in elementary 1D CA. In particular, we showed for the threeclass II rules 61, 99 and 111 that the sensitivity towards initial conditions, measured in terms of the divergenceof the Hamming distance between initially close configurations HðtÞ / tg, and the associated entropic index q,shows pronounced transitions at critical values ac slightly above one. Interestingly, the behavior of qðaÞ is

ARTICLE IN PRESST. Rohlf, C. Tsallis / Physica A 379 (2007) 465–470470

strikingly different for rules 61 and 111: whereas the former shows a step-like transition from q ¼ 0 to �1 at ac

with increasing a, the latter has q ¼ 0 above ac and a gradual divergence to �1 when alpha approaches zero.Let us first comment on the fact that the transition takes place at ac41, although the memory function in Eq.(2) becomes nonsummable for ap1; this is probably due to two effects: first, the present CA are embedded in atwo-dimensional space–time, however, they dynamically evolve into a complex fractal-like structure whichexhibits a dimensionality typically slightly above unity; second, there is a nontrivial interplay between thememory and the update rules, leading e.g. to considerable fluctuations of qðaÞ at ac that appear to be conservedeven in the limits of large N and T (compare Fig. 5). Considering the fact that, without long-range memory,both rules converge to very similar dynamical attractors these striking differences are really a surprise.Remarkably, the different behavior with regard to q is still conserved in the limit of large a, where our new,generalized scheme maps on the conventional CA without memory. This may indicate that the application ofthese concepts derived from nonextensive statistical mechanics could help to refine existing CA classificationschemes, at least for a subset of class II and IV rules where, in our simulations, similar behavior was found asfor the three rules discussed in this paper. In particular, complex dynamical behavior, as e.g. found for thefamous ‘universal computer rule’ 110, depends on a delicate interplay between memory and informationspreading; however, this is poorly understood for most CA. Our generalized scheme may help to address thisinteresting reverse problem, i.e., the question of how memory emerges for certain CA rules, and why it does notfor others (we will report details elsewhere).

We acknowledge interesting discussions with J.M. Gomez Soto. Partial financial support by SIInternational and AFRL (USA Agencies), and Pronex/MCT, Faperj and CNPq (Brazilian Agencies) isacknowledged as well. Finally, we have benefited from the proverbially warm hospitality of the Santa FeInstitute, New Mexico, where most of this work was done.

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