CELLULAR AUTOMATA WITH ACCUMULATIVE MEMORY: LEGAL RULES STARTING FROM A SINGLE SITE SEED

October 18, 2003 11:55 WSPC/141-IJMPC 00486

International Journal of Modern Physics CVol. 14, No. 5 (2003) 695–719c© World Scientific Publishing Company

CELLULAR AUTOMATA WITH ACCUMULATIVE MEMORY:

LEGAL RULES STARTING FROM A SINGLE SITE SEED

RAMON ALONSO-SANZ

ETSI Agronomos (Estadıstica)C. Universitaria. 28040, Madrid, Spain

[email protected]

MARGARITA MARTIN

F. Veterinaria (Bioquımica y Biologıa Molecular IV, UCM)C. Universitaria. 28040, Madrid, Spain

Received 29 July 2002Revised 25 November 2002

Standard Cellular Automata (CA) are ahistoric (memoryless), i.e., the new state of acell depends only on the neighborhood configuration at the preceding time step. Thisarticle introduces an extension of the standard framework of CA by considering automataimplementing memory capabilities. While the update rules of the CA remains the same,each site remembers a weighted mean of all its past states, with a decreasing weight ofstates farther back in the past. The historic weighting is defined by a potential series of

coefficients, tk , k acting as a forgetting factor. This paper considers the time evolutionof one-dimensional, legal CA rules with accumulative memory.

Keywords: Cellular automata; memory.

1. Introduction

Standard Cellular Automata (CA) are ahistoric (memoryless), i.e., no memory of

previous iterations except the last one is taken into account to decide the next

one. Thus, if a(Ti is taken to denote the value of cell i in a one-dimensional cellular

automaton at time step T , the site values evolve by iteration of the mapping:

a(T+1i = φ(N (a

(Ti )), where φ is a rule operating on the neighborhood (N ) of the

cell i. The subject has been reviewed recently by Wolfram1 and Ilachinski.2

This paper considers a variation of conventional one-dimensional CA, featuring

cells by a weighted mean of all their past states. We propose to maintain the rules

(φ) unaltered, while making them act over cells featured by a weighted mean of all

their past states: a(T+1i = φ(N (f

(Ti )), f

(Ti being the weighted mean corresponding

to cell i after time step T .

We refer these automata, considering historic memory, as historic and the stan-

dard ones as ahistoric. Some authors (for example Wolf-Gladrow3) define rules

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696 R. Alonso-Sanz & M. Martın

with memory as those with dependence in φ of the state of the cell to be up-

dated. So in the nearest-neighbor scenario, rules with no memory take the form:

a(T+1i = φ(a

(Ti−1, a

(Ti+1). Our use of the term memory clearly is not this.

This paper deals with one-dimensional CA with two possible values at each site

and range (r = 1): the value of a given site depends on the nearest neighbors. These

rules, following Wolfram’s4 notation, are characterized by a sequence binary values

(β) associated with each of the eight possible triplets (ai−1, ai, ai+1):

111 110 101 100 011 010 001 000

(β1, β2, β3, β4, β5, β6, β7, β8)binary ≡8

∑

i=1

βi28−idecimal = R .

The rules are conveniently specified by a decimal integer, to be referred as their

rule number, R. The rule number of elementary CA will range from 0 to 255.

Legal rules4 are reflection symmetric (so that 100 and 001 as well as 110 and 011

yield identical values), and quiescent : i.e., they do not transform a dead cell with

dead neighbors into a live cell (so β8 = 0). These restrictions leave 32 possible legal

rules of the form: β1, β2, β3, β4, β2, β6, β4, 0. Only legal rules are to be examined

in this work.

2. Full History

Historic memory can be implemented in CA just by featuring every cell by its most

frequent state. In case of a tie, the cell is featured by its last state. We refer this

model as fully historic.

We exemplify the full memory mechanism using an easy rule, rule 254 (11111110)

which assigns a live state to any cell in whose neighborhood there would be at least

one live cell. Rule 254 progresses as fast as possible (i.e., at the speed of light): in the

ahistoric model a single site live cell will grow monotonically, generating segments

with size increasing by two units with every time step. This is slower in the historic

model (see Table 1): the outer live cells are not featured as live cells until the number

of times they “live” is equal to the number of times they were “dead”. Then, at the

powers of two time steps T = 2, 4, 8, 16, . . . , 2n, . . . , the automaton fires two new

outer live cells. The dynamics of rule 254 in the fully historic model, again the speed

of light in this scenario, might be described in terms of punctuated equilibrium, i.e.,

long periods of stability (equilibrium or stasis) in the (simple) pattern of live cells,

altered by changes that take place by well-defined steps (the punctuation marks).

The duration of the stable periods tends to increase as T increases.

As long as the local rules of both historic and ahistoric automata remain un-

altered, and both the latest and most frequent states coincide after the two first

time steps, the historic and ahistoric evolution patterns are the same till T = 3.

Starting with a single site live cell, the evolution dynamics will start as: → ,

for any quiescent rule with β1 = 1. At time step T = 3, the cells distant two sites

from the initial seed will be live (recall that β1 = 1); the cells in between these

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Cellular Automata with Accumulative Memory 697

Table 1. Rule 254 (11111110) starting with a single site live cell till T = 8. Fully historic andahistoric model. Live cells: last, most frequent.

Table 2. Rule 90 (01011010) starting with a single site live cell till T = 4. Historic and ahistoricmodel. Live cells: last, most frequent.

two, will be dead or alive (*) depending on the CA rule considered. Schematically,

at T = 3: ***. But these outer live cells are to be featured as dead in the fully

historic model as they were two time steps dead and only one live. Consequently

the patterns for the fully historic and ahistoric automata diverge at T = 4.

This is the case in Table 1 and in Table 2 with rule 90 (01011010), where after

T = 3 the most frequent state for all the cells is the dead one and consequently the

pattern dies out.

In the standard (ahistoric) scenario, rules 90 and 150 (together with the trivial

0 and 204) are the only additive legal rules, i.e., any initial pattern can be de-

composed into the superposition of patterns from a single site seed. Each of these

configurations can be evolved independently and the results superposed (module

two) to obtain the final complete pattern. Additivity of rules 90 and 150 is lost in

the historic model (see Ref. 5).

Every configuration in a standard (ahistoric) cellular automaton has a unique

successor in time (although it may have several distinct predecessors). On the con-

trary, a last (or current) configuration in the historic scenario may have multiple

successorsa: the transition rules operate on the most frequent state (mfs) configu-

ration, not on the last, so the successor of a given configuration depends on the

underlying mfs configuration. We envisage that this will notably alter the appear-

ance of the state transition diagrams in the historic scenario as compared to those

of ahistoric CA.6 A considerable number of rules generate oscillators that appear

fairly soon in the fully historic model (this is illustrated in Fig. 1). Thus the number

of different configurations generated by evolution is smaller in the historic model.

aA primary example is given with rule 254 (Table 1). In the fully historic model, live configurationsreproduce themselves for several iterations ( for example, twice) but at a critical time step(T = 5 for the example), generate a new () one.

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This would lead to conjecture the existence of a higher number of “Garden-of-

Eden” nodes (configurations for an automaton which could only exist initially) in

the historic scenario.

Fig. 1. Evolving patterns of legal rules affected by memory when starting from a single site seed(). Evolution up to 62 time steps for different values of the forgetting factor k. The evolutionof the cellular automata at successive time steps is shown at successive horizontal lines. Ahistoricmodel for k = 49, fully historic model for k = 0.

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Fig. 1 (Continued)

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3. Weighting Memory

In a previous workb we implemented a historic memory mechanism intended to

ponder in a high degree in the recent states, based on a geometric discounting

process in which the state a(T−τ , obtained τ time steps before the last round, is

updated to the value: ατa(T−τ , α being the memory factor (0 ≤ α ≤ 1). A given cell

is featured by the rounded weighted mean of all its past states. This well known

mechanism fully ponders the last round (α0 = 1), and tends to forget the older

rounds. The mechanism is implemented in two stages:

(i) after the time step T , the weighted mean of the states of a given cell will be:

m(Ti (a

(1i , a

(2i , . . . , a

(Ti ) =

∑T

t=1 αT−ta(ti

∑T

t=1 αT−t. (1)

(ii) the weighted mean state (f) is obtained by comparing its unrounded mean m

to 0.5,c assigning the last state in case of an equality:

f(Ti =

1 if m(Ti > 0.5 ,

a(Ti if m

(Ti = 0.5 ,

0 if m(Ti < 0.5 .

The weighting memory mechanism just described is holistic in its information

demands: for evaluating the weighted mean of a cell, it is necessary to know its

whole states series in time. To avoid this demand, previous states can be weighted

in the following way15:

m(Ti (a

(1i , a

(2i , . . . , a

(Ti ) =

∑T

t=1 tka(ti

∑T

t=1 tk≡

ω(ai, T )

Ω(T ), (2)

where k is a free parameter.

The rounded weighted mean state (f) is obtained, again, by comparison to 0.5.

This mechanism is not holistic but accumulative in its demand of knowledge of

past history, i.e., for evaluating ω(ai, T ) it is not necessary to know the whole a(ti

series because it is determined by the accumulation of the contribution of the last

state (T ka(Ti ) to the already accumulated ω(ai, T − 1): ω(a

(ti , t ≤ T ) = ω(a

(ti , t ≤

T − 1) + T ka(Ti .

For k = 0 the full memory model is recovered: every state contributes equally

to the mean state, regardless of the time step at which it occurred: m(Ti =

(1/T )∑T

t=1 a(ti and cells will be featured by their most frequent state (the last

one in case of a tie). For k = 1 it is Ω(T ) =∑T

t=1 t = T (T + 1)/2. The larger the

value of k, the more heavily the recent past is taken into account, and consequently

bReferences 5 and 7–9 refer to proper CA while Refs. 10–14 to the spatial formulation of theprisoner’s dilemma.cProvided that states a are coded as 0 (dead) and 1 (live).

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the closer the scenario to the ahistoric one. Consequently k can be termed as a

forgetting factor.

Choosing integer k values allows working only with integers (a la CA), a clear

computational advantage of the accumulative model over the holistic (1).

Comparing m(Ti to 0.5 is equivalent to studying the sign of:

∂Ti (a

(1i , a

(2i , . . . , a

(Ti ) = 2

T∑

t=1

tka(ti −

T∑

t=1

tk =

T∑

t=1

(2a(ti − 1)tk ,

f(Ti =

1 if ∂(Ti > 0 ,

a(Ti if ∂

(Ti = 0 ,

0 if ∂(Ti < 0 .

Historic memory has no effect after T = 2, so f (2 = a(2.d After T = 3 it is

∂(3i (a

(1i , a

(2i , a

(3i ) = (2a

(1i − 1) + (2a

(2i − 1)2k + (2a

(3i − 1)3k. Of course, if a

(1i =

a(2i = a

(3i , history does not alter the series and will be f

(3i = a

(3i .e Neither does

history take effect till T = 3 if a(2i = a

(3i ,f nor if a

(1i = a

(3i .g If a

(1i = a

(2i 6= a

(3i it is

∂(0, 0, 1) = −∂(1, 1, 0) = −1 − 2k + 3k ≥ 0 if k > 0, so that, again, f(3i = a

(3i .h In

short, accumulative memory does not affect the scenario after T = 3. This contrasts

with what happens when applying the geometrical memory mechanism, in which

memory switches the featuring state of a cell after T = 3 if a(1i = a

(2i 6= a

(3i and

α > 0.61805.i

But the scenario may change after T = 4. Thus, if a(1i = a

(2i = a

(3i 6= a

(4i it

is: ∂(0, 0, 0, 1) = −∂(1, 1, 1, 0) = −1 − 2k − 3k + 4k. If k = 1 is ∂ < 0 and then

f(0, 0, 0, 1) = 0. This scenario is that of the two outer live cells at T = 4 in Table 3.

For k = 2 it is ∂ > 0 and memory does not alter the cell characterization to any other

except the last one. After T = 5 time steps, ∂(0, 0, 0, 0, 1) = −1− 2k − 3k − 4k +5k,

which is negative for k = 1 and for k = 2 but not for k = 3. The two outer live

cells live again at T = 5 in Table 3 are featured as live after this time step because

∂(0, 0, 0, 1, 1) = −1 − 2 − 3 + 4 + 5 > 0, or comparing the ω and Ω values as in

Table 3, because 2ω = 18 > 15 = Ω.

d∂(2i

(a(1i

, a(2i

) = (2a(1i

− 1) + (2a(2i

− 1)2k ⇒

∂(1, 1) = −∂(0, 0) = 1 + 2k > 0 ,

∂(0, 1) = −∂(1, 0) = −1 + 2k > 0 .e∂(1, 1, 1) = −∂(0, 0, 0) = 1 + 2k + 3k > 0.f∂(0, 1, 1) = −∂(1, 0, 0) = −1 + 2k + 3k > 0.g∂(1, 0, 1) = −∂(0, 1, 0) = 1 − 2k + 3k > 0.hIf k = 1 it is ∂ = 0 and cells are to be featured by their last state applying the criterion ofassigning the last state in case of an equality.iIn the fully historic memory model (α = 1) for example. From the accumulative memory per-spective, k = 0 leads to ∂(0, 0, 1) = −1 and consequently f (3 = 1 − a(3.

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Table 3. Evolving patterns generated from a single site live cell using φ = 254. Historic model

with k = 1. Evolution up to T = 5. Live cells: last, most frequent.

4. Evolution Patterns

Figure 1 shows the spatio-temporal patterns of the legal rules significantly affected

by memory when starting from a single live site. The k factor varies in Fig. 1 from

0 (fully historic model) to 6 by one interval, from 10 to 25 by five intervals and

finally k = 49, a high value which virtually means no memory effect. The patterns

shown are symmetric due to the symmetry of legal rules.

Two main conclusions can be derived from Fig. 1:

(i) as an overall rule, the patterns become more expanded as less historic memory

is retained (higher k),

(ii) the transition from the fully historic (k = 0) to the ahistoric scenario (k = 49)

is gradual in most cases.

Rules 50, 122, 178, 250, 94, 222, and 254 are paradigmatic of this smooth evo-

lution. Rules 126 and 182 present also a gradual evolution, although the historic

patterns do not resemble the ahistoric at all.

In some cases the above transition presents a notable discontinuity. This applies

for (i) rules such as 22 and 54 which tend to die out in 1 < k < 10 but generate

oscillators in the fully historic model, and (ii) the group of rules for 18, 90, 146,

and 218 which exhibit extinctionj till k = 25 (with the exception of k = 20) and,

abruptly, the typical ahistoric expansion for k = 49. The equivalent16 rules 146 and

182 are not affected by memory in a similar way.

Figure 2 shows the differences in pattern (DP) produced by evolution from

a randomk initial configuration resulting from change in the value of its initial

center site. The pictures show the damaged region as black squares corresponding

to the site values that differed among the patterns generated with the two initial

configurations.

The perturbations in the ahistoric model propagate very rapidly to the right

and left at great velocity at any time. This behavior illustrates the butterfly effect :

a small perturbation grows, and finally rules the whole system. The velocity in the

jThese are the only rules for which extinction is found in the fully historic model.kThe value of each site is initially uncorrelated, and is taken to be 0 or 1 with probability 1/2.

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Fig. 2. Differences in patterns produced by evolution of the legal rules when the values of sitesin the initial configuration are chosen at random to be 0 or 1 with probability 1/2. After reversingthe central site, the pictures show the damaged region as black squares corresponding to the sitevalues that differed among the patterns generated with the two initial configurations. Evolutionup to 62 time steps in a lattice with 133 sites. Periodic boundary conditions are imposed on theedges.

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Fig. 2 (Continued)

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damage spreading is quantified by means of the left and right Lyapunov exponents

(λL, λR) which measure the rate at which perturbations spread to the left and right,

and are given by the slopes of the left and right boundaries of the growth of the

difference patterns.

As an overall rule, historic memory produces, again, a preserving effect regard-

ing the damage induced by the reversal of a single site value. This result can be

expected from Fig. 1, which can be considered as a particular case of the same ini-

tial alteration of a unique (central) site value. In fact there is notable parallelism in

the qualitative evolution of patterns in the rules included in both figures, though in

Fig. 2 things are much more sophisticated: (i) there are no groups of rules evolving

in an identical way, and (ii) rules such as the additive rules 90 and 150 exhibit their

high degree of “chaoticity” even for low k values. The equivalent rules 146 and 182

do evolve now in a similar way.

5. Reversibility

The second order in time mechanism proposed by Fredkin17:

a(T+1i = φ(N (a

(Ti )) + a

(T−1i mod 2

is reversible on account of the algebraic properties of the addition module two.

Thus, backtracking is possible by defining a(T−1i in the above equationl:

a(T−1i = φ(N (a

(Ti )) + a

(T+1i mod 2 .

The reversibility mechanism operates independently of the first summand of the

second member. Thus, incorporating memory to it, the rules:

a(T+1i = φ(N (f

(Ti )) + a

(T−1i mod 2

reverse asm:

a(T−1i = φ(N (f

(Ti )) + a

(T+1i mod 2 .

Table 4 shows the evolving patterns generated (→) from a single site live cell

at T = 1 and at T = 0 when using the reversible formulation with φ = 254 in the

fully historic model (k = 0) up to T = 5.

For reversing from T in the fully historic model it is necessary to know not only

a(T and a(T+1 but also ω(a, T ) =∑T

t=1 a(t. The reversing mechanism progresses as

follows in the fully historic model. First ω(a, T ), is compared to T in order to obtain

f (T =

1 if 2ω(a, T ) > T ,

a(T+1 if 2ω(a, T ) = T ,

0 if 2ω(a, T ) < T .

lA catalogue of their evolution patterns in the r = 1 scenario can be found in Ref. 16.mWe have studied two-dimensional reversible rules with memory in Ref. 18.

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Table 4. Reversible evolving patterns generated from a single site live cell using φ = 254. Fully

historic model (k = 0). The forward (→) evolution is shown up to T = 5, the backward (⇐) fromT = 4. Live cells: last, most frequent.

Then, to obtain f (T−1, the contribution of the last pattern to ω(a, T ), thus a(T , is

subtracted, giving: ω(a, T − 1) = ω(a, T ) − a(T ; these values all across the lattice

are to be compared to T − 1:

f (T−1 =

1 if 2ω(a, T − 1) > T − 1 ,

a(T if 2ω(a, T − 1) = T − 1 ,

0 if 2ω(a, T − 1) < T − 1 .

Continuing in the reversing process: ω(a, T − 2) = ω(a, T − 1) − a(T−1 is to be

compared to T − 2. In general: ω(a, T − τ) = ω(a, T − τ + 1) − a(T−τ+1, and

f (T−τ =

1 if 2ω(a, T − τ) > T − τ ,

a(T−τ+1 if 2ω(a, T − τ) = T − τ ,

0 if 2ω(a, T − τ) < T − τ .

Table 4 shows also schematically the reversing mechanism (⇐) from T = 4.

Note that at even time steps (T = 2 and T = 4) it has been necessary to resort to

the last configuration regenerated to decide the most frequent, because for all the

last live cells, it is 2ω = T .

Reversing in the accumulative memory model with a generic k value mimics the

algorithm described above for k = 0.n The general form is given below with the

weight corresponding to time step t (tk or any other possible accumulative choice)

denoted as δ(t.

(1) Departing from ω(a, T ) =∑T

t=1 δ(ta(t, calculate Ω(T ) =∑T

t=1 δ(t ⇒

f (T =

1 if 2ω(a, T ) > Ω(T ) ,

a(T+1 if 2ω(a, T ) = Ω(T ) ,

0 if 2ω(a, T ) < Ω(T ) .

nIn the holistic memory scenario the reversing process is not so straightforward (see Ref. 18).

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(2) ω(a, T − 1) = ω(a, T ) − δ(T a(T , Ω(T − 1) = Ω(T ) − δ(T ⇒

f (T−1 =

1 if 2ω(a, T − 1) > Ω(T − 1) ,

a(T if 2ω(a, T − 1) = Ω(T − 1) ,

0 if 2ω(a, T − 1) < Ω(T − 1) .

(3) ω(a, T − 2) = ω(a, T − 1)− δ(T−1a(T−1, Ω(T − 2) = Ω(T − 1)− δ(T−1 ⇒ f (T−2

· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

τ)ω(a, T − τ) = ω(a, T − τ + 1)− δ(T−τ+1a(T−τ+1, Ω(T − τ) = Ω(T − τ + 1)−

δ(T−τ+1

f (T−τ =

1 if 2ω(a, T − τ) > 2Ω(T − τ) ,

a(T−τ+1 if 2ω(a, T − τ) = 2Ω(T − τ) ,

0 if 2ω(a, T − τ) < 2Ω(T − τ) .

Table 5 shows the evolving patterns generated (→) from a single site live cell

at T = 1 and at T = 0 when using the reversible formulation with φ = 254 in

the historic model, with k = 1 up to T = 5. Table 5 also schematically shows the

reversing mechanism (⇐) from T = 4. It has been necessary to resort to the last

configuration regenerated to decide the most frequent (2ω = Ω scenario) at time

steps T = 4 (three central cells) and T = 3 (two outer cells).

A catalogue of the evolution patterns for the reversible formulation of one-

dimensional rules in the ahistoric model starting from a single live cell at initial

times is given in Ref. 15, Table 16.

Figure 3 shows the evolving patterns of the reversible formulation of legal rules

starting with a single site live cell (at T = 1 and T = 0). Again, the confinement

of the disruption generated by a single live cell becomes very clear. Only rule 94

dies out, and only in the fully historic model (k = 0). Configuration oscillators are

frequent when full history is considered and for some rules and higher k values;

k = 1 in particular, as shown, for example, in rules 18 or 54.

Table 5. Reversible evolving patterns generated from a single site live cell using φ = 254. Historicmodel with k = 1. The forward (→) evolution is shown up to T = 5, the backward (⇐) from T = 4.Live cells: last, most frequent.

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Fig. 3. Evolving patterns of reversible legal rules affected by memory, starting from a single siteseed (). Evolution up to 62 time steps for different values of the forgetting factor k.

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Fig. 3 (Continued)

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Fig. 3 (Continued)

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Fig. 3 (Continued)

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Fig. 4. Evolving patterns of reversible legal rules affected by memory, starting from a single site

seed (). Evolution up to 62 time steps for different values of the memory factor α. Ahistoricmodel for α = 0.5, fully historic model for α = 1.0.

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Fig. 4 (Continued)

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Fig. 4 (Continued)

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Fig. 4 (Continued)

Figure 4 adopts the same scenario as Fig. 3, but with the geometric memory

discount (1). These evolving patterns are unpublished and serve as a reference

for Fig. 3. Again, when full history is considered, the evolution dynamics tends to

generate oscillators, and only rule 94 dies out, only in the fully historic model. Some

rules show unexpected similar evolving patterns: rules 126 and 254 for example.

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6. Other Accumulative Memories

This work deals only with the accumulative memory mechanism defined by the

weight δ(t = tk. But other accumulative weights are possible.

The weight δ(t = kt, has the same appealing feature as tk of allowing operation

only with integer values (again provided that k is integer), but it is not useful here.

For k = 1 the fully historic model is recovered, but for any other k > 1 the evolving

patterns are those of the ahistoric model because cells are always featured by their

last value. This is so even in the most unbalanced scenario (a(1 = a(2 = · · · =

a(T−1 = 0, a(T = 1), in which 2ω = 2kT is greater than Ω = (kT+1 − k)/(k − 1).

For example, for k = 2 it is 2ω = 2T+1 > 2T+1 − 2 = Ω.

Inverse memory

The weighting mechanism may be designed in an inverse way, so that the older

rounds are remembered more than the more recent ones.o The idea is implemented

with weights δ(t) = tk by choosing negative k values.p The inhibitor effect of inverse

memory is illustrated in Table 6.

Parity memory

Cells can be featured by the parity on the sum of previous states: p(Ti =

(∑T

t=1 a(ti ) mod 2.

Interest memory

In the interest memory scenario, a cell will be featured as dead if all its previ-

ous states are equal (zero interest means boring), and as alive if any of them are

different: f (T (a(1i , a

(2i , . . . , a

(Ti ) = 0 iff a

(1i = a

(2i = · · · = a

(Ti .

Table 6. Evolving patterns generated from a single site live cell with inverse memory (k = −1).Evolution patterns of legal rules affected by memory are shown up to T = 13.

oThe first author of this manuscript begins to experience this aging symptom in details of ordinarylife. The first author.pAlso with δ(t) = kt and 0 < k < 1.

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Toffoli and Margolus19 use the interest idea when considering the time-tunnel,

a reversible mechanism based on the two-dimensional rule: φ(N (a(Ti,j )) = 0 if all the

states in N are equal, 1 in the contrary case.

Continuous valued memory

Historic memory can be embedded in continuous CA (or coupled map lattices),

in which the state variable ranges in R, just considering m instead of a in the

application of the updating rule: a(T+1i = φ(N (m

(Ti )).

Fuzzy rules

Fuzzy CA, with states ranging in the real [0, 1] interval, may be obtained by fuzzi-

fication of the disjunctive normal form of Boolean CA rules by replacing: a ∨ b by

min(1, a+b), a∧b by ab, and ¬a by 1−a. Featuring cells by the unrounded weighted

mean of their past states, i.e., m, will provide fuzzy historic CA. An illustration of

the effect of memory in fuzzy CA, starting with a single crisp live cell, is given in

Ref. 5. The illustration operates on rule 90q: a(T+1i = m

(Ti−1 + m

(Ti+1 − 2m

(Ti−1m

(Ti+1.

Imperfect memory transmission

The standard models of CA and the CA with memory can be combined by consider-

ing two types of cell characterization of the neighborhood N . Thus for a subset of N ,

let it be S, the cells can be featured by their last state and the remaining S = N−S

by their weighted mean states. For example, history can feature the cells of the strict

neighborhood but not the cell to be updated: a(T+1i = φ(f

(Ti−1, a

(Ti , f

(Ti+1), or, the

contrary, the cell to be updated but not the outer cells: a(T+1i = φ(a

(Ti−1, f

(Ti , a

(Ti+1).

The different memories mentioned can be considered to implement new re-

versible mechanisms. For example, by using partial memory transmission: a(T+1i =

φ(a(Ti−1, f

(Ti , p

(Ti+1) ⊕ a

(T−1i .

7. Conclusion

The consideration of historic memory of past states has an inertial (or conservation)

effect. According to this principle, historic memory produces a preserving effect

regarding the evolving patterns from a single site or the damage induced by the

reversion of a single site value. On increasing the role of memory from the standard

(ahistoric) model to the fully historic (k = 0), there is usually a gradual effect.

Nevertheless, an appreciable number of exceptions (and discontinuities) to such a

gradual effect have been found.

qThe fuzzification process when applied to Boolean rule 90 given in its disjunctive normal form:

(ai−1 ∧ (¬ai+1)) ∨ ((¬ai−1) ∧ ai+1), yields a(T+1i

= a(Ti−1 + a

(Ti+1 − 2a

(Ti−1a

(Ti+1.

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The accumulative memory model allows working only with integers and with

minimal computer memory demands, two obvious computational advantages over

the holistic model (1). However, the accumulative memory mechanism has a seri-

ous drawback: the weight tk explodes, even for k = 2, when t grows.r A limited

trailing memory would be interesting to avoid both the excessive computer mem-

ory demands in the holistic model and the numerical explosion in the accumulative

memory model.

In general, there are two main approaches to CA: forward and backward. The

forward (theoretical) approach implies the study of transition rules of a given cel-

lular space in order to establish its intrinsic properties (dynamic behavior, pattern

growth, and so on). The backward (practical) approach involves the design of sets

of transition rules to match the “correct” behavior of the CA system of a given

complex system (physical, biological, social or so on).

This paper adopts the forward approach: it introduces the kind of CA with

Accumulative Memory (CAM) in the elementary context (1D, two states, nearest

neighbors), and surveys its properties in a fairly qualitative (pictorial) form. Bor-

rowing a Vichniac20 expression, this paper deals with the CAM “Zoology”, i.e., the

study of CAM for its own sake.s Much remains to be done: entropy, fractal proper-

ties, the dynamics of CAM in the state (phase) space being issues to come under

scrutiny. When extending the study to more than two states, it is conjecturable

that Langton’s λ parameter will play some part in explaining the foreseeable di-

verse effect of memory on different rules. So a more complete analysis of the class

CAM, its quantitative characterization in particular, is planned for future work.

The forward and backward (or inverse) approaches are obviously interrelated. A

major impediment in the backward approach stems from the difficulty of utilizing

the CA complex behavior to exhibit a particular behavior or perform a particu-

lar task. This has severely limited their applications and hindered computing and

modeling with CA.

CAM structurally exhibit a growth inhibition feature found exceptionally in

standard CA.4 The extent of the growth inhibition can be modulated by varying the

forgetting factor k. This could mean a potential advantage of CAM over standard

CA as a new tool for modeling slow diffusive growth from small regions, a common

phenomenon in nature.

The question of how errors spread and propagate in cooperative systems has

been studied in a variety of fields. Given the difficulty of creating analytical mod-

els for any but the simplest systems, most investigations have been conducted by

computer simulations, especially in the area of statistical physics of many-body

systems. In CAM, the damage induced by a single error is generally confined to

the proximity of the site where it occurred. From this point of view, CAM can

rComputing in this work has been implemented with MATLAB, which allows for integers up to17977e308.sFollowing this metaphor, CAM increase the CA-(bio)diversity.

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be featured as resilient in the face of errors. Fault tolerance is an important issue

when considering systems with a large number of components, in which faults will

be highly probable. The robust CAM could play a role in this (nanotechnological)

scenario.

Acknowledgments

This work was supported by Comunidad de Madrid (08.8/006/2001.1) and

INIA(MCYT) RTA02-010. Computing was performed using the computing facil-

ities of the Department of Mathematics and Statistics, University of Edinburgh.

References

1. S. Wolfram, A New Kind of Science (Wolfram Media, 2002).2. A. Ilachinski, Cellular Automata. A Discrete Universe (World Scientific, 2001).3. D. A. Wolf-Gladrow, Lattice-Gas Cellular Automata and Lattice Boltzman Models

(Springer, 2000).4. S. Wolfram, Rev. Mod. Phys. 55, 601 (1983).5. R. Alonso-Sanz and M. Martin, Int. J. Bifurcation and Chaos 12, 205 (2002).6. A. Wuensche and M. Lesser, The Global Dynamics of Cellular Automata, Santa Fe

Institute Studies in the Sciences of Complexity, Vol. 1 (Addison-Wesley, 1992).7. R. Alonso-Sanz, M. C. Martin, and M. Martin, Int. J. Bifurcation and Chaos 11, 1665

(2001).8. R. Alonso-Sanz and M. Martin, Int. J. Mod. Phys. C 13, 49 (2002).9. R. Alonso-Sanz and M. Martin, Complex Systems 14, 99 (2003).

10. R. Alonso-Sanz, Int. J. Bifurcation and Chaos 9, 1197 (1999).11. R. Alonso-Sanz, M. C. Martin, and M. Martin, Int. J. Bifurcation and Chaos 10, 87

(2000).12. R. Alonso-Sanz, M. C. Martin, and M. Martin, Int. J. Bifurcation and Chaos 11, 943



(2001).15. S. Smale, Econometrica 48, 1617 (1980).16. S. Wolfram, Cellular Automata and Complexity (Addison-Wesley, 1994).17. E. Fredkin, Physica D 45, 254 (1990).18. R. Alonso-Sanz, Physica D 165, 1 (2003).19. T. Toffoli and N. Margolus, Cellular Automata Machines (MIT Press, 1987).20. G. Vichniac, Physica D 45, 63 (1990).

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Documents

CELLULAR AUTOMATA WITH ACCUMULATIVE MEMORY: LEGAL RULES STARTING FROM A SINGLE SITE SEED