Com plex Sy stem s 6 (1992) 287-300

Effect of Noise on Long-term Memory in Cellular Automata with Asynchronous Delays between the Processors R ez a Ghar avi Venkat A nantharam School of Electrical Engineering, Cornell University, Ith aca, N Y 14853, USA

Abst r act. We consider monotonic binary cellular automa ta on t he lat tice ZP as models of systems capa ble of long-term memory, t hat is, capable of admitting multiple invarian t configurations. Long-term memory in cellular aut omata can be robust in t he presence of noise, so t hat in a noisy environment t he automaton may admit more t han one stationary distribution on configurations. We examine t he effects of asyn chronous communication delays between th e pro cessors on longterm memory in cellular automata and describe when asynchronism can cause t he eras ure of long-term memory in t he presence of noise. Our main result is a simple generalization to t he asynchronous compu tation model of a deep result of Toom charac terizing th e invariant configurat ions of monoto nic binary cellular aut omata t ha t are robust to noise. Several qualitat ive consequences of asynchronism are illustr ated t hrough examples. 1.

Introduction

Cellula r autom a t a are sim ple com putational models t hat ar e capable of exhibiting a wid e range of complex dyna mi cal b eh avior (see [7]) . The comput at ion is conside re d as pro ceeding synchro nous ly via iden ti cal processor s at ea ch sit e on a regular lat ti ce-usu all y ZP- and t he comp utationa l rule is as sume d t o b e spat ia lly hom ogenou s. In ot her wor ds, at every t ime st ep t he state at a sit e is updat ed as a function of cer tain previous st at es of some of t he neighb oring sites wit h a spa t ially hom ogen ou s up da ting rule. The interest in studying such a uto mat a comes from several p oint s of view. For exam ple, t here is a b elief t hat t he complex dynami cal b eh avior exhibite d by t hes e aut om a t a is a good mo del for t he nat ur al stat ist ical beh avior of physical syst em s such as ga ses, which cons ist of large numbers of int eracting eleme nts. Another powerful source of reawaken ed int er est in cellula r auto mat a has b een the development of par allel compu tat ional syste ms t ha t

employ different types of regular architect ures (e.g., [1]) .

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A pro cessor at a site can be in one of a finit e set of states , and the ent ire collect ion of states at t he sites is called a configuration. On e of t he remarkable properties of certain cellular automaton updating rules is that t hey can admit more t han one invariant configuration, representing t he ability to maint ain long-t erm memory. Further , it is known [6] that t here are automata whose long-term memory persist s under noise. Even in environments in which the st at e of a site may change du e to noise or computatio nal erro r , the evolut ion of t he automa to n is such that one can give a t est that distinguishes between initial cond itions even afte r arbit rarily long period s of t ime . T his ability is particularly import an t from t he point of view of the automaton as a computational device, where t he initi al configurat ion is the in pu t on which the pro cessors perform their calcula tions (see [2]). For the op eration to be reliabl e in a noisy or unreliab le environment , we would require t he syste m t o remember enoug h relevant information about its initial configurat ion over arbitrar ily long period s of time . For a gene ra l discussion of t he sub ject of reliable computation , see the surv ey pap er [5]. Our interest in this pap er is in the persist ence of long-term memory in automata ope rat ing in an unreliab le or noisy environment when there is also unr eliability in t he data transfer bet ween t he process ors. We examine this quest ion in an asy nchro nous comp utat ional model, where the computations are synchrono us but there are unknown delays in t he t ransmission of dat a between pr ocessors. In other words, a processor carrying out a compu t at ion may only have available t o it delayed versions of t he data on which it dep end s. Such a sit uation may arise when dat a is lost or delayed in tran sit between t he pr ocessors. We ass ume that t he delays are bo unded by some int eger d; in ot her words, the sys tems we consider are parti ally asynchronous (see [1]). We are par ticula rly int erested in discussing how lar ge a delay in data transfer can be to lerat ed before t he ability of the aut omat on to rememb er it s initi al configuration br eaks down. Our resul ts ar e for a class of automata called monotonic binar y tessellati ons (MBT s), defined in section 2. In an MBT t he state of each pro cessor is 1 or 0, and the all-zero and t he all-one configur at ions are invariant under t he updating rul e. Our main result is a simple generalization t o t he par ti ally asynchronous computat ional model of a deep resul t of Toom [6], which gives necessary and sufficient condit ions for the stability of the all-zero configurat ion of an MBT t o no ise. Toom 's characterizat ion is widely acknowledge d to be one of t he deep est result s to date abo ut cellular auto mata- it is a simple geometric characte rization based on the geomet ry of the dat a set that a processor depend s upon for its computation. We discuss this in section 2 afte r giving a formal definition of MBTs . Our res ult is state d an d pr oved as T heorem 2 in sect ion 3. Our characterizat ion of the stability of t he states un der asyn chronous comput at ion allows us to discuss the effects of asy nchronism on the st abil ity of states . Several qu alit ative asp ect s of t his ar e brought out through examples in sect ion 4.

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Effect of Noise on Lon g-t erm Memory

2.

Problem definition

Before defining an MB T , we will need the followin g it ems: 1. A function U : Z p+1

f-+

z(p+1 )r

defined by

where T and p are po siti ve integers and We deno t e tw = maxi=l ,...,r Iti l.

u;

E

Z P,

t,
(V'yd and ¢' : {O, 1}Td f-> {O, I} as follows:

U'(v) cjg (v

¢' (XU'(v) )

+ (Ul' td , v + (Ul, t 1 - 1), . . . , v + (Ul, t 1 . .. , v + (UT,tT - d)),

~f ¢ C~ XV+(Ul.tI- i), .. . , i~ Xv+(Ur,tr -i) )

d),

.

Clearl y, ¢' is monotonic. We deno t e a t ra ject ory of (U', ¢' ) with base Xw' by T' (x w' ) and call it a dom inant trajectory. In order to apply Theorem 1 to (U', ¢'), we will need to know it s minimal zero sets. Lemma 1. If Zl ,"" ZQ are all of th e minimal zero sets of (U, ¢), th en C( Zl) "'" C(Zq) are all of th e minimal zero sets of (U' , ¢'). Proof. By virtue of the V's in t he definition of ¢' , it is clear that the C(Zi)'S are zero sets , and furtherm ore that they ar e indeed minimal. Now let A' be a min imal zero set of (U' , ¢'). Since it is minimal, it mu st have the form C(A) for some A E W. It is then eas y t o see that A mu st be a minimal zero set of

(U, ¢).• Lemma 2. Given any base XW', th ere exists a T E S such that T'( xw') = TT( XW')' Proof. Let x = T' (xw') and y = T T(xw' ). We will const ruct the desired T inductively. Assume that we know t he values of T(V) for all v E {(s, t) : t ::::: to} for some to ?: o. Letting a = (s , to + 1), where s E ZP, we define T(a) as follows. If vi; xa+(uj,tj -i) = 0, then Tj(a) can be arbit ra ry. Else, if the express ion equals one, then for some k , xv+(uj ,tj-k) = 1, and we let T(a) = k, Simple verification shows that ¢' (XU'(a) ) = ¢ (XUT (a) )' • Proof of Theorem 2. We will first show that (a) imp lies (b) . Let Xw' be any base such that II( xw') 1 is finite. By T heor em 1, (a) imp lies that

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Effect of Noise on Lon g-t erm Memory

t he zero state is at tract ive for t he dominant MBT, that is, II (T ' (xw,))1 is finit e. Now (T'( xw')) v 2: (TT(XW' ))v for all v E V' and T E 5 , a fact easily est ab lish ed by induction. Thus, II (T' (xw,))1 2: II (T T(xw' ))1 for all T E 5 , which implies (b ). Now we will prove t hat (a) im plies (c). Let z b e the zero traj ect ory and M~ b e the M< set of the dominan t MBT . Again by Theorem 1, (a) implies that lim

sup

oD n , -

Effect of Noise on Long-term Memory In ot her words, t he zero state is not a stable

295 7

tr ajecto ry. •

Thus, in ord er to det ermine if th e zero traj ectory of an MBT is stable under asynchronism , it is enough to apply Toom 's crite rion to the domi nant MBT.

Remark 1. Dtial argument s can be used t o st udy the stability of the one t rajectory under asy nchronism. We are particularly int erest ed in t he effects of asy nchr onism on MBTs where bot h t he zero and the one trajectories are stable. We will discuss t hese effects through a series of examples in section 4. Remark 2 . The concept of st abilit y used in Theorems 1 and 2 is rat her strong, namely, an MBT is called st able only if equa tion (1) holds wit h the supremum over all M E ME' In particul ar , t his requires stability under asymmet ric noise where O's can become 1's, bu t not vice versa. It would be int eresting to charact erize st ability under more symmetric noise mo dels (e.g., with positive probabilities for each kind of error ). See, however , Example 3 of sect ion 4, which shows that asynchronism can cause MBTs wit h multiple st able t raject ories t o become ergodic even und er symm et ric noise. 4.

Examples

We conclude our discussion with several examples and a theorem.

Example 1. Here we not e that there are MBTs such t hat for a given d, t he zero traj ect ory is attractive for all of the homogenous scenarios bu t not at t rac t ive for all scenarios. For exa mple, consider a one-dimensional (U,