Graphical limit sets for general cellular automata

Theoretical Computer Science 580 (2015) 14–27

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Theoretical Computer Science www.elsevier.com/locate/tcs

Graphical limit sets for general cellular automata Johannes Müller a,b,∗ , Hengrui Jiang a a b

TU München, Centre for Mathematical Sciences, Boltzmannstr. 3, D-85747 Garching, Germany Helmholtz Center Munich, Institute for Computational Biology, Ingolstädter Landstr. 1, D-85764 Neuherberg, Germany

a r t i c l e

i n f o

Article history: Received 24 April 2014 Received in revised form 22 January 2015 Accepted 12 February 2015 Available online 3 March 2015 Communicated by J. Kari Keywords: Cellular automata Graphical limit sets

a b s t r a c t The existing theory of graphical limit sets for cellular automata relies on algebraic structures and applies only to certain classes of cellular automata that possess this structure. We extend this theory to general cellular automata using topological methods. The starting point is the observation that the rescaled space-time diagrams, intersected with an appropriately chosen compact set, form sequences in a compact, metric space. They necessarily possess converging subsequences. In the present paper we define graphical limit sets as the collection of the accumulation points. The main result is that for a large class of cellular automata the graphical limit set defined in this way carries a group structure, which is either the trivial group consisting of one element only, or is homeomorphic to S 1 . The well known self-similar, graphical limit sets are representatives of the second class. © 2015 Elsevier B.V. All rights reserved.

1. Introduction A central problem in the theory of cellular automata is the classification of these objects in sensible subsets. Cellular automata are – similarly to e.g. finitely generated groups, or partial differential equations – much too diverse to allow for many strong theorems covering all of them. The Curtis–Lyndon–Hedlund theorem [5,1] may be one of the very few examples for such general theorems. The general feeling is that an appropriate classification yields subgroups that can be well understood. Since the seminal work of Wolfram [19], where a purely phenomenological classification by visual inspection has been proposed, there have been many attempts to find formal justifications and methods to introduce classification schemes. Examples of these approaches are topological dynamical systems (Hurley classification [6,7]), or formal grammars ˚ (Gilman classification [2]) or continuity (Kurka classification [9]). We aim to contribute some ideas that are perhaps close to Wolfram’s original, phenomenological concept. This concept has been formalised by Wilson for some cellular automata. The space-time diagrams of Wolfram automata yield in some cases structures resembling self-similar patterns. Taking an appropriate limit, Willson [14,15] has been able to show that the space-time patterns of certain cellular automata indeed tend to self-similar sets. This observation gained a lot of attention [17,16,12,11,10]. The core assumption has always been the existence of certain algebraic properties of the cellular automaton. Recent developments attempt to relax the conditions, using more general algebraic structures [4]. In the present work, we propose to define graphical limit sets in a purely topological way. No algebraic structures are required. We simply use the fact, that the set of non-void, compact sets which are uniformly bounded together with the Hausdorff metric form a compact metric space. This fact indicates that the sequence of graphical representations of

*

Corresponding author at: TU München, Centre for Mathematical Sciences, Boltzmannstr. 3, D-85747 Garching, Germany. E-mail addresses: [email protected] (J. Müller), [email protected] (H. Jiang).

http://dx.doi.org/10.1016/j.tcs.2015.02.022 0304-3975/© 2015 Elsevier B.V. All rights reserved.

J. Müller, H. Jiang / Theoretical Computer Science 580 (2015) 14–27

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space-time patterns evolving during simulations of cellular automata have accumulation points. The graphical limit set of a cellular automaton is defined as the collection of these accumulation points. We investigate the structure of this set. 2. Definition of graphical limit sets We recall that a cellular automaton is defined by a tuple, such as (, U 0 , E , f 0 ), where  is a finitely generated Abelian group (e.g., Z), U 0 ⊂  is a finite set denoting the neighborhood of the neutral element (called “origin” in the present context) 0 in  , E is a finite set of local states. Let E U 0 (resp. E  ) denote the set of all maps from U 0 to E, { g : U 0 → E } (resp. { g :  → E }); the elements in E U 0 (E  ) are called configurations or states. f 0 : E U 0 → E is the local function. The global function of the cellular automaton is defined by f : E  → E  , u → f (u ) where f (u )( z) = f 0 (u |z+U 0 ). Definition 2.1. Given a cellular automaton (, U 0 , E , f 0 ) with  = Z, E = {0, . . . , Q }. The state 0 ∈ E U 0 denotes 0(x) = 0 for all x ∈ U 0 . If f 0 (0) = 0, then 0 is called quiescent or resting state. The support of u 0 ∈ E  is defined by supp(u 0 ) = {x ∈  | u 0 (x) = 0}. Denote by ( E  )c the states with finite (or compact) support. We define

P (u 0 ) = {( z, t ) ∈ Z × N0 | f t (u 0 )( z) = 0}. Notation: For a, b ∈ Z, we denote by [a, b] = {a, . . . , b} ⊂ Z; if [a, b] ⊂ R is meant, this becomes clear from the context. Let U 0 ⊂ [−d, d] and supp(u 0 ) ⊂ [− K , K ]. Assume that the cellular automaton possesses a quiescent state 0. Then,

P (u 0 ) ⊆



[−(d t + K ), (d t + K )] × {t } =: .

t ∈N

In any case, also without the assumption of a quiescent state, the only part of P (u 0 ) influenced by the non-trivial information stored in the initial state u 0 is contained in . Therefore,  is also called the “light-cone” of the cellular automaton – the information contained in u 0 only spreads within this set. Outside this set, the automaton cycles through states that are constant in space; to be more precise, we find the same pattern that can be observed if we start with the initial value 0 ∈ E  (the state that is identically zero). Rescaling P (u 0 ) yields the definition of the graphical representation. Definition 2.2 (Recall notation of Definition 2.1). Let I = [−1/2, 1/2] × [0, 1], and for ( z, t ) ∈ Z2 define I z,t = ( z, t ) + I . Let henceforth the set J be defined as

J = [−(d + K + 1), d + K + 1] × [0, 1]. The rescaled graphical representation of the space-time pattern (or simply the space-time pattern) is defined as

⎛ F n (u 0 ) = ⎝



1 n

⎞ I z,t ⎠ ∩ J .

( z,t )∈ P (u 0 )

Remark 2.3. Due to the scaling by 1/n, F n (u 0 ) only takes into account the first n − 1 time steps. The region of F n (u 0 ) influenced by the non-zero pattern in the initial condition of the initial state u 0 is contained in

n (u 0 ) =



[−(d m + K + 1/2)/n, (d m + K + 1/2)/n] × [m/n, (m + 1)/n)] ⊂ J .

m∈[0,n−1]

Outside of n (u 0 ), the set F n (u 0 ) is either empty or solid, or consists of horizontal strips. We will use this fact later. Let us denote the part of J that carries information for n → ∞ by ∞ , i.e. ∞ = {(x, y ) | − d y ≤ x ≤ d y }. We aim at graphical limit sets, which means that we aim at an understanding of the convergence of subsequences of the sequence ( F n (u 0 ))n∈N . We recall the definitions of Hausdorff metric and Kuratowski convergence (see also [8]). Definition 2.4. Let A, B be compact sets in Rn . If either A or B is empty, define d H ( A , B ) = ∞. Else, define

d H ( A , B ) = inf{ε > 0 | B ⊆ A ε , and A ⊆ B ε }, where A ε = {x | ∃ y ∈ A : x − y < ε }. d H (., .) is called Hausdorff metric. The Hausdorff metric introduces a topology on the set of all compact subsets of Rn . A sequence of sets A n converges to a set B, if d H ( A n , B ) → 0.

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Definition 2.5. Let ( A m )m∈N denote a sequence of compact subsets of Rn . A m converges to a set A in the sense of Kuratowski, if lim sup A m = A = lim inf A m , where

lim inf A m = {x ∈ Rn | ∃xm ∈ A m , xm → x} and

lim sup A m = {x ∈ Rn | ∃mi ∈ N, mi → ∞ monotonously, xmi ∈ A mi , xmi → x}. Theorem 2.6. Let X ⊂ Rn be compact, H( X ) denote the set of all non-empty, compact subsets of X , together with the Hausdorff metric. Then, (H( X ), d H ) is a compact, metric space. The topology induced by the Kuratowski limit on H( X ) is identical with the topology induced by the metric on H( X ). The proof of this theorem can be found e.g., in [8]. We will later use the following lemma, that is straightforward to prove. Lemma 2.7. Let X ⊂ Rn be a compact set, A n ⊂ X a converging sequence of compact sets with limit point B, and θi ∈ R a converging sequence of real numbers, θi → θ . Then, θi A i → θ B. After these preparing remarks, we introduce the graphical limit set of a cellular automaton. Definition 2.8. Given a cellular automaton (Z, U 0 , E , f 0 ), and an initial state u 0 ∈ ( E  )c , u 0 = 0, denote by S (u 0 ) all accumulation points of the sequence ( F n (u 0 ))n∈N in H( J ). We call S (u 0 ) the graphical limit set of the CA (w.r.t. u 0 ). To shorten notation, we will in general suppress the initial state u 0 and simply write S . We assume from now on that u 0 = 0, and that u 0 has a compact support. Remark 2.9. As F n ⊂ J and J is compact, H( J ) is a compact metric space, and hence S = ∅. Furthermore, if u 0 = 0, then (0, 0) ∈ A for all A ∈ S . An immediate consequence of Remark 2.3 is the following observation: outside of ∞ , the graphical limit set is either empty (in case of a quiescent state) or fills the region (in all other cases). Corollary 2.10. If the CA possesses a quiescent state 0, then for S ∈ S we find S \ ∞ = ∅. Otherwise, not only S \ ∞ = J \ ∞ , but also S = J . As this second case is rather trivial, we make the following assumption. Assumption. From now on, let 0 be a quiescent state. As 0 is a quiescent state, any ( z, t ) ∈ Z ×N with f t (u 0 )( z) = 0 has a predecessor ( z , t − 1) ∈ Z ×N0 with f t −1 (u 0 )( z ) = 0, where z ∈ z + [−d, d]. Finite induction yields a discrete path connecting ( z, t ) and supp(u 0 ) × {0} (see Fig. 1). Some obvious properties of this path are collected in the following corollary. Corollary 2.11. If f t (u 0 )( z) = 0, then there is a path ( zi , i ) ∈ Z × N0 , i ∈ [0, t ], with f i (u 0 )( zi ) = 0, zt = z, and | zi +1 − zi | ≤ d. The path is contained in the downward cone − ,

( z i , i ) ∈ ( z , t ) + − ,

− =

t 

[−d (t − t  ), d (t − t  )].

t  =0

Moreover, ( zi , i ) ∈ . Though these facts are simple implications from the existence of a quiescent state, they have far-reaching consequences. For example, this corollary implies that S ∈ S is pathwise connected. Proving this fact is our next aim. Therefore, we start with the investigation of closed sets that satisfy a certain cone condition. Definition 2.12. Let A ⊂ R2 be a closed set. For k > 0, let Kk = {(x, y ) | y ≤ 0, |x| ≤ k| y | } denote the downwards directed, closed cone in R2 with an aperture angle controlled by k. The set A satisfies the cone condition (with constant k), if

∀ w ∈ A , r ≥ 0 ∃ w  ∈ A ∩ ( w + Kk ) : w − w  = r .

J. Müller, H. Jiang / Theoretical Computer Science 580 (2015) 14–27

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Fig. 1. Visualization of Corollary 2.11. In the lowest row, the support of u 0 is indicated; upward, the cone  (the “light-cone”) is drawn. The black upper cell is ( z.t ). The grey cells form the path from ( z, t ) to the support of u 0 (at time zero). The downward directed cone ( z, t ) + − forms a barrier for the path. Indeed, if ( z, t ) ∈ , then the path stays in the intersection of the two cones. For this example we chose d = 1.

Notation: As usual, C 0 denotes the Banach space of continuous functions, and C 0,α that of Hölder continuous functions (with Hölder exponent α ); in particular, C 0,1 are the Lipshitz continuous functions. Lemma 2.13. Let A ⊂ R2 be a set that satisfies the cone condition with constant k, (x0 , y 0 ) ∈ A, and y 1 < y 0 . Then, there is a continuous function h : [ y 1 , y 0 ] → R, such that (h( y ), y ) ∈ A ∩ (( y 0 , x0 ) + Kk ) for y ∈ [ y 1 , y 0 ]. Proof. Let n ∈ N. Using the cone condition, we find by finite induction a finite sequence of points w i ∈ A, i = 0, . . . n with the properties (1) w 0 = (x0 , y 0 ), ( w i ) y = y 0 + i ( y 1 − y 0 )/n (2) w i +1 ∈ w i + Kk . In particular, w i ∈ w 0 + Kk . Linear interpolation yields a function hn : [ y 1 , y 0 ] → R, with hn ∈ C 0,1 ([ y 1 , y 0 ]) (with Lipschitz constant k),

(hi ( y 0 + i ( y 1 − y 0 )/n), y 0 + i ( y 1 − y 0 )/n) = w i ∈ A , and (h i ( y ), y ) ∈ ( y 0 , x0 ) + Kk for y ∈ [ y 1 , y 0 ]. As the embedding of C 0,1 in C 0 is compact, we find a subsequence h i j converging to a continuous function h : [ y 1 , y 0 ] → R. Since the graph of h i are contained in the closed set (x0 , y 0 ) + Kk , so is the graph of h. We show that the graph of h is also contained in A: Let x = h( y ), y ∈ [ y 1 , y 0 ]. There are breakpoints y˜ i j of the functions h i j that tend to y for j → ∞. Thus, (h i j ( y i j ), y i j ) ∈ A and (h i j ( y i j ), y i j ) → (h( y ), y ) for j → ∞. As A is closed, (h( y ), y ) ∈ A. 2 Remark 2.14. C 0,1 is compactly embedded in C 0,α ,

α ∈ (0, 1). Hence, our path (h( y ), y ) belongs to C 0,α for any α ∈ (0, 1).

Lemma 2.15. For S ∈ S , the set Sˆ = S ∪ {(x, y ) | y ≤ 0} satisfies the cone condition with k = d.

ˆ We Proof. Sˆ is – as union of two closed sets – closed. Let w = (x0 , y 0 ) ∈ Sˆ and r ≥ 0. If y 0 − r ≤ 0, then w  = w − (0, r ) ∈ S. assume y 0 > r, in particular w ∈ S. Then, there are sequences n → ∞, ( zn , tn ) ∈ Z × N0 with tn ≤ n , f n (u 0 )( zn ) = 0, (zn , tn )/n → w. Since y 0 > r, tn /n → y 0 , we know that tn > n r for all but finitely many indices. For  given, there is a path described in Corollary 2.11 from ( zn , tn ) to supp(u 0 ) × {0}. As the distance between two neighboring points within this path is less than or equal to d + 1, we find elements (˜zn , t˜n ) in this path with n r − (d + 1) ≤ (˜zn , t˜n ) − ( zn , tn ) ≤ n r + (d + 1). Since (˜zn , t˜n )/n ∈ J and J is compact, there is a converging subsequence,

(˜zn , t˜n )/n → w  ∈ S . All in all, we have w  ∈ w + Kk for k = d (this is a consequence of Corollary 2.11), and

r = lim r − (d + 1)/n ≤ lim (˜zn , t˜n )/n − ( zn , tn )/n = w − w  ≤ lim r + (d + 1)/n = r . n →∞

→∞









n →∞

2

If we start in the proof above with w = (x, y ) ∈ S, we even find a path (h(s), s) ∈ S ∩ ∞ ∩ w + Kd for s ∈ [0, y ] with h ∈ C 0,α , α ∈ (0, 1). In particular, this path hits (0, 0) as this is the only point in ∞ ∩ (R × {0}).

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Theorem 2.16. Any S ∈ S is pathwise connected; for any point w = (x, y ) ∈ S, there is a path (h(s), s) ∈ S ∩ ∞ ∩ w + Kd for s ∈ [0, y ] with h ∈ C 0,α for any α ∈ (0, 1). The following lemma, which appears to be rather technical but will be useful later on, is an immediate consequence of the shape of the paths that we constructed above. Lemma 2.17. Let J  = int( J ) ∪ {(0, 0)}. Then, ∀ S ∈ S , t ≥ 1 : t S ∩ J = t S ∩ J  . Proof. We know that t S ∩ J ⊂ ∞ , hence any point in t S ∩ J has an x-component in (−d, d) and cannot hit the leftor right boundary of J . If w ∈ (t S ∩ J ) \ J  , then the y-component of w is 1, w y = 1. However, there is a path (h(s), s) connecting w and (0, 0) as described above. In particular, (h(s), s) ∈ t S ∩ J  for s ∈ [0, w y ). Hence, w is an accumulation point of t S ∩ J  . 2 We know by now that S ∈ S is (pathwise) connected. Next we focus on the connectivity of S itself. Therefore we first estimate the distance between F n and F n+1 . In the following statement we use d and K as introduced before Definition 2.2. Proposition 2.18. Let n > 1. Then, d H ( F n , F n+1 ) ≤ 2(d + K + 4)/n. 1 Proof. First of all, from the definition of F n we find F n = n+ F n+1 ∩ J ⊂ (1 + 1/n) F n+1 . Let n ( F n )ε ⊃ F n+1 and ( F n+1 )ε ⊃ F n .

ε = 2(d + K + 4)/n. We show

(a) ( F n+1 )ε ⊃ F n : Let (x, y ) ∈ F n . Since F n ⊂ (1 + 1/n) F n+1 , we know (1 + 1/n)−1 (x, y ) =: (˜x, y˜ ) ∈ F n+1 , and hence (x, y ) − (˜x, y˜ ) = (˜x, y˜ )/n. The maximal Euclidean distance between (0, 0) and a point in J is given by (d + K + 1/2)2 + 1 < 2(d + K + 2). Thus,

(x, y ) − (˜x, y˜ ) ≤ (˜x, y˜ ) /n < 2(d + K + 4)/n, and ( F n+1 )ε ⊃ F n . (b) ( F n )ε ⊃ F n+1 : Let (x, y ) ∈ F n+1 . If (1 + 1/n)(x, y ) ∈ F n we argue as in (a), and find (x, y ) ∈ ( F n )ε . Let (1 + 1/n)(x, y ) ∈ / F n . As (x, y ) ∈ 1 /(n + 1) and F n = n+ F n+1 ∩ J , this implies that y ≥ 1/(1 + 1/n) = 1 − 1/(n + 1). Assume y > 1 − 1/(n + 1). n There is ( z, n + 1) ∈ Z × N0 with f n+1 (u 0 )( z) = 0 and (n + 1)(x, y ) ∈ I z,n+1 . Since 0 is a quiescent state, we find ( z , n) with f n (u 0 )( z ) = 0 and | z − z | ≤ d. Hence (x , y  ) = ( z , n)/n ∈ F n and

(x , y  ) − (x, y ) ≤ ( z , n)/n − ( z, n + 1)/(n + 1) + ( z, n + 1)/(n + 1) − (x, y )  1  

( z , n) − ( z, n + 1) + ( z , n) /n + ( z, n + 1) − (n + 1)(x, y ) ≤ n+1 √ (d + 1) + 2(d + K + 2) + 2 < 2(d + K + 4)/n = ε . ≤ n+1 If y = 1 − 1/(n + 1), we can either argue as in the case y < 1 − 1/(n + 1), or as in the case y > 1 − 1/(n + 1).

2

ω-limit sets: Given u ∈ E Z , the ω-limit set ω(u ) is defined by (see e.g. [3])

 ω (u ) = { f i (u )}.

Recall the definition of

n∈N i >n

It is well known that ω -limit sets of time-continuous dynamical systems on compact subsets of Rn are compact and connected. We find that S shares similarities with these ω -limit sets. In particular, in the proof of the next theorem we precisely use the arguments that show that ω -limit sets are compact and connected. The deeper reason is the fact that { F n }n∈N resembles a trajectory of a time-continuous dynamical system in H( J ), as the distance between F n and F n+1 becomes arbitrarily small for n large, and S can be defined in a similar way as ω -limit sets,

S (u ) =



{ F i (u )}d H .

n∈N i >n

Theorem 2.19. S is a compact and connected set in H( J ). (n)

Proof. S is closed: let S n ∈ S , S n → S. There are sequences F (n) → S n , where m are N-valued, strictly increasing sem quences. Using an appropriate subsequence, we are allowed to assume that d H ( S , S n ) ≤ 1/n, resp. d H ( S n , F (n) ) ≤ 1/m. (k)

Consider the diagonal sequence, k = k . For

m

ε > 0 given, we find that for k > 2/ε

J. Müller, H. Jiang / Theoretical Computer Science 580 (2015) 14–27

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d H ( S , F k ) ≤ d H ( S , S k ) + d H ( S k , F (k) ) ≤ 2/k ≤ ε . k

The diagonal sequence converges to S. The indices of the diagonal sequence may not be monotonously increasing, but there is an appropriate subsequence with monotonously increasing indices converging to S. Hence, S ∈ S . S is connected: Assume that this is not the case. Then, there are two open sets U 1 , U 2 ∈ H( J ), such that

U 1 ∩ U 2 = ∅,

S1 = ∅ = S2 ,

S = S1 ∪ S2 with Si = U i ∩ S .

Note that Si are compact as S is compact, and S1 ∩ S2 = ∅. Hence, δ = d H (S1 , S2 ) > 0. Recall that (Proposition 2.18) d H ( F n , F n+1 ) ≤ 2(d + K + 4)/n. Furthermore there are subsequences F 1 (F 2 ) of F n that converge to elements in S1 (S2 ). i

i

That is, the sequence F n repeatedly “moves” arbitrarily close to S1 and then arbitrarily close to S2 , always back and forth. For indices larger than 4(d + K + 4)/δ , it is not possible for F n to jump over the gap of size δ ; hence, there are again and again elements F n ∈ X = H( J ) \ (S1 ∪ S2 ). As X is closed, we find a converging subsequence of these elements with a limit S ∈ X . This finding contradicts X ∩ S = ∅. 2 3. Structure of graphical limit sets

Remark 3.1. For n > m, the set F n contains more information than the set F m . We already used that the sets F n and F m are related by

Fm =

n



m

Fn ∩ J .

The following definitions and lemmas state the consequences forced on S by this simple and obvious relation. Notation: All N-valued (sub)sequences of indices are assumed to be strictly monotonously increasing. For a point w ∈ J we denote by w x the x-component, and by w y the y-component, w = ( w x , w y ). Definition 3.2. Let A, B ∈ S . We say that A is λ-larger than B for λ ∈ [1, ∞) (in symbols: A ;λ B), iff there are sequences F ni → A, F mi → B, such that the limit of ni /mi exists, with

λ = lim

i →∞

ni mi

.

A ; B indicates that A ;λ B for some λ ≥ 1. Furthermore, we introduce [ A ] = { B ∈ S | A ; B }. Definition 3.3. For λ ≥ 1, define the family of maps L λ : H( J ) → H( J ), A → (λ A ) ∩ J and for t ∈ R≥0 the family of maps T t : H( J ) → H( J ) , A →  T t A = L exp(t ) A. Remark 3.4. We have L t s A = L t ( L s A ) for s, t ≥ 1, and accordingly T s+t A = T s ( T t A ) for all t , s ∈ R≥0 . Furthermore, T 0 A = A. Hence, T t is a semi-group acting on H( J ). However, as we will see below, this semi-group is not continuous on H( J ). Lemma 3.5. For a given A ∈ H( J ) let φ A : {λ | λ ≥ 1} → H( J ), λ → L λ ( A ). Then, φ A is left continuous, but in general not right continuous. Proof. φ A is left continuous: If A = ∅, then φ A (λ) ≡ A, and φ A is continuous. Let A = ∅, and assume that φλ is not left continuous at λ0 . Since λ ≥ 1, λ0 > 1. Hence there is a sequence λi with λi < λ0 , λi → λ0 , and η > 0 such that d H (φ A (λi ), φ A (λ0 )) > η . Let

εi(1) = inf{ε > 0 | (φ A (λi ))ε ⊃ φ A (λ0 )}, εi(2) = inf{ε > 0 | φ A (λi ) ⊂ (φ A (λ0 ))ε }. (1)

(2)

(∗)

Then, max{εi , εi } ≥ η . We first figure out which of the two εi does not tend to 0 if λi tends to λ0 . Due to the definition of φ A we know that φ A (λ) = ((λ/λ )φ A (λ )) ∩ J for λ ∈ [1, λ]. Since the maximal distance between (0, 0) and a point in J is less or equal to 2( K + d + 2), we have (φ A (λ ))ε ⊃ φ A (λ) for ε = 2( K + d + 2) (λ − λ )/λ . (1) Therefore, εi tends to 0 if λi tends to λ. (2)

Hence, εi ≥ η for i sufficiently large; there is a sequence of points w i ∈ φ A (λi ) with d H ({ w i }, φ A (λ0 )) > η . As w i /λi ∈ A, there is a convergent subsequence w in /λin → w ∈ A. Since λi < λ0 , we know that λi w ∈ φ A (λ0 ), and hence η < d H ({ w in }, φ A (λ0 )) ≤ w in − λin w → 0. The assumption that φ A is not left continuous leads to a contradiction. φ A is in general not right continuous: Let A = {(0, 1/2)}. Then, φ A (λ) = {(0, λ/2)} for λ ∈ [1, 2], but φ A (λ) = ∅ for λ > 2. 2

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J. Müller, H. Jiang / Theoretical Computer Science 580 (2015) 14–27

Fig. 2. Example of Remark 3.6 for K + d + 1 = 4. The dashed rectangle indicates J , the bold lines represent φ A (λ).

Remark 3.6. The map φ A (λ) is, in general, also for a (pathwise) connected set A not continuous in λ. Even if we require that L λ A is (pathwise) connected for all λ ≥ 1 is not sufficient to force φ to be continuous in λ (see Fig. 2): Let

˜ 2 = {(x, y ) | x = 1/2}, A

˜ 1 = {(x, y ) | y = x/(2( K + d + 1)}, A

˜ 1 ∪ A˜ 2 ). A = J ∩ (A

Then, φ A (λ) = φ A˜ (λ) for λ ≤ K + d + 1, but φ A (λ) = φ A˜ (λ) for λ > K + d + 1, and φ A (λ) is not right continuous at 1 1 λ = K + d + 1. The next statements investigate the relation between L t A and A ;t B for A , B ∈ S . Theorem 3.7. Let F ni → A, F mi → B, and ni /mi → t with t > 1. Then, F mi = L ni /mi F ni for all but finite many indices, and B = L t A. Proof. As the limit of ni /mi is larger than 1, for all but finite many indices ni > mi . Hence, F mi = L ni /mi F ni for all but finite many indices. Without restriction, we assume ni > mi for all indices. Under the condition



X :=

lim

i →∞

ni mi



F ni



∩ J = lim

we obtain

i →∞



B = lim F mi = lim i →∞

i →∞

ni mi



ni mi

F ni ∩ J

F ni ∩ J

 =

lim

i →∞

=: Y

ni mi

F ni

∩ J = t A ∩ J = Lt A .

All we need to prove is X = Y . We know that (ni /mi ) F ni as well as F mi = (ni /mi ) F ni ∩ J converge in Hausdorff- as well as in the Kuratowski-sense. Here, we use Kuratowksy’s approach. Y ⊆ X : Let w ∈ Y . There is a sequence of points w i ∈ (ni /mi ) F ni ∩ J and w i → w. Therefore, w i ∈ (ni /mi ) F ni , and hence n w ∈ limi →∞ mi F ni . As w ∈ J , we conclude w ∈ X . i

X ⊆ Y : Let w ∈ X . If w ∈ J  (with the notation of Lemma 2.17), then the y-component of w is less than 1, w y < 1. We know that there is a sequence w i ∈ (ni /mi ) F ni converging to w. As w y < 1, also ( w i ) y < 1 for all but finitely many indices. Hence, w i ∈ J (for all but finite many indices), and w ∈ Y . We have X ∩ J  ⊂ Y . As X = t A ∩ J , we may write t A ∩ J  ⊂ Y . Since Y is closed, we find with the help of Lemma 2.17 that

X = t A ∩ J = t A ∩ J ⊆ Y = Y .

2

Lemma 3.8. Let A , B ∈ S , A ;t B for t ≥ 1. Then, B = L t A. Furthermore, A ;1 B implies A = B. Proof. As A ;t B, there are sequences (ni )i ∈N , (mi )i ∈N , such that F ni → A, F mi → B, and ni /mi → t. If t > 1, there is a subsequence with ni ≥ mi . We restrict ourself to this subsequence. Then, F mi = L ni /mi F ni . According to Lemma 3.7 we find in the limit B = L t A. Now assume t = 1. Case 1: ni ≥ mi for an infinite subsequence. We can use the construction above and find B = L 1 A. As L 1 is the identity, this implies A = B. Case 2: If case 1 is not given, there is an infinite subsequence mi ≥ ni , and B ;1 A. I.e., exchanging the role of A and B, we again find A = B = L 1 A = L 1 B. 2 Corollary 3.9. A ;t B 1 and A ;t B 2 for some t ≥ 1 implies B 1 = L t A = B 2 . Lemma 3.10. Let A ∈ S , B = L t A for t ≥ 1. Then, B ∈ S and A ;t B. Proof. As A ∈ S , there is a sequence (ni )i ∈N such that F ni → A. Define mi = ni /t  + 1. Then, ni /mi → t. Furthermore, there ˆ i.e. Bˆ ∈ S . As ni /m ˆ Since L t A = B, we know ˆ and hence L t A = B. ˜ ik → t, A ;t B, is a subsequence mik such that F mi → B, k Bˆ = B.

k

2

In particular, we find that L t : S → S for all t ≥ 1.

J. Müller, H. Jiang / Theoretical Computer Science 580 (2015) 14–27

21

Lemma 3.11. Fix A ∈ S , and consider the set [ A ] = { B ∈ S | A ; B } ⊆ S . The map

ϕ A : [1, ∞) → [ A ], t → Lt A is well defined, surjective, satisfies ϕ A (1) = A, and ϕ A (t 2 ) = L t2 /t1 ϕ A (t 1 ) for t 1 ≤ t 2 . Proof. Due to Lemma 3.10, ϕ A is well defined. The equations ϕ A (1) = A, and ϕ A (t 2 ) = L t2 /t1 ϕ A (t 1 ) for t 1 ≤ t 2 are immediate consequences of the definition of L t . In order to show surjectivity of ϕ A , we select B ∈ [ A ], i.e. there is t ≥ 1 s.t A ;t B. Due to Lemma 3.10, B = L t A, and hence ϕ A (t ) = B. 2 Recall that L t is not continuous in general. The next lemma utilizes the fact that the sets we are interested in are graphical representations of cellular automata. Lemma 3.12. Let A ∈ S . The map ϕ A : [1, ∞) → [ A ] constructed above is Lipschitz-continuous.

d H (ϕ A (t 1 ), ϕ A (t 2 )) ≤ 2(d + K + 4)|t 1 − t 2 |. Proof. Let t 1 > t 2 ≥ 1, B i = (1)

quence mk

ϕ A (t i ) for i = 1, 2. Let F n → A. Using the construction of Lemma 3.10, we obtain a se(1)

and a subsequence nk such that mk /nk → t 1 , F m(1) → B 1 . Repeating this construction – this time starting k

(i )

with nk , and renaming the subsequences again – we finds three sequences n , m , i = 1, 2, such that F n → A, F m(i) → B i ,  (i )

(1)

(2)

and m /n → t i . Hence, m /m Proposition 2.18, we obtain (1)

(1)

(2)

k

k

(2)

(1)

mk −1

d H ( F m(1) , F m(2) ) ≤

(1)

= (m /n )/(n /m ) → t 1 /t 2 . Without restriction, we assume that mk > mk . Using mk −1



(1 ) (2 )  2(d + K + 4) (mk − mk ) d H ( F  , F +1 ) ≤ ≤ 2(d + K + 4) (2 )  m (2) (2)

=mk

= 2(d + K + 4)



k

l=mk



(1 )

mk

(2 )

mk

−1

Passing to the limit yields d H ( B 1 , B 2 ) ≤ 2(d + K + 4)(t 1 − t 2 )/t 2 ≤ 2(d + K + 4)(t 1 − t 2 ).

2

Next we exploit the fact that T t = L exp(t ) , and that the exponential function is a homeomorphism from (R, +) to (R>0 , ·). The lemmata above can be summarized in the following theorem. Theorem 3.13. The family of operators T t : S → S , t ∈ R≥0 , is well defined, and satisfies (1) T 0 S = S (2) T s+t S = T s ( T t S ) (3) t → T t S is locally Lipschitz. The orbit of S ∈ S under T t coincides with [ S ]. T t is a semigroup (or semiflow) acting on S . It is straightforward to show that – for t fixed – T t is continuous on a larger set, namely

Gk = { A ∈ H( J ) | A ⊂ ∞ , A ∪ {(x, y ) | y ≤ 0} has the cone condition with constant k}. There is a well developed theory of semigroups, which could be used to investigate T t . However, as Gk is rather general and covers all graphical limit sets and more, we cannot hope to find specific information about the graphical limit sets of a given cellular automaton with a given initial condition, if we focus on the semiflow T t . We need to concentrate on properties of a sequence F n . 4. Exponential graphical limit set We know that ϕ A is surjective. Now we investigate aspects related to the injectivity of to well behaved elements A ∈ S .

ϕ A . However, we restrict ourselves

Definition 4.1. An element S ∈ S is called exponential, if there is a sequence (ni )i ∈N such that ni +1 /ni ≤ k for some k ∈ R, and F ni → S. The set Sexp ⊆ S is the set of exponential elements in S .

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J. Müller, H. Jiang / Theoretical Computer Science 580 (2015) 14–27

Fig. 3. The three semigroups and the homomorphisms defined (proof of Lemma 4.2).

Notation: As usual, we identify ( S 1 , ·) with the topological group formed by the complex unit circle, U (1) = {e ix | x ∈ R}, together with the complex multiplication and trace topology. Lemma 4.2. Let ( M , ∗) be a compact topological semigroup, and θ : (R≥0 , +) → ( M , ∗) a continuous surjective homomorphism. If there is t 0 > 0 with θ(0) = θ(t 0 ), then ( M , ∗) already is a topological group that is homeomorphic with either the trivial group ({e }, ·), or with ( S 1 , ·). Proof. Let τ = inf{t > 0 | θ(t ) = θ(0)}. Then, τ ∈ [0, t 0 ]. Case 1: τ = 0. There is a sequence tn → 0 with tn > 0, and θ(tn ) = θ(0). For k ∈ N, we also find θ(k tn ) = θ((k − 1) tn ) ∗ θ(tn ) = θ((k − 1) tn ) = . . . = θ(0). Hence, θ is constant on the set {k tn | k, n ∈ N} ⊂ R≥0 . As this set is dense in R≥0 , and θ continuous, θ(t ) ≡ θ(0). Thus, M = {θ(0)}. The semigroup with only one element already is the trivial group. Since there is only one topology on a set with only one element, ( M , ∗) is homeomorphic to the trivial topological group ({e }, ·). Case 2: τ > 0. Define (see Fig. 3) ψ : (R≥0 , +) → (U (1), ·) by

x → e i 2π x/τ and

η : (U (1), ·) → ( M , ∗) by e ix → θ(ψ −1 (e ix )).

Step 1. η is well defined and a homomorphism. We first show that ker(θ) = τ N0 . Clearly, ker(θ) ⊃ τ N0 . If t ∈ ker(θ) \ τ N0 , there is k ∈ N0 with t = kτ + t, t ∈ (0, τ ). Hence, θ(0) = θ(t + kτ ) = θ(t ). This is a contradiction to the definition of τ (note that t < τ ), and therefore ker(θ) = τ N0 . Also ker(ψ) = τ N0 . Now, ψ −1 (e ix ) is not a unique element in R≥0 , but only defined up to ker(ψ). However, as ker(ψ) = ker(θ), η is well defined and a homomorphism. Step 2: ( M , ∗) is isomorphic to ( S 1 , ·). Assume that η is not injective. Then, we find x, y ∈ R with 0 ≤ x < y < 2π , and η(e ix ) = η(e i y ). Since 0 ≤ x < y < 2π , there are t x , t y ∈ [0, τ ) with t x < t y and ψ(t x ) = e ix and ψ(t y ) = e i y . Hence, θ(t x ) = η(e ix ) = η(e i y ) = θ(t y ), and

θ(0) = θ(τ ) = θ(τ − t x + t x ) = θ(τ − t x ) ∗ θ(t x ) = θ(τ − t x ) ∗ θ(t y ) = θ(τ − t x + t y ) = θ(t y − t x ). Since t y − t x ∈ (0, τ ), this is a contradiction to the definition of τ . Hence, η is injective. The surjectivity of η follows from the surjectivity of θ . Thus, ( M , ∗) is isomorphic to ( S 1 , ·). Step 3: η is a homeomorphism. We select an element t ∈ ψ −1 (e ix ) with t ∈ [0, τ ). η = θ ◦ ψ −1 is obviously continuous if t = 0. Since θ(t ) has the same limes for t → τ − and t → 0+, η is continuous and bijective. As M and S 1 are compact, η already is a homeomorphism. 2 Lemma 4.3. If S ∈ Sexp , then there is t ≥ 2 with L t S = S. Proof. Step 1: Let F ni → S, ni +1 /ni ≤ k for some fixed k ∈ R. We first show that there is a subsequence n j i of (ni )i ∈N such that 2 ≤ n j i+1 /n j i ≤ 2k. Therefore, define j 1 = 1, and

j i +1 = min{ | n ≥ 2n j i }. As ni tends to infinity, the numbers j i are well defined. Obviously, n j i+1 /n j i ≥ 2. Fix i. Let nm sequence (ni )i ∈N with nm < 2 n j i . Then, n j i+1 = nm+1 ≤ knm ≤ 2kn j i . Hence, n j i+1 /n j i ≤ 2k. Step 2: In the present step, we show that there is tˆ ≥ 2 such that L tˆ S = S. According to step 1, restriction that 2 ≤ ni +1 /ni ≤ 2k. Define m1 = 1, mi = ni −1 for i > 1. Then, F ni → S, F mi → S, and for i > 1. Therefore, there is a converging subsequence such that S ;tˆ S, where tˆ ∈ [2, 2k], which in We combine the last two statements into one statement for the semigroup T t .

the largest entry of the we may assume without 2 ≤ ni /mi = ni /ni −1 ≤ 2k turn implies L tˆ S = S. 2

J. Müller, H. Jiang / Theoretical Computer Science 580 (2015) 14–27

23

Fig. 4. Examples for cellular automata that yield a graphical limit set consisting of one point only (time runs downwards).

Theorem 4.4. If S ∈ Sexp , then the dynamical system ( T t , [ S ]) is either topologically conjugated to the trivial singleton system, or to a non-trivial rotation of S 1 . Note that Sexp ⊆ S , but Sexp may be empty. The next proposition shows that Sexp = ∅ already implies S = Sexp . Lemma 4.5. If S ∈ Sexp , then [ S ] = Sexp = S . Proof. We basically use the same construction as we already did in Step 1 of Lemma 4.3. Let A ∈ S . Then, there are sequences (ni )i ∈N , (mi )i ∈N , such that ni +1 /ni ≤ k, F ni → S, and F mi → A. Now define j (i ) = min{ j | n j > 2 mi }. Then, n j (i ) /mi ≥ 2 and

n j (i ) ≤ kn j (i )−1 ≤ 2 k mi due to the definition of j (i ) and the condition ni /ni −1 ≤ k. Hence, n j (i ) /mi ∈ [2, 2 k]. There is a subsequence such that n j (il ) /mil → t ∈ [2, 2 k], and hence S ;t A. ˜ i → t, and ˜ i = ni /t  + 1. Then, ni /m Now define m



F m˜ i =

ni

˜i m



F ni

∩ J → (t S ) ∩ J = Lt S = A .

As

˜ i +1 m ˜i m

=

ni +1 /t  + 1 ni +1 /t + 2 2t ≤ ≤k+ ≤ k + 2t ni /t  + 1 ni /t ni

we obtain A ∈ Sexp .

2

5. Examples In this section, we study some examples for the different cases. First, we consider the case Sexp = ∅, where we have two possibilities: |S | = 1 or S ≡ S 1 . We will consider several Wolfram-automata, i.e. the grid is Z, the local states E = {0, 1}, and the local function depends only on the states in the sites U 0 = {−1, 0, 1}. We will also use the Wolfram-enumeration of these automata [18]. Let δ0 ∈ E Z the state that is one at the origin, and zero elsewhere. We will only consider space-time patterns that are triggered by the initial state u 0 = δ0 . In the third part, we present an example with Sexp = ∅, and discuss the structure of the S for this specific example. 5.1. Sexp = S , and S consist of one point In this case we know that the only element S of S is invariant under L t . Therefore, if x ∈ S, also {λx | λ ≥ 0} ∩ J ⊂ S. The set possesses the structure of a compact cone (intersected with J ). The graphical limit sets of the next three examples are sketched in Fig. 4. Example 5.1. Rule 240. The right-shift, f 0 (a, b, c ) = a. This clearly yields one straight line with slope one.

24

J. Müller, H. Jiang / Theoretical Computer Science 580 (2015) 14–27

Fig. 5. Sierpinski gasket as the graphical limit set of the cellular automaton with rule 30. As depicted, this is an example for a graphical limit set homeomorphic to S 1 .

Example 5.2. Rule 254. The cellular automaton with a local rule that renders a cell black if any cell in the local neighbourhood is black. The resulting graphical limit set is a black triangle. Example 5.3. Rule 50. A cell becomes black, if it is white, and the left, the right, or both neighbors are black. This cellular automaton yields, similarly to rule 254, a black triangle. To distinguish between Rule 254 and Rule 50 using graphical limit sets only, it is possible to define cellular automata resembling spatial derivatives of other cellular automata [13]. The graphical limit sets of the derivatives of rule 254 and rule 50 will be different. 5.2. Sexp = S , and S is homeomorphic to S 1 The classical example for a graphical limit set, already discussed by Willson [14] in 1978, is the Sierpinski gasket resulting from Wolframs rule 90. Example 5.4. Rule 90. The graphical limit set of this rule is the Sierpinski gasket. The homeomorphsim with S 1 is depicted in Fig. 5. Willson is able to show even more: as long as the initial state u 0 has a compact support, the graphical limit set does not depend on u 0 . There are more, and more complex examples, together with a method to compute the box-counting dimension for these examples in the work of van Haeseler [12,11,10]. The complete theory developed in those papers is based on algebraic properties of cellular automata connected with polynomials over finite fields. The next example shows that algebraic properties (at least obvious algebraic properties) are not necessary for a cellular automaton to have a limit set in the present class. This limit set does not have a fractal (box-counting) dimension, but dimension 1. The idea of the automaton is based on the definition of two states L and R spreading with velocity 1 to the left respectively to the right. If we start with .. L .. R .., two straight lines are forming (see Fig. 6), where the L-line is going to the left and the R-line to the right. Between these two lines, a fast particle runs back and forth at velocity 2. The particle assumes one of the two states  and r; if the particle runs left its state is , if it runs right, its state is r. It toggles its state if it is neighboring L or R; in this way, the particle bounces back at the boundaries of the region defined by L and R. Example 5.5. We consider a cellular automaton with states E = {0, L , R , , r }, and U 0 = {−3, −2, −1, 0, 1, 2, 3}. We start with u 0 = . . . 00L00r R00 . . . . We do not state the complete local function, but only indicate the transitions necessary for us: u (−3)

u (−2)

∗ ∗ ∗L

∗ ∗ ∗L 

L 0

∗

u (−1) 0 R

u (0) 0 0

∗L ∗

∗L ∗ ∗R ∗

r

0

∗

∗

u (1) L 0 0

∗ ∗R ∗

u (2)

u (3)

∗ ∗  ∗ ∗R

∗ ∗

f0 L R

0



∗ ∗R

r r

r

R



remark left shift of L right shift of r left shift of  (fast particle) bounce back of  at L right shift of r (fast particle) bounce back of r at R

where ∗ ∈ E, ∗ R ∈ {0, R }, and ∗ L ∈ {0, L }. All configurations of E U 0 that do not appear in the table can be taken to an arbitrary element, e.g., to 0. The trajectory of u 0 is depicted in Fig. 6. There is a cone, formed by L and R; between the walls of the cone, a line of  resp. r bounces back and forth. A short computation shows that the i’th rebound of the fast particle takes place at time t i = 3i − 1. If we consider F t i , we find clearly a converging sequence. The limit of this sequence is shown in Fig. 6. For this example, Sexp = ∅, and S is homeomorphic with S 1 . Each element of the graphical limit set has box counting dimension 1.

J. Müller, H. Jiang / Theoretical Computer Science 580 (2015) 14–27

25

Fig. 6. Left: Example 5.5, trajectory of u 0 (where we suppressed the local state 0). Right: One element of the graphical limit set for this example.

Fig. 7. (a) Example 5.6, trajectory of u 0 . (b)–(d) Three elements of S .

5.3. Sexp = ∅ The example for this class of cellular automata is an extension of Example 5.5. A vertical line of an immobile, additional symbol s may appear if the fast particle hits the left boundary (see Fig. 7). This line disappears again, if the symbol  is located in the neighborhood of s. Apart of L, we introduce an additional symbol  that also defines the left boundary of our region, similar to L. There is only one difference between L and : If the fast particle hits L, it just rebounds, as before. If it hits a , the particle also rebounds, but additionally an s appears. The type of the left boundary (L or ) is controlled by a two-component counter attached to the left side of L resp. . This two-component counter stores (1) the number (plus one) of changes between L and  that took place since time zero, and (2) the number of rebounds (at the left hand side) of the fast particle since the last transition between L and . If both counters coincide, the next transition from L to  or from  to L is triggered. In consequence, the number of rebounds grows linearly in-between two transitions between L and . Let us estimate the number of bounces between time zero and the i’th transition. Since the number of bounces between two transitions grow linearly, the number of bounces that happen up to the i’th transition is a function of order i 2 . The times for the bounces grow with order 3i , hence the time points for the transitions between time intervals in which no vertical line is created (L) and time intervals, where a vertical line is 2 created () grow with order 3i . This growth is faster than an exponential function, and leads in this way to Sexp = ∅. Example 5.6. We consider a cellular automaton with local states E˜ = E ∪ {, s, a, A , b, c }, where E are the local states from Example 5.5, and U 0 = {−3, −2, −1, 0, 1, 2, 3}. We start with u 0 = . . . 0a00r R00 . . . , and extend the local function indicated above. We do not formally introduce the local function; it is more instructive to show how the counters actually work. In principle, the configuration

26

J. Müller, H. Jiang / Theoretical Computer Science 580 (2015) 14–27

Fig. 8. Structure of the graphical limit set S of Example 5.6. There are two periodic orbits (under T t ), and two orbits connecting these two periodic orbits. Note that it is not possible for a set to leave a periodic orbit under the action of T t , though the orbit tending away from the periodic orbit gets (backward in time) topologically arbitrary close.

· · · 00 b · · · b a · · · a L · · · n

· · · 00 b · · · b a · · · a  · · ·

respectively

m

n

m

indicates that n + m − 1 transitions between L and  already took place, and that the fast particle did rebound m times since the last transition. Let us consider the timer in action in two examples. In the first example, we start with n = m = 2. We suppress the state 0.

b

b a

b a a

b a a a

b b a a

b b A a

b b a a

b b a A

b a a

a a



s s s s s

a













 

start with n = 2, m = 2 counter moves to the left fast particle hits state s is created; state A signals a hit the signal moves left, s doesn’t move, b, A are neighbors counter one is increased, A destroyed done; result: n = 1, m = 3

The second situation (n = 0, m = 2) shows how the transition is triggered, and how the first part of the counter is reset.

b

b b

b b b

b b b

c c



A a L

a a L

a A L

a a L

a a L

a L



L

start with n = 0, m = 2 counter moves to the left fast particle hits state A signals a hit the signal moves left; configuration 0 A indicates a transition state c is a signal for the transition, and moves to the right configuration cL indicates that L is to replace by  transition finished; result: n = 3, m = 0



The number of rebounds between two transitions are 2, 3, 4, . . . , i.e., the i’th transition takes place at time

t i = 3i (i +1)/2 − 1. The sequence F t2i tend to the set depicted in Fig. 7(b), while F t2i+1 tends to the set displayed in Fig. 7(d). In order to discuss these figures, let us call the time interval between two rebounds at the left boundary of the cone a “period”. The sets in subfigures (b) and (d) cannot be transformed into each other by the semigroup T t , as in each period of (d) there is a vertical line, which is missing in (b). This fact proves that Sexp = ∅. If we consider the sequence F 3(t i +1)−1 , we find the set shown in Fig. 7(c). It resembles that in subfigure (b), but only one vertical line is located in the last period. If we apply T t with t large enough to subfigure (c), we obtain subfigure (b). In a similar way, it is possible to find sets with vertical lines in the last n sections, n ∈ N. Also sets with vertical lines in all sections but the last n sections are possible to construct. If we apply T t , t ≥ 0, to all of these sets, we obtain the complete graphical limit set S . Thus, S consists of two periodic orbits (under T t ), and two lines connecting these two periodic orbits (see Fig. 8). These connecting orbits are necessary, as we know that S is connected. However, it is not possible for a set within a periodic orbit to leave this periodic orbit again by the action of T t . Hence, the S is divided into two disjoint parts with respect to T t . However, the complete set S forms one single chain component.

J. Müller, H. Jiang / Theoretical Computer Science 580 (2015) 14–27

27

6. Discussion In this paper, we developed an approach to define graphical limit sets that is not based on algebraic properties. In this way, we are able to handle general cellular automata defined on Z. The extension to general, finitely generated Abelian groups is straight forward; an extension also to non-Abelian groups using Cayley graphs seems to be possible if these graphs can be embedded into Rn . We followed the ideas developed for cellular automata with algebraic properties in using sets to characterize space-time diagrams; a more refined approach could be based on Radon measured, which would, for example, lead to different results for Wolframs rule 254 and rule 50. Though our approach is possible for general cellular automata, it only yields sensible results for cellular automata with a quiescent state. If there is no quiescent state, we just obtain one solid rectangle. Also this draw-back could be removed by the usage of Radon measures instead of sets. The graphical limit set we obtain strongly depends on the initial state. This is different to the classical results for cellular automata with appropriate algebraic properties, where the graphical limit set is the same for every non-trivial initial state with compact support. In general, we cannot expect similar structures for graphical limit sets of different initial states (but one cellular automaton). Nevertheless, we expect the graphical limit sets to code central features of cellular automata. Elements (sets) S in graphical limit sets have the following properties: (1) S is a compact subset of J intersected with a positive cone {|x| < d| y |}; (2) S ∪ {(x, y ) | y ≤ 0} satisfies the cone condition. These two properties are incorporated by many sets, for example all sets {λ(x, y ) | λ ≥ 0} ∩ J for (x, y ) ∈ {|x| < d| y |} satisfy these conditions. This is an uncountable family of sets. As there are only countable cellular automata and countable initial states with finite support, clearly some properties of graphical limit sets are missing. It is an appealing task to reveal more features of graphical limit sets, and to characterize properties of cellular automata in terms of graphical limit sets. Acknowledgement We thank Peter Massopust for discussions, in particular concerning self-similar sets. We also thank two anonymous referees for their careful review and many hints that greatly improved readability of the paper. Moreover, one of the referees pointed out Example 5.5 and Example 5.6, that clarified the questions (which have been open in the first version of the manuscript), if cellular automata with non-exponential graphical limit sets exist, and if all cellular automata with cyclic graphical limit sets possess an obvious intrinsic algebraic property. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]

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