The goal of this paper is to show why the framework of communication complexity seems suitable for the study of cellular automata. Researchers have tackled different algorithmic problems ranging from the complexity of predicting to the decidability of different dynamical properties of cellular automata. But the difference here is that we look for communication protocols arising in the dynamics itself. Our work is guided by the following idea : if we are able to give a protocol describing a cellular automaton, then we can understand its behavior.

syntactical representation. Moreover, an important part of the paper (Section 4) focuses on elementary CA, which is a fixed syntactical framework.

After n time steps the value of a cell depends on its own initial state together with the initial states of the rn left and rn right neighbouring cells. More precisely, we define the n-th iteration of local rule f n : f0; 1g2rn+1 ! f0; 1g recursively: f 1 = f and, for n 2,

f n(z rn : : : z1; z0; z1 : : : zrn) = f n 1( f (z rn; : : : ; z rn+2r) : : : f (zrn 2r; : : : ; zrn)):

Finally, we call P-complete a cellular automaton such that the problem of predicting Fn on all configurations of size 2rn + 1 is P-complete.

Our work is motivated by the following idea: if we are able to give a simple explicit description of f n (for arbitrary n), then we can understand the behavior of the corresponding CA. 2

Communication complexity is a model introduced by A. C.-C. Yao in [

For instance, let f : X Y ! Z be a function taking pairs as input. If we give first elements of pairs to Alice, and second to Bob, the question communication complexity asks is ?how much information do they have to communicate to each other in the worst case in order to compute f ??.

More precisely, we define protocols, which specify, at each step of the communication between Alice and Bob, who speaks (Alice or Bob), and what he says (a bit, 0 or 1), as a function of their respective inputs.

This simple framework, and some of its variants we discuss in this article, appear to us as a relevant way to study CA. The tools of communication complexity suggest experiments to test hypothesis about properties of CA (see Section 4).

Definition 1. A protocol P over domain X Y and range Z is a binary tree where each internal node v is labeled either by a map av : X ! f0; 1g or by a map bv : Y ! f0; 1g, and each leaf v is labeled either by a map Av : X ! Z or by a map Bv : Y ! Z.

The value of protocol P on input (x; y) 2 X Y is given by Av(x) (or Bv(y)) where Av (or Bv) is the label of the leaf reached by walking on the tree from the root, and walking left if av(x) = 0 (or bv(y) = 0), and walking right otherwise. We say that a protocol computes a function f : X Y ! Z if for any (x; y) 2 X Y , its value on input (x; y) is f (x; y).

Intuitively, each internal node specifies a bit to be communicated either by Alice or by Bob, whereas at leaves either Alice or Bob determines the final value of f since she (or he) has received enough information from the other.

Remark. In our formalism, we don?t ask both Alice and Bob to be able to give the final value. We do so to be able to consider protocols where communication is unidirectional (see below). Definition 2. We denote by cc( f ) the deterministic communication complexity of a function f : X Z. It is the minimal depth of a protocol tree computing f .

Y !

We study functions with the help of their associated matrices. In such matrices, rows are indexed by elements in X , columns by elements in Y . They are defined by Mi; j = f (i; j) 2 Z. For elementary CA, we represent the n-th iteration function of f n : f0; 1g2n+1 ! f0; 1g as f n : f0; 1gn f0; 1gn+1 ! f0; 1g. For instance, Figure 1 represents the matrix of elementary CA rule 178, when we give n bits to Alice (rows) and n + 1 bits to Bob (columns); i.e. when X = f0; 1gn and Y = f0; 1gn+1. We denote as M1n78 such a matrix.

From the study in [

Moreover, we call one-round communication complexity, denoted by cc1, the communication complexity when restricted to protocols where only one person (Alice or Bob) can speak. Precisely, a oneround protocol is a tree where either all internal nodes have labels of type av and all leaves labels of type Bv (Alice speaking to Bob who then gives the final answer), or all internal nodes have labels of type bv and all leaves labels of type Av (Bob speaking to Alice who gives the final answer).

Definition 3. The one-round deterministic communication complexity of a function f : X noted by cc1( f ), is the minimal depth of a one-round protocol tree computing f .

Y ! Z, de

This restriction is justified by the ease of experimental measures on the communication complexity of cellular automata it allows. More precisely, according to Fact 1, simply counting the number of different rows in a matrix gives the exact one-round communication complexity of a rule, while measuring the deterministic communication complexity of a function implies being able to find an optimal partition of its matrix into monochromatic rectangles.

Fact 1 (from [

When several rounds are allowed, the communication complexity is connected to the rank of
matrices. In fact, for an arbitrary boolean function f , we have the following bounds (see [

Open Problem 1. Is there a constant c > 1 verifying, for any function f : rank(Mf ) cc( f )

log(rank(Mf )) cc( f ) 2 O log(rank(Mf ))c :

Experimentally, the rank of matrices is the only parameter we computed in order to evaluate the multi-round communication complexity of CA. But it did not give tight bounds and the matrices to be considered are exponentially large.

A theorem by J. Hromkovic? and G. Schnitger [

TA(n) 2 W cc(cn(L))2 Where TA(n) is the time required by A to recognize L and cn is the characteristic function of L restricted to length n.

The proof uses the crossing sequence argument, introduced by Cobham [

We are interested in the sequence of iterations ( f n)n of the local rule of CA. So we won?t consider the communication complexity of a single function but the sequence of complexities associated to the family ( f n)n.

Another important point is the choice of how the input is split into 2 parts. We consider any possible splitting into 2 connected parts and take the worst case. Formally, given a CA local rule f : S2r+1 ! S, we denote by fi (with 0 i 2r + 1) the function fi : Si S2r+1 i ! S. We also define fin for all n 1 and all i with 0 i 2rn + 1.

Definition 4. The communication complexity cc(A ) of A is the function

n 7! 0 im2arxn+1 cc( fin); where f and r are the local rule and radius of A . We define in a similar way the one-round communication complexity cc1(A ).

Remark. This definition with arbitrary splitting of input is a slight modification of the definition
proposed by E. Goles and I. Rapaport in [

Maximal communication complexity can be reached by cellular automata.

3.1

Separation results
One could ask whether counting the number of different rows is a really accurate measure, and how large
is the gap between the cost of one-round protocols and the cost of protocols where several rounds are
allowed. We already know from [

Proposition 2. There exists a CA A such that cc1(A ) is an exponential in cc(A ). Proof sketch. For general functions in f0; 1g f0; 1g ! f0; 1g, there is one canonical function satisfying this relation between cc1 and cc : consider the complete binary tree with height h and label all its leaves and nodes with 0 or 1. The path associated to such a labeling is defined as follows : upon arriving on a node labeled with a 0 (resp. a 1), define the next node of the path as the root of the left (resp. right) subtree. The final value of the function is the label of the last node of the path (i.e. a leaf).

In this tree, give all odd levels to Alice and even ones to Bob. An easy multi-round protocol solves it
in communication complexity h : in each turn, either Alice or Bob tells each other the label of the current
node, giving to the other one the direction to follow (left or right) to get to the next node. It is a known
fact from [

We describe how a CA can encode this problem on figure 2, in terms of signals. The top of the tree is encoded on the sides of the initial configurations, and the final values (leaves) are at the center. The squares delimit the levels of the tree. Clearly, odd levels are on the left, while even ones are on the right side of the configuration.

The general behavior of this CA is to select data from the bottom of the tree. The green signals represent the data, the dashed ones represent data already selected, and the black ones are the selectors. All of them can carry the values 0 or 1. dotted signals separates the levels, transforming dashed signals into green ones. Black signals are selectors, they carry the values 0 (resp. 1) and transform into red signals carrying the value of the first (resp. second) green signal crossed.

At each step, the set of ?selected? leaves is halved by selections by black signals. Since these signals select the ?correct? (i.e. left of right, depending on their label) subtree, the last leaf remaining is the actual value of this instance of the tree problem.

0A A1A0A0A0A 1B0B0B0B0B1B1B1B B1B0B 0 0

1 1 0 0

0 The problem with the previous proof is that we build an artificial and complicated CA, with many states and an unclear local rule. A more accurate question is : Are there elementary CA with a low multi-round communication complexity, but a high lower bound for one-round protocols? We leave this as an open problem.

Considering the recent results by D. Woods and T. Neary [

Proposition 3. For any k

1, there exists a P-complete CA A such that cc(A ) 2 O(n1=k).

Proof sketch. Consider any Turing machine M . We construct a CA A able to simulate M only slowly but still in polynomial time: it takes nk steps of A to simulates n steps of M . Hence, by a suitable choice of M , A is P-complete.

First it is easy to construct a CA simulating M in real time. We encode each symbol of the tape alphabet of the Turing machine by a CA state, and add a ?layer? for the head, with ?!? symbols on its left and ? ? symbols on its right. We guarantee this way that there can be only one head : if a ?!? state is adjacent to a ? ? state without head between them, we propagate an ?error? state destroying everything.

We then add a new layer to slow down the simulation: it consists in a single particle (we use the same trick to ensure that there is only one particle) moving left and right inside a marked region of the configuration. More precisely, it goes right until it reaches the end of the marked region, then it adds a marked cell at the end and starts to move left to reach the other end, doing the same thing forever. Clearly, for any cell in a finite marked region, seeing n traversals of the particle takes W(n2) steps. Then, the idea is to authorize heads moves in the previous construction only at particle traversals. This way, n steps of M require n2 time steps of the automaton. By adding another layer, one can also slow down the above particle with the same principle and it is not difficult to finally construct a CA A such that n steps of M require nk time steps of A .

Now, the communication complexity of A is O(n1=k) because on any input of size n, either the ?error? appears, or a correct computation of O(n1=k) steps of M occurs. Distinguishing the two cases takes only constant communication. Moreover, in the case of a correct computation, it is sufficient to determine the initial position of particles, the sizes of marked regions (cost O(log(n)), and the initial position of the Turing heads as well as the O(n1=k) surrounding states. 3.2

Upper bounds We propose here a first scheme of complexity classes in cellular automata, based on their communication complexity. What we actually measure is 2D( f ). This is mainly justified by experiments : the protocols we got for cellular automata with D( f ) = 2 log n seemed much more sophisticated than those in log n. This is also justified by the fact that what we actually compute is either the number of different rows or columns, or the number of rectangles in the matrices. Thus, in the rest of this article, we will use the terms bounded or constant for CA with communication complexity bounded by a constant, linear for CA with communication complexity log n + O(1), quadratic for CA with communication complexity 2 log n + O(1), and so on.

In this section, we give some well-known properties of CA that induce a bounded communication
complexity. The results below are adaptations of ideas of [

Proposition 4. Let A be any CA of local function f . If there is a function g : N ! N such that f n depends on only g(n) cells, then cc(A ) g(n)=2.

Following the work of M. Sablik [

Another set of CA with that property is the set of linear CA. A CA A with state set S, radius r and local global rule G is linear if there is an operator such that (S; ) is a semi-group with neutral element e and for all configurations c and c0 we have:

G(c c0) = G(c)

G(c0); where is the uniform extension of

to configurations. Proposition 5. If A is linear then cc1(A ) is bounded.

The proof appears in [

The base ingredient is the relation of sub-automaton. A CA A is a sub-automaton of a CA B, denote A v B, if there is an injective map i from SA to SB such that i GA = GB i , where i : SAZ ! SBZ denotes the uniform extension of i .

A CA A simulates a CA B if some rescaling of A is a sub-automaton of some rescaling of B. The ingredients of the rescalings are simple: packing cells into blocs, iterating the rule and composing with a translation. Formally, given any state set Q and any m 1, we define the bijective packing map Qm Z by: 8z 2 Z : bm(c) (z) = c(mz); : : : ; c(mz + m 1) for all c 2 QZ. The rescaling A <m;t;z> of A by parameters m (packing), t (shifting) is the CA of state set Qm and global rule: 1 (iterating) and z 2 Z bm sz

GtA b 1 m : With these definitions, we say that A simulates B, denoted A 4 B, if there are rescaling parameters m1, m2, t1, t2, z1 and z2 such that A <m1;t1;z1> v B<m2;t2;z2>.

We can now naturally define the notion of universality associated to this simulation relation. Definition 5. A is intrinsically universal if for all B it holds B 4 A .

This definition of universality may seem very resctrictive. In fact, many so-called universal CA (i.e.
Turing-universal CA) are also intrinsically universal (see [

But, most importantly, by completely formalizing1 the notion of universality, we facilitate the proof of negative results.

We are going to show that the tool of communication complexity is precisely a good candidate to obtain negative results. The idea is simple: if A simulates B then the communication complexity of A must be ?greater? than the communication complexity of B.

More precisely, we consider the following relation of comparison between functions from N to N:
f1
f2 () 9a ; b ; g
1; 8n 2 N : f1(a n)
b f2(g n):
1There is actually no consensus on the formal definition of Turing-universality in CA (see [

Iterating: the complexity function of A t is n 7! f (t n) if f is the complexity function of A . Shifting: this operation only affects the splitting of inputs. Since we always take in each case the splitting of maximum complexity, this has no influence on the final complexity function. Packing: let A be CA with local rule f and states set S. Consider any sequence of valid protocols (Pj), one for each splitting of inputs of f n, and denote by h : (Sm)i (Sm)k i ! Sm some splitting of the nth iteration of the local rule of A <m;1;0>. By definition of packing map bm, a valid protocol for h is deduced by simultaneous application of protocols Pj; : : : ; Pj+m 1 (for a suitable choice of j), each being used to determined one component of the resulting value of h which belongs to Sm. It follows that cc(h) m cc( f n).

cc (A <m;t;z>), cc (A <m;t;z>) cc(A ) and if A v B then cc(A ) cc(B).

From Proposition 1, we derive the following necessary condition for intrinsic universality. It is one of the main motivations to study communication complexity of CA, both theoretically and experimentally.

4

In this section we concentrate on elementary cellular automata (ECA) : dimension one, two states, and radius r = 1. And we split the input as follows : f : f0; 1gn f0; 1gn+1 ! f0; 1g. Since any ECA has the same (one-round) communication complexity as its reflex and its conjugate, we propose here a classification of the 88 nonisomorphic ECA. Since we only consider one-round communication complexity here, Fact 1 allows us to consider matrices associated to functions and study the number of their different rows or columns.

Therefore, for the sake of clarity, the name we give to classes of ECA is related to the number of different rows and columns (instead of the one-round communication complexity, which is the logarithm of the previous). 4.1

Bounded (by a constant) As shown above, several results allow us to bound the (one-round) communication complexity of many CA.

The ECA proved to be in this class are the following 44 ones: 0, 1, 2, 3, 4, 5, 7, 8, 10, 12, 13, 15, 19, 24, 27, 28, 29, 32, 34, 36, 38, 42, 46, 51, 60, 72, 76, 78, 90, 105, 108, 128, 130, 136, 138, 140, 150, 156, 160, 162, 170, 172, 200, 204 (and all their reflexes, conjugates, and reflex-conjugates). 4.2

Linear
Consider for instance rule 178, which has been studied recently by D. Regnault [

It is not difficult, using the methods of [

The ECA Rule 178 is given by the following local rule :

Proposition 7. Protocol P is correct for ECA Rule 178.

Proof. First remark that configurations 01 and 10 map to each other for any values of their right or left neighbour (which we can see in figure 3 where undetermined cells are represented in gray), and thus stay stable. So once Alice knows where the first 01 or 10 occurs, she can assume w.l.g. that the rest of Bob?s part are only zeros (the final result is the same). But then, she also knows the beginning of Bob?s part, so she can compute the final result of Rule 178.

Proposition 8. Protocol P is optimal even as a multi-round protocol.

Proof. To show this, we use the results of [

C = f(0n 2k 1102k; c2kccn 2k)j0 2k n 1; c 2 f0; 1gg

0n 2k 1102kc2kccn 2k is always n mod 2, while for any i 6= j, the result of 0n 2i 1102ic2 jccn 2 j is n + 1 mod 2.

For the case c = 1, this is shown by our previous remark on stable configuration. For c = 0, this is a simple remark on the space-time diagrams of rule 178.

Then jCj = 2 bn=2c, and thus no deterministic protocol, even multiround, could predict rule 178 in less than log n + 1 rounds. Remark. The same argument can be used for rule 50 (and thus also 179).

We believe that the linearity of Rule 178 and the fact that it is amenable to other types analysis is not
a coincidence.
As soon as we move up in our hierarchy the underlying protocols become rather sophisticated. In fact,
for Rule 218, we prove in [

We encountered Rule 218 when trying to find a (kind of) double-quiescent palindrome-recognizer. Despite the fact that it belongs to class II (according to Wolfram?s classification), it mimics Rule 90 (class III) for very particular initial configurations.

Behind the following ?proofs? there are lots of lemmas that we are not even stating. Therefore, the purpose here is just to give an idea of how we proceed. We are considering the case when the central cell is 0. We write f instead of f218.

Definition 6. We say that a word in f0; 1g is additive if the 1s are isolated and every consecutive couple of 1s is separated by an odd number of 0s.

Notation 1. Let a be the maximum index i for which xi : : : x10 is additive. Let b be the maximum index j for which 0y1 : : : y j is additive. Let x0 = xa : : : x1 2 f0; 1ga and y0 = y1 : : : yb 2 f0; 1gb . Notation 2. Let l be the minimum index i for which xi = 1. If such index does not exist we define l = 0. Let r be the minimum index j for which y j = 1. If such index does not exist we define r = 0. Proposition 9. There exists a one-round f -protocol P0 with cost 2dlog(n)e + 1.

Proof. Recall the Alice knows x and Bob knows y. P0 goes as follows. Alice sends to Bob a, l, and a = f a (x0; 0; 0a ). The number of bits is therefore 2dlog(n)e + 1.

If l = 0 then Bob knows (by definition of l) that x = 0n and he outputs f n(0n; 0; y). If r = 0 his output depends on a. If a = n he outputs a and if a < n he outputs 1. We can assume now that neither l nor r are 0. The way Bob proceeds depends mainly on the parity of jl + r 1j.

1 Bob outputs 1. If a = b = k he outputs a + f k(0k; 0; y0).

Case jl + r 1j is even. Bob compares r with l. If l r 1 then Bob outputs f n(1n l+10l 1; 0; y) if l r + 3 and 1 otherwise. If l r 3 then he outputs a = f a (x0; 0; 0a ) if r = a + 1 and 1 otherwise.

Now we exhibit lower bounds for the number of different rows of the corresponding matrix. If these bounds appear to be tight then, from Fact 1, they can be used for proving the optimality of our protocol. Proposition 10. The cost of any one-round f -protocol is at least 2dlog(n)e 5.

Proof. Consider the following subsets of f0; 1gn. First, S3 = f1n 3000g. Also,

S5 = f1n 500000; 1n 501000g:

2 such that 2k + 1

n, we define S2k+1 = f1n 2k 102k+1g [ f1n 2k 10a10bj a odd, b odd, b

Let xn : : : x1 2 S2k+1 and x?n : : : x?1 2 S2k?+1 with k 6= k?. It follows that the rows of Mcf;n indexed by xn : : : x1 and x?n : : : x?1 are different.

Let x = xn : : : x1, x? = x?n : : : x?1 2 S2k+1 with x 6= x?. It follows that there exists y = y1 : : : yn 2 f0; 1gn such that f n(x; 0; y) 6= f n(x?; 0; y). 4.4

Non-polynomial Our experiments suggested the existence of (at least) two subclasses of this class of ?hard? ECA.

Automata with a high one-round communication complexity but a low matrix rank (suggesting a low multi-round communication complexity), meaning they are easy to predict with several actors and a protocol between them, but the exact influence of each cell of the initial state is hard to determine. We do not know whether this class really exists among ECA, but our experiments suggest that rule 30 may be a candidate.

Automata that are ?intrinsically hard?, meaning that they do not have a deterministic protocol in the previous classes. 5

Input splitting. When defining the communication complexity in CA we consider the worst case for splitting the inputs for each n. We believe that the sequence (sn)n of such worst-case splittings is meaningful and raises several interesting questions: Is sn unique for each n? If it is the case, what is the function n 7! sn? Is it linear, thus showing a direction of maximal ?information exchange? along time? What is the meaning of such a direction?

Higher dimensional CA and multi-party protocols. We focused our study on the model where Alice and Bob need to communicate to predict a given CA. There are also other models of protocols with k players, but the difficulty of experimentation would probably not be the same. A greater number of players seems more natural for dimension 2 or more, since we can partition the set of dependant cells into adjacent regions. But the two-player framework could also be applied to higher dimensional CA.

Nondeterministic protocols. A possible generalization of our definitions of protocols is to allow
Alice and Bob to take nondeterministic steps in the protocol tree. This gives us other interesting tools
and measures, for instance the notion of a cover of a matrix, which seems linked to circuits. We can find
in [

Probabilistic protocols. Another relevant generalization of communication complexity for the study
of CA is randomized complexity, where errors are allowed. In this model, Alice and Bob are allowed
to toss a coin before communicating (see [