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Periodic Points and Entropies for Cellular Automata Complex Systems 3 (1989) 117-128 Periodic Points and Entropies for Cellular Automata Rui M.A. Dillio CERN, SPS Division, CH-1211 Geneva, Switzerland and CFMC, Av. Prof Gama Pinto 2,1699 Lisbon Codex, Port ugal Abstract. For the class of permutive cellular automata the num­ ber of periodic points and the topological and metrical entropies are calculated. 1. Introduction Cellular automata (CA) are infinite sequences of symbols of a finite alphabet that evolve in time according to a definite local rule. We can take, for ex­ ample, a lattice where each vertex contains the value of a dynamical variable ranging over a finite set of numbers. At a certain t ime to the values on the vertices of the lattice take a definite value, but at time to + 1 , the values of the dynamical variable can change according to a law defined locally. This kind of system, as shown by von Neumann [8] can simulate a Turing ma­ chine, and specific examples exist where the cellular automaton is capable of uni versal computation [3]. The simplest examples of CA ru les are obtained when we restrict to one-dimensional lattices and the dynamic variable over each site of the lattice can take values on the finite field Z2 = {O,I} [10]. There is an extensive literature on the dynamics of one-dimensional CA and on the patterns generated by the evolution of the CA ru les in the ex­ tended phase space Z x Z+ [12]. These patterns appear to be organized in several complexity classes according to the behavior of the iterates of random initial conditions [11]. In Wolfram's classification [11], we have four classes of CA maps. Class 1: Evolution leads to a homogeneous state. Class 2: Evolution leads to a set of separated simple or periodic structures. Class 3: Evolution leads to chaotic patterns. Class 4: Evolution leads to complex localized structures. The concept of metrical entropy, introduced by Kolmogoroff [9], and its topological analog, the topological entropy [1] are invariants that quantify the intuitive notions of mean complexity and global complexity of a dynamical system. (For a discussion of this concepts see [2].) It has been shown by Brudno [4] that the complexity of a trajectory of a dynamical system is strictly related to the mean complexity, that is, if there exists an ergodic © 1989 Compl ex Systems Publications, Inc. 118 Rui M.A. Dil5,o measure u, the complexities of fl-almost all trajectories equals the metrical entropy. Under these conditions the classification of Wolfram can be made more precise by calculating the ergodic invariant measures and the metrical entropy for classes of CA maps. The topological entropy gives an equal weight to the different kinds of complex behaviors of the traj ectories of a dynamical system, and so, we can find systems with a positive topological entropy but with complex trajectories localized on a subset of phase space with zero Lebesgue measure. However, if the invariant measure associated to a compact dynamical system is unique, the topological entropy equals the metrical entropy. We can define a third quantity to quantify the complexity of a dynamical system. Let f be a self map of a compact space X. Let Per(r) denote the number of fixed points of I". We define the periodic complexity of the map f by · logz Per(r) P (f) = 1lm sup . n e-e oo n If f has the property of separability of trajectories, P(f) is a lower bo und for the topological entropy [5]. In topological Markov chains P (f) equals the topological entropy [2] . This paper is organized as follows. In Section 2 we calculate th e number of fixed points and the periodic complexity for the class of permutive CA maps (see the definit ions below), and for the other classes an upper bound is given. It is also shown that orbits of points, under the evolution of a pe rm ut ive CA map, are associated uniquely to orbits of points of a topological Markov chain. In Section 3 we calculate the entropies for permutive maps. Section 4 is devoted to the discussion of the results and some examples are presented. In the rest of the intro du cti on we give some definitions that will be used in the subsequent sections. Cellular automata are maps T : E -t E where E is the space of doubly infinite sequences of °and 1, E = {O,1}t. We restrict our study to the class of finite breadth CA, that is, those CA whose value at site number i, i E Z, at time t + 1 depends on the values of site numbers i + m , .. , i + n , with m < n , at time t. The number I = n - m + 1 is the breadth of the CA rule T. Giving to Zz the discrete topology and endowing E with the product topology, E is a compact topological space and a finite breadth CA ma p is a cont inuous function of E. A simple representation of a CA map can be obtained t hrough the shift map a : E -t E and a Boolean function of I ~ 2 variables. Given f : {O, 1P -t {O, 1} we can define a cellular automaton map T by (T(X))i = f(xi+m, ... , Xi+n), for all i E Z" = n - m + 1, or, T(X) = f (am(x ), ... ,an(x)) where (a(X))i = Xi+l an d x = (... ,X-I, Xo, Xl , .. .) E E. f is the generating function of the finite breadth map T. An example of a CA map is given by T(X) = a -I(X) EB al(x), where EB means addition modulo 2. Periodic Points and Entropies for Cellular Automata 119 There are 22~ Boolean functions f(x}, . .. , x'"( ) of, variables and some of them do not depend of X l or xT An easy calculation shows that the numb er of Boolean functions that depends of Xl and x'"( is N(,) = 22 ~ - 2 · 22r 1 + 2 2 ~-2 , for any v > 2. In the following, C(,;m, n) , with m < n , is the class of CA map s whose generating functions depend of Xl and x '"( and , = n - m + 1. For each fixed pair of integers m and n, C(2; m, n) and C(3; m, n) have respectively 10 and 228 elements. Following Milnor [7], we say that T is a right-permutive CA map (RP), iff, its generating function f verifies to the condition f(xl, ... , x'"() = f( Xl" '" x'"() , for all ( Xl ," " X'"( ) E Zl. T is left-permutive (LP), iff, f( Xl" ' " x'"() = f( Xl" ' " x'"( ), for all (Xl, ... , x'"() E Zl, where 0 = 1 and I = O. T is permutive (P) iff both the above condit ions are verified. Denoting by P(, ;m , n) the number of permutive CA maps and by Q(,;m,n) the number of right- or left-p ermutive CA maps in C(,;m ,n), it can be found easily that P('; m,n) Q(,;m,n) where N(I) = 2. 2. P er iodic points Let {AJ with 0 ~ j ~ 2'"( - 1 represent th e elements of Zl, where j = i'"( + 2 . i'"(-l + 4 · i'"( _2 + .. .+ 2'"(- 1 . il and (il , ... , i '"( ) E ZJ, This defines an invertible map Jr'"( : Zl -t {A j } . For example, if , = 2, we have the symbolic representation, Ao f--> (0,0), Al f--> (0,1), A 2 f--> (1,0), A3 f--> (1,1). We can now identify the set L; = Z~ with a subset of I:'"( = {Ajri by taking X = (oO"X_l ,XO,Xl,' '') E L; and identifying consecutive and overlapping ,-blocks of elements of Z2 with th e elements of t he set {A j } . For, = 2, we have, for example, ... 01101001110 ... {} .. AlA3A2Al A2AoAlA3A3A2.. Hence, we have constructed a bijection h'"( : L; -> S'"( c I:'"( such that 0"'"( ( h'"( (x)) = h'"( (0"( x)), where 0"'"( is th e shift map over I:T The map h.; L; -t S'"( is defined by (h'"(( X))i := Jr'"((Xi+m,' " ,Xi+n) = bi , for all i E Z and, = n - m + 1, 120 Rui M .A. Dil8,o where bi = A j and j = Xi +n + 2 . X i+n-l + ... + 21'-1 . Xi+m' Let M1' = [Mi j ] , 0 ~ i,j ~ 21' - 1, be a 21' X 21' matrix. With i = i1'+2· i1'-1 +...+ 21'-1 .i1 we define M1' by Mi j = 1 if j = 2· i1'+...+21'-1 .i2 or j = 1 + 2 . i.; + ... + 21'- 1 . i2 and M i j = 0 otherwise. We say that Ai --t Aj is an allowed or compatible transition for M1' iff Mij = 1. The set 51' is completely characterized by the Markov transition matrix M1" i.e., b = (.. .,b_1 , ba, b1 , . •• ) E 51' iff b, --t bi +! is an allowed transition of M 1" for all i E Z. For, = 2, we have The characteris tic polynomial of M 2 is ),3(), - 2) and th e number of periodic n configurations of 51' with perio d n is 2 . For each class Chim,n) we call M1' the space transition matrix to the right (STMR). The space transition matrix to th e left (STML) is the transposed of the STMR, M~. We can now associate to a CA rule -r E Chim,n), with , ~ 2, and m ~ 0 ~ n , a time transit ion matrix. Let (bm , . .. , bn ) be a , -block of elements of {Aj }, 0 ~ j ~ 21' - 1. We define the map f* : { A j} ---t { Aj} by where c E {AiJ and f is the generating function of r .
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