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Periodicity and transitivity for cellular automata in Besicovitch topologies F. Blanchard1 , J. Cervelle2 , and E. Formenti3 1

Institut de Math´ematique de Luminy, CNRS, Campus de Luminy, Case 907 - 13288 MARSEILLE Cedex 9,France, [email protected] 2 Laboratoire d’informatique Institut Gaspard-Monge, 5 Bd Descartes, Champs-sur-Marne, F-77454 Marne-la-Vall´ee cedex 2 France [email protected] 3 Laboratoire d’Informatique Fondamentale de Marseille (LIF), 39 rue Joliot-Curie, 13453 Marseille Cedex 13, France, [email protected]. Corresponding author: Enrico Formenti.

Abstract. We study cellular automata (CA) behavior in Besicovitch topology. We solve an open problem about the existence of transitive CA. The proof of this result has some interest in its own since it is obtained by using Kolmogorov complexity. At our knowledge it if the first result on discrete dynamical systems obtained using Kolmogorov complexity. We also prove that every CA (in Besicovitch topology) either has a unique fixed point or a countable set of periodic points. This result underlines that CA have a great degree of stability and may be considered a further step towards the understanding of CA periodic behavior.



In the last twenty years CA received a growing and growing attention as formal models for complex systems with applications in almost every scientific domain. They consists in an infinite lattice of finite automata. All automata are identical. Each automaton updates its state according to a local rule on the basis of its actual state and of the one of a fixed finite set of neighboring automata. The state of all automata is updated synchronously. A configuration is a snapshot of the state of all automata in the lattice. The simplicity of the definition of this model is in contrast with the wide variety of different dynamical behaviors most of them are not completely understood yet. Dynamical behavior of CA is studied mainly in the context of discrete dynamical systems by putting on configurations the classical Cantor topology (i.e. the one obtained by putting the product topology when the set of states of the automata is equipped with the discrete topology). Deterministic chaos is one of the most appealing (and poorly understood) dynamical behavior. Among CA, one can find many interesting example of this kind of behavior. The problem is that shift map is chaotic according to most popular chaos definitions in litterature (see [7] for example).

The shift map is very a simple CA which shifts left the content of configurations. The chaoticity of this map is somewhat counter-intuitive (see [4, 1] for a discussion on this topic). In fact, the chaoticity of the shift is mostly due to the structure of the topology than to the intrinsic complexity of the automaton [8, 5]. In [4], to overcome the drawbacks of the Cantor topology (in the context of chaotic behavior), the authors proposed to substitute it with the Besicovitch topology. In [10], the authors proved that this new topology better links the classical notion of sensibility to initial conditions with the intuitive notion of chaotic behavior. As usual, the introduction of a new result discloses a series of new questions (some of them are reported in [6]). In this paper we solve the problem of finding a transitive CA in Besicovitch topology (Theorem 2) which is qualified of challenging open problem in [12]. This result has deep implications in CA dynamics. First, it states that they are unable to vary arbitrarily the density of differences between two configurations during their evolutions. In its own turn this fact implies that the information contained in configurations cannot spread too much during evolutions. Second, the proof technique is of some interest in its own since we used Kolmogorov complexity to prove a purely topological property about discrete dynamical systems. The low degree of complexity from a chaotic behavior point of view is underlined by the second main result of the paper: a CA either has a unique fixed point or an uncountable set of periodic points. These two results open the quest for new more appropriate properties for describing the “complex” behavior of CA dynamics. Some very interesting proposals along this line of thoughts may be found in [13]. The authors are currently investigating this subject.


Cellular automata

Formally, a CA is a quadruple hd, S, N, λi. The integer d is the dimension of the CA and controls how the cells of the lattice are indexed. Indeed, indexes of cells take values in ZZd . The symbol S is the finite set of states of cells and λ : SN → S is the local rule which updates the state of a cell on the basis of a (finite) neighborhood N ⊂ ZZd . A configuration c is a function from ZZd to S and may be viewed as a snapshot of the content of each cell in the lattice. Denote by X the set SZZ of all configurations. The local rule induces naturally a global d d rule on the space of configurations fA : SZZ → SZZ as follows d ∀c ∈ SZZ ∀i ∈ ZZd , fA (c)(i) = λ(i + n1 , . . . , i + nt ) ,

where N = (n1 . . . , nt ) and + is addition in ZZd . In the sequel, when no misunderstanding is possible, we will often make no distinction between a CA and its local rule. Moreover we will denote fA simply by f.

Some sets of configurations play a special role in the study of dynamical behavior such as finite and spatial periodic configurations. A configuration c ∈ X is finite if it has a finite number of non-zero cells. A configuration is spatial periodic c ∈ X if there exists p ∈ IN such that ∀i ∈ ZZ, ci = cp+i . The least p with the above property is the (spatial ) period of c. Denote P the set of spatial periodic configurations. A point x ∈ X is ultimately periodic for f if there exist p, t ∈ IN such that ∀h ∈ IN, fph+t (x) = x. The least integer p with such a property is called the (temporal ) period of x and t is its pre-period. A point is fixed if it has period 1. In this paper, we mainly study one-dimensional CA (d = 1) with S = {0, 1}. For any configuration c ∈ X, ca:b is the word ca ca+1 . . . cb if b > a, ε (the empty word) otherwise. The pattern of size 2n+1 centered at index 0 of a configuration c ∈ X is denoted Mx (n) = x−n:n . For any word w ∈ S? , |u| denotes its length. Finally, if {0, 1} ⊂ S, 0 is the configuration in which all cells are in state 0, and, similarly, 1 is the configuration in which all cells are in state 1. In the sequel, configurations like 0 or 1 are said to be homogeneous configurations. For any c ∈ X, the set Of (c) = {fi (c), i ∈ IN} is the orbit of initial condition c for f (we assume f0 (x) = x). The space-time diagram of initial configuration c ∈ X is a graphical representation of Of (c) and it is obtained by superposing the configurations c, f( c), . . ., fn (c), . . . This representation is very useful in the visualization of simulations of cellular automata evolutions on a computer. Our main interest is to study CA in the context of discrete dynamical systems i.e. structures hU, Fi, where U is a topological (possibly metric) space and F a continuous function from X to itself. When S is endowed with the discrete topology and SZZ the induced product topology, then any global rule f of a CA can be considered as a discrete dynamical system hSZZ , fi. The product topology on SZZ is usually called Cantor topology since it is a compact, totally disconnected and perfect space. One can easily verify that the following metric on configurations ∀x, y ∈ SZZ , d(x, y) = 2min{|i|,xi 6=yi ,i∈ZZ} . induces exactly the Cantor topology on SZZ .


The Besicovitch topology

The topology with which the space of a dynamical system hX, fi is endowed plays a fundamental role in the study of the asymptotic behavior. In particular it can filter out special intrinsic behaviors and hide unimportant marginal phenomena. Besicovitch topology has been introduced in [4] in order to refine the study of sensitivity to initial conditions for CA. In contrast with Cantor topology, this one greatly decreases the “importance” of errors near the cell of index zero by giving to all cells the same weight. Let dB be the following function dB (x, y) = lim sup n→∞

∆(x−n:n , y−n:n ) , 2n + 1

where ∆(x−n:n , y−n:n ) is the number of positions which the words x−n:n and y−n:n differ at. The function dB is a pseudo-distance ([4]), and is called Besicovitch pseudodistance. Besicovitch topology is obtained by taking the quotient space w.r.t. the ˙ this relation. In this equivalence relation “being at zero dB -distance”. Denote ≡ way, dB becomes a metric on classes. In the sequel, when no misunderstanding is possible, we will denote dB simply by d. This topology is suitable for the study of CA. The following proposition allows us to lift a global function from X to itself into a global function from X˙ to itself. ˙ i.e. x≡y ˙ =⇒ f(x)≡f(y). ˙ Proposition 1 ([4]). Any CA is compatible with ≡ For all CA f, denote f˙ the function which transform a class c of X˙ to the class which contains all the images of the configurations of c by f. Such a f˙ is called a CA on (Besicovitch) classes. 3.1

Fixed and periodic points

Some recent works pointed out that CA are not complex from a purely algorithmic complexity point of view [8, 3, 5]. This fact seems to originate from an intrinsic stability of such systems. It is well-known that in Cantor topology, if a CA has a non-homogeneous fixed point then it has at least a countable set of fixed points. Besicovitch topology allows to go even further. In this section we prove that either a CA has one unique fixed point or it has uncountably many periodic points. Clearly these “new” periodic points are due to the special structure of the Besicovitch space but if we analyze more in details how they are built one can see that they are made of larger and larger areas in which the system is periodic even in the Cantor sense. Before introducing the main result of this section (Theorem 1) we need some technical lemma and notation. If p is an integer and u a word of size at least 2p, then p |u|p is the word up+1:|u|−p i.e. the word u in which the first and last p letters are deleted. Lemma 1. For all CA of global rule f and radius r, it holds that ∆(r |ab|r , f(ab)) 6 ∆(r |b|r , f(b)) + ∆(r |a|r , f(a)) + 2r, where a and b have length bigger than 2r. Previous result comes from the fact that the image of the concatenation of two words is the concatenation of the images of these words separated by 2r cells of perturbation. An iterated application of this lemma gives an inequality for the concatenation of h words. Lemma 2. Let (ai )i∈J1,hK be a finite sequence of h words of length greater than 2r. Let x = a1 . . . ah be the Phconcatenation of these words. Then, for all CA f of radius r, ∆(r |x|r , f(x)) 6 i=1 ∆(r |ai |r , f(ai )) + 2r(h − 1). Proof. By induction on the number of concatenated words and Lemma 1.

t u

We will also make use of the well-known Cesaro’s lemma on series. Lemma 3 (C´ esaro’s lemma). Let (an )n∈IN and (un )n∈IN be such that lim


un =l , an

where an is divergent and positive. Then, limn→∞

Pn ui Pi=0 n i=0 ai

= l.

Proposition 2. In Besicovitch space, any CA with two distinct periodic points of period p1 and p2 has an uncountable number of periodic points whose period is lcm(p1 , p2 ). Proof. Let f be a CA on classes with radius r. Let x and y be two periodic points of f with periods p1 and p2 , respectively. Let p = lcm(p1 , p2 ). Let x 0 a member of the class x and y 0 a member of the class y. Since x and y are distinct, d(x 0 , y 0 ) = δ > 0. Hence, there exists a sequence of integers (un )n∈IN such that, for all integer n > 0 : 0 0 ∆(x−u , y−u ) > δun . n :un n :un


Let σ be an increasing function such that uσ(0) > 4r and uσ(n+1) > 2uσ(n) . Let vn = uσ(n) . By a simple recurrence on n, it holds that : n X

vi 6 2vn .



˙ such that the Let us construct an injection g from {0, 1}ZZ into X (not X) ˙ Let α be sequence of classes of the image set of g are all periodic points of f. {0, 1}IN . For all positive integers i, define the sequences kα and k0α as follows 0 0 x−vi :−1 if αi = 0 x1:vi if αi = 0 0α α and ki = ki = 0 0 y1:v if α = 1 y−v if αi = 1 i i i :−1 Define the configuration g(α) as follows 0α 0α 0α 0α 2rp+1 α α α α g(α) = . . . k0α k0 k1 k2 . . . kα n kn−1 . . . k2 k1 k0 0 n−1 kn . . .

Let us prove that the class containing g(α) is a periodic point for f with period p, i.e. d(g(α), fp (g(α))) = 0. One has to prove that lim sup n→∞

∆(g(α)−n:n , fp (g(α))−n:n ) =0 2n

Since x and y are periodic points of period p1 and p2 , respectively, and both p1 and p2 are divisors of p, we have that d(x 0 , fp (x 0 )) = d(y 0 , fp (y 0 )) = 0, 0 0 ∆(y−n:n ,fp (y 0 )−n:n ) ∆(x−n:n ,fp (x 0 )−n:n ) and then, that limn→∞ = limn→∞ = 0. 2n 2n Hence, it holds that 0 0 ∆(x1:n , fp (x 0 )1:n ) ∆(y1:n , fp (y 0 )1:n ) = lim = 0. n→∞ n→∞ n n



0α Fix an integer n > r. Let us decompose g(α)−n:n in its kα i and ki factors

−→ 0α ← − 0α 0α 0α 2rp+1 α α α α g(α)−n:n = k0α k0 k1 k2 . . . k α h kh−1 . . . k2 k1 k0 0 h−1 kh −→ ← − α 0α where h depends on n, and k0α h [resp. kh ] is the suffix [resp. prefix] of kh [resp. kα ] of g(α) . −n:n h Since fp is a CA of radius rp, Lemma 2 gives Ph−1 p α ∆(g(α)−n:n , fp (g(α))−n:n ) 6 ∆(0, fp (02rp+1 )) + i=0 ∆(rp |kα i |rp , f (ki )) Ph−1 0α p 0α + i=0 ∆(rp |ki |rp , f (ki ))+ −→ −→ p 0α ∆(rp |k0α h |rp , f (kh )) ← − − p ← α +∆(rp |kα h |rp , f (kh )) + 2rp(2h). ← − α 0 0 By the definition of kα i , one have that kh is a prefix of x or of y . Hence, ← − − p ← α 0 p 0 0 p 0 ∆(rp |kα h |rp , f (kh )) 6 ∆(rp |x1:n |rp , f (x1:n )) + ∆(rp |y1:n |rp , f (y1:n )) . Similarly −→ −→ p 0α 0 p 0 0 p 0 ∆(rp |k0α h |rp , f (kh )) 6 ∆(rp |x−n:−1 |rp , f (x−n:−1 )) + ∆(rp |y−n:−1 |rp , f (y−n:−1 )).

Then one has

Ph−1 p α ∆(g(α)−n:n , fp (g(α))−n:n ) 6 1 + i=0 ∆(rp |kα i |rp , f (ki ))+ Ph−1 0α p 0α i=0 ∆(rp |ki |rp , f (ki ))+ 0 0 ))+ |rp , fp (x−n:n ∆(rp |x−n:n p 0 0 )) + 4rph ∆(rp |y−n:n |rp , f (y−n:n


By Equation (3), one finds lim

i→∞ αi =1

p α ∆(rp |kα i |rp ,f (ki )) vi

And similarly, lim

i→∞ αi =0

0 |rp ,fp (x−v )) i :vi vi 0 ∆(x−n:n ,fp (x 0 )−n:n ) limn→∞ = 0. n

6 limi→∞

p α ∆(rp |kα i |rp ,f (ki )) vi

0 ∆(rp |x−v

i :vi

= 0.

p α ∆(rp |kα i |rp ,f (ki )) = vi |kα |rp ,fp (kα )) i i i=0 ∆(rp Ph 0 and by Cesaro’s Lemma we have limh→∞ = 0, since the i=0 vi series (vi )i∈IN is positive and divergent. The same argument applies to k 0 , and Ph Ph α p α 0α ∆( |k | ,f (k )) ∆( |k | ,fp (k0α rp rp rp i i )) therefore one has limh→∞ i=0 Ph i v + i=0 Phi vrp = 0. i=0 i i=0 i Ph Since i=0 vi 6 2n, one gets

Summing the two previous equations, we obtain limi→∞ Ph





p α ∆(rp |kα i |rp , f (ki )) + 2n



p 0α ∆(rp |k0α i |rp , f (ki )) =0 2n


Using Equation (3) again, we obtain that 0 0 0 0 ∆(rp |x−n:n |rp , fp (x−n:n )) ∆(rp |y−n:n |rp , fp (y−n:n )) = lim =0 n→∞ n→∞ 2n 2n



Finally 4rph + 1 =0 n→∞ 2n lim


since h 6 ln2 (n). Using Equations (5), (6) and (7) inside Equation (4), one finds that p limn→∞ ∆(g(α)−n:n f2n(g(α))−n:n ,) = 0 which implies that g(α) is a periodic point of f of period p. Let ∼ be the equivalence relation such that x ∼ y if and only if y and x differ only at a finite number of positions, and  the converse relation. Let α and β be two sequences of {0, 1}IN such that α  β. Let (ai )i∈IN be the increasing sequence of indices where they differ. Then, ∆(kα , kβ ∆(g(α)−n:n , g(β)−n:n ) an ) > lim sup Pan an 2n 2 v + 2r + 1 n∈IN i∈IN i=0 i 0 0 ∆(xv , yvan ) > lim sup Pan an . i∈IN 2 i=0 vi + 2r + 1

d(g(α), g(β)) > lim sup

Applying Equation (1) and (2), we obtain d(g(α), g(β)) > lim supn∈IN

δvan 2van +1

> δ2 . Hence, g(α) and g(β) are in different classes. Finally, we prove that all configurations in g(X) are periodic points of period ˙ g(β). Let E be a set containing a member of each p, and that α  β =⇒ g(α)6≡ equivalence class of ∼. Since {0, 1}IN / ∼ is not countable, so is E. Using previous equation, g|E is injective and hence, g(E) is a non countable set of periodic points of f of period p. t u This result has two easy corollaries. The first one is obtained simply recalling that a fixed point is a periodic point of period 1. The second one comes from the fact that if p is a periodic points of period greater of equal to 2, then there are at least two distinct periodic points. Corollary 1. If a CA f has two fixed points, it has an uncountable number of fixed points. Corollary 2. If a CA has a periodic point of period p > 1, then its set of periodic points is not countable. Putting together the results of the two previous corollaries we have the main result of this section. Theorem 1. Any CA has either one and only one fixed point or an uncountable number of periodic points. Proof. Let f be a CA. There are several cases. First f can have one and only one fixed point. Second, if f can has two fixed points, then it has a uncountable set of periodic points (which are in this case fixed points); finally, if f has no fix points, then f(0) = 1 and f(1) = (0), and 0 is a periodic point of period 2, and therefore, using Corollary 2, there are uncountably many periodic points for f. t u

Proposition 3. If a surjective CA has a blocking word (or, equivalently, an equicontinuity point for Cantor topology), then its set of periodic points is dense in Besicovitch topology. Remark that the same argument can be used to prove a similar result for the Cantor topology, without making use of measure theory, as it is the case in [2]. Proof. Let f be a CA and w a blocking word for f. One has to prove that for all configurations x, and all real numbers ε > 0, there exists a periodic configuration at distance less than ε from x. Let y be the following configuration wl if l < |w| ∀n, l < k ∈ IN, ynk+l = xnk+l otherwise, where k = 2ε|w|. The configuration y is everywhere equal to the x except that periodically we put w. The number of differences between x−n:n and y−n:n is bounded by the product of |w| and the number of times w is written within y−n:n . Hence, it holds that n d 2n e|w| +|w| ,y−n:n ) 6 limn→∞ k2n 6 limn→∞ ε 2n < ε. d(x, y) 6 limn→∞ ∆(x−n:n 2n+1 Now we are going to prove that y is a periodic point of f. Since w is a blocking word, for all integers i and n, the pattern fi+1 (y)nk+d|w|/2e:n(k+1)+b|w|/2c depends only on the corresponding word of the same size in the pre-image fi (y)nk+d|w|/2e:n(k+1)+b|w|/2c . (i)

For all i, n ∈ IN, let un = fi (y)nk+d|w|/2e:n(k+1)+b|w|/2c . For any fixed (i) is has finite range and each term depends only on n, the sequence un i∈IN

its predecessor. Hence, it is ultimately periodic. By the hypothesis the CA is surjective, this implies that y is periodic (for if the sequence is periodic of period (k+p−1) (k−1) (k) , which and un p with pre-period k, then un has two pre-images: un is impossible since a surjective CA is pre-injective – see [9] for more on preinjectivity). Since the number of possible values for this sequence is 2k , the period of each column is at most 2p . Hence, the configuration is periodic and its period is at most lcm{2i , 1 6 i 6 k}. t u 3.2

Transitive cellular automata

As already recalled, Besicovitch topology was introduced in order to further study CA chaotic behavior and, in particular, sensitivity to initial conditions. In [1], the authors wondered about the existence of transitive CA in this topology. The same problem has been qualified as “challenging” in [12]. In this section we prove that the question has a negative answer. Former researches tried to prove or disprove the existence of transitive CA either by looking for counter-examples or by complicated combinatorial proofs. Here we have drastically diminished the complexity of the problem by making use of Kolmogorov complexity and the classical approach of the “incompressibility method” (see [11] for more on this last subject). Clearly, giving a glance to our

proof, now one can find a pure combinatorial proof by doing some “reverseengineering”. For any two words u, w on {0, 1}? , denote K(u) the Kolmogorov complexity of u and K(u|w), the Kolmogorov complexity of u conditional to w. We make reference to [11] for the precise definitions of these quantities and all the related well-known inequalities. The intuition behind the proof of next result is that CA cannot increase the algorithmic complexity of configurations. If a CA would be transitive in Besicovitch topology then, given two configurations x, y which differs on a sequence of places with relatively large complexity, it should be able to take a point arbitrarily near to x, arbitrarily near to y, but this implies a great change in complexity contradicting our initial intuition. Theorem 2. In Besicovitch topological space there is no transitive CA. Proof. By contradiction, suppose that there exists a transitive CA f of radius r with C states. Let x and y be two configurations such that for all integers K(x−n:n |y−n:n ) > n 2 . One can prove that configurations x and y exist by a simple counting argument. Since f is transitive, there are two configurations x 0 and y 0 such that 0 ∆(x−n:n , x−n:n ) 6 4εn


0 ∆(y−n:n , y−n:n ) 6 4δn


and an integer u (which only depends on ε and δ) such that fu (y 0 ) = x 0 ,



where ε = δ = (4e10 log2 C ) . In the sequel of the proof, only n varies, while C, u, x, y, x 0 , y 0 , δ and ε are fixed and independent of n. 0 from the following items: By Equation (9), one may compute the word x−n:n 0 0 0 and which are y−n:n , f, u, n and the twice ur bits of y which surrounds y−n:n 0 missing to compute x−n:n with Equation (9). We obtain that 0 0 K(x−n:n |y−n:n ) 6 2ur + K(u) + K(n) + K(f) + O(1) 6 o(n)


(the notations O and o are defined with respect to n). Now, let us evaluate 0 |y−n:n ). Let a1 , a2 , a3 , . . . , ak be the positive positions which y−n:n and K(y−n:n 0 y−n:n differ at, sorted increasingly. Let b1 = a1 and bi = ai −ai−1 , for 2 6 i 6 k. Pk Using Equation (8), we know that k 6 4δn. Remark that i=1 bi = ak 6 n. By contradiction, let a10 , a20 , a30 , . . . , ak0 0 the absolute value of the strictly negative 0 positions which y−n:n and y−n:n differ at, sorted increasingly. Let b10 = a10 and 0 0 0 0 bi = ai − ai−1 , where 2 6 i 6 k . Equation (8) statesPthat k 0 6 4δn. Since the P ln bi bi logarithm is a concave function, one has 6 ln n k 6 ln k k and hence X n ln bi 6 k ln (11) k which also holds for bi0 and k 0 . The knowledge of the bi , the bi0 , and of the 0 0 k + k 0 states of the cells of y−n:n where y−n:n differs from y−n:n is enough to

P P 0 0 compute y−n:n from y−n:n . Hence, K(y−n:n |y−n:n ) 6 ln(bi ) + ln(bi0 ) + 0 0 (k + k ) log2 C + O(1). Equation (11) states that K(y−n:n |y−n:n ) 6 k ln n k + n k 0 ln kn0 +(k+k 0 ) log2 C+O(1). The function k 7→ k ln n is increasing on [0, ]. k e As n n 4n n 10 log2 C , we have that k ln 6 4δn ln 6 ln e 6 k 6 4δn 6 e10 log 2C k 4δn e10 log2 C 2 log2 Cn 40 0 0 0 n and that C(k + k ) 6 e10 log2 C . Replacing a, b and k by a , b and e10 log2 C k 0 , the same sequence of inequalities leads to the same result. One may deduce that (2 log2 C + 80)n 0 K(y−n:n |y−n:n ) 6 + O(1) (12) e10 log2 C 0 Similarly, Equation (12) is also true with K(x−n:n |x−n:n ). The triangular inequality of the Kolmogorov complexity gives 0 0 0 0 K(x−n:n |y−n:n ) 6 K(x−n:n |x−n:n ) + K(x−n:n |y−n:n ) + K(y−n:n |y−n:n ) + O(1) . 2 C+80)n By Equations (12) and (10) one concludes that K(x−n:n |y−n:n ) 6 (2 log + e10 log2 C n o(n). The hypothesis on x and y was K(x−n:n |y−n:n ) > 2 . This implies that (2C+80)n n + o(n). Last inequality is false for a big enough n. t u 2 6 e10C

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