Communications in cellular automata
Eric Goles, Pierre-Etienne Meunier, Ivan Rapaport, Guillaume Theyssier
TT. Neary, D. Woods, A.K. Seda and N. Murphy (Eds.):The Complexity of Simple Programs 2008.EPTCS 1, 2009, pp. 81–92, doi:10.4204/EPTCS.1.8 c (cid:13)
E. Goles, P.-E. Meunier, I. Rapaport and G. Theyssier
Communications in cellular automata ∗ Eric Goles
Facultad de Ingenier´ıa y Ciencias, Universidad Adolfo Ib´a˜nez, Santiago, ChileCentro de Modelamiento Matem´atico (UMI 2807 CNRS), Universidad de Chile, Chile
Pierre-Etienne Meunier
LAMA, Universit´e de Savoie, CNRS, France
Ivan Rapaport [email protected]
Centro de Modelamiento Matem´atico (UMI 2807 CNRS), Universidad de Chile, ChileDepartamento de Ingenier´ıa Matem´atica, Universidad de Chile, Chile
Guillaume Theyssier
LAMA, Universit´e de Savoie, CNRS, France
The goal of this paper is to show why the framework of communication complexity seems suitable forthe study of cellular automata. Researchers have tackled different algorithmic problems ranging fromthe 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 cellularautomaton, then we can understand its behavior . Throughout this paper we restrict our study to one-dimensional cellular automata. These are infinitecollections of cells arranged linearly, each having a state from a finite set. The dynamics of the system isgoverned by a local rule applied uniformly and synchronously to the lattice of cells.A cellular automaton (CA) is a triple A = ( S , r , f ) where: • S is a (finite) state set , • r is the neighborhood radius , • f : S r + → S is the local transition function .A coloring of the lattice Z with states from S ( i.e. an element of S Z ) is called a configuration . To A we associate a global function G acting on configurations by synchronous and uniform application of thelocal transition function. Formally, G : S Z → S Z is defined by: G ( x ) z = f ( x z − r , . . . , x z + r ) for all z ∈ Z . Several CA can share the same global function although there are syntactically different(different radii and local functions). However, as we will see below (section 3), the main property weare interested in (namely, communication complexity) is independant of the particular choice of the ∗ Partially supported by Programs Fondap, Basal-CMM, Fondecyt 1070022 (E.G), Fondecyt 1090156 (I.R.), and InstitutoMilenio ICDB a r X i v : . [ c s . CC ] J un Communications in cellular automata syntactical representation. Moreover, an important part of the paper (Section 4) focuses on elementaryCA, 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 statesof the rn left and rn right neighbouring cells. More precisely, we define the n -th iteration of local rule f n : { , } rn + → { , } recursively: f = f and, for n ≥ f n ( z − rn . . . z , z , z . . . z rn ) = f n − ( f ( z − rn , . . . , z − rn + r ) . . . f ( z rn − r , . . . , z rn )) . Finally, we call
P-complete a cellular automaton such that the problem of predicting F n on all con-figurations of size 2 rn + if we are able to give a simple explicit description off n (for arbitrary n), then we can understand the behavior of the corresponding CA . Communication complexity is a model introduced by A. C.-C. Yao in [16], and designed at first forlower-bounding the amount of communication needed in parallel programs. In this model we considertwo players, namely Alice and Bob, each with arbitrary computational power and talking to each otherto decide the value of a given function.For instance, let f : X × Y → Z be a function taking pairs as input. If we give first elements of pairsto Alice, and second to Bob, the question communication complexity asks is “how much information dothey 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 Aliceand Bob, who speaks (Alice or Bob), and what he says (a bit, 0 or 1), as a function of their respectiveinputs.This simple framework, and some of its variants we discuss in this article, appear to us as a relevantway to study CA. The tools of communication complexity suggest experiments to test hypothesis aboutproperties 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 vis labeled either by a map a v : X → { , } or by a map b v : Y → { , } , and each leaf v is labeled eitherby a map A v : X → Z or by a map B v : Y → Z.The value of protocol P on input ( x , y ) ∈ X × Y is given by A v ( x ) (or B v ( y ) ) where A v (or B v ) isthe label of the leaf reached by walking on the tree from the root, and walking left if a v ( x ) = (orb v ( y ) = ), and walking right otherwise. We say that a protocol computes a function f : X × Y → Z if forany ( x , y ) ∈ 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, whereasat leaves either Alice or Bob determines the final value of f since she (or he) has received enoughinformation 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 soto 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 × Y → Z. It is the minimal depth of a protocol tree computing f .
We study functions with the help of their associated matrices. In such matrices, rows are indexed byelements in X , columns by elements in Y . They are defined by M i , j = f ( i , j ) ∈ Z . For elementary CA, we . Goles, P.-E. Meunier, I. Rapaport and G. Theyssier n -th iteration function of f n : { , } n + → { , } as f n : { , } n × { , } n + → { , } . Forinstance, Figure 1 represents the matrix of elementary CA rule 178, when we give n bits to Alice (rows)and n + X = { , } n and Y = { , } n + . We denote as M n such amatrix. Figure 1: Matrices of rule 178, for n = n = X and Y ). Soa lower bound for the deterministic communication complexity of a function φ is given by log C P ( φ ) ,where C P ( φ ) stands for the partition number of φ , i.e. the number of rectangles needed in a minimalpartition of the matrix into monochromatic rectangles.Moreover, we call one-round communication complexity , denoted by cc , the communication com-plexity when restricted to protocols where only one person (Alice or Bob) can speak. Precisely, a one-round protocol is a tree where either all internal nodes have labels of type a v and all leaves labels of type B v (Alice speaking to Bob who then gives the final answer), or all internal nodes have labels of type b v and all leaves labels of type A v (Bob speaking to Alice who gives the final answer). Definition 3.
The one-round deterministic communication complexity of a function f : X × Y → Z, de-noted by cc ( f ) , is the minimal depth of a one-round protocol tree computing f . This restriction is justified by the ease of experimental measures on the communication complexity ofcellular automata it allows. More precisely, according to Fact 1, simply counting the number of differentrows in a matrix gives the exact one-round communication complexity of a rule, while measuring thedeterministic communication complexity of a function implies being able to find an optimal partition ofits matrix into monochromatic rectangles.
Fact 1 (from [7]) . Let f be a binary function of n variables and M f ∈ { , } n × n its matrix represen-tation, defined by M f ( x , y ) = f ( xy ) for x , y ∈ { , } n . Let d ( M f ) be minimum between the number ofdifferent rows and the number of different columns in M f . We have cc ( f ) = (cid:6) log (cid:0) d ( M f ) (cid:1)(cid:7) . When several rounds are allowed, the communication complexity is connected to the rank of matri-ces. In fact, for an arbitrary boolean function f , we have the following bounds (see [7]):rank ( M f ) ≥ cc ( f ) ≥ log ( rank ( M f )) Moreover, the following conjecture appears in [11] :
Open Problem 1.
Is there a constant c > verifying, for any function f : cc ( f ) ∈ O (cid:0) log ( rank ( M f )) c (cid:1) . Communications in cellular automata
Experimentally, the rank of matrices is the only parameter we computed in order to evaluate themulti-round communication complexity of CA. But it did not give tight bounds and the matrices to beconsidered are exponentially large.A theorem by J. Hromkoviˇc and G. Schnitger [5] upper bounds the communication complexity ofTuring computations:
Theorem 1.
For a language L ⊆ { , } ∗ and a nondeterministic TM A recognizing this language, wehave T A ( n ) ∈ Ω (cid:0) cc ( χ n ( L )) (cid:1) Where T A ( n ) is the time required by A to recognize L and χ n is the characteristic function of L restrictedto length n. The proof uses the crossing sequence argument, introduced by Cobham [1]
We are interested in the sequence of iterations ( f n ) n of the local rule of CA. So we won’t consider thecommunication 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 possiblesplitting into 2 connected parts and take the worst case. Formally, given a CA local rule f : S r + → S ,we denote by f i (with 0 ≤ i ≤ r +
1) the function f i : S i × S r + − i → S . We also define f ni for all n ≥ i with 0 ≤ i ≤ rn + Definition 4.
The communication complexity cc ( A ) of A is the functionn (cid:55)→ max ≤ i ≤ rn + cc ( f ni ) , where f and r are the local rule and radius of A . We define in a similar way the one-round communica-tion complexity cc ( A ) . Remark.
This definition with arbitrary splitting of input is a slight modification of the definition pro-posed by E. Goles and I. Rapaport in [3], where the central cell is fixed, and Alice and Bob recieveexactly the same number of input cells.
Maximal communication complexity can be reached by cellular automata.
Proposition 1 ([3]) . There is a CA A such that cc ( A ) ∈ Ω ( n ) . One could ask whether counting the number of different rows is a really accurate measure, and how largeis the gap between the cost of one-round protocols and the cost of protocols where several rounds areallowed. We already know from [7] that the gap between one-round protocols and multi-round protocolscan be exponential. The following fact shows that we get the same exponential gap if we restrict ourselvesto functions predicting CA.
Proposition 2.
There exists a CA A such that cc ( A ) is an exponential in cc ( A ) . . Goles, P.-E. Meunier, I. Rapaport and G. Theyssier Proof sketch.
For general functions in { , } ∗ × { , } ∗ → { , } , there is one canonical function satis-fying this relation between cc and cc : consider the complete binary tree with height h and label all itsleaves and nodes with 0 or 1. The path associated to such a labeling is defined as follows : upon arrivingon 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 itin communication complexity h : in each turn, either Alice or Bob tells each other the label of the currentnode, giving to the other one the direction to follow (left or right) to get to the next node. It is a knownfact from [7] that this problem cannot be solved with a one-round protocol in o ( n ) rounds.We describe how a CA can encode this problem on figure 2, in terms of signals. The top of the treeis encoded on the sides of the initial configurations, and the final values (leaves) are at the center. Thesquares delimit the levels of the tree. Clearly, odd levels are on the left, while even ones are on the rightside of the configuration.The general behavior of this CA is to select data from the bottom of the tree. The green signalsrepresent 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 signalsinto green ones. Black signals are selectors, they carry the values 0 (resp. 1) and transform into redsignals 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 signalsselect the “correct” (i.e. left of right, depending on their label) subtree, the last leaf remaining is theactual value of this instance of the tree problem. A (cid:3) A A A A A (cid:3) B B B B B B B B (cid:3) B B B Figure 2: A cellular automaton computing the tree problem, and the corresponding treeThe problem with the previous proof is that we build an artificial and complicated CA, with manystates and an unclear local rule. A more accurate question is : Are there elementary CA with a lowmulti-round communication complexity, but a high lower bound for one-round protocols? We leave thisas an open problem.Considering the recent results by D. Woods and T. Neary [8], a very natural question one could askis the following: What do computational properties of CA, such as P-completeness, imply on the itscommunication complexity? As shown by the following proposition, one can build P-complete cellularautomata with arbitrarily low communication complexity.
Proposition 3.
For any k ≥ , there exists a P-complete CA A such that cc ( A ) ∈ O ( n / k ) .Proof sketch. Consider any Turing machine M . We construct a CA A able to simulate M only slowlybut still in polynomial time: it takes n k steps of A to simulates n steps of M . Hence, by a suitable choice6 Communications in cellular automata 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 tapealphabet of the Turing machine by a CA state, and add a “layer” for the head, with ’ → ’ symbols on itsleft and ’ ← ’ symbols on its right. We guarantee this way that there can be only one head : if a ’ → ’ state isadjacent 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 thesame trick to ensure that there is only one particle) moving left and right inside a marked region of theconfiguration. More precisely, it goes right until it reaches the end of the marked region, then it addsa 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 Ω ( n ) steps. Then,the idea is to authorize heads moves in the previous construction only at particle traversals. This way, n steps of M require n time steps of the automaton. By adding another layer, one can also slow down theabove particle with the same principle and it is not difficult to finally construct a CA A such that n stepsof M require n k time steps of A .Now, the communication complexity of A is O ( n / k ) because on any input of size n , either the“error” appears, or a correct computation of O ( n / k ) steps of M occurs. Distinguishing the two casestakes only constant communication. Moreover, in the case of a correct computation, it is sufficient todetermine the initial position of particles, the sizes of marked regions (cost O ( log ( n ) ), and the initialposition of the Turing heads as well as the O ( n / k ) surrounding states. We propose here a first scheme of complexity classes in cellular automata, based on their communicationcomplexity. What we actually measure is 2 D ( f ) . This is mainly justified by experiments : the protocolswe got for cellular automata with D ( f ) = 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 rowsor columns, or the number of rectangles in the matrices. Thus, in the rest of this article, we will usethe terms bounded or constant for CA with communication complexity bounded by a constant, linear for CA with communication complexity log n + O ( ) , quadratic for CA with communication complexity2 log n + O ( ) , and so on.In this section, we give some well-known properties of CA that induce a bounded communicationcomplexity. The results below are adaptations of ideas of [3] to the formalism adopted in the presentpaper. 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 ) / . Following the work of M. Sablik [14], one can characterize the set of CA having a bounded numberof dependant cells ( i.e. a bounded function g ( n ) ): they are exactly these CA which are equicontinuous insome direction (theorem 4.3 of [14]). This set contains the nilpotent CA (a CA is nilpotent if it convergesto a unique configuration from any initial configuration, i.e. f n is a constant for any large enough n ). Corollary 1. If A is equicontinuous in some direction then cc ( A ) is bounded. Another set of CA with that property is the set of linear CA. A CA A with state set S , radius r andlocal 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 c (cid:48) we have: G ( c ⊕ c (cid:48) ) = G ( c ) ⊕ G ( c (cid:48) ) , where ⊕ is the uniform extension of ⊕ to configurations. . Goles, P.-E. Meunier, I. Rapaport and G. Theyssier Proposition 5. If A is linear then cc ( A ) is bounded. The proof appears in [3] in a different setting. The idea is that there is a simple one-round protocol tocompute linear functions: Alice and Bob can each compute on their own the image the function wouldproduce assuming the other party has only the neutral element as input, then Alice or Bob communicatethis result to the other who can answer the final result by linearity.
Since the pioneering work of J. von Neumman [9], universality in CA has received a lot of attention(see [12] for a survey). Historically, the notion of universality used for CA was more or less an adaptationof the classical Turing-universality. Later, a stronger notion called intrinsic universality was proposed:a CA is intrinsically universal if it is able to simulate any other CA. This definition relies on a notion ofsimulation which is formalized below.The base ingredient is the relation of sub-automaton. A CA A is a sub-automaton of a CA B , denote A (cid:118) B , if there is an injective map ι from S A to S B such that ι ◦ G A = G B ◦ ι , where ι : S Z A → S Z B denotes the uniform extension of ι .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 composingwith a translation. Formally, given any state set Q and any m ≥
1, we define the bijective packing map b m : Q Z → (cid:0) Q m (cid:1) Z by: ∀ z ∈ Z : (cid:0) b m ( c ) (cid:1) ( z ) = (cid:0) c ( mz ) , . . . , c ( mz + m − ) (cid:1) for all c ∈ Q Z . The rescaling A < m , t , z > of A by parameters m (packing), t ≥ z ∈ Z (shifting) is the CA of state set Q m and global rule: b m ◦ σ z ◦ G t A ◦ b − m . With these definitions, we say that A simulates B , denoted A (cid:52) B , if there are rescaling parameters m , m , t , t , z and z such that A < m , t , z > (cid:118) B < m , t , z > .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 (cid:52) 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 [12] and [2] for the particular case of Game ofLife), although there is still a gap for one-dimensional CA (the elementary CA 110 is Turing-universaland no elementary CA is known to be intrinsically universal). Moreover, intrinsic universality appears tobe very common in some classes of CA (see [15]).But, most importantly, by completely formalizing the notion of universality, we facilitate the proofof negative results.We are going to show that the tool of communication complexity is precisely a good candidate toobtain 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 : φ ≺ φ ⇐⇒ ∃ α , β , γ ≥ , ∀ n ∈ N : φ ( α n ) ≤ β φ ( γ n ) . There is actually no consensus on the formal definition of Turing-universality in CA (see [2] for a discussion about encod-ing/decoding problems). Communications in cellular automata
Proposition 6. If A (cid:52) B then cc ( A ) ≺ cc ( B ) .Proof sketch. We consider successively each ingredient involved in the simulation relation:
Sub-automaton: if A (cid:118) B then each valid protocol to compute iterations of B is also a valid protocolto compute iterations of A (up to state renaming). Iterating: the complexity function of A t is n (cid:55)→ φ ( t · n ) if φ is the complexity function of A . Shifting: this operation only affects the splitting of inputs. Since we always take in each case thesplitting 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 ( P j ) ,one for each splitting of inputs of f n , and denote by h : ( S m ) i × ( S m ) k − i → S m some splitting of the n th iteration of the local rule of A < m , , > . By definition of packing map b m , a valid protocol for h is deduced by simultaneous application of protocols P j , . . . , P j + m − (for a suitable choice of j ),each being used to determined one component of the resulting value of h which belongs to S m . Itfollows that cc ( h ) ≤ m · cc ( f n ) .Therefore we have: cc ( A ) ≺ cc ( A < m , t , z > ) , cc ( A < m , t , z > ) ≺ cc ( A ) and if A (cid:118) B then cc ( A ) ≺ cc ( B ) .From Proposition 1, we derive the following necessary condition for intrinsic universality. It is one ofthe main motivations to study communication complexity of CA, both theoretically and experimentally. Corollary 2. If A is intrinsically universal then cc ( A ) ∈ Ω ( n ) . In this section we concentrate on elementary cellular automata (ECA) : dimension one, two states, andradius r =
1. And we split the input as follows : f : { , } n × { , } n + → { , } . Since any ECA hasthe same (one-round) communication complexity as its reflex and its conjugate, we propose here a clas-sification of the 88 nonisomorphic ECA. Since we only consider one-round communication complexityhere, Fact 1 allows us to consider matrices associated to functions and study the number of their differentrows or columns.Therefore, for the sake of clarity, the name we give to classes of ECA is related to the number ofdifferent rows and columns (instead of the one-round communication complexity, which is the logarithmof the previous). As shown above, several results allow us to bound the (one-round) communication complexity of manyCA.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). . Goles, P.-E. Meunier, I. Rapaport and G. Theyssier Consider for instance rule 178, which has been studied recently by D. Regnault [13] using percolationtheory. The author considered the case where each cell has an independent probability ρ to be updatedin each step. He studied Rule 178 because it “exhibited rich behavior such as phase transition”. Despiteits complexity, this CA was amenable to formal analysis: the proofs were based on a coupling betweenits space-time diagram and oriented percolation on a graph.It is not difficult, using the methods of [7], to prove that the communication complexity of CA 178grows as Θ ( n ) . Notice that in order to get such a result we must find, on one hand, a communicationprotocol (upper bound) of complexity ≈ log ( n ) and, on the other hand, to exhibit a “fooling set” (i.e. aset C of configurations such that for any couple ( x , y ) of configurations of C , x and y are necessarily intwo distinct monochromatic rectangles) of size in Ω ( n ) .The ECA Rule 178 is given by the following local rule :
00 0 0 10 0 1 00 1 0 00 1 1 11 0 0 11 0 1 01 1 0 11 1 1
There is a very simple protocol P in log n + c the value of the central cellat the beginning (Bob knows it), then Bob sends the length of the longest string of cells with value c ,starting from the left of his part, to Alice. 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 leftneighbour (which we can see in figure 3 where undetermined cells are represented in gray), and thus staystable. So once Alice knows where the first 01 or 10 occurs, she can assume w.l.g. that the rest of Bob’sFigure 3: Evolution of 01 and 10 for rule 178part are only zeros (the final result is the same). But then, she also knows the beginning of Bob’s part, soshe 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 [7] and exhibit a fooling set. Let C = { ( n − k − k , c k cc n − k ) | ≤ k ≤ n − , c ∈ { , }} First remark that the result of Rule 178 on configurations of the form0 n − k − k c k cc n − k is always n mod 2, while for any i (cid:54) = j , the result of 0 n − i − i c j cc n − j is n + c =
1, this is shown by our previous remark on stable configuration. For c =
0, this is asimple remark on the space-time diagrams of rule 178.Then | C | = · (cid:98) n / (cid:99) , and thus no deterministic protocol, even multiround, could predict rule 178 inless than log n + Communications in cellular automata
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 nota coincidence.
As soon as we move up in our hierarchy the underlying protocols become rather sophisticated. In fact,for Rule 218, we prove in [4] that if c = ( n ) bits). The difference in the difficulty between sending 1 position ( Θ ( n ) behavior) and 2 positions( Θ ( n ) behavior) is huge.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 (classIII) for very particular initial configurations.Behind the following “proofs” there are lots of lemmas that we are not even stating. Therefore, thepurpose here is just to give an idea of how we proceed. We are considering the case when the central cellis 0. We write f instead of f . Definition 6.
We say that a word in { , } ∗ is additive if the 1s are isolated and every consecutive coupleof 1s is separated by an odd number of 0s. Notation 1.
Let α be the maximum index i for which x i . . . x is additive. Let β be the maximum indexj for which y . . . y j is additive. Let x (cid:48) = x α . . . x ∈ { , } α and y (cid:48) = y . . . y β ∈ { , } β . Notation 2.
Let l be the minimum index i for which x i = . If such index does not exist we define l = .Let r be the minimum index j for which y j = . If such index does not exist we define r = . Proposition 9.
There exists a one-round f -protocol P with cost (cid:100) log ( n ) (cid:101) + .Proof. Recall the Alice knows x and Bob knows y . P goes as follows. Alice sends to Bob α , l , and a = f α ( x (cid:48) , , α ) . The number of bits is therefore 2 (cid:100) log ( n ) (cid:101) + l = l ) that x = n and he outputs f n ( n , , y ) . If r = α . If α = n he outputs a and if α < 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 | l + r − | . Case | l + r − | is odd. If | α − β | ≥ α = β = k he outputs a + f k ( k , , y (cid:48) ) . Case | l + r − | is even. Bob compares r with l . If l ≥ r − f n ( n − l + l − , , y ) if l ≥ r + l ≤ r − a = f α ( x (cid:48) , , α ) if r = α + Proposition 10.
The cost of any one-round f -protocol is at least (cid:100) log ( n ) (cid:101) − .Proof. Consider the following subsets of { , } n . First, S = { n − } . Also, S = { n − , n − } . In general, for every k ≥ k + ≤ n , we define S k + = { n − k − k + } ∪ { n − k − a b | a odd, b odd, b ≥ a + b = k } . . Goles, P.-E. Meunier, I. Rapaport and G. Theyssier x n . . . x ∈ S k + and ˜ x n . . . ˜ x ∈ S k + with k (cid:54) = ˜ k . It follows that the rows of M c , nf indexed by x n . . . x and ˜ x n . . . ˜ x are different.Let x = x n . . . x , ˜ x = ˜ x n . . . ˜ x ∈ S k + with x (cid:54) = ˜ x . It follows that there exists y = y . . . y n ∈ { , } n such that f n ( x , , y ) (cid:54) = f n ( ˜ x , , y ) . 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 alow multi-round communication complexity), meaning they are easy to predict with several actorsand a protocol between them, but the exact influence of each cell of the initial state is hard todetermine. We do not know whether this class really exists among ECA, but our experimentssuggest that rule 30 may be a candidate. • Automata that are “intrinsically hard”, meaning that they do not have a deterministic protocol inthe previous classes.
Input splitting.
When defining the communication complexity in CA we consider the worst case forsplitting the inputs for each n . We believe that the sequence ( s n ) n of such worst-case splittings is mean-ingful and raises several interesting questions: Is s n unique for each n ? If it is the case, what is thefunction n (cid:55)→ s n ? 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 whereAlice 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 ofplayers seems more natural for dimension 2 or more, since we can partition the set of dependant cellsinto 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 allowAlice and Bob to take nondeterministic steps in the protocol tree. This gives us other interesting toolsand measures, for instance the notion of a cover of a matrix, which seems linked to circuits. We can findin [7] a link between nondeterministic protocols and the minimal number of rectangles needed to covera matrix with possible intersections between rectangles.
Probabilistic protocols.
Another relevant generalization of communication complexity for the studyof CA is randomized complexity, where errors are allowed. In this model, Alice and Bob are allowedto toss a coin before communicating (see [6] regarding one-round randomized complexity and [10] formany-round). Allowing randmoness just changes the notion of complexity and can be applied to deter-ministic CA, but it may make sense to use this framework for stochastic CA (see for instance [13]).
References [1] A. Cobham (1964):
The intrinsic computational difficulty of functions . In:
Congress for Logic, Mathematicsand Philosophy of science . pp. 24–30.[2] B. Durand & Z. R´oka (1999):
Cellular Automata: a Parallel Model , Mathematics and its Applications. Communications in cellular automata [3] Christoph D¨urr, Ivan Rapaport & Guillaume Theyssier (2004):
Cellular automata and communication com-plexity . Theor. Comput. Sci. http://dx.doi.org/10.1016/j.tcs.2004.03.017 .[4] Eric Goles, Cedric Little & Ivan Rapaport (2008):
Understanding a non-trivial cellular automaton by findingits simplest underlying communication protocol . In: Seok-Hee Hong & Hiroshi Nagamochi, editors:
ISAAC , Lecture Notes in Computer Science
Communication Complexity and Sequential Compuation . In:
MFCS ’97: Proceedings of the 22nd International Symposium on Mathematical Foundations of ComputerScience . Springer-Verlag, London, UK, pp. 71–84.[6] Ilan Kremer, Noam Nisan & Dana Ron (2001):
On randomized one-round communication complexity . Com-putational Complexity
Communication complexity . Cambridge university press.[8] Turlough Neary & Damien Woods (2006):
P-completeness of cellular automaton Rule 110 . In:
In Interna-tional Colloquium on Automata Languages and Programming (ICALP), volume 4051 of LNCS . Springer,pp. 132–143.[9] John von Neumann (1967):
The theory of self-reproducing cellular automata . University of Illinois Press,Urbana, Illinois.[10] Noam Nisan & Avi Wigderson (1993):
Rounds in Communication Complexity Revisited . SIAM J. Comput.
On Rank vs. Communication Complexity . Electronic Colloquium onComputational Complexity (ECCC) http://eccc.hpi-web.de/eccc-reports/1994/TR94-001/index.html .[12] Nicolas Ollinger (2008):
Universalities in Cellular Automata: a (short) survey . In: B. Durand, editor:
Symposium on Cellular Automata Journ´ees Automates Cellulaires (JAC’08) . MCCME Publishing House,Moscow, pp. 102–118.[13] Damien Regnault (2008):
Directed Percolation Arising in Stochastic Cellular Automata Analysis . In: EdwardOchmanski & Jerzy Tyszkiewicz, editors:
MFCS , Lecture Notes in Computer Science
Directional dynamics for cellular automata: A sensitivity to initial condition ap-proach . Theor. Comput. Sci.
How Common Can Be Universality for Cellular Automata?
In:
STACS . pp.121–132. Available at http://dx.doi.org/10.1007/b106485 [16] Andrew Chi-Chih Yao (1979):
Some Complexity Questions Related to Distributive Computing (PreliminaryReport) . In: