The paper proposes a simple formalism for dealing with deterministic, non-deterministic and stochastic cellular automata in a unifying and composable manner. Armed with this formalism, we extend the notion of intrinsic simulation between deterministic cellular automata, to the non-deterministic and stochastic settings. We then provide explicit tools to prove or disprove the existence of such a simulation between two stochastic cellular automata, even though the intrinsic simulation relation is shown to be undecidable in dimension two and higher. The key result behind this is the caracterization of equality of stochastic global maps by the existence of a coupling between the random sources. We then prove that there is a universal non-deterministic cellular automaton, but no universal stochastic cellular automaton. Yet we provide stochastic cellular automata achieving optimal partial universality.

studied [

In this paper we propose a simple formalism to deal with general stochastic CA. The formalism relies on
considering a CA F(c; s) fed, besides the current configuration c, with a new fresh independent uniform
random configuration s at every time step. This allows any kind of local correlations and includes in
particular all the examples of [

The choice of making expicit the random source allows us to show some no-go results. Any stochastic CA may only simulate stochastic CA with a compatible random source (where compatibility is expressed as a simple arithmetic equation, Theorem 4). It follows that there is no universal stochastic CA (Corollary 1). Still, we show that there is a universal CA for the non-deterministic dynamics (Theorem 5), and we are able to provide a universal stochastic CA for every class of compatible random source (Theorem 6).

Plan. Section 2 recalls the vital minimum about probability theory. Section 3 states our formalism. Section 4 gives tools to prove (or disprove) equality of stochastic global functions. Section 5 extends the notions of intrinsic simulations to the non-deterministic and stochastic settings. Section 6 provides the no-go results in the stochastic setting, the universality constructions. Section 7 concludes this article with a list of open questions. 2

Even if this article focuses mainly on one-dimensional CA for the sake of simplicity, it extends naturally to higher dimensions.

For any finite set A we consider the symbolic space AZ. For any c 2 AZ and z 2 Z we denote by cz the
value of c at point z. AZ is endowed with the Cantor topology (infinite product of the discrete topology
on each copy of A) which is compact and metric (see [

We denote by M (AZ) the set of Borel probability measures on AZ. By Carathe´odory extension theorem, Borel probability measures are characterized by their value on cylinders. Concretely, a measure is given by a function m from cylinders to the real interval [0; 1] such that m(AZ) = 1 and 8u 2 Q ; 8z 2 Z; m([u]z) = å m([ua]z) = å m([au]z 1)

a2A a2A

We denote by nA the uniform measure over AZ (s.t. nA([u]z) = jA1jjuj ). We shall denote it as n when the underlying alphabet A is clear from the context.

We endow the set M (AZ) with the compact topology given by the following distance:
D(m1; m2) = ån>0 2 n maxu2A2n+1 m1([u] n) m2([u] n) . See [

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

Definition 1. A stochastic cellular automaton A = (Q; R;V;V 0; f ) consists in: a finite set of states Q a finite set R called the random symbols two finite subsets of Z: V = fv1; : : : ; vrg and V 0 = fv01; : : : ; v0r0 g, called the neighborhoods; r and r0 are the sizes of the neighborhoods and r = maxv2V [V 0 jvj is the radius of the neighborhoods. a local transition function f : Qr Rr0 ! Q A function c 2 QZ is called a configuration; c j is called the state of the cell j in configuration c. A function s 2 RZ is called a R-configuration.

In the particular case where V 0 = f0g (i.e., where each cell uses its own random symbol only), we say that A is a plain probabilistic cellular automaton (PlainPCA for short).

Definition 2 (Explicit Global Function). To this local description, we associate the explicit global function F : QZ RZ ! QZ defined for any configuration c and R-configuration s by: F(c; s)z = f (cz+v1 ; : : : ; cz+vr ); (sz+v01 ; : : : ; sz+v0r0 ) : Given a sequence st t of R-configurations and an initial configuration c, we define the associated space-time diagram as the bi-infinite matrix ctz t>0;z2Z where ct 2 QZ is defined by c0 = c and ct+1 = F(ct ; st ). We also define for any t > 1 the tth iterate of the explicit global function Ft : QZ RZ t ! QZ by F0(c) = c for all configuration c and

Ft+1(c; s1; : : : ; st+1) = F Ft (c; s1; : : : ; st ); st+1 so that ct = Ft (c; s1; : : : ; st ).

In this paper, we adopt the convention that local functions are denoted by a lowercase letter (typically 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 not matter in this paper.

The explicit global function capture all possible actions of the automaton on configurations. This function allows to derive three kinds of dynamics: deterministic, non-deterministic and stochastic. 3.2

Deterministic. The deterministic global function DF : QZ ! QZ of A = (Q; R;V;V 0; f ) is defined by DF (c) = F(c; 0Z) where 0 is a distinguished element of R. A is said to be deterministic if its local transition function f does not depend on its second argument (the random symbols). Non-Deterministic. The non-deterministic global function NF : QZ ! P(QZ) of A is defined for any configuration c 2 QZ by NF (c) = fF(c; s) : s 2 RZg.

Dynamics. The deterministic dynamics of A is given by the sequence of iterates (DFt )t>0. Similarly the non-deterministic dynamics of A is given by the iterates NFt : QZ ! P(QZ) defined by NF0(c) = fcg and NFt+1(c) = Sc02NFt(c) NF (c0). 3.3

The stochastic point of view consists in taking the R-component as a source of randomness. More precisely, the explicit global function F is fed at each time step with a random uniform and independent R-configuration. This defines a stochastic process for which we are then interested in the distribution of states across space and time. By Carathe´odory extension theorem, this distribution is fully determined by the probabilities of the events of the form ?starting from c, the word u occurs at position z after t steps of the process?. Formally, for t = 1, this event is the set:

E c;[u]z = s 2 RZ : F(c; s) 2 [u]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) 2 [u]z? only depends of the cells of s from position a = z r to position b = z + r + juj 1. Therefore, if J = fv 2 Rb a : F(c; [v]a) [u]zg, then E c;[u]z = [v2J[v]a and hence E c;[u]z is a measurable set of probability: nR(E c;[u]z ) = åv2J nR([v]a) = jJj=jRjb a (recall that nR is the uniform measure over RZ).

More generally to any CA A we associate its stochastic global function SF : QZ ! M (QZ) defined for any configuration c 2 Q by: 8u 2 QZ; 8z 2 Z;

SF (c) ([u]z) = nR(E c;[u]z ) = the probability of event E c;[u]z .

Example. For instance, consider the stochastic function Parity that maps every configuration c over the alphabet f0; 1; #g to a random configuration in which every f0; 1g-word of length ` delimited by two consecutive # in c is replaced by a random independent uniform word of length ` with even parity. This cannot be realized by a PlainPCA. Still, one can realize the stochastic function Parity as a stochastic CA by means of a local transition function f of the above type, as follows. Given the configuration c and a uniform random f0; 1g-configuration s, f (c 1; c0; c1; s 1; s0) is: # if c0 = #; else s0 if c 1 = #; else s 1 if c1 = #; and (s 1 + s0 mod 2) otherwise. One can easily check that this local correlation ensures that every word delimited by two consecutive # is indeed mapped to a uniform independent random word of even parity. 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 extending the map SF to a map from M (QZ) to itself. Both rely crucially on the continuity of F. In particular, we want to make sure that the definition of the measure after t steps matches t iterations of the one-step map, and hence, is independent of the explicit mechanics of F but depends only on the map SF defined by F.

The easiest one to present is the first approach. For any t > 1, the event E tc;[u]z that the word u appears at position z at time t from configuration c consists in the set of all t-uples of random configurations (s1; : : : ; st ) yieldings u at position z from c, i.e.:

E tc;[u]z = (s1; : : : ; st ) 2 RZ t : Ft (c; s1; : : : ; st ) 2 [u]z As before E tc;[u]z is a measurable set in RZ t because it is a product of finite unions of cylinders by the locality of F. We therefore define SFt : QZ ! M (QZ), the iterate of the stochastic global function, by:

SFt (c) ([u]z) = nRt (E tc;[u]z ) = the probability of event E tc;[u]z where nRt denotes the uniform measure on the product space RZ t . For similar reasons as above, SFt (c) is a well-defined probability measure.

The following key technical fact ensures that two automata define the same distribution over time as soon as their one-step distributions match.

Fact 1. 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 SF = SG then for all t > 1 we have SFt = SGt Proof. Consider a CA of stochastic global function F. Consider a word u and a position z. Let f : QZ ! P(RZ) be function that associates to a configuration c the event E c;[u]z . f (c) is entirely determined by the states of the cells from positions a = z r to b = z + juj + r in c (locality of F). Therefore f is constant over every cylinder [v]a with v 2 Qb a. If we distinguish some cv 2 [v]a for every v 2 Qb a, we obtain by definition of Ft and continuity of F:

E tc+;[u1]z =

[ v2Qb a

t E c;[v]a

E cv;[u]z : Then, since sets E tc;[v]a v2Qb a are pairwise disjoint (because F is deterministic and cylinders [v]a are pairwise disjoint), we have

SFt+1(c) ([u]z) = nRt+1 (E tc+;[u1]z ) =

å v2Qb a = å

v2Qb a = å v2Qb a nRt+1 E tc;[v]a

E cv;[u]z nRt (E tc;[v]a ) nR(E cv;[u]z ) SFt (c) ([v]a)

SF (cv) ([u]z) The value of StF (c) over cylinders can thus be expressed recursively as a function of a finite number of values SF over a finite number of cylinders. It follows that if for some pair of CA A and B with explicit global functions F and G we have SF = SG, then SFt = SGt for all t.

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 symbolic spaces.

Fact 2. Given two CA with same set of states and explicit global functions FA and FB, if SFA = SFB then NFA = NFB .

Proof. Given some stochastic CA of explicit global function F, some configuration c and some cylinder [u]z we have

NF (c) \ [u]z 6= ? , E c;[u]z 6= ? ,

SF (c) ([u]z) > 0 by definition of NF and SF . Since NF (c) is a closed set (continuity of F) it is determined by the set of cylinders intersecting it (compacity of the space). Hence NF (c) is determined by SF (c). The lemma follows. 4

An undecidable task for dimension 2 and higher. In the classical deterministic case, it is easy to
determine whether two CA have the same global function. Equivalently determining whether two
stochastic CA, as syntactical objects, have the same explicit global functions F and G is easy. However, given
two stochastic CA which have possibly different explicit global function F and G, it still happen that
NF = NG or SF = SG, and determining whether this is the case turns out to be a difficult problem. In
fact, Theorem 1 states that these two decision problems are at least as difficult as the surjectivity problem
of classical CA, which is undecidable in dimension 2 and higher [

Theorem 1. Let PN (resp. PS) be the problem of deciding whether two given stochastic CA have the same non-deterministic (resp. stochastic) global function. The surjectivity problem of classical deterministic CA is reducible to both PN and PS.

Proof. Consider a classical CA F : QZ ! QZ and define mF as the image by F of the uniform measure m0 on QZ:

mF ([u]z) = n F 1([u]z)
It is well-known that F is surjective if and only if mF = n (this result is true in any dimension, the proof
for dimension 1 is in [

Now let us define the stochastic CA A = (Q; Q;V;V 0; g) such that G(c; s) = F(s). With this definition, A is such that, for all c, SG(c) = mF . Hence, SG(c) is the uniform measure for any c if and only if F is surjective. We have also NG(c) = QZ for all c if and only if G is surjective. The theorem follows since A is recursively defined from F.

Explicit tools for (dis)proving equality. Even if testing the equality of the non-deterministic or stochastic dynamics of two stochastic CA is undecidable for dimension 2 and higher, Theorem 2 states that equality, when it holds, can always be certified in terms of a stochastic coupling. Indeed the stochastic 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 is easy to prove or disprove in many concrete examples. Again the result heavily relies on the continuity of the explicit global function F.

Let us first recall the standard notion of coupling. Definition 3. Let m1 2 M (Q1Z) and m2 2 M (Q2Z). A coupling of m1 and m2 is a measure g 2 M (Q1Z Q2Z) such that for any measurable sets E 1 and E 2, g(E 1 Q2Z) = m1(E 1) and g(Q1Z E 2) = m2(E 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 bound 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 f1; 2g and f1; 2; 3; 4g so that with probability 1, both numbers will have the same parity (g gives a probability 1=4 to each pair (1; 1), (2; 2), (1; 3) and (2; 4) and 0 to the others). This demonstrates that the parity function is identically distributed in both cases.

Theorem 2 states that the dynamics of two stochastic CA are identical if and only if their is a coupling of their random configurations so that their stochastic global functions become almost surely identical. This is one of our main results.

Definition 4. Two stochastic cellular automata, A1 = (Q; R1;V1;V10; f1) and A2 = (Q; R2;V2;V20; f2), with the same set of states Q are coupled on configuration c 2 QZ by a measure g 2 M (R1Z R2Z) if 1. g is a coupling of the uniform measures on R1Z and R2Z; 2. g f(s1; s2) 2 R1Z R2Z : F1(c; s1) = F2(c; s2)g = 1, i.e. F1 and F2 produce almost surely the same image when fed with the g-coupled random sources.

Note that the set of pairs (s1; s2) defined above is measurable because it is closed (F1 and F2 are continuous).

Theorem 2. Two stochastic CA with the same set of states have the same stochastic global function if and only if, on each configuration c, they are coupled by some measure gc (which depends on c). Outline of the proof. We fix a configuration c. By continuity of the explicit global functions, we construct a sequence of partial couplings (gcn) matching the random configurations of finite support of radius n. We then extract the coupling gc from (gcn) by compacity of M (RAZ RBZ).

Proof. Details. First, if A1 = (Q; R1;V1;V10; f1) and A2 = (Q; R2;V2;V20; f2) are coupled by gc on configuration c 2 QZ, consider for any cylinder [u]z the sets

E 1 = fs 2 R1Z : F1(c; s) 2 [u]zg E 2 = fs 2 R2Z : F2(c; s) 2 [u]zg X = f(s1; s2) 2 R1Z

SF1 (c) ([u]z) = n1(E 1) = gc(E 1 R2Z) where n1 is the uniform measure on R1Z. But gc(E 1 R2Z) = gc (E 1 R2Z) \ X since gc(X ) = 1. Symmetrically we have

SF2 (c) ([u]z) = gc (R1Z

E 2) \ X : But, by definition of sets E 1, E 2 and X , we have R1Z E 2 \X = E 1 R2Z \ X . We conclude that SF1 = SF2 . For the other direction of the theorem, suppose SF1 = SF2 and fix some configuration c. We denote by m the measure SF1 (c) = SF2 (c). Without loss of generality we can suppose that A1 and A2 have same raidii r: r1 = r2 = r. We construct a sequence (gn) of measures from which we can extract a limit point (by compacity of the space of measures) which is a valid coupling of A1 and A2 on configuration F1 and F2 being of radius r we can write Sui as a finite union of centered cylinders of length 2(n + r) + 1: where Pui

R2(n+r)+1. Define the following partition Iui of the real interval [0; 1) by:

i c. To simplify the proof we focus on centered cylinders: for any word w of odd length, we denote by [w] = [w]zw where zw = jwj2 1 . Let?s fix n. For any word u 2 Q2n+1 we define: where rank(v) 2 f0; : : : ; #Pui 1g is the rank of v in some arbitrarily chosen total ordering of Pui (the lexicographical order for instance). Since the sets Pui form a partition of Ri2(n+r)+1 when u ranges over all words of Q2n+1, we have for any v 2 Ri2(n+r)+1: Now, for every v1 2 R12(n+r)+1 and (recall that ni stands for the uniform measure over RiZ). v2 2 R22(n+r)+1, we construct gn as: gn([v1]; [v2]) = (jIu1(v1) \ Iu2(v2)j mQ([u]) 0 if 9u s.t. vi 2 Pui for both i = 1; 2 otherwise.

Furthermore, if 0i is a distinguished element of Ri, we extend the definition of gn to any pair v1 2 R12m+1 and v2 2 R22m+1 with m > n + r by: gn([v1]; [v2]) = (gn([w1]; [w2]) 0 if vi = 0im n r wi0im n r for i = 1; 2 else.

By s -additivity gn is thus defined on any cylinder and by extension theorem is a well-defined measure. Now by construction, we have for any v1 2 R2(n+r)+1:

gn([v1]; R2Z) = jIu1(v1)j mQ([u]) for some u such that v1 2 Pu1. Hence, gn([v1]; R2Z) = n1([v1]). By s -additivity of gn and m, this equality holds for any v1 2 R2m+1 with m 6 n + r. Symmetrically we have gn(R1Z; [v2]) = mU2 ([v2]) for any v1 2 R2m+1. Moreover, by definition, gn([v1]; [v2]) = 0 if there is no u such that vi 2 Pui for i = 1; 2. We deduce that the set:

Xn = [ Su1

Su2 u2Q2n+1 has measure 1. More precisely, since Xn+1 Xn, for any m 6 n, gn(Xm) = 1.

To conclude the proof, let g be any limit point of the sequence (gn)n. By the definition of the distance on the space of measures, we have: 1. 8m; 8w 2 R2m+1; g([w]; R2Z) = n1([w]) and symmetrically for the R2 component, hence g is a cou1 pling of uniform measure on R1Z and R2Z; 2. 8n; g(Xn) = 1 hence g(\nXn) = 1 where \ Xn = X = f(s1; s2) 2 R1Z n

Notice that the proof of this theorem is non-constructive (recall that equality of stochastic global maps is undecidable in dimension 2 and higher). Moreover, it is easy to get convinced on a simple example that the coupling must depend on the configuration. Consider the two following automata with states Q = R = f0; 1g and neighborhoods V = V 0 = f0g: A with explicit global function F(c; s) = s and B with explicit global function G(c0; s0) = c0 + s0 mod 2. Clearly, both A and B define the same blank noise CA and the coupling proving this fact is defined for all z 2 Z and all a; b 2 f0; 1g by gc([a]z; [b]z) = 1=2 if and only if a = b + cz mod 2, and = 0 otherwise. This coupling demonstrates indeed that gc f(s; s0) : F(c; s) = G(c; s0)g = 1 yielding that the dynamics are identical; but note that gc must depend on c.

The purpose of this section is to give a precise meaning to the sentence ?A is able to simulate B? or equivalently ?A contains the behavior of B?.

Our approach follows a series of works on simulations between classical deterministic CA [

In each case (the deterministic, the non-deterministic, and the stochastic global functions), we will define simulation as an equality of dynamics up to some local transformations. 5.1

The transformations we consider are natural stochastic extensions of the transformation defined in [

Trimming operations They are based on three ingredients: 1) renaming states; 2) restricting to a stable subset of states; and 3) merging compatible states. These ingredients are synthetized into two definitions (state renaming is implicit in both definitions).

Definition 5. Let A = (Q; R;V;V 0; f ) be a stochastic CA. if i : Q0 ! Q is an injective function such that Y = (i(Q0))Z is F-stable (i.e. F(Y; RZ) i-restriction of A is the stochastic CA:

Y ) then the iA = (Q0; R;V;V 0; i f ) where i f is the local function associated with the explicit global function iF such that, 8c 2 I, 8s 2 RZ, iF(c; s) = I 1 F(I(c); s) where I : Q0Z ! QZ denotes the cell-by-cell extension of i; if p : Q ! Q0 is surjective and F-compatible (s.t. P F(c; s) = P F(c0; s) for all s and all c; c0 such that P(c) = P(c0), where P : QZ ! Q0Z is the cell-by-cell extension of p), then the p-projection of A is the stochastic CA:

p A = (Q0; R;V;V 0; p f ) where p f is the local function associated with the explicit global function p F such that p F(c0; s) = P F(c; s) where c is any configuration in P 1(c0).

If i : Q0 ! Q and p : Q0 ! Q00 verify the required stability and compatibility conditions, we denote by ip A the p-projection of the i-restriction of A .

Definition 6. Let A1 = (Q1; R1;V1;V10; f1) and A2 = (Q2; R2;V2;V20; f2) be two arbitrary stochastic CA. We define the following relations:

S A1 v A2, A1 is a stochastic subautomaton of A2, if there is some i-restriction of A2 such that SF1 = SiF2 ;

S A1 E A2, A1 is a stochastic factor of A2, if there is some p-projection of A2 such that SF1 = Sp F2 ;

N N D D Similarly, we define v and E (for non-deterministic global maps) and v and E (for deterministic global

D N S maps). We also define the three relations Ev, Ev and Ev using projections of restrictions. For instance:

S A1EvA2 if there are i and p such that SF1 = Sip F2 . 5.1.2

Rescaling transformations.

The transformations defined so far only allow to derive a finite number of CA from a given CA (up to renaming of the states) and thus induce only a finite number of dynamics. In particular, the size of the set of states, and the size of the neighborhood, can only decrease. Following the approach taken for classical deterministic CA, we now consider rescaling transformations, which allow to increase the set of states, the neighborhood, etc. Rescaling transformations consist in: composing with a fixed translation, packing cells into fixed-size blocks, and iterating the rule a fixed number of times. Notice that since stochastic CA are composable, they are stable under rescaling operations, whereas PlainPCA are not.

The translation sk (for k 2 Z) is the deterministic CA whose deterministic global function verifies: 8c; 8z; Dsk (c)z = cz+k.

Given any finite set S and any m > 1, we define the bijective packing map bm : SZ ! Sm Z by bm(c)z = (cmz; cmz+1; : : : ; cmz+m 1) for all c and z.

Definition 7. Let A = (Q; R;V;V 0; f ) be any stochastic CA. Let m;t > 1 and k 2 Z. The rescaling of A with parameters (m;t; k) is the stochastic CA A hm;t;ki = Qm; (Rm)t ;V+;V +0; f hm;t;ki whose explicit global function Fhm;t;ki is defined by:

Fhm;t;ki(c; s) = bm sk Ft (bm1(c); bm1(s1); : : : ; bm1(st )) where s1; : : : ; st 2 (Rm)Z are the t components of s (s.t. sij = (s j)i), and V+;V +0 the modified neighbourhoods following bm. 5.2

Definition 8. For each local relation < among the nine relations of Definition 6, we define the associated simulation relation 4 by

A1 4 A2 , 9m1; m2; t1; t2; k1; k2; A1hm1;t1;k1i < A2hm2;t2;k2i We therefore define nine simulation relations 4iS, 4Sp , 4Sm, 4iN , 4pN , 4Nm, 4iD, 4pD and 4mD, where the subscript denotes the kind of local relation used (injection, projection or mixed) and the superscript denotes the kind of global functions which are compared (Stochastic, Non-deterministic or Deterministic). Lemma 1. A restriction (resp. projection) of a restriction (resp. projection) of some stochastic CA A is a restriction (resp. projection) of A . Moreover, any restriction of a projection of A is the projection of some restriction of A .

Proof. This is a straightforward generalization of the corresponding result in the classical deterministic
settings. A detailed proof for the deterministic case appears in Theorem 2.1 of [

The lemma above implies that any sequence of admissible restrictions and projections can be expressed as the projection of some restriction.

From Lemma 1 it follows that all local relations defined are transitive and reflexive. Moreover,

D D
the deterministic relations v and E are exactly the same as those defined in the classical setting of
deterministic CA [

Fact 3. All simulation relations 4iS, 4Sp , 4Sm, 4i , 4p , 4m, 4iD, 4pD, 4mD are pre-orders.

N N N Proof. It is sufficient to verify that for any local comparison relation <: 1. < is compatible with rescalings, i.e. 2. rescalings are commutative with respect to <, i.e.

A1 < A2 ) A1hm;t;ki < A2hm;t;ki

A1hm;t;kihm0;t0;k0i < A1hm0;t0;k0ihm;t;ki Both properties are straightforward from the definitions. Then, the transitivity of any simulation relation follows from the transitivity of the corresponding local comparison relation <.

Each stochastic pre-order is a refinement of the corresponding non-deterministic pre-order as shown by the following fact (straightforward corollary of Fact 2).

Fact 4. If A14iSA2 then A14iN A2. The same is true for pre-orders 4p , 4Sm and the corresponding S (non-)deterministic pre-orders.

Note that for any simulation relation 4, A1 4 A2 means that two global functions are equal where one is obtained by applying only space-time-diagram-preserving rescaling transformations to A1 (the simulated CA) and the other is obtained by applying both rescaling transformations and trimming operations to A2 (the simulator). 5.3

Simulation pre-orders can be seen as a tool to classify the behaviors of CA [

Fact 5. Let 4 be any non-deterministic or stochastic pre-order. Let A1 and A2 be stochastic CA such that A1 4 A2. If A2 is deterministic (resp. noisy) then A1 is deterministic (resp. noisy). Proof. By Fact 4 it is sufficient to prove this for non-deterministic simulations. The property that the explicit global function is deterministic or noisy (i.e. surjective on each configuration) is preserved by rescaling transformation. Hence it is sufficient to check that being deterministic or noisy is preserved by restriction and projection. This is straightforward for projection (because a projection is an onto map). Determinism is clearly preserved by restriction. Moreover, a noisy stochastic CA does not admit any non-trivial restriction because no subset of states is stable under iteration. Hence, the restriction of a noisy CA is necessarily itself (up to renaming of states) or the trivial CA with only one state. Both are noisy and the fact follows.

Simulation of stochastic CA by a PlainPCA. Even if some stochastic CA cannot be expressed as a PlainPCA (because of potential local random correlation), each can be simulated by a particular PlainPCA. Each step is simulated by two steps: 1) each cell first copies its random symbol in its state so that 2) its neighbors read in its state its random symbol to complete the transition.

Theorem 3. For any stochastic CA A = (Q; R;V;V 0; fA) there is a PlainPCA B such that A 4iSB.

1. generate a random symbol locally and copy it to a component of states; 2. simulate a stochastic transition of A reading states only and ignoring random symbols. Formally, let B = (QB; R;V;V 0; fB) where QB = Q [ Q associated explicit global function FB verifies: 1. for any c 2 QZ

QBZ and any s 2 RZ, FB(c; s) z = (cz; sz) 2. for any c 2 (Q R)Z QBZ and any s 2 RZ, FB(c; s) = FA(pQ(c); pR(c)) where pQ and pR are cellby-cell projections on Q and R respectively.

S It is straightforward to check that A v B2 with the restriction induced by the identity injection i : QZ ! QZ QBZ.

Note that the restriction is essential in the above construction since the behavior is not specified (and no correct behavior can be specified) on configurations where states of type Q and states of type Q R are mixed. In particular it is false that the stochastic CA is the square of the PlainPCA; it is a restriction of that.

Still, one could think that we might achieve a simpler simulation by taking QB = Q R and doing the two steps simultaneously so that FB(c; s) would be the cell by cell product of FA(pQ(c); pR(c)) and

s. But this does not work: for such a B there is generally no restriction nor projection nor combination of both able to reproduce the stochastic global function of A . Indeed, if some c and s1; s2 are such that FA(c; s1) 6= FA(c; s2) there is no valid way to define a corresponding configuration for c in FB because the Q-component of states in FB depends only on the previous deterministic configuration, not on the random configuration. Then, one might see this impossibility as an argument against our formalism of simulation. Of course, many extensions of our definitions might be considered to allow more simulations between stochastic CA. However, we think that the random component of the simulated CA should never be used to determine which deterministic configuration of the simulator CA corresponds to which deterministic configuration of the simulated CA. Doing so would be like predicting the noise of a system to prepare the state of another system. In particular, we do not see any reasonable formal setting where FB defined as above would be able to simulate FA. FA and FB might look like two syntactical variants of essentially the same object, but, as stochastic dynamical systems, they are very different. For instance, not every configuration can be reached from any configuration in FB whereas FA could have this property (i.e. be a noisy stochastic CA).

We believe that a better understanding of the relationship between stochastic CA and PlainPCA should go through the following questions: is there a PlainPCA in any equivalence class induced by the pre-order 4iS? is any stochastic CA 4Sp -simulated by some PlainPCA? 6

The quest for universal CA is as old as the model itself. Intrinsic universality has also a long story as
reported in [

Indeed, one of the main by-product of each simulation pre-order defined above is a notion of intrinsic
universality. Formally, given some simulation pre-order 4, a stochastic CA A is 4-universal if for any
stochastic CA B we have B 4 A . When considering deterministic pre-orders, we recover the notions
of universality already studied in literature for classical deterministic CA [

When considering non-deterministic or stochastic global functions, the random symbols are hidden. Still, the choice of the set of random symbols plays an important role in the global functions we can possibly obtain. We denote by PF(n) the set of the prime factors of n. By extension, for a stochastic CA A with set of random symbols R, we denote by PF(A ) the set PF(jRj). We have the following result: Lemma 2. Let A1 = (Q; R1;V1;V10; f1) and A2 = (Q; R2;V2;V20; f2) be two stochastic CA with same set of states. If they are not deterministic and SF1 = SF2 then PF(A1) \ PF(A2) 6= ?. Proof. If A1 is not deterministic, then there must exist some configuration c 2 QZ and two configurations y 6= y0 such that fy; y0g NF1 (c). So there are two disjoint cylinders [u]z \ [u0]z = ? with y 2 [u]z and y0 2 [u0]z. Therefore 0 < SF (c) ([u]z) < 1. Besides, by definition of SF , we have p SF1 (c) ([u]z) = n1(E c1;[u]z ) = q < 1 for some relatively prime numbers p and q (recall that n1 is the uniform measure over R1Z and that E c1;[u]z = fs 2 R1Z : F1(c; s) 2 [u]zg). Moreover PF(q) PF(jR1j) = PF(A1) since E c1;[u]z is a finite union of cylinders and since the n1-measure of any cylinder is a rational of the form jRa1jb for some integers a; b > 1. Now, by hypothesis, we have also and by a similar argument as above we deduce that PF(q) ? (because qp < 1).

PF(A2). The lemma follows since PF(q) 6=

From Lemma 2 it follows, surprisingly perhaps, that the random symbols of a stochastic CA limit its simulation power to stochastic CA that have compatible random symbols.

Theorem 4. Let 4 be any stochastic simulation pre-order, and A1 and A2 two stochastic CA which are not deterministic. If A1 4 A2 then PF(A1) \ PF(A2) 6= ?.

Proof. Trimming operations (restrictions and projections) do not modify the set of random symbols. Rescaling transformations modify the set of random symbols in the following way: R 7! Rn for some integer n. Therefore such transformations preserve the set of prime factors PF(A ) of the considered CA A . Moreover, rescaling transformations do not affect determinism: the rescaled version of a CA which is not deterministic cannot be deterministic. Hence, the relation A1 4 A2 implies an equality of stochastic global functions of two CA which have the same prime factors as A1 and A2 and one of which is not deterministic. Therefore none of them is deterministic and the theorem follows from lemma 2.

The consequence in terms of universality is immediate and breaks our hopes for a stochastic universality construction.

Corollary 1. Let 4 be any stochastic simulation pre-order. There is no 4-universal stochastic CA. 6.2

Still, the negative result of Corollary 1 leaves open the possibility of partial universality constructions. We will now describe how to construct a stochastic CA which is 4iN -universal (hence also 4Nm-universal; however note that the existence of a 4p - or even of a 4pD-universal is still open), and then draw the

N consequences.

Since we are not concerned with size optimization, we will use simple construction techniques using
parallel Turing heads and table lookup as described for classical deterministic CA in [

The blocks of m cells have the following structure (the restriction in the pre-order handles the trimming of any invalid block):

SYNC transition table

Q-state

R-symbol

Q-states of neighbors

R-symbols of neighbors where each part uses a fixed alphabet (independent of Q and R) and only the width of each part may depend on A . To each such block is attached a Turing head which will repeat cyclically a sequence of 4 steps (sub-routines) described below. On a complete configuration made of such blocks there will be infinitely many such heads (one per block) executing these steps in parallel. Execution is synchronized at the end of each step (SYNC part) and such that two Turing heads never collide. Precisely, for some steps (2 and 4) the moves of all heads are rigorously identical (hence synchronous and without head collision). For some other steps (1 and 3), the sequence of moves of each head depend on the content of its corresponding block but these steps are always such that the head don?t go outside the block (hence no risk of head collision) and they are synchronized at the end by the SYNC part which implements a small time countdown initialized to the maximum time needed to complete the step in the worst case. The parts holding R-symbols are initially empty (uniformly equal to some symbol) for each block. The 4 steps are as follows: 1. generate a string representing a random R-symbol in the R-symbol part using (possibly several) random RU -symbols present in that part of the block; 2. copy the R-symbol part to the appropriate position in the R-symbols of neighbors part of each neighboring block. Do the same for Q-state; 3. using information about Q-states and R-symbols in the block, find the corresponding entry in the transition table and update the Q-states part of the block accordingly; 4. clean R-symbol and R-symbols of neighbors parts (i.e. write some uniform symbol everywhere).

This construction scheme is very similar to the one used for classical deterministic CA but two points are important in our context: step 4 is here to ensure that each configuration of A has a canonical corresponding configuration of U made of blocks where the parts holding R-symbols is clean; (step 4 is required for the existence of the injection i) depending on the way we generate strings representing a R-symbols from strings of RU -symbols in step 1, we will obtain or not a uniform distribution over R (recall Theorem 4).

In the general case, we can always fix (by the means of the injection i) a width large enough for parts containing the R-symbols so that all R-symbols can be obtained (but with possibly different probabilities). We therefore obtain a universality result for non-deterministic simulations.

Theorem 5. Let 4 be either 4iN or 4Nm. There exists a 4-universal CA.

Note that this 4-universal CA is a PlainPCA, and we obtain thus a stronger version of the simulation mentioned in Section 5.3 page 219.

Now, if we are in a case where PF(A ) PF(U ) then it is possible to choose a generation process in step 1 such that each R-symbol is generated with the same probability. We therefore obtain an optimal partial universality construction for stochastic simulations.

Theorem 6. Let 4 be either 4iS or 4Sm. For any finite set P of prime numbers, there is a stochastic CA UP such that for any stochastic CA A : PF(A ) P ) A 4 UP: Moreover UP is a PlainPCA. 7

Intrinsic simulations has been proven to be a powerful tool to hierarchize behaviors in the deterministic
world. In particular, the notion universal CA allows to formalize the concept of ?most complex? CA
as the ones concentrating ?all the possible behaviors? within a given class [

Is there for any stochastic A , a PlainPCA B which is 4iS-equivalent to A ? Is there for any stochastic A , a PlainPCA B such that A 4Sp B? Are there 4pN -universal cellular automata?

Are universal CA the same for pre-order 4iN and 4pN ?

Our setting can also be generalized by taking any Bernouilli measure on the R-component (instead of the uniform measure). We believe that positive and negative results about universality essentially still hold but under a different form.

As noticed by an anonymous referee, there is an easy algorithm to decide whether to 1D CA have the same non-deterministic global function. We are currently working on an adaptation to decide equality of global stochastic functions.