Infinite Number of Chaotic Generalized Sub-shifts of Cellular Automaton Rule 180

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1 Infinite Number of Chaotic Generalized Sub-shifts of Cellular Automaton Rule 180 Wei Chen, Fangyue Chen, Yunfeng Bian, Jing Chen School of Science, Hangzhou Dianzi University, Hangzhou, Zhejiang, P. R. China Abstract This aer is devoted to an in-deth study of cellular automaton rule 180 under the framework of symbolic dynamics. Rule 180, a member of Wolfram s class IV Chua s hyer Bernoulli shift rules, defines infinite number of generalized sub-shifts. An effective method of constructing the shift invariant sets of the rule s global ma is roosed. It is noted that this method is also alicable to studying the dynamics of other rules. Furthermore, the rich comlex dynamical behaviors on these sub-shifts, such as ositive toological entroies, toologically mixing, chaos in the sense of Li-Yorke Devaney, are revealed. Keywords: cellular automata; chaos; generalized sub-shift; symbolic dynamics; toologically mixing. 1. Introduction Cellular automata (CA) are among the oldest models of natural comuting, dating back over half a century [1]. The first CA studied by von Neumann in the late 1940s were biologically motivated: the goal was to design selfrelicating artificial systems that are also comutationally universal. Following suggestions by S. Ulam, he envisioned a discrete universe consisting of a two-dimensional mesh of finite state machines, called cells, interconnected locally with each other. The cells change their states synchronously deending on the states of some nearby cells, the neighbors, as determined by a local udate rule. All cells use the same udate rule so that the system is homogeneous like many hysical biological systems. These cellular universes are now known as CA. CA have been widely used to model a variety of dynamical systems in hysics, biology, chemistry, comuter science in the recent decades. The toological dynamics of CA began in 1969 with Hedlund who viewed one-dimensional CA in the context of symbolic dynamics as endomorhisms of the shift dynamical system. His main results are the characterizations of surjective oen CA [2]. In the 1980s, Wolfram roosed cellular automata as models for hysical systems exhibiting comlex or even chaotic behaviors. In his work, he divided the 2 23 = 256 elementary cellular automata (ECA) rules informally into four classes using dynamical concets like eriodicity, stability, chaos [3-5]. In 2002, he introduced his monumental work A New kind of Science [6]. Based on this work, Chua et.al rovided a nonlinear dynamics ersective to Wolfram s emirical observations from the viewoint of mathematical analysis via the concets like characteristic function, forward time-τ ma, basin tree diagram, Isle-of-Eden digrah so on [7-11]. Although there are 256 ECA rules, only 88 rules are globally indeendent from each other [12]. These 88 global indeendent ECA rules are also organized into 4 grous with distinct qualitative dynamics; 40 eriod-k (k = 1, 2, 3, 6), 30 toologically distinct Bernoulli shift rules, 10 comlex Bernoulli shift rules 8 hyer Bernoulli shift rules [8-11]. Due to the fact that many roerties of the temoral evolution of CA, such as toological entroy, toologically sensitivity toologically mixing are undecidable [13], one should, in rincile, searately analyze time-asymtotic dynamics for each class of rules. It will be seen in the next sections that rule 180, a member of Wolfram s class IV Chua s hyer Bernoulli shift rules, has infinite number of generalized sub-shifts with rich comlex dynamics. Precisely, it has ositive entroies, is toologically mixing on these generalized sub-shifts. This imlies that rule 180 is chaotic in the sense of Li-Yorke Devaney. The rest of this aer is organized as follows. Section 2 resents the reliminaries of symbolic dynamical systems CA. Section 3 exlores infinite number of f ositively invariant subsets, which are hyer generalized sub-shifts of f 180. Section 4 demonstrates some comlex dynamics of the rule. Finally, Section 5 concludes this aer. 2. Preliminaries A word over S = {0, 1} is a finite sequence a = (a 0,, a n 1 ). The length of a is denoted by l(a) = n. If a is a finite or infinite word I = [i, j] is an interval of integers on which a is defined, then denote a [i,j] = (a i,, a j ) a [i,j) = (a i,, a j 1 ). b is a subword of a, denoted by b a, if b = a I for some interval I Z; otherwise, denoted by b a. A bi-infinite word is called a configuration, the collection of all configurations is Σ 2 = S Z = {(, x 1, x 0, x 1, ) x i S, i Z} the roduct d induced by Hamming distance is defined as 1 d(x, y) = 2 x i i y i, i=

2 Table 1: Truth table of Boolean function of rule 180 (x i 1, x i, x i+1 ) ˆf180 (x i 1, x i, x i+1 ) (0, 0, 0) 0 (0, 0, 1) 0 (0, 1, 0) 1 (0, 1, 1) 0 (1, 0, 0) 1 (1, 0, 1) 1 (1, 1, 0) 0 (1, 1, 1) 1 for any x, y Σ 2. It is easy to know that Σ 2 is a Cantor comlete metric sace. The left-shift ma σ L right-shift ma σ R are defined by [σ L (x)] i = x i+1, [σ R (x)] i = x i 1, for any x Σ 2, i Z resectively, where [σ(x)] i sts for the i-th symbol of σ(x). By a theorem of Hedlund [2], a ma f : Σ 2 Σ 2 is a cellular automaton iff it is continuous commutes with σ, i.e., σ f = f σ, where σ is left-shift or right-shift. Moreover, any CA f defines a dynamical system (Σ 2, f). A subset X Σ 2 is f-invariant if f(x) X, strongly f-ositively invariant if f(x) = X. If X is a closed f-invariant, then (X, f) or simly X is called a subsystem of (Σ 2, f). Each ECA rule can be exressed by a logical truth table, the one of rule 180 is shown in Table 1. It is clear that its binary outut sequence is Let ˆf180 be the local ma defined by the truth table, f 180 : Σ 2 Σ 2 with f 180 (, x 1, x 0, x 1, ) = (, y 1, y 0, y 1, ) be the global ma induced by ˆf 180, where y i = ˆf 180 (x i 1, x i, x i+1 ). The local ma ˆf180 also defines a ma from S n+1 to S n 1 (n 2) with ˆf 180 (a 0, a 1,, a n ) = (b 0, b 1,, b n 2 ), where b i = ˆf 180 (a i, a i+1, a i+2 ) (i = 0, 1, 2,, n 2). The n times iteration of ˆf 180 is a ma ˆf 180 n from S 2n+1 to S with ˆf n 180(a n,, a 0,, a n ) = n 1 ˆf 180 ( ˆf 180 (a n 1 [ n,n 2]), ˆf 180 (a n 1 [ n+1,n 1]), ˆf 180 (a [ n+2,n])). 3. Generalized sub-shifts Let x S n be a finite word, x = ( x 0, x 1,, x n 1 ) with boundary condition x 0 = x n 1 = 1. Denote x : Z S with i x i of the following form: { xi, if 0 i (n 1); m Z, x m+i = 0, otherwise. These configurations are called bi-infinite extensions of the block x in the background of 0 s [14, 15]. Define F n = {x Σ 2 x S n, x 0 = x n 1 = 1} (1) F = n N F n. (2) Any configuration of this set is said to be 0-finite [14]. Therefore a configuration x is 0-finite iff it is of the form x = (, 0, 0, x m,, x, 0, 0, ), with m the minimum the maximum site resectively for which x m = x = 1. To each 0-finite configuration x we assign its length l(x) = m + 1. articularly, for the quiescent configuration 0 = (, 0, 0, 0, ), l(0) = 0. Definition 1: (1) [14] A dynamical system (Σ 2, f) (or simly f, non-necessarily induced from a CA rule) is said to be a generalized shift iff there exists a ma M : Σ 2 N such that for any α Σ 2, any t N, f M(α) (f t (α)) = σ M(α) (f t (α)); (2) [14] f is said to be a generalized sub-shift on the subset X of Σ 2 iff X is f-ositively invariant (f(x) X), there exists a ma M : X N such that for any x X, any t N, f M(x) (f t (x)) = σ M(x) (f t (x)); (3) f is said to be a hyer generalized sub-shift on the subset X of Σ 2 iff X is f-ositively invariant (f(x) X), there exists a number M N such that for any x X f M (x) = σ M (x). Remark 1: If f is a (hyer) generalized sub-shift on the subset X, then simly say X is a (hyer) generalized subshift of f. Proosition 1: If X is a hyer generalized sub-shift of CA f, then there exists a set A of words of length n = (2M + 1) such that X = X A = {x Σ 2 x [i n,i+n] A, i Z}, where A is said to be the determinative block system of X. Proof: Without loss of generality, σ is chosen as the right shift with [σ(x)] i = x i 1 for x X. Since the local ma ˆf : S 3 S, it can lead out of its M times iteration ˆf M : S 2M+1 S. Thus, f M (x) = σ M (x), x X, if only if ˆf M (x [i M,i+M] ) = [f M (x)] i = [σ M (x)] i = x i M, for all i Z. Let A = {(a 0,, a 2M ) S 2M+1 (a 0,, a 2M ) = x [i M,i+M], x X, i Z}, A is a finite set since A < 2 2M+1. Then it follows that X = X A. It was known that f 180 is not a generalized shift, but is a generalized sub-shift hyer generalized sub-shift on the subset F [14, 15].

3 Lemma 1: [14, 15] For rule 180, there exists a ma M : {0, 1} N such that for any x {0, 1}, f M( x) 180 (x) = σ M( x) (x), where {0, 1} is the set of all blocks of finite length defined over S = {0, 1} the ma M is a ower of two, i.e., for any x {0, 1} n, M( x) = 2 E( x) for a suitable ma E : {0, 1} N. Where as usual x( F Σ 2 ) denotes any bi-infinite extension of block x in a background of 0 s. Lemma 2: [14, 15] The ma which satisfies Lemma 1 is such that for every n N, R M(1 n ) = 2 [log(n+1)], where 1 n is the 1-constant block of length n. The satio-temoral evolution of the bi-infinite extensions of blocks in a background of 0 s is shown in Figure 1. (a) (b) Fig. 1: Satio-temoral evolution of the bi-infinite extension of block 1 n in a background of 0 s, (a) n = 14; (b) n = 30. Unfortunately, the f 180 -ositively invariant subset F defined in (2) is only a countable infinite set, the dynamics of f 180 on the generalized sub-shift F is not enough to reveal the dynamics on the total hase sace Σ 2. In the following discussion in this section, infinite number of generalized subshifts of f 180 will be exressed. For given N, let n satisfies 2 1 n (i.e., log(n + 1) < + 1), construct a finite word A (n) i.e., A (n) = ( 0,, 0, 1 n, 0,, 0 ). }{{}}{{} a (n) denotes the ( )-sequence set aeared in a (n), = {(a 0, a 1,, a 2 +1) (a 0, a 1,, a 2 +1) a (n) }. It is easily known that the number of symbols of a (n) is ( n + 1), thus, the number of sequences of A (n) is ( n + 1), ˆf 180(a 0, a 1,, a 2 +1) = a 0, for any (a 0, a 1,, a 2 +1) A (n). Now let Λ (n) = {x Σ 2 x [i 2,i+2 ] A (n), i Z} (3) Λ = n=2 1 Λ (n). (4) In (3), the ( )-sequence set A (n) is the determinative block system of Λ (n). Proosition 2: For given N, n satisfies 2 1 n , Λ is a f180-ositively 2 invariant set, is a hyer generalized sub-shift of f Proof: In fact, for any x Λ, if x Λ (n), where n satisfies 2 1 n , thus, x [i 2,i+2 ] A (n) 2, so ˆf 180(x [i 2,i+2 ]) = x i 2, this imlies f180(x) 2 = σr 2 (x) Λ(n) Λ. Moreover, let Λ = Λ f 180 ( Λ ) f 2 180( Λ ) f ( Λ ). (5) Proosition 3: Λ is a f 180 -ositively invariant set, is a hyer generalized sub-shift of f 180. Proof: The result is obvious, the details are omitted here. Based on above roositions, the following interesting result is obtained. Theorem 1: For rule 180, there exists infinite number of f 180 -ositively invariant subsets Λ ( = 1, 2, 3, ) such that f180 2 Λ = σr 2 Λ, i.e., Λ ( = 1, 2, 3, ) are the generalized sub-shifts of f 180. Examle 1: The structure of the generalized sub-shift Λ 1 : when = 1, then n = 1, 2. Thus, a (1) 1 = ( ), a (2) 1 = ( ), where Λ 1 = Λ (1) 1 Λ (2) 1, Λ (1) 1 = {x Σ 2 x [i 2,i+2] A (1) 1, i Z}, Λ (2) 1 = {x Σ 2 x [i 2,i+2] A (2) 1, i Z}, A (1) 1 = {(a 0, a 1,, a 4 ) (a 0, a 1,, a 4 ) a (1) 1 } = {00000, 00001, 00010, 00100, 01000, 10000} A (2) 1 = {(a 0, a 1,, a 4 ) (a 0, a 1,, a 4 ) a (2) 1 } = {00000, 00001, 00011, 00110, 01100, 11000, 10000}. It follows that Λ 1 = {x Σ 2 x [i 2,i+2] A 1, i Z}, where A 1 = A (1) 1 A (2) 1 = {00000, 00001, 00010, 00100, 01000,

4 10000, 00011, 00110, 01100, 11000}. Let Λ 1 = Λ 1 f( Λ 1 ), where f( Λ 1 ) = {x Σ 2 x [i 2,i+2] A 1, i Z}, A 1 = {(b 0, b 1,, b 4 ) (b 0, b 1,, b 4 ) = ˆf 180 (a 0, a 1,, a 6 ), (a i, a i+1,, a i+4 ) A 1, i = 0, 1, 2} = A 1. Remark 2: Generally, the f 180 -ositively invariant set Λ in (5) may be one of subset of the set which also is a generalized sub-shifts with f180 2 = σr 2. For examle, = 1, it is easily investigated that there exists a f 180 -ositively invariant set 1 = {x Σ 2 x [i 2,i+2] A, i Z} such that f = σr 2 1, where A = {00000, 00001, 00010, 00011, 00100, 00101, 00110, 01000, 01001, 01011, 01100, 01101, 10000, 10001, 10010, 10011, 10110, 11000, 11001, 11011}. Obviously, Λ Comlex dynamics If Λ Σ 2 is a hyer generalized sub-shift of a CA f, Λ = Λ A, A is the determinative block system of Λ, then Λ A can be described by a finite directed grah G A = {A, E }, where each vertex is labeled by a sequence in A, E is the edge set. Two vertices a = (a 0,, a n 1 ) b = (b 0,, b n 1 ) are connected by an edge of E if only if a k = b k 1, k = 1, 2,, n 1. Every edge (a 0,, a n 1 ) (b 0,, b n 1 ) of E is labeled by b n 1. One can think of each element of Λ A as a bi-infinite ath on the grah G A. Whereas a directed grah corresonds to a square transition matrix A = (A ij ) m m with A ij = 1 if only if there is an edge from vertex b (i) to vertex b (j), where m = A is the number of elements in A, i (or j) is the code of the vertex in A, i, j = 0, 1,, m 1. Thus, Λ A is recisely defined by the transition matrix A. Remarkably, a square matrix A is irreducible, if for any i, j, there exists an n such that A n ij > 0; aeriodic if there exists an n, such that A n ij > 0, for all i, j, where An ij is the (i, j) entry of A n. If Λ A is a sub shift of finite tye of the shift ma σ, then the ma is toological transitive if only if A is irreducible; the ma is toologically mixing if only if A is aeriodic. Equivalently, A is irreducible if only if for every ordered air of vertices b (i) b (j) in A there is a ath in the grah G A starting at b (i) ending at b (j) ; A is aeriodic if only if it is irreducible the numbers of the length of any two different closed aths in the grah G A are corime [19-21]. Lemma 3: Let Λ be a hyer generalized sub-shift of a CA f with f M (x) = σ M (x), x Λ, A be the determinative block system of Λ, A be transition matrix corresonding to the finite directed grah G A = {A, E }, if A is aeriodic, then σ f are both toological mixing on Λ [19, 20]. Lemma 4: Let Λ be a hyer generalized sub-shift of a CA f with f M (x) = σ M (x), x Λ, then (1) the toological entroy of f on Λ is ent(f Λ ) = log(ρ(a)), where ρ(a) is the sectral radius of the transition matrix A corresonding to the finite directed grah G A = {A, E }; [19, 20] (2) ent(f) ent(f Λ ), where ent(f) is the toological entroy of f on total symbolic sace Σ 2. [16-20] Lemma 5: For a hyer generalized sub-shift (Λ, f), if f is toological mixing on Λ, then (1) f is chaotic in the sense of Devaney on Λ [16, 19, 20]; (2) f is chaotic in the sense of Li-Yorke. [19, 27] Theorem 2: (1) f 180 is chaotic in the sense of Devaney on the hyer generalized sub-shift Λ 1 in Examle 1; (2) the toological entroy of f 180 on Λ 1 is ositive. Proof: (1) At the time, f180(x) 2 = σr 2 (x), x Λ 1. It is already known that the determinative block system of Λ 1 is A 1 = {00000, 00001, 00010, 00100, 01000, 10000, 00011, 00110, 01100, 11000}, the matrix corresonding to the finite directed grah G A1 = {A 1, E } is A 1 = It follows that A n 1 > 0 (n 10), so A is aeriodic, σ R σr 2 are mixing on Λ 1, it imlies f 180 is mixing on Λ 1. Thus, f 180 is chaotic in the sense of Devaney on Λ 1 based on Lemma 5. (2) in fact, ent(f 180 Λ1 ) = log(ρ(a 1 )) log(1.4196) > 0. Remark 3: The result of the theorem can also be obtained from the corime roerty of the numbers of the length of any two different closed aths in the finite directed grah G A1 = {A 1, E }. Theorem 3: f 180 is chaotic in the sense of Devaney on each hyer generalized sub-shift Λ ( = 1, 2, 3, ) in (3), (4) (5). Proof: Recall the structure of Λ ( = 1, 2, 3, ): Λ = Λ f 180 ( Λ ) f 2 180( Λ ) f ( Λ ). It is obvious that Λ is f 180 -ositively invariant set. Since Λ is a hyer generalized sub-shift of f 180, by Proosition 1, there exists its determinative block system A. It is clear that the ( ) word belongs to A, this imlies

5 there exists a closed ath from the word to self whose length is 1 in the finite directed grah GA = { A, E }, so it is easy to know that σ R f180 2 is toologically mixing on Λ. Moreover, f 180 is a homeomorhic ma from Λ to f 180 ( Λ ), so it is easily roved that belongs to each A ( = 2, 3,, 2 1), thus, σ R f180 2 is also toologically mixing on f180( Λ i ) (i = 2, 3,, 2 1). Finally, let A denote the determinative block system of Λ, then there still holds that belongs to A, it imlies that σ R f 180 is toologically mixing on Λ, so f 180 is chaotic in the sense of Devaney on Λ. Of course, it is chaotic in the sense of Li-Yorke. 5. Conclusion As a articular class of dynamical systems, CA have been widely used for modeling aroximating many hysical henomena. Desite their aarent simlicity, CA can dislay rich comlex evolutions [22-26, 28, 29], many roerties of their satio-temoral evolutions are undecidable [13]. This aer is devoted to an in-deth study of cellular automaton rule 180 in the framework of symbolic dynamics. It has been rigorously roved that its global ma defines infinite number of generalized sub-shifts with rich comlex dynamical behaviors, such as toologically mixing, ositive toological entroies chaos in the sense of Li-Yorke Devaney. Indeed, the dynamics of rule 180 have not been comletely revealed, therefore new effective analytical methods should be exloited to investigate it as well as other CA rules in future studies. Acknowledgments This research was jointly suorted by the NSFC (Grant No No ). References [1] von Neumann, J. Theory of self-reroducing automata (edited comleted by A. W. Burks), University of Illinois Press, Urbana London, [2] Hedlund, G. A. Endomorhisms automorhism of the shift dynamical system, Theory of Comuting Systems, 3: , [3] Wolfram, S. Statistical mechanics of cellular automata. Rev. Mod. Phys., 3: , [4] Wolfram, S. 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