CELLULAR AUTOMATA Math118, O. Knill

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Dec 1, 2013 (3 years and 17 days ago)

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CELLULARAUTOMATAMath118,O.Knill
ABSTRACT.AshiftinvariantcontinuousmaponthesequencespaceAZ
overanitealphabetAiscalled
acellularautomatonorshortaCA.Thesedynamicalsystemscanbeconsideredasdiscretizedcousinsof
dierentialequations,forwhichtime,space,aswellasthecongurationspacearediscretized.
THENAMECELLULARAUTOMATON.Interactionsbetweendierentsci-
enticeldsisalwaysproductive.Historically,itseemsthatcellularautomata
wereintroducedinthelate40ieswhilesomeappliedMathematicianswere
dealingwithproblemsfrombiology.Theetymologyofthename"CA"could
conrma"bonmot"ofStanUlam:
Asknotwhatmathematicscandoforbiology.
AskwhatbiologycandoforMathematics.
Source:citedfromDavidCampbell,whoreceivedhisB.A.inchemistryandphysics
fromHarvardin1966andworkedinnonlinearscience.UlamhimselfwasatHar-
vardfrom1936-1939,eatingatAdamshousewhere"theluncheswereparticularly
agreeable"andwasalsoteachingtheMath1Ahere(Source:Ulam:Adventuresofa
mathematician).
Anyway,itwouldnotsurpriseif"cellularautomaton"hadbeenderivedfrom
"cellularspaces"becauseofmathematicalresearchonbiologicalproblems.
SEQUENCESPACES.LetAbeanitesetcalledthealphabetandletAZ
denotethesetofallsequencesand
(x)n
=xn+1
theshiftonX.Adistancebetweentwosequencesisgivenbyd(x;y)=1=(n+1),wherenisthe
largestnumbersuchthatxi
=yi
forjijn.Example:LetA=f1;2;3;4g.For
...x3
x2
x1
x0
x1
x2
x3
...
...1143211...
...y3
y2
y1
y0
y1
y2
y3
...
...1233411...
wehaved(x;y)=1=3,becausexi
=yi
ifjij3butx2
6=y2.
LEMMA:Xisacompactmetricspace(X;d).
PROOF.Tohaveametricspace,showd(x;x)=0;d(x;z)d(x;y)+d(y;z);d(x;y)=d(y;x).Tohavecom-
pactness,everysequencex(k)inXmusthaveanaccumulationpoint.Thatis,theremustexistasubsequence
x(kl)inXwhichconvergesfork!1.Seehomework.
1D-CELLULARAUTOMATA.AcontinuousmapTonXwhichcommutes
withiscalledacellularautomaton.AtheoremofCurtis,Hedlundand
Lyndon,whichwewillprovelaterimpliesthatthereisafunctionfrom
A2R+1
!AsuchthatT(x)i
=(xiR;xik+1;:::;xi+R).TheintegerRis
calledtheradiusoftheCA.ItisassumedthatRisthesmallestnumberfor
whichtheCAstillcanbedenedlikethat.Onecanvisualizethedynamics
ofonedimensionalCAbycodingeachletterinasequencewithacolor.The
rstrowistheinitialcondition.Applyingthemapgivesthesecondrow,etc.
Drawingafewiteratesproducesaphasespacediagram.Theexampleshows
theautomatonoverthealphabetf0;1g,wherexn
=xn+xn1
mod2andwhere
0isblack.Ifinitiallyxn(0)=0forn6=0andx0
(0)=1,wehaveanexplicit
solutionformulawithbinomialcoecientsxn(t)=

n+t
n

mod(2).
CANTORSDIAGONALARGUMENT.
THEOREM(Cantor)ThesetX=A
Z
isuncountable.
PROOF.IfXwerecountable,onecouldenumerateallsequencesx(k)using
integerindicesk.Denethe"Diagonal"sequenceyn
=(1+xn(jnj))
(herea+1is
thenextinthealphabetA,orthefirstelementinA,ifawasthelast).Thesequenceyisdierent
fromanyofthesequencesx(k)becauseyandx(k)dieratthek'thentry.The
assumptionabouttheenumerabilitywasnotpossible.
WOLFRAMSNUMBERINGOF1DCA.Anyone-dimensionalcellular
automatawithradius1andalphabetf0;1gcanbelabeledbyarulenumber.
Becausethereare23
=8possiblemaps,wehave28
=256possiblerules.
TheWolframnumberisw=
P8
k=1
f(k)2k,wherey0
=f(k)isthenew
colorfork=4x1
+2x0
+x1.
Forexample,let(a;b;c)=a,thenthenewmiddlecellis1fortheneighbor-
hoods111;110;101;100whichcodetheintegers7;6;5;4.So,f(7)=f(6)=
f(5)=f(4)=1,andf(k)=0otherwise.Theruleoftheautomatonis
w=27
+26
+25
+24
=240.Indeed,rule240istheshiftautomaton.Letus
lookatanotherexample.
EXAMPLES.ThebinominalCAdiscussedabovehasrule90.Oneofthemost
studiedCAisrule18.Since18=24
+21
,whichis10010tothebase2,we
obtainthefollowingfunction:
neighborhood(dec)
neighborhood(bin)
newmiddlecell
factor
7
111
0
128
6
110
0
64
5
101
0
32
4
100
1
16
3
011
0
8
2
010
0
4
1
001
1
2
0
000
0
1
SPEEDOFACA.EveryCAhasamaximalspeedcwithwhichsignalscan
propagate.Thismeansifwetakeaninitialconditionsxwhichisconstant
outsideanintervalI,thenthenT
k(x)willstillbeconstantoutsideaninterval
Ik
ofsizejIkjjIjj2c
LEMMA.ThespeedofaCAisboundedabovebytheradiusR.
PROOF.EachtimestepcanchangeonlycellsmaximallyRunitstotheleftor
totheright.
Example:The"Takahashi-SusamaSolitonautomaton"isdenedon
pointsx2f0;1gZ
forwhichonlynitelymanycellsare1.TheruleforT
istostartfromtheleftandmoveeach1tothenext0position.Sinceapack
ofnadjecent10smoveswithspeedn,themapTisnotacellularautomaton.
EXAMPLES.
a)ThecellularautomatonT=
c
shiftingc2Nentriestotherighthasthe
speedc.Sincecisalsotheradius,thisshowsthatthespeedcannotbefaster
thentheradiusR.Thespeedratioc=Rsatisesc=R1.
b)TheCAT(x)n
=(:::;a;a;a;a;:::)isobtainedbyafunctionwhichis
constant.Everyorbitofthisautomatonisattractedtothexedpoint.The
speediszero.Thepicturetotherightshowsrule-100cellularautomaton.
POSSIBLESPEEDS.Notethatwecanenumeratethesetofcellularautomata:
itisacountableset.Becausethesetofrealnumbersintheinterval[0;1]is
uncountable,wecannotobtainallthespeeds.
PROPOSITION.FixA.Forevery0<a<b<1,thereisaCAwithradiusR
overthealphabetAforwhichthespeedcsatisesac=Rb.
Youexplorethisfactabitinahomework.Theideaisrsttousealarger
alphabetinordertoslowdownthemotionusinginternal"colorswapping".For
dierentalphabetsA;B,aA-automatoncanbesimulatedbyaBautomaton,
possiblychangingtheradius.