coherent structures in cellular automata

Volume147, number7 PHYSICSLETFERSA 23 July 1990

Coherentstructuresin cellularautomata

A.S. Fokas,E. PapadopoulouandY. SaridakisDepartmentofMathematicsandComputerScienceandInstitutefor NonlinearStudies,Clarkson University,Potsdam,NY13676, USA

Received7 September1989; revisedmanuscriptreceived12 February1990; acceptedfor publication30April 1990Communicatedby A.P. Fordy

Thefilter cellularautomaton(CA) introducedby Park,SteiglitzandThurstonis generalizedto takevaluesin anarbitraryfinitegroup. InterestingexamplesincludeZn, thecyclic groupof ordern, andSn, thegroupof permutationsof n elements.ThesenewCA, cansupportcoherentperiodicstructures,andcanbeanalyzedby thetheoryrecentlydevelopedby theauthors.Furthermore,severalmultidimensionalversionsoftheaboveCA arealsopresented.

1. Introduction

Parket a!. [1] introduceda particularone-dimensionalcellularautomaton,calledfilter cellularautomatonandshowednumericallythat it could supportsolitons.Namely, if oneconsidersa given initial configurationasa collectionof particles,thenfor mostof the initial dataconsideredin the numericalexperimentsof refs.[1,21, the particlesemergingafter interactionwere identical to the original ones.Papatheodorouet al. [3]foundan equivalentbut moreexplicit rule than thatof ref. [1], which they calledfast rule theorem(FRT).Usingthe FRT severalinterestinganalyticalresultswerederivedin refs. [3—5].In derivingsomeof thesere-sultsone finds it usefulto considera particleas a collectionof basicstrings [4,51.

Following the abovedevelopmentswe begana systematicstudyof filter cellularautomata(CA). In order

to describethe evolutionof an arbitraryconfiguration,we havestudiedsingleparticles[6] aswell asthe in-teractionof particles[7,8]. In particularwe haveshownthat: (a) Theinteractionpropertiesof the particlesof CA arericherthan theusualsolitons: it is possiblefor two periodicparticlesnot to interactsolitonicallybutinsteadto recombineandform newperiodic.Also, evenif two particlesinteractsolitonically,theymayinteractseveraltimesbeforetheygetseparatedwith the fasterparticlemovingin frontof theslowerone. (b) Arbitraryinitial configurationswill decomposeas t-+ ~ intoa numberof periodicparticles,andinto a numberof certainnewperiodic coherentstructureswhich aregeneralizedbreathers[8]. (c) Thereexists a differenceequationformulation of the aboveCA [8].

In this Letterwe show that the filter CA introducedin ref. [1] canbe generalizedsubstantially,withoutaffectingits basicproperties.Namely,while the CA of ref. [1] is formulatedonly in termsof the elements0and1, theCA introducedhereis formulatedin termsof the elementsof anarbitrary finite group.Interestingexamplesinclude 74, (the cyclic groupof order n) andS,, (the groupof permutationsof n elements).Fur-thermore,weprovethatthetheorydevelopedin refs. [6—8],suitablygeneralized,isalsoapplicableto theaboveCA. In particularwecomputethe periodof a periodicparticle,andshow that if prematuresplittingdoesnotoccurduringthecollisionof two periodicparticles,thentheir interactionis solitonic(seeref. [8] for thenotionof prematuresplitting).

It is quite interestingthat, usingtheaboveideas,one is ableto constructcertainmultidimensionalCA sup-portingcoherentstructures.However,theseCAhavecertainlimitations,inparticularthereisminimal coupling

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between the space dimensions. More general multidimensional CA will be presented elsewhere.

2. A soliton cellularautomaton

Thefilter CA introducedin ref. [1] consistsof a collectionof zerosor ones,which evolve in timeaccordingto the following rule: let

at ~ L<oo, (2.1)

be the stateat time t, wherea~=0or 1 for all i anda~= 1, ak = 1 are the first andthe last I respectivelyina’. Thenthe next stateis calculated,sequentiallyfrom the left to right, as follows:

a~=l, seven,

=0, soddorzero, (2.2)

where

s~~ a~iJ+~ a~J_—r j=O

(oneassumesthat a = 0 for i far enoughto theleft). In the abover is a fixed integercalledthe radius andr>~2. For example,if r= 2 andat is given as below,thena’~’is asfollows:

a’: •~S0l 10101101110000001 io•••a’~’: •~o l000lOOlOlOl00000lOlOSS•. (2.3)

The FRI consistsof thefollowing four steps:(i) Placethefirst I in the box. (ii) Putboxesevery r+ 1 bits,unlessthereexist at leastr+ 1 0’s aftera given box; in thiscaseputa box at the first available 1 afterthe 0’s.(iii) Changethe boxesto their complementsandleave the restunchanged.(iv) Displaceeverythingby r tothe left. As an illustration of the FRT consideragainthe exampleof (2.3). a’ is givenby

at: S..0~J10~01E0lJl0I~30000i10E~J..S,

which implies the samea’~’as (2.3).An arbitraryconfigurationis collectionof particles anda particleis a collectionof basic strings (BS) [8].

Forexample, ‘aboveconsistsof two particlesA andB separatedby five zeros;A consistsof fourbasicstringsA’ = 110, A2=101, A3 101, A4 110, while B consistsof the basicstring 110.

Wehavefound [6—81thefollowingconvenientmathematicalcharacterizationof the FRT. Let B1denotethe

BS obtainedfrom B by leaving the part of B up to the jth 1 unchangedandby replacingthe remainingpartof B with zeros.Let “exclusiveor” denotetheoperation~ definedby b~0= b, b~1=b, whereb is 0 or 1 andbis I or 0. Thenif

at: AIA2...ALAOOBIB2...BLBOO...,

then

a’~’: A(AIA2...ALA)A~OB~(BIB2...BLH)B~O...,

where0 denotesthe null BS. Usingthe abovemathematicalexpressionof the FRT we haveshownthat: (i)If a singleparticleA is periodicthen its periodequalsthe sum of the l’s in the BSs A’, A’~A2,A2~A3AL_I SAL, AL. (A particleceasesto be periodic in two cases,namelywhensplitting occursor whena BS atthe beginningor at the end of the particledisappears[61.) (ii) Theinteractionof two particlesis so!itonic,providedthat prematuresplittingdoesnot occur [8].

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In what follows we generalizethe aboveresultsfor a filter CA with elementsbelongingto any finite group.As in refs. [6,8] we find it convenientto usea movingframeof referenceby omittingthe step (iv) of theFRT.

3. A generalizedsoliton cellular automaton

Definition 3.1. Let G= {0, g,, ... , g~}be a finite groupwith its associatedcompositiondenotedby ®. Let0 bethe identity element,i.e. 0®g=g®0=g, andlet g be the inverseof g, i.e. gøg=g®g=0.

Definition 3.2. (A generalizedsoliton CA.) Let 0 by any finite group,and let

a’: ~ L<cc, (3.1)

0 anda~�0.Then,

a~’= 0, if all factorsof Skare0,

Sk®C®~~+r, otherwise, (3.2)

whereSk= ~ ‘(aj~,~®a5) andthe carryc:=a~÷~wheneverall factorsof Sk are0; otherwisec remainsun-changed(it is assumedthat all factorsof Sk, k—p — ~, are all zero.)

The aboverule providesa generalizationof the sumrule (2.2).Remark3.1. If 0 = (®, 74,) with Zn= {0, 1, ... , n—i) and ® denotingtheadditionmodulusn thentheabove

sumrule is simplified: sinceG is Abelian

r r—IC, — t+l~C,~ ~-t‘-‘k ak...J¼Y ak+J

j=O®

and

ifallfactorsofSkareO,

= (Sk®d~+r®c), otherwise, (3.4)

where c is as in definition 3.1.Furthermore,due to the propertiesof the modulusadditionoperationthe aboverule canalsobewritten as

r r—1— t+, -t

k ak_f a,~,.1,

j=1 j~O

a~7’= 0, if all factorsof Sk are0,

(Sk+d’k+r+C)mOdfl, otherwise, (3.6)

wherec is as in definition 3.1.Definition 3.3. A basicstring (BS) B is a collectionof r+ 1 elementsof 0. B1 denotesthe BS obtainedvia

the followingoperation:replacethe elementsof B up to the ith nonidentity(nonzero)elementwith their in-verses,andreplacetheremainingpart of B with 0’s. For example,let B=g,0g2g3(i.e. r= 3); then B, =g,000,

B2=g10g20,B3=,~,0g~3.Definition3.4. (An alternativedefinitionofa generalizedCA.) Let 0 beanyfinite group.Thena generalized

soliton CA is definedby the following evolution. If

a’: AIA2...ALAOO...OBIB2...BLBOO...0 FIF2...FLFOO..., (3.7)

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where the first elements of A’, B’, ... , F’, ... are nonzero, and ALA, BLB, ..., FLF are differentfrom 0 andarerr~

different from a trivial basic string T= gO—0, then

a’~’: ~ (3.8)

In the above0 denotesthenull BS (i.e. r+ 1 consecutive 0’s). The above definition provides a generalizationof the FRT. It canbe shownthat the CA definedby definitions3.2 and3.4 areequivalent,providedthat theconfigurationobtainedby definition3.4 is shifted by r to the right at eachtime step.

Example3.1. Let G=14f0, 1, 2, 3), i.e. 1=3, 2=2, ~=l. Consider the evolutionof the singleparticle(r=3) 0102311010,consistingof the two BSs A’ = 1023, A2= 1101. Using a moving framewe find

10231101002301013000000301213020000001223021

002233210300000233010320000033030322

030313221000000310221100000010231101

Example3.2. As an illustration of the sumrule on a non-Abeliangroup we considerS3, the groupof per-

mutationof threeelements.We choosethe following representationof S3:

(1 2 3\ (1 2 3\ (1 2 3\2 3)’ 0~~2 ~ i)’ 02(\3 1 2)’

/1 2 3’\ (1 2 3\ (1 2 3a3=~2 1 3)’ ~ 3 2)’ 05=~3 2 1

wherea,means1 -.~2,2-+3, 3—+ 1 and0 denotestheidentityelement.Using (aJ•®ak)(y) = cr~(ak(Y)), for example

1! 2 3\(l 2 3\ /1 2 3\3 1 3)~l 3

we find

0 a, 02 a3 04 05

0 0 01 02 03 ~

0~ 0~ 02 0 04 05 03

02 ~2 0 a, a~ 03 04

03 a~ 0~ 04 0 a2 a,04 04 03 0~ a~ 0 a2a5 0~ 04 03 02 a, 0

Thus

0~=02, ~ =a3, 04 04, 05=05

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Considernow the following statea’ andr=2,

a’=...000o,o2a3000...,a~,=o1,a’1=o2, a’2=a3,

S_2=0®O®0®O=Owithallelements0 a~=0andc:=aG=o,,

S_,=0®0®0®d1=d, a~11=(S_,®c®aç)=(a,®a,®a

2)=a2,

So=0®ã,®a2®cJ2=d1 ~. a~=(S0®c®d~)=(ã1®o,®ã3)=o3,

S1=o2®ã2®a3®d3=0 =‘ a~~’=(0®o,O)=d1=o2,

S2=o3®ã3®o2®O=a2 ~. a~~’=(o2®o1®O)=O=0,

S3=a2®0®0®0=o2 a’j’’=(o2®o1®0)=0=0,

S4=0ØO®0®O=Owithallelements0 =. a’~’ =Oandc=0.

Hence

a’: 000 0,0203000a’~: 00203020 00

or consideringthe movingframeof reference

a’: 0000102030 00a’~’: 0000 02030200

If we continuethe evolutionof the aboveparticlewe find that it is periodicwith p=6, i.e.

a’: 0000,a2a3000000a’~’: 0000 0203020 0 0 0 0a’~

2: 0000003020,0000a’~3: 000000020,03000a’~4: 0000 0 0 0 0~03010 0a’~5: 0000 0 0 0 0 0302020a’~6: 0000 0 0 0 0 0 010203

Theorem 3.1. (Evolutionof singleparticles.)Considerthe particle

t=0: OAIA2...ALO, (39)

consistingof L basicstringsA’, ... , AL with elementsin0. Let l~,I, ‘L bethenumberof non-zeroelementsin the following BSs,

AI,Al®A2,A2®A3,...,AL_I®AL,A~~, (3.10)

whereif A containselementsa, A containselementsa. If theparticle (3.10) is periodic(i.e. if neithersplittingnor loss of a BS occurs),then its evolutionis givenby

1k]

t=10+l1 + ... +lk_I +i,

0<~~<’k• 0—0 (Ak®Ak ).®A”Ø(A”~’ ...AL®AI ... Ak). (3.11)In particularat

~k+I]

t=lo+lI+...+lk: ~ (3.12)

andat

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[L+ Iit=p~lo+l,+...lL: 0_OAIA2...AL. (3.13)

[kiIn the above0—0 denotes k zero BSs.

Proof

t=i, 0<i~<l0: A,I®(AIA

2...ALO).

Since A/0 =A’ andA’®A’ = 0, it follows that

tl~: OAI®(A2...AL®Al),

t=10+i: 0(Al®A

2),®Al®(A2...ALOA1).

Since (A’®A2)l,=A’®A2=A2®A’, it follows that

t=10+l,: 00A

2®(A3A4...ALOAIA2), etc.

Remark3.2. In deriving the aboveresult we haveusedthat ® is associative,andthatA®A= 0, thusweneedan arbitrary (andnot necessarilyAbelian) group structure.

Consideragainexample3.1: A’=l023, A’=(302l)®(llOl)=0122, A2=3303,thusp=9.Theorem3.2. (Interactionof periodicparticles.)Considertwo periodicparticlesA andB which begin in-

teractingat t = 1. If prematuresplitting doesnot occur (i.e. if splittingdoesnot occur in t=p, wherep is theperiodof theparticleon theleft), thenthe interactionof A, B is solitonic. (Seerefs. [6—8]for the definitionsof splittingandinteractionof particles.)

Proof Supposethat A andB are separatedby r+ 1 + m zerosat t = 0, andthat they begintheir collision att= 1:

t=0 A OOBO

t=l A~®A A~ OB~®B~

tPA 0 0 A~®(oB~®B~) A

tPA+l 0 0 OB~®B~ A~®A A~

In the above0 denotesm zerosandA denotesthe part of the BS A up to r+ 1— m position.Remark3.3. In a similar way it is straightforwardto generalizeall the resultsobtainedin refs. [6—8].Example3.3. As an exampleof solitonic interactionof two particlesconsideragainthe groupdefinedin

example3.2 andthe particlesA=cr,0o4 andB=a,o2a3with r=2. Their interactionis shown in fig. 1.

4. Two-dimensionalsoliton cellular automata

Usingtheaboveideasit is possibletoconstructcertainmultidimensionalCA thatcansupportcoherentstruc-tures.For simplicity we considertwo spacedimensions,the extensionto arbitrarynumberof dimensionsisstraightforward.In what follows we presenttwo classesof two-dimensional(2-D) CA.

4.1. Matrix valued2-D CA

To constructeda 2-D CA, weconsiderthenewcellularautomatonintroducedin definitions3.2 and3.4 andwe allow a givensite to takevaluesin anarbitrarymatrix.The resulting2-D CA hasthe limitation that it can

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t=0 a,0a40 000 a,a2a3

1=1 0 0a4c200 0 0 a2a3a2

1=2 0 00 a20a40 0 0 a3a2a,t=3 00000o,u,000a2a,a3t=4 0 00 a,0 t740 a2a3a30 a4t=5 0 00 0 0 a4a2a2a3a50a4a21=6 0 a2 a2 ~Tj a50 0 a20 a4t=7 Oa2a,a30000a4a1t=8 Oa,a3a,0000a,0a4

1=9 Oa3a,a200000a4a2t=10 Oa,a2a300000a20a4t=ll 0 a2a3a20 0 0 0 0 Oa4a,t=12 Ocr3a2a,000000ci,0a41=13 Oa2a,a30000000a4a2t=14 0 a,a3a20 000000 a20a4t=15 a3a,a20 00 0 00 0 Oa4a,t=16 a,a2a300000000 a,0a4

Fig. I.

move only in a specificdirection (sayin the x-direction) andthereis minimal couplingbetweenthe x- andy-directions.The theorydevelopedin section3 is sufficientto coverthis 2-D CA.

Example4.1. (A periodicparticle.) Let r= 2, 0 = Mat3 2(Z2),theadditivegroupof 3 x 2 matriceswith ele-mentsin 74. Let the groupoperationbe the matrix additionmodulo2. The configuration

100010110110000001 111101010011001011

canbethoughtof asa singleparticle,consistingof the two BSs A = a,a2a3,B = b,b2b3,where

a,=10, a2=0, a3=10, b,=1l, b2=0l, b3=a3.00 01 11 1101 11 00 10

Then

A,=a,00, A2=A, AØB=Ol 0100, (A®B),=OlOO, (A®B)2=A®B.111100 11011000 01

Hence,the evolutionof the aboveparticleis givenby

t=0 100010110110000001111101010011001011

t=l 0000lOOlOllOl00000000111110100000000110110110100

t=2 00010100100010001111000000010001100001001 1

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1=3 000100110010010011001100011100100000001101

1=4 000011011001010000001111011111000000001011011000

t=5 00000110100100110000110100110011

0000101101100000

1=6 001010000011010001000000111100110100000010

1=7 001000101101100000000111110100010011001011

In particular,theaboveparticleis periodicwith periodequalto thesumof thenonzero“elements”of A, A®B,B, i.e. p=2+2+3=7.

Example4.2. (Interactionof two periodicparticles.)An exampleof a soliton collision is givenby (comparethe statesat 1=0 and t= 10, r=2)

t=0 100011000000001101001 l000l0000000000lOl 101001000000000101101

1=1 000011100000000001001100000lll00000000lOl 1000000lOOl00000000l 10110

t=2 OOl000ll00000000llOlOOll000l000000llOOlO000lOOl00000000llOll

1=3 0000lll0000000llOl000000000lll00000000lOllOO0000lOOl000000lOllOlOO

t=4 OOl000llOOlll00000llOOll000l0000llll000l000lOOl000lOOlOlOOlO

t=5 0000lllOlllOl000lll00000000lllOOll00000lllOO0000lOOllOOl0000lOOlOO

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t=6 OOlOllOll00000l000ll001100100000001100010001 1011000000010010

1=7 OOllOl0000000000lll0000000101 l000000000lllOOOOlOllOl00000000lOOlOO

t=8 0001001 100000000100011001011000000000011000100110l1000000000010010

1=9 0000llOl0000000000lll000001 lOOl000000000000l 11000001101 10000000000100100

1=10 OOllOl000000000000l000ll00001011000000000011000100101 lOl00000000000lOOlO

4.2. A direct product ofI-D CA

Givena group0, andtwo radii r~and ry, we constructa 2-D CAas follows: First we use 0, r~(see definitions3.2, 3.4) to evolve a given configuration in the x-direction andthen we use0, r,, to evolvethe resulting con-figurationin the y-direction. It turns out [9] that this 2-D CA not only cansupportcoherentstructures,butit alsoexhibits interestingnewfeaturesnot found in the 1 -D CA thatgeneratedit. However,the theoreticaltools introducedby the authors,suitablyextended,arequite effectivefor the analysisof the multidimensionalCA. For example,thebasicstringB~of definition 3.3 is now replacedby a suitablematrix B = B~®B1, whereB-’, B~’arematricesassociatedwith the evolutionin the x- andy-directions.From this it follows that this CAis merely a “direct problem” of two I -D CA. The analysisof theseCA is presentedin ref. [9].

In examples4.3.—4.5we takeG=74, r~=3,r~=2.Example4.3. (A periodicparticle.) W considerthe evolutionof

B°=111000111

Theperiodof 111 is P~=3 while theperiodof 101 is P,= 2. Thetheory developedin ref. [9] predictsthatB°is a periodicparticlewith periodP~P,,= 6. This canbe verified by following the evolutionof B°:

B’=llOl, B2=lOll, B3=lll, B4=llOl, B5=lOll, B6=B°.1101 0000 111 0000 lOll

lOll 1101

It is possiblefor a particleto evolveto a newparticleF, whereF is a periodicparticle,without splittingorloosinga basicstring.

Example4.4. TheparticleB°= A evolvesto B4= F, whereF is a periodicparticle,

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A=lOll, F=l1.1111 111111 11

Example4.5. (Solitonic interactionof two periodicparticles.)Fig. 2 showsthe solitonic interactionof theperiodicparticles~fl and ~ At 1=2 a solitonicinteractionbeginswhich lastsfor six stepsaspredictedbythe theory in ref. [9] (the periodof the slowerparticleis P~P~=6).At t=8 the particleshaveshifted II bitsin the x-directionand 13 bits in the y-direction (againwe use a movingframeof reference,i.e. eachconfig-

1=0: 10110000011110000000000 1=6: 00000000000000000000000lOll00000lIll0000000000 0000000000000000000000000000000000000000000000 00000000000000000000000

000000000000000000000001=1: 00000000000000000000000 00000000000000000000000

00111000001111000000000 0000000000000000000000000000000000000000000000 00000000000000000000000OO1II00000I11I000000000 00000000000000000000000

000000000000000000000001=2: 00000000000000000000000 00000000101101001011000

00000000000000000000000 00000000101101001011000

0000000000000000000000000011010000111100000000 1=7: 00000000000000000000000

00011010000111100000000 0000000000000000000000000000000000000000000000

1=3: 00000000000000000000000 0000000000000000000000000000000000000000000000 00000000000000000000000

00000000000000000000000 0000000000000000000000000000000000000000000000 0000000000000000000000000001011000011110000000 0000000000000000000000000000000000000000000000 0000000000000000000000000001011000011110000000 00000000000000000000000

000000000011110000111001=4: 00000000000000000000000 00000000000000000000000

00000000000000000000000 000000000011110000111000000000000000000000000000000000000000000000000 1=8: 0000000000000000000000000000000000000000000000 00000000000000000000000

00000000000000000000000 0000000000000000000000000000011100001111000000 00000000000000000000000000000lll0000llll000000 00000000000000000000000

000000000000000000000001=5: 00000000000000000000000 00000000000000000000000

00000000000000000000000 0000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000 0000000000011110000110100000001101001011010000 000000000001111000011010000000000000000000000000000001 10100101 1010000

Fig. 2.

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uration is displacedby r~bits to the right and by r~bits downwardsat eachtime step).

Acknowledgement

It is our pleasureto thank Vas Papageorgioufor many importantdiscussions,andin particularfor his sug-gestionregardingnon-Abeliangroups.Thiswork waspartiallysupportedby theOfficeof NavalResearchunderGrantNumberN0001 4-88K-0447,NationalScienceFoundationunderGrantNumberDMS-88O347I, andAirForceOffice of Scientific ResearchunderGrantNumber87-0310and 88-0073.

References

[1] J.K.Park,K. SteiglitzandW.P.Thurston,PhysicaD 19 (1976)423.(2] K. Steiglitz,I. KamalandA. Watson,IEEETrans.Comput.37 (1988) 138.[3] T.S.Papatheodorou,MJ. Ablowitz andY.G.Saridakis,Stud.AppI. Math.79 (1988) 173.[4] T.S. PapatheodorouandA.S. Fokas,Stud.AppI. Math.80 (1989) 165.[5] C.H. Goldberg,ComplexSyst.2 (1988)91.[6] A.S. Fokas,E.P.PapadopoulouandY.G. Saridakis,Particlesin solitoncellularautomata,preprint,ClarksonUniversity, INS 107

(1989),to bepublished.[7] A.S. Fokas,E.P.Papadopoulou,Y.G. SaridakisandM.J. Ablowitz,Stud.Appi. Math.81(1989) 153.[8] A.S. Fokas,E.P.PapadopoulouandY.G. Saridakis,PhysicaD41(1990)297.[9] AS. Fokas,E.P.PapadopoulouandY.G.Saridakis,Coherentstructuresin multidimensionalcellularautomata,preprint,Clarkson

University,INS 128 (1989).

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