P. Flocchini and F. Geurts. UCL, INFO, Place Sainte Barbe 2, 1348 Louvain-la-Neuve, Belgium

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1 Searching for Chaos in Cellular Automata: New Tools for Classication P. Flocchini and F. Geurts DSI, Via Comelico 39, Milano, Italy UCL, INFO, Place Sainte Barbe 2, 1348 Louvain-la-Neuve, Belgium Abstract. We present new tools allowing a formal classication of cellular automata, i.e. trans- nite attraction and a kind of Hamming distance. We also investigate a class of complex aperiodic systems. This brings new insights in understanding spatio-temporal chaos for the class of discrete-time multi-dimensional systems. 1. Introduction Among all discrete-time discrete-space multi-dimensional systems, cellular automata (CA) are very interesting to study because they oer a rich variety of behaviors allowing to model specic physical systems, as well as universal computational devices. In the theory of CA, classication of behaviors is a central theme. The goal is to impose a structure in the space of CA rules, grouping together CA related to equivalent properties. Many dierent behaviors are possible, from very simple (destruction of information) to very complex (propagation of information following complex rules). In general simple behaviors are easy to understand and to characterize a priori, which is not the case when analyzing complex behaviors. Dierent tools have been introduced, leading to dierent classication schemes. The main problem appearing in (almost) every classication scheme is the qualitative denition: several classes are not formally dened. There is thus an ambiguity inside each classication. The goal of this paper is twofold: to propose a new classication of CA, formally and precisely dened; to investigate the class of complex behaviors (particularly \aperiodic" behaviors). We propose new tools, i.e. transnite attraction and shifted hamming distance, giving us a way of dening a new classication of CA in which every class is formally dened. We also analyze three dierent ways of grouping these classes, which brings new insights in understanding chaotic behaviors. Finally, we present related works and draw some conclusions 2. Basic Denitions 2.1. Transnite Iterations Let us consider a space E. Classically, an iteration scheme is dened as follows: starting from an initial value x 0 2 E, f 0 (x 0 ) = x 0 and f n+1 (x 0 ) = f(f n (x 0 )):

2 It can be shown that strong conditions on function (or relation) f are to be assumed to prove convergence theorems in this classical framework. However, if we relax the hypothesis of nite (and their limit, innite) iterations, then it is possible to get more general results: the relations involved only need to be monotonic. Therefore, we need to introduce transnite iterations. The class of ordinal numbers (denoted O) is well ordered by the classical. The expression [ i2i i is used to denote the upper bound of the ordinals family f i ji 2 Ig. A limit ordinal is such that [ < =. A successor ordinal is such that [ < =?1, where the predecessor of is denoted by? 1. Let us assume that L(; ;; E; [; \) is a complete lattice, with ordering relation, inmum ;, supremum E, least upper bound operator [, and greatest lower bound \. With the same notations, (L) denotes the smallest ordinal number such that #fj 2 (L)g > #L. Denition 1 Decreasing iteration The decreasing iteration starting from E in the complete lattice L(; ;; E; [; \), and dened by a monotonic relation f, is a sequence (x ) 2(L) of elements of L, dened by transnite recurrence: x 0 = E, x = f(x?1 ) for all successor ordinal, and x = \ < x for all limit ordinal. Denition 2 Stationary sequence The sequence (x ) 2 of elements of the complete lattice L(; ;; E; [; \) is stationary i 9" 2 : ( ") ) (x " = x ). The limit of this sequence is x ". Theorem 3 Greatest xed-point ([1] Corollary ) Let L(; ;; E; [; \) be a complete lattice. A decreasing iteration (x ) 2(L) starting from E, and dened by a monotonic relation f, is a stationary decreasing sequence and its limit is the greatest xed-point of f Predicate transformers Let us take a relation f E E, and a predicate (or set of states) P included in its domain. We dene the pre-image of P, f? :P, as the set of states from which a state in P can be reached by an application of f, and the post-image of P, f + :P, as the set of states which can be reached from a state in P by an application of f [2, 3]. Denition 4 Pre-image, post-image f? :P = fuj(u 2 Dom(f)) ^ 9v : (v = f(u)) ^ (P:v)g f + :P = fvj9u : (u 2 Dom(f)) ^ (P:u) ^ (v = f(u))g These operators are called predicate transformers and can be seen as inverse (resp. direct) extensions of relations from point-to-point to set-to-set. It is easy to see that we have f + :E E, f? :E E, and monotonicity of these predicate transformers, i.e. 8X; Y E; (X Y ) ) (f :X f :Y ): 2.3. Cellular Automata Finally, let us recall the denition of cellular automata. We consider one-dimensional cellular automata, the cells of which being arranged on a linear bi-innite lattice.

3 Denition 5 Cellular automaton Any linear cellular automaton is a structure C = (G; r; g) where G = f0; 1; : : : ; k? 1g is the set of the states, r 2 N is the radius of the neighborhood, g : G 2r+1! G is the local transition function. A conguration of a CA is a function that species a state for each cell x : Z! G and can be represented by a doubly-innite sequence x = (: : : ; x i?n ; : : : ; x i ; : : : ; x i+n ; : : :): So the set of the congurations of the CA is G Z. The neighborhood of a cell i 2 Z is the vector (i? r; : : : ; i; : : : ; i + r): The global transition function of the CA, f : G Z! G Z ; species the next state of each cell as the local function applied to the states of the neighborhood (8i 2 Z): f i (x) = g(x i?r ; : : : ; x i ; : : : ; x i+r ): In the next sections, we will only use elementary cellular automata, i.e. with r = 1 and k = 2. We will also use C instead of 2 Z to denote the conguration space. 3. New tools 3.1. Transnite Attraction We now extend denition 2 and split it into three forms, each making use of theorem 3 very easily. Working in a space or set E, we consider the set of its subsets, namely PE. As it is well known, this set is a complete lattice. We denote it as above: L(; ;; E; [; \). Classically, one says that a state x is attracted by p under iteration of f i (9U : 8n 2 N; f n (x) 2 U) ^ (lim n!1 f n (x) = p). We can extend this denition as follows. A set P is asymptotically attracted by a set Q i lim n!1 f n + :P Q. In general we consider the smallest such Q. We can also give a nite version of this notion. P is nitely attracted by Q i 9n 2 N : f n + :P Q. The previous denitions are not new but we can extend them in a third way, making use of transnite iterations. Denition 6 Transnite attraction P is transnitely attracted by Q i 9 an ordinal number n 2 O such that f n + :P Q. In general, the symbol 1 used in the rst denitions is equivalent to the rst transnite ordinal!. We extend the notion to all ordinal numbers (nite,!, and all other transnite ones) to simplify our next developments. To nd the global attractor of conguration space C, we have to compute, for a certain ordinal number n 2 O, the negative invariant of the system [2, 3]: f n + :C : This expression is computable by successive approximations, and leads to the attractor, thanks to monotonicity of f + assumed by theorem 3. Let us try to justify transnite iterations intuitively. We consider a single deterministic automaton with a nite number of possible states. We let the system iterate and we look at the orbits generated from dierent initial states. If we only take a smaller orbit than the total number of states, dierent behaviors are observable: xed-point attraction, periodicity, seemingly random orbits. Otherwise, since the system is deterministic, random orbits vanish. Everything becomes eventually xed or periodic. We consider now bigger and bigger state spaces, until we reach some kind of innity. For example, we take a state space of positive integers N of 0, the rst transnite cardinal, also equal to!. The same behaviors appear and we have to allow more than! iterations to see only periodic behaviors.

4 3.2. Shifted Hamming Distance We introduce here a notion of distance that is vey close to the very well known \Hamming distance". We work with a nite alphabet isomorphic to = f0; 1; :::; n?1g N. On this alphabet we dene a distance (x; y) = 0 if x = y and 1 if x 6= y: For two strings of symbols a; b 2 m, the Hamming distance between a and b, H(a; b) is de- ned as the number of places (or indices) where a and b dier: H(a; b) = P m i=1 (a i; b i ): For two bi-innite sequences a and b of symbols, H(a; b) = P i2z (a i ; b i ): Let us now introduce the new notion. Denition 7 Shifted Hamming distance The shifted Hamming distance between two bi-innite sequences x and y of C is dened by H (x; y) = min j2z u(h(x; j (y))), where is the shift function: 8i 2 Z; (x) i = x i+1, u(+1) = 1, and u(i) = 1?e?i if i is nite. 1+e?i 4. Classication w.r.t. Attraction Our motivation to classify the dierent behaviors w.r.t. transnite attraction is to provide a formal denition of behaviors with a technical tool intuitively close to our observation. Analytic studies and simulation reveal three kinds of observable behaviors: nitely regular (null, xed-point, or periodic rules), completely irregular (our type A ), and \chaotic" behaviors (subshift rules). Let us discuss the third class. These rules are chaotic in the sense of Devaney's denition. However, when we observe their long term behavior, what we see is a beautiful regularity. The problem is a bit technical: the proof is based on a metric assigning a weighting to the cells of the cellular automaton, which strongly inuences the consideration made, whereas we don't have this weighting in mind when we observe the successive congurations generated. Thus, we have to nd a technical aspect closer to our own observation. We propose to use attraction as technical tool but, since we work with doubly innite lattices of cells, we need to consider transnite iterations and attraction. We consider bi-innite congurations in zero backgrounds. We study the successive iterations of each system f over a certain amount of time (nite, innite, or transnite), f n + :P, from two kinds of initial conditions: random congurations (#P = 1), or the whole conguration space C (P = C). Let us now examine the dierent classes separately Type N cellular automata They fastly evolve to homogeneous congurations. Any conguration is nitely attracted to the same conguration, homogeneously composed of quiescent cell states, i.e. without information. The homogeneous state is a function of the rule itself: 9h 2 C : (8x 0 2 C; 9n 2 N : f n :fx + 0g = fhg) ^ (f(h) = h): More globally, we have f! + :C = fhg : This class is called N 0 because another version is possible, with several possible quiescent congurations: 9H C : 8x 0 2 C; 9h 2 H : (9n 2 N : f n :fx + 0g = fhg)^(f(h) = h): More globally, we have f! :C = H : We call this subclass N + 1.

5 4.2. Type F cellular automata They evolve to xed congurations after nite transients. This class contains the rst one, which is a particular case. The nal xed conguration is in general dependent on the initial one. We have here a nite attraction, too: 8x 0 2 C; 9s(x 0 ) 2 C : (9n 2 N : f n + :fx 0g = fsg) ^ (f(s) = s): More globally, we have f! + :C = [ x 0 2Cfs(x 0 )g : For the sake of simplicity, we will include N 1 in F and keep N equal to N Type P cellular automata They evolve to cycles of congurations after nite transients. This class contains the two previous ones. The limit cycle is dependent on the initial condition. We have a nite attraction to a set of points rather than to a single xed-point: 8x 0 2 C; 9C(x 0 ) C; y 2 C; m 2 N; n 2 N : (f n + :fx 0g = fyg) ^ (f m (y) = y) ^ (8y 0 2 C; f m (y 0 ) = y 0 ): More globally, we have f! + :C = [ x 0 2CC(x 0 ) : 4.4. Type S cellular automata They behave like generalized alternating subshifts. Here is a denition of this behavior, generalizing [4]. Denition 8 Generalized alternating subshift rule A CA is a gen. alt. subshift rule if its global function f is such that there is a closed invariant subset 1 of C on which it operates as follows: 8x 2 1 ; f n (x) = m (x), where n 2 N and m 2 Z, and : C! C is the shift: 8i 2 Z; (x) i = x i+1. It is possible to prove that this kind of behavior leads to chaos in the sense of Devaney (topological transitivity, density of periodic points) [5, 6]. When observing a specic cellular automaton starting from a random initial con- guration, what we see is the initial conguration progressively shifting to the right or to the left, together with a kind of periodic behavior. If we take an initial - nite conguration in a zero background, for example, we will see our conguration escaping the nite observation domain, unless this domain can indenitely grow. If we iterate more times than the total amount of cells composing the CA, even if this lattice possesses a bi-innite number of cells, then, starting from any initial conguration, we observe an attraction to a homogeneous conguration, exactly as type N cellular automata behave. From this point of view, the behavior of type S cellular automaton becomes more regular than chaotic, if we accept transnite iterations: 9h 2 C : (8x 0 2 C; 9n 2 OnN : f n :fx + 0g = fhg) ^ (f(h) = h) or, more generally, 9H C : 8x 0 2 C; 9m 2 O; n 2 OnN : (9y 2 H : f n :fx + 0g = fyg) ^ (8y 0 2 H; f m (y 0 ) = y 0 ) where H is a cycle of homogeneous congurations. It is also possible to write 9n 2 OnN : f n + :C = H : Here, we see a dierence regarding the use of nite/transnite iterations. Finite iterations lead to a typical shift behavior which can be seen as chaotic (classical definition). Transnite iterations show a simple behavior of attraction to homogeneous congurations Type A cellular automata They have an aperiodic behavior which is responsible for the observable (spatiotemporal) chaos. This is the same class as type \c" of [7]. We prefer calling it \aperi-

6 odic" because it is not really chaotic in the sense of Devaney's denition. Actually, there does not exist any analytical denition for these phenomena. Aperiodicity seems to be the least restrictive denition for the moment: a conguration is aperiodic if it is not eventually periodic (neither periodic nor one of its forward iterations). Phenomenologically, we observe patterns growing, vanishing, and moving towards space-time. There is a kind of regularity (these forms are far from noise) but also a diversity (dierent forms). There is no stabilization as such but there is no real disorder. We remark that aperiodicity entails that almost the whole domain is visited through successive iterations. Back to attraction, we have here an \attraction" to a huge cycle containing (almost) the whole conguration space: 8x 0 2 C; 9C 0 C; m 2 N; n 2 OnN : (9y 2 C 0 : f n :fx + 0g = y) ^ (8y 0 2 C 0 ; f m (y 0 ) = y 0 ) where the symbol \ " means \dense subset of" but is still to be precised more formally. It is important because it makes the dierence with type S cellular automata. It is also possible to write 9n 2 OnN : f n + :C C : Here also, we have a dierence between nite and transnite iterations. Finite iterations show irregular behaviors, spatio-temporal chaotic patterns, aperiodic evolutions. The problem is that it is dicult to give an explicit characterization of this kind of behavior. On the other hand, transnite iterations allow us to give a very simple denition, saying that the system involved is periodic with a huge period very close to the cardinality of the conguration space itself. 5. Classes Organization 5.1. Linear Periodicity Hierarchy Though we have the following inclusions: N F P; it is dicult to compare the rst classes with type S and type A. However, if we try to see all classes with respect to transnite iterations, we can see a hierarchy of periodic systems. From type N to type A the period grows from one to an ordinal \close" to the cardinality of our conguration space, and the resulting attractor grows from a homogeneous xed conguration to almost the whole conguration space. Hence, we have a linear hierarchy, where we volontary do not give a precise denition to \": N F P S A Periodicity Clusterization In this second organization, we introduce two criteria: (in)dependence to initial conditions, and (trans)nite iterations, permitting to build a classication table of our dierent sources of behaviors: Periodicity Dep. to I.C. Indep. to I.C. Finite F P N Transnite A S 5.3. Organization wrt Shifted Hamming Distance With the help of the tool previously introduced, we can classify our dierent behaviors in a very simple way. Thanks to shifted Hamming distance we are able to show that subshifts behaviors are simple.

7 If the system is n-periodic or if it has a generalized subshift behavior including a n-periodicity, they both appear very simple through the eye of our shifted Hamming distance: for any x in the orbit of an initial condition, after the transient, H (x; f n (x)) = 0: The opposite behavior is aperiodicity. This gives us a new characterization: there is no x in the conguration space C for which the previous condition applies and thus, for all state x, and all n, we have H (x; f n (x)) 6= 0: We summarize this last organization as follows: Null SHD N [F [P [S Positive SHD A Under \center-rst" metrics, subshifts can be considered as chaotic (Devaney's denition [5]). Under SHD, subshifts are very simple, just as periodic behaviors. 6. Related Work In this section, we do not want to treat everything but we would like to cite some other works and compare them with our approach. Two themes are important: classication and aperiodicity. Classication. This is one of the central themes in the theory of CA. A structure is imposed on the space of CA, grouping together CA with related properties. Several authors have proposed dierent classications, starting with Wolfram in 1983 [8]. Two problems appear when classication is studied: it is dicult to give a formal denition of each class of CA (in particular, spatio-temporal chaos is not precisely dened in this context); these denitions are often based on undecidable properties. Our classication is strongly inuenced by [7]'s. We present new tools allowing us to give a precise formal denition of each class. These tools can be related to the characterization appearing in [9]. Our classication is not decidable for all CA rules but only for simple basis ones. We are trying to extend these results to more complex rules, with the help of composition operators [10]. Aperiodicity. The notion of \chaos" is still not well dened in the context of discrete-time discrete-space multi-dimensional dynamical systems such as, for example, cellular automata. Several authors propose ways of dening complex behaviors in CA. This is one of the goals of classication. In [11], the author presents a classication of chaotic behaviors, based on notions of randomness, complexity measures, computability of initial conditions, and (non)determinism of rules. Finally, in [12], the author studies aperiodicity of some CA analytically. We take this point of view in our classication scheme because it is easier to dene than complex or chaotic rules. However, we do not make use of linearity and injectivity notions presented by Jen. This point of view is interesting because aperiodicity includes complex and chaotic behaviors, in some sense. 7. Conclusion Our goal was to nd a classication of elementary cellular automata in which each class is dened by a mathematical expression. In particular, we wanted to characterize the most \chaotic" classes. We have rened a given classication and we have added new tools to go deeper: predicate transformers, transnite attraction, and shifted hamming distance. With

8 these tools we gave an explicit characterization of each class and we have seen that spatio-temporal or aperiodic systems are in fact periodic systems with huge periods. In a future work, we will use this explicit denition of CA classes to nd equivalent classes of more general dynamical systems. In parallel, we will also investigate dierent composition operators and analyze how they behave in this framework, regarding properties of invariance, attraction, etc. 8. Acknowledgements This paper has been partially supported by grant CT12, C.N.R. (Italy), and by the National Funds for Scientic Research (Belgium). We wich to thank Carla Quaranta Vogliotti and Petr Kurka for their valuable comments. References [1] Cousot P., (1978). Methodes Iteratives de Construction et d'approximation de Points Fixes d'operateurs Monotones sur un Treillis, Analyse Semantique des Programmes. PhD thesis, Univ. Scient. et Med., Inst. Nat. Polytechnique de Grenoble. [2] Sintzo M., (1992). Invariance and contraction by innite iteration of relations. In Ban^atre J.P. & Le Metayer D. (ed.), Research Directions in High-Level Parallel Programming Languages, LNCS 574, pp. 349{373. Springer-Verlag. [3] Sintzo M. & Geurts F., (1994). Analysis of dynamical systems using predicate transformers: Attraction and composition (to appear). In Andersson S.I. (ed.), Proc. of the 1993 Summer University of Southern Stockholm, Analysis of Dynamical and Cognitive Systems. Springer-Verlag. [4] Cattaneo G., Flocchini P., Mauri G. & Santoro N., (1993). Chaos and subshift rules in neural networks and cellular automata. In Proc. of 1993 International Symposium on Nonlinear Theory and its Applications, Hawaii, Vol. 4, pp. 1153{1156. [5] Devaney R.L., (1989). An Introduction to Chaotic Dynamical Systems. Addison-Wesley, 2nd edition. [6] Banks J., Brooks J., Cairns G., Davis G. & Stacey P., (1992). On devaney's denition of chaos. The American Mathematics Monthly, 99(4):332{334. [7] Cattaneo G., Flocchini P., Mauri G. & Santoro N., (1993). A new classication of cellular automata and their algebraic properties. In Proc. of 1993 International Symposium on Nonlinear Theory and its Applications, Hawaii, Vol. 1, pp. 223{226. [8] Wolfram S., (1986). Theory and Applications of Cellular Automata. World Scientic. [9] Kaneko K., (1986). Attractors, basin structures and information processing in cellular automata. In Wolfram S. (ed.), Theory and Applications of Cellular Automata, pp. 367{ 399. World Scientic. [10] Flocchini P. & Geurts F., (1994). Searching for chaos in cellular automata: Compositional approach. In these proceedings. [11] Svozil K., (1990). Constructive chaos by cellular automata and possible sources of an arrow of time. Physica D, 45:420{427. [12] Jen E., (1990). Aperiodicity in one-dimensional cellular automata. Physica D, 45:3{18.