Stochastic Cellular Automata

37 Pages·2017·0.86 MB·English

Stochastic Cellular Automata: Correlations, Decidability
and Simulations
Pablo Arrighi, Nicolas Schabanel, Guillaume Theyssier
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Pablo Arrighi, Nicolas Schabanel, Guillaume Theyssier. Stochastic Cellular Automata: Correlations,
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FundamentaInformaticaeXX(2013)1–36 1
DOI10.3233/FI-2012-0000
IOSPress
Stochastic Cellular Automata:
Correlations, Decidability and Simulations
PabloArrighi∗
Universite´ deGrenoble(LIG,UMR5217),France
Universite´ deLyon(LIP,UMR5668),France
NicolasSchabanel†
CNRS,Universite´ ParisDiderot(LIAFA,UMR7089),France
Universite´ deLyon(IXXI),France
GuillaumeTheyssier
CNRS,Universite´ deSavoie(LAMA,UMR5127),France
Abstract. This paper introduces a simple formalism for dealing with deterministic, non-
deterministicandstochasticcellularautomatainanunifiedandcomposablemanner.Thisformalism
allows for local probabilistic correlations, a feature which is not present in usual definitions. We
showthatthisfeatureallowsforstrictlymorebehaviors(forinstance,numberconservingstochastic
cellularautomatarequiretheselocalprobabilisticcorrelations). Wealsoshowthatseveralproblems
whicharedeceptivelysimpleintheusualdefinitions,becomeundecidablewhenweallowforlocal
probabilisticcorrelations,evenindimensionone. Armedwiththisformalism,weextendthenotion
ofintrinsicsimulationbetweendeterministiccellularautomata,tothenon-deterministicandstochas-
ticsettings. Althoughtheintrinsicsimulationrelationisshowntobecomeundecidableindimension
two and higher, we provide explicit tools to prove or disprove the existence of such a simulation
between any two given stochastic cellular automata. Those tools rely upon a characterization of
equalityofstochasticglobalmaps,showntobeequivalenttotheexistenceofastochasticcoupling
between therandom sources. We applythem toprove thatthere is nouniversal stochasticcellular
automaton. Yet we provide stochastic cellular automata achieving optimal partial universality, as
wellasauniversalnon-deterministiccellularautomaton.
∗ThisworkhasbeenpartiallyfundedbytheANR-10-JCJC-0208CausaQgrant.
†ThisworkhasbeenpartiallyfundedbytheANR-2010-BLAN-0204MagnumandANR-12-BS02-005RDAMgrants.
1. Introduction
A motivation: stochastic simulation. Cellular Automata (CA) are a key tool in simulating natural
phenomena. This is because they constitute a privileged mathematical framework in which to cast the
simulated phenomena, and they describe a massively parallel architecture in which to implement the
simulator. Oftenhowever,thesystemthatneedstobesimulatedisanoisysystem. Moreembarrassingly
even, it may happen that the system that is used as a simulator is again a noisy system. The latter is
uncommon if one thinks of a classical computer as the simulator, but quite common for instance if one
thinksofusingasmallscalemodelofasystemasasimulatorforthatsystem.
Fortunately, when both the simulated system and the simulating system are noisy, it may happen that
botheffectscancelout,i.e. thatthenoiseofthesimulatorismadetocoincidewiththatofthesimulated.
Insuchasituationamodelofnoiseisusedtosimulateanother,andthesimulationmayeventurnoutto
be...exact. This paper begins to give a formal answer to the question: When can it be said that a noisy
systemisabletoexactlysimulateanother?
This precise question has become crucial in the field of quantum simulation. Indeed, there are many
quantum phenomena which we need to simulate, and these in general are quite noisy. Moreover, only
quantumcomputersareabletosimulatethemefficiently,butinthecurrentstateofexperimentalphysics
these are also quite noisy. Could it be that noisy quantum computers may serve to simulate a noisy
quantum systems? The same remark applies to Natural Computing in general. Still, the question is
challengingenoughintheclassicalsetting.
A challenge: the need for local probabilistic correlations. The first problem that one comes across
isthatstochasticCAhaveonlyreceivedlittleattentionfromthetheoreticalcommunity. Whentheyhave
beenconsidered,theywereusuallydefinedastheapplicationofaprobabilisticfunctionuniformlyacross
space[33,14,8,3,29,9]. InthispaperwewillrefertothismodelaslocalCorrelation-FreeCA(CFCA).
Indeed, this particular class of stochastic CA has the unique property that, starting from a determined
configuration, the cell’s distributions remain uncorrelated after one step. This was pointed out in [1],
whichprovidesanexample(cf. PARITYstochasticCAwhichwewilluselater)whichcannotberealized
asCFCA,inspiteofthefactthattheyrequireonlylocalprobabilisticcorrelationsandhencefitnaturally
intheCAframework. Moreover,[1]pointsoutthatthecompositionoftwoCFCAisnotalwaysaCFCA.
Thelackofcomposablityofamodelisanobstaclefordefiningintrinsicsimulation, becausethenotion
mustbedefineduptogroupinginspaceandintime. In[1]acomposablemodelissuggested,butitlacks
formalization.
InthispaperweproposeasimpleformalismtodealwithgeneralstochasticCA.Theformalismrelieson
consideringaCAF(c,s)fed,besidesthecurrentconfigurationc,withanewfreshindependentuniform
random configuration s at every time step. This allows any kind of local probabilistic correlations and
includes in particular all the examples of [1]. As it turns out, the definition also captures deterministic
and non-deterministic CA (non-deterministic CA are obtained by ignoring the probability distribution
overtherandomconfiguration).
Results on stochastic simulation. This formalism allows us to extend the notions of simulation de-
veloped for the deterministic setting [5, 6], to the non-deterministic and stochastic settings. The choice
of making explicit the random source in the formalism has turned out to be crucial to tackle the second
problem,asitallowsapreciseanalysisoftheinfluenceofrandomness,intermsofsimulationpower.
The second problem that one comes across is that the question of whether two such stochastic CA are
equal in terms of probability distributions is highly non-trivial. In particular, we show that testing if
two stochastic CA define the same random map becomes undecidable in dimension 2 and higher (The-
orem 4.1). Still, we provide an explicit tool (the coupling of the random sources of two stochastic CA)
thatallowstoprove(ordisprove)theequalityoftheirprobabilitydistributions. Moreprecisely,weshow
thattheexistenceofsuchacouplingisstrictlyequivalenttotheequalityofthedistributionoftherandom
mapsoftwostochasticCA(Theorem4.5).
Thechoiceofmakingexplicittherandomsourceallowsustoshowsomeno-goresults. Anystochastic
CAmayonlysimulatestochasticCAwithacompatiblerandomsource(wherecompatibilityisexpressed
asasimplearithmeticequation,Theorem8.2). ItfollowsthatthereisnouniversalstochasticCA(Corol-
lary8.3). Still,weshowthatthereisauniversalCAforthenon-deterministicdynamics(Theorem8.4),
andweareabletoprovideauniversalstochasticCAforeveryclassofcompatiblerandomsource(The-
orem8.5).
Resultsonquestionsofcomputabilityversusallowingforlocalprobabilisticcorrelations. Thefact
thattestingiftwostochasticCAdefinethesamerandommapisundecidableindimension2andhigher,
whichisnot thecaseintheparticularcase ofCFCA,suggestedthatmany problemsappeardeceptively
simple in the CFCA formalism. And indeed, their difficulty comes back as soon as one iterates the
CFCA: for instance, we show that in dimension 2 and higher, it is undecidable whether the squares of
two CFCA define the same random map (Corollary 5.8). Worse even, it is undecidable whether the
squareofaCFCAisnoisy(Theorem5.7).
In dimension one, these problems, and many others, are shown to be decidable for general stochastic
CA(Corollary6.2and6.8). Thus,theycannotservetopointoutaseparationwithCFCA.Yet,weshow
that the Pattern-Probability-Threshold problem (i.e. the question whether some pattern can appear with
aprobabilityhigherthatsomethreshold)isagainundecidableforstochasticCA(Theorem4.9),whereas
it is decidable for CFCA (Theorem 6.5). We also show that some behaviors are out-of-reach of CFCA.
Indeed,CFCAcannotbenumber-conservingunlesstheyaredeterministic(Lemma5.3). Moreover,even
the iterates of a CFCA cannot be surjective number-conserving unless they are deterministic (Theorem
5.4). Iteratesoftwo-stateCFCAcannotreproducebehaviorsalikethePARITYexampleeither(Theorem
5.5).
Plan. Section 2 recalls the vital minimum about probability theory. Section 3 states our formalism.
Section4explainsthatsomeglobalpropertiesoftheSCA(suchasequalityofstochasticglobalfunctions,
theprobabilityofappearanceofcertainpatterns,injectivityandsurjectivity)cannotbedecidedfromthe
localrules,butmaybeapproachedthroughsomeotherprooftechniques. Theseproblemstendtosimplify
for the particular case of local Correlation-Free SCA, which corresponds to the more usual definition,
and for the one-dimensional case, as explained in Sections 5 and 6. This shows that local Correlation-
FreeSCAarefundamentallysimpler: somebehaviorscannotbereached. Section7extendsthenotionof
intrinsicsimulationtothenon-deterministicandstochasticsettings. Section8providestheno-goresults
in the stochastic setting, the universality constructions. Section 10 concludes this article with a list of
openquestions.
2. Standard Definitions
Evenifthisarticlefocusesmainlyonone-dimensionalCAforthesakeofsimplicity,itextendsnaturally
tohigherdimensions. Eachtimearesultissensitivetodimension,itwillbeexplicitedinthestatement.
For any finite set A we consider the symbolic space AZ. For any c ∈ AZ and z ∈ Z we denote by
Z
c the value of c at point z. A is endowed with the Cantor topology (infinite product of the discrete
z
topologyoneachcopyofA)whichiscompactandmetric(see[19]fordetails). Abasisofthistopology
is given by cylinders which are actually clopen sets: given some finite word u and some position z, the
cylinder[u] istheset[u] = {c ∈ AZ : ∀x,0 (cid:54) x < |u|−1,c = u }.
z z z+x x
Z Z
We denote by M(A ) the set of Borel probability measures on A . By Carathe´odory extension
theorem,Borelprobabilitymeasuresarecharacterizedbytheirvalueoncylinders. Concretely,ameasure
Z
isgivenbyafunctionµfromcylinderstotherealinterval[0,1]suchthatµ(A ) = 1and
(cid:88) (cid:88)
∀u ∈ A∗,∀z ∈ Z, µ([u] ) = µ([ua] ) = µ([au] )
z z z−1
a∈A a∈A
Wedenotebyν theuniformmeasureoverAZ (s.t. ν ([u] ) = 1 ). Weshalldenoteitasν when
A A z |A||u|
theunderlyingalphabetAisclearfromthecontext.
Z
We endow the set M(A ) with the compact topology given by the following distance:
D(µ1,µ2) = (cid:80)n(cid:62)02−n·maxu∈A2n+1(cid:12)(cid:12)µ1([u]−n)−µ2([u]−n)(cid:12)(cid:12). See[27]forareviewofworksoncel-
lularautomatafromthemeasure-theoreticpointofview.
3. Stochastic Cellular Automata
Non-deterministic and stochastic cellular automata are captured by the same syntactical object given
in the following definition. They differ only by the way we look at the associated global behavior.
Moreover, deterministic CA are a particular case of stochastic CA and can also be defined in the same
formalism.
3.1. TheSyntacticalObject
Definition3.1. AstochasticcellularautomatonA = (Q,R,V,V(cid:48),f)consistsin:
• afinitesetofstatesQ
• afinitesetRcalledtherandomsymbols
• twofinitesubsetsofZ: V = {v ,...,v }andV(cid:48) = {v(cid:48),...,v(cid:48) },calledtheneighborhoods;ρand
1 ρ 1 ρ(cid:48)
ρ(cid:48) arethesizesoftheneighborhoodsandk = max |v|istheradiusoftheneighborhoods.
v∈V∪V(cid:48)
• alocaltransitionfunctionf : Qρ×Rρ(cid:48) → Q
Z
A function c ∈ Q is called a configuration; c is called the state of the cell j in configuration c. A
j
Z
functions ∈ R iscalledanR-configuration.
In the particular case where V(cid:48) = {0} (i.e., where each cell uses its own random symbol only), we
saythatAisaCorrelation-FreeCellularAutomaton(CFCAforshort).
Definition3.2.(ExplicitGlobalFunction)
Z Z Z
To this local description, we associate the explicit global function F : Q ×R → Q defined for any
(cid:0) (cid:1)
configurationcandR-configurationsby: F(c,s) = f (c ,...,c ),(s ,...,s ) .Given
z z+v1 z+vρ z+v1(cid:48) z+vρ(cid:48)(cid:48)
asequence(cid:0)st(cid:1) ofR-configurationsandaninitialconfigurationc,wedefinetheassociatedspace-time
t
diagramasthebi-infinitematrix(cid:0)ct(cid:1) wherect ∈ QZ isdefinedbyc0 = candct+1 = F(ct,st).
z t(cid:62)0,z∈Z
We also define for any t (cid:62) 1 the tth iterate of the explicit global function Ft : QZ ×(cid:0)RZ(cid:1)t → QZ by
F0(c) = cforallconfigurationcand
Ft+1(c,s1,...,st+1) = F(cid:0)Ft(c,s1,...,st),st+1(cid:1)
sothatct = Ft(c,s1,...,st).
In this paper, we adopt the convention that local functions are denoted by a lowercase letter (typ-
ically f) and explicit global functions by the corresponding capital letter (typically F). Moreover, we
will often define CA through their explicit global function since details about neighborhoods often do
notmatterinthispaper.
The explicit global function captures all possible actions of the automaton on configurations. This
functionallowstoderivethreekindsofdynamics: deterministic,non-deterministicandstochastic.
3.2. DeterministicandNon-DeterministicDynamics
Deterministic. The deterministic global function D : QZ → QZ of A = (Q,R,V,V(cid:48),f) is defined
F
Z
by D (c) = F(c,0 ) where 0 is a distinguished element of R. A is said to be deterministic if its local
F
transitionfunctionf doesnotdependonitssecondargument(therandomsymbols).
Z Z
Non-Deterministic. Thenon-deterministicglobalfunctionN : Q → P(Q )ofAisdefinedforany
F
Z Z
configurationc ∈ Q byN (c) = {F(c,s) : s ∈ R }.
F
Dynamics. The deterministic dynamics of A is given by the sequence of iterates (DFt )t(cid:62)0. Similarly
the non-deterministic dynamics of A is given by the iterates Nt : QZ → P(QZ) defined by N0(c) =
F F
{c}andNt+1(c) = (cid:83) N (c(cid:48)).
F c(cid:48)∈Nt(c) F
F
3.3. StochasticDynamics
The stochastic point of view consists in taking the R-component as a source of randomness. More
precisely,theexplicitglobalfunctionF isfedateachtimestepwitharandomuniformandindependent
R-configuration. Thisdefinesastochasticprocessforwhichwearetheninterestedinthedistributionof
states across space and time. By Carathe´odory extension theorem, this distribution is fully determined
bytheprobabilitiesoftheeventsoftheform“startingfromc,theworduoccursatpositionzaftertsteps
oftheprocess”. Formally,fort = 1,thiseventistheset:
(cid:8) Z (cid:9)
E = s ∈ R : F(c,s) ∈ [u] .
c,[u]z z
In order to evaluate the probability of this event, we use the locality of the explicit global function
F. The event “F(c,s) ∈ [u] ” only depends of the cells of s from position a = z−k to position
z
b = z+k+|u|−1. Therefore, if J = {v ∈ Rb−a : F(c,[v] ) ⊆ [u] }, then E = ∪ [v] and
a z c,[u]z v∈J a
henceE isameasurablesetofprobability: ν (E ) = (cid:80) ν ([v] ) = |J|/|R|b−a (recallthat
c,[u]z R c,[u]z v∈J R a
Z
ν istheuniformmeasureoverR ).
R
Z Z
MoregenerallytoanyCAAweassociateitsstochasticglobalfunctionS : Q → M(Q )defined
F
foranyconfigurationc ∈ Q∗ by: ∀u ∈ QZ,∀z ∈ Z,
(cid:0) (cid:1)
S (c) ([u] ) = ν (E )=theprobabilityofeventE .
F z R c,[u]z c,[u]z
Example. Forinstance,considerthestochasticfunctionPARITYthatmapseveryconfigurationoverthe
alphabet {0,1,#} to a random configuration in which every {0,1}-word delimited by two consecutive
#isreplacedbyarandomindependentuniformwordoflengthwithevenparity. Thiscannotberealized
byaCFCA.Still,onecanrealizethestochasticfunctionPARITYasastochasticCAwithQ = {#,0,1},
R = {0,1} and local rule f : Q{−1,0,1}×R{−1,0} → Q given by: for all c ,c ,c ,s ,s ∈ {0,1}
−1 0 1 −1 0
anda,b ∈ {#,0,1},
f(a#b,s s ) = # f(#c #,s s ) = 0 f(#c c ,s s ) = s
−1 0 0 −1 0 0 1 −1 0 0
f(c c #,s s ) = s f(c c c ,s s ) = s +s
−1 0 −1 0 −1 −1 0 1 −1 0 −1 0
One can easily check that this local probabilistic correlations ensures that every word delimited by two
consecutive#isindeedmappedtoauniformindependentrandomwordofevenparity.
Dynamics. As opposed to the deterministic and non-deterministic setting, defining an iterate of this
map is a not so trivial task. There are two approaches: defining directly the measure after t steps or
Z
extending the map S to a map from M(Q ) to itself. Both rely crucially on the continuity of F. In
F
particular, we want to make sure that the definition of the measure after t steps matches t iterations of
theone-stepmap,andhence,isindependentoftheexplicitmechanicsofF butdependsonlyonthemap
S definedbyF.
F
Theeasiestonetopresentisthefirstapproach. Foranyt (cid:62) 1,theeventEt thattheworduappears
c,[u]z
at position z at time t from configuration c consists in the set of all t-uples of random configurations
(s1,...,st)yieldinguatpositionz fromc,i.e.:
Et = (cid:8)(s1,...,st) ∈ (cid:0)RZ(cid:1)t : Ft(c,s1,...,st) ∈ [u] (cid:9)
c,[u]z z
AsbeforeEt isameasurablesetin(cid:0)RZ(cid:1)t becauseitisaproductoffiniteunionsofcylindersbythe
c,[u]z
localityofF. WethereforedefineSt : QZ → M(QZ),theiterateofthestochasticglobalfunction,by:
F
(cid:0)St (c)(cid:1)([u] ) = ν (Et )=theprobabilityofeventEt
F z Rt c,[u]z c,[u]z
whereν denotestheuniformmeasureontheproductspace(cid:0)RZ(cid:1)t. Forsimilarreasonsasabove,St (c)
Rt F
isawell-definedprobabilitymeasure.
For all t (cid:62) 0 and all words u ∈ Qn with n (cid:62) 2kt+1, we will also denote by Ft(u) the random
variable for the random image v ∈ Qn−2kt of u by Ft, defined formally as: for all u ∈ Qn and v ∈
Qn−2kt,
Pr{Ft(u) = v} = (cid:0)St (c)(cid:1)([v] ), foranyc ∈ [u] .
F kt 0
Thefollowingkeytechnicalfactensuresthattwoautomatadefinethesamedistributionovertimeas
soonastheirone-stepdistributionsmatch.
Fact3.3. Let A and B be two stochastic CA with the same set of states Q (and possibility different
random alphabet) and of explicit global functions F and G respectively. If S = S then for all t (cid:62) 1
F G
wehaveSt = St
F G
Proof:
Theproofiswrittenfor1DCAtosimplifynotationsbutitextendstoanydimensioninastraighforward
Z
way. Consider a CA of explicit global function F. Consider a word u and a position z. Let φ : Q →
Z
P(R ) be function that associates to a configuration c the event E . φ(c) is entirely determined by
c,[u]z
the states of the cells from positions a = z −k to b = z +|u|+k in c (locality of F). Therefore φ is
constantovereverycylinder[v] withv ∈ Qb−a. Ifwedistinguishsomec ∈ [v] foreveryv ∈ Qb−a,
a v a
weobtainbydefinitionofFt andcontinuityofF:
(cid:91) (cid:16) (cid:17)
Et+1 = Et ×E .
c,[u]z c,[v]a cv,[u]z
v∈Qb−a
Then, since sets (cid:0)Et (cid:1) are pairwise disjoint (because F is deterministic and cylinders [v] are
c,[v]a v∈Qb−a a
pairwisedisjoint),wehave
(cid:0)St+1(c)(cid:1)([u] ) = ν (Et+1 ) = (cid:88) ν (cid:0)Et ×E (cid:1)
F z Rt+1 c,[u]z Rt+1 c,[v]a cv,[u]z
v∈Qb−a
(cid:88)
= ν (Et )·ν (E )
Rt c,[v]a R cv,[u]z
v∈Qb−a
= (cid:88) (cid:0)St (c)(cid:1)([v] )·(cid:0)S (c )(cid:1)([u] )
F a F v z
v∈Qb−a
The value of St (c) over cylinders can thus be expressed recursively as a function of a finite number of
F
valuesS overafinitenumberofcylinders. ItfollowsthatifforsomepairofCAAandB withexplicit
F
globalfunctionsF andGwehaveS = S ,thenSt = St forallt. (cid:116)(cid:117)
F G F G
In our setting one can recover the non-deterministic dynamics from the stochastic dynamics of a
given stochastic CA. This heavily relies on the continuity of explicit global functions and compacity of
symbolicspaces.
Fact3.4. GiventwoCAwithsamesetofstatesandexplicitglobalfunctionsF andF ,ifS = S
A B FA FB
thenN = N .
FA FB
Proof:
GivensomestochasticCAofexplicitglobalfunctionF,someconfigurationcandsomecylinder[u] we
z
have
N (c)∩[u] (cid:54)= ∅ ⇔ E (cid:54)= ∅ ⇔ (cid:0)S (c)(cid:1)([u] ) > 0
F z c,[u]z F z
by definition of N and S . Since N (c) is a closed set (continuity of F) it is determined by the set
F F F
ofcylindersintersectingit(compacityofthespace). HenceN (c)isdeterminedbyS (c). Thelemma
F F
follows. (cid:116)(cid:117)
4. From Local to Global
It is well-known in deterministic CA that determining global properties from the local representation is
a generally hard problem. The purpose of this section is to show that the situation is even worse for
stochasticCA.
4.1. Equalityofrandommaps: undecidabilityandexplicittools
Anundecidabletaskfordimension2andhigher. Intheclassicaldeterministiccase,itiseasytode-
terminewhethertwoCAhavethesameglobalfunction. Equivalentlydeterminingwhethertwostochastic
CA,assyntacticalobjects,havethesameexplicitglobalfunctionsF andGiseasy. However,giventwo
stochastic CA which have possibly different explicit global function F and G, it still may happen that
N = N or S = S , and determining whether this is the case turns out to be a difficult problem.
F G F G
In fact, Theorem 4.1 states that these two decision problems are at least as difficult as the surjectivity
problemofclassicalCA.
Theorem4.1. Let P (resp. P ) be the problem of deciding whether two given stochastic CA have
N S
the same non-deterministic (resp. stochastic) global function. The surjectivity problem of classical
deterministicCAisreducibletobothP andP .
N S
Proof:
Theproofiswrittenfor1DCAtosimplifynotationsbutitextendstoanydimensioninastraighforward
Z Z
way. Consider a classical CA F : Q → Q and define µ as the image by F of the uniform measure
F
Z
µ onQ :
0
µ ([u] ) = ν(cid:0)F−1([u] )(cid:1)
F z z
Itiswell-knownthatF issurjectiveifandonlyifµ = ν (thisresultistrueinanydimension;theproof
F
for dimension 1 is in [16] and follows from [22] for higher dimensions, but we recommend [27] for a
modernexpositioninanydimension).
Now let us define the stochastic CA A = (Q,Q,V,V(cid:48),g) such that G(c,s) = F(s). With this
definition, A is such that, for all c, S (c) = µ . Hence, S (c) is the uniform measure for any c if and
G F G
Z
only if F is surjective. We have also N (c) = Q for all c if and only if G is surjective. The theorem
G
followssinceAisrecursivelydefinedfromF. (cid:116)(cid:117)
SurjectivityofclassicalCAisanundecidablepropertyindimension2andhigher[18]. Asanimme-
diatecorollary,wegetundecidabilityofequalityofglobalmapsindimension2andhigher.
Corollary4.2. Fixd (cid:62) 2. ProblemsP andP areundecidableforCAofdimensiond.
N S
Explicit tools for (dis)proving equality. Even if testing the equality of the non-deterministic or
stochasticdynamicsoftwostochasticCAisundecidablefordimension2andhigher,Theorem4.5states
thatequality,whenitholds,canalwaysbecertifiedintermsofastochasticcoupling. Indeedthestochas-
tic coupling, by matching their two source of randomness, serves as a witness of the equality of the
stochastic CA. This provides us with a very useful technique, because the existence of such a coupling
iseasytoproveordisproveinmanyconcreteexamples. Againtheresultheavilyreliesonthecontinuity
oftheexplicitglobalfunctionF.
Letusfirstrecallthestandardnotionofcoupling.
Z Z
Definition4.3. Let µ ∈ M(Q ) and µ ∈ M(Q ). A coupling of µ and µ is a mea-
1 1 2 2 1 2
Z Z Z
sure γ ∈ M(Q ×Q ) such that for any measurable sets E and E , γ(E ×Q ) = µ (E ) and
1 2 1 2 1 2 1 1
Z
γ(Q ×E ) = µ (E ).
1 2 2 2
Concretely, a coupling couples two measures so that each is recovered when the other is ignored. The
motivation in defining a coupling is to bind the two distributions in order to prove that they induce the
same kind of behavior: for instance, one can easily couple the two uniform measures over {1,2} and
{1,2,3,4}sothatwithprobability1,bothnumberswillhavethesameparity(γ givesaprobability1/4
toeachpair(1,1),(2,2),(1,3)and(2,4)and0totheothers). Thisdemonstratesthattheparityfunction
isidenticallydistributedinbothcases.
Theorem 4.5 states that the dynamics of two stochastic CA are identical if and only if there is a
coupling of their random configurations so that their stochastic global functions become almost surely
identical. Thisisoneofourmainresults.
Definition4.4. Two stochastic cellular automata, A = (Q,R ,V ,V(cid:48),f ) and A =
1 1 1 1 1 2
(Q,R ,V ,V(cid:48),f ), with the same set of states Q are coupled on configuration c ∈ QZ by a measure
2 2 2 2
Z Z
γ ∈ M(R ×R )if
1 2
Z Z
1. γ isacouplingoftheuniformmeasuresonR andR ;
1 2
(cid:0) Z Z (cid:1)
2. γ {(s ,s ) ∈ R ×R : F (c,s ) = F (c,s )} = 1, i.e. F andF producealmostsurelythe
1 2 1 2 1 1 2 2 1 2
sameimagewhenfedwiththeγ-coupledrandomsources.
Notethatthesetofpairs(s ,s )definedaboveismeasurablebecauseitisclosed(F andF arecontin-
1 2 1 2
uous).
Theorem4.5. TwostochasticCAwiththesamesetofstateshavethesamestochasticglobalfunctionif
andonlyif,oneachconfigurationc,theyarecoupledbysomemeasureγ (whichdependsonc).
c
Outlineoftheproof. Wefixaconfigurationc. Bycontinuityoftheexplicitglobalfunctions,weconstruct
a sequence of partial couplings (γn) matching the random configurations of finite support of radius n.
c
Wethenextractthecouplingγ from(γn)bycompacityofM(RZ ×RZ).
c c A B
Proof:
Theproofiswrittenfor1DCAtosimplifynotationsbutitextendstoanydimensioninastraighforward
way. First,ifA = (Q,R ,V ,V(cid:48),f )andA = (Q,R ,V ,V(cid:48),f )arecoupledbyγ onconfiguration
1 1 1 1 1 2 2 2 2 2 c
Z
c ∈ Q ,considerforanycylinder[u] thesets
z
Z
E = {s ∈ R : F (c,s) ∈ [u] }
1 1 1 z
Z
E = {s ∈ R : F (c,s) ∈ [u] }
2 2 2 z
Z Z
X = {(s ,s ) ∈ R ×R : F (c,s ) = F (c,s )}
1 2 1 2 1 1 2 2
Then,bythepropertyofthecouplingbyγ ,wehave
c
(cid:0) (cid:1) Z
S (c) ([u] ) = ν (E ) = γ (E ×R )
F1 z 1 1 c 1 2