,
1
2D Cellular Automata: new constructions and dynamics
I
Alberto Dennunzioa , Enrico Formenti∗,b a Universit` a
degli studi di Milano-Bicocca, Dipartimento di Informatica Sistemistica e Comunicazione, via Bicocca degli Arcimboldi 8, 20126 Milano (Italy) b Universit´ e de Nice-Sophia Antipolis, Laboratoire I3S, 2000 Route des Colles, 06903 Sophia Antipolis (France)
Abstract In this paper the dynamical behavior of two–dimensional cellular automata (2D CA) is studied. The notion of closingness is generalized to the 2D case and it is linked to permutivity and openness. The major contributions of this work are two deep constructions which have been fundamental in order to prove our new results. The authors strongly believe they will be a valuable tool for proving other new results in the near future. Key words: cellular automata, symbolic dynamics.
1. Introduction Cellular automata (CA) are a well-known formal model for complex systems and, at the same time, a paradigmatic model of massive parallel computation. Indeed, a CA is made of an infinite number of identical finite automata arranged on a regular lattice (Z2 or Z in this paper). Each automaton assumes a state chosen from a set A, called the set of states or the alphabet. A configuration is a snapshot of all states of the automata. A local rule updates the state of each automaton on the basis of its current state and the ones of a finite set of neighboring automata. All the automata are updated synchronously. Behind the simplicity of the CA definition stands the huge complexity of different dynamical behaviors which captured the attention of researchers all over these last thirty years. We refer the reader to [16, 13, 19, 8, 17, 23] for a review of the main results and for a comprehensive bibliography. Historically, Von Neumann introduced CA to study formal models for cells self-reproduction. These were two-dimensional models. Paradoxically, the study I This
is an extended and improved version of [9]. author. Email addresses:
[email protected] (Alberto Dennunzio),
[email protected] (Enrico Formenti) ∗ Corresponding
Preprint submitted to Elsevier
July 15, 2009
of the CA dynamical behavior concentrated essentially on the one-dimensional case except for additive CA and a few others (see [24], for example). The reason of this gap is maybe twofold: from one hand, there is a common feeling that most of results involving dynamical properties are “automatically” transferred to higher dimensions; from the other hand, researchers mind the “complexity gap”. Indeed, many CA properties are dimension sensitive i.e. they are decidable in dimension 1 and undecidable in higher dimensions [2, 11, 15, 12, 3]. In this paper, in order to overcome this complexity gap, we use two deep constructions which allow to see a 2D as a 1D CA (see Section 3). In this way, well-known properties of one-dimensional CA can be lifted to the twodimensional case. The constructions can be generalized to any dimension n. The idea is to “cut” the configuration space of a CA in dimension n into slices of dimension n−1. Hence, the former CA (in dimension n) can be seen as a new CA (in dimension n − 1) operating on configurations made of slices (Theorem 1). The only inconvenient is that this latter CA has an infinite set of states. Theorem 2 proves that this is not always a problem. However, even if the constructions are of help for proving 1D–like results, most of their proofs are significantly different from the ones of the 1D case (cf. Theorem 7 in this paper and the corresponding 1D version in [18] or its one–sided version in [4]). Moreover, in the present work, we generalize the notions of closingness and permutivity to the 2D settings. The further generalization to any dimension is straightforward. We choose to consider the 2D case for simplicity sake. By the aforementioned constructions, we characterize the dynamical behavior of closing and permutive 2D CA. The paper is structured as follows. Next section recalls basic notions and some known results about CA and discrete dynamical systems. Section 3 presents the slicing constructions. Sections 4 and 5 contain the results on the CA dynamical behavior. 2. Basic notions In this section, we briefly recall standard definitions about CA as dynamical systems. For introductory matter and recent results see [18, 10, 1], for instance. For all i, j ∈ Z with i ≤ j, let [i, j] = {i, i + 1, . . . , j}. Let N+ be the set of positive integers. For a vector ~x ∈ Z2 , denote by |~x| the infinite norm (in R2 ) of ~x. Let r ∈ N. Denote by Mr the set of all the two-dimensional matrices with values in A and entry vectors in the square [−r, r]2 . For any matrix N ∈ Mr , N (~x) ∈ A represents the element of the matrix with entry vector ~x. 1D CA. Let A be a possibly infinite alphabet. A 1D CA configuration is a function from Z to A. The 1D CA configuration set AZ is usually equipped with the metric d defined as follows ∀c, c0 ∈ AZ , d(c, c0 ) = 2−n , where n = min i ≥ 0 : ci 6= c0i or c−i 6= c0−i . When A is finite, the set AZ is a compact, totally disconnected and perfect topological space (i.e., AZ is a Cantor space). For any pair i, j ∈ Z, with i ≤ j, 3
and any configuration c ∈ AZ we denote by c[i,j] the word ci · · · cj ∈ Aj−i+1 , i.e., the portion of c inside the interval [i, j]. A cylinder of block u ∈ Ak and position i ∈ Z is the set [u]i = {c ∈ AZ : c[i,i+k−1] = u}. Cylinders are clopen sets w.r.t. the metric d and they form a basis for the topology induced by d. Formally, a 1D CA is a structure h1, A, r, f i, where A is the alphabet, r ∈ N is the radius and f : A2r+1 → A is the local rule of the automaton. The local rule f induces a global rule F : AZ → AZ defined as follows, ∀c ∈ AZ , ∀i ∈ Z,
F (c)i = f (xi−r , . . . , xi+r ) .
Recall that F is a uniformly continuous map w.r.t. the metric d. A 1D CA with global rule F is right (resp., left) closing iff F (c) 6= F (c0 ) for any pair c, c0 ∈ AZ of distinct left (resp., right) asymptotic configurations, i.e., c(−∞,n] = c0(−∞,n] (resp., c[n,∞) = c0[n,∞) ) for some n ∈ Z, where a(−∞,n] (resp., a[n,∞) ) denotes the portion of a configuration a inside the infinite integer interval (−∞, n] (resp., [n, ∞)). A CA is said to be closing if it is either left or right closing. A rule f : A2r+1 → A is righmost (resp., leftmost) permutive iff ∀u ∈ A2r , ∀β ∈ A, ∃α ∈ A such that f (uα) = β (resp., f (αu) = β). A 1D CA is said to be permutive if its local rule is either rightmost or leftmost permutive. 2D CA. Let A be a finite alphabet. A 2D CA configuration is a function 2 from Z2 to A. The 2D CA configuration set AZ is equipped with the following metric which is denoted for the sake of simplicity by the same symbol of the 1D case: 2 ∀c, c0 ∈ AZ , d(c, c0 ) = 2−k where k = min |~x| : ~x ∈ Z2 , c(~x) 6= c0 (~x) . The 2D configuration set is a Cantor space. A 2D CA is a structure h2, A, r, f i, where A is the alphabet, r ∈ N is the radius and f : Mr → A is the local rule 2 2 of the automaton. The local rule f induces a global rule F : AZ → AZ defined as follows, 2 ∀c ∈ AZ , ∀~x ∈ Z2 , F (c)(~x) = f Mr~x (c) , where Mr~x (c) ∈ Mr is the finite portion of c of reference position ~x ∈ Z2 and radius r defined by ∀~k ∈ [−r, r]2 , Mr~x (c)(~k) = c(~x + ~k). For any ~v ∈ Z2 the shift 2 2 2 map σ~v : AZ → AZ is defined by ∀c ∈ AZ , ∀~x ∈ Z2 , σ~v (c)(~x) = c(~x + ~v ). A 2 2 ~ ~ function F : AZ → AZ is said to be shift-commuting if ∀~k ∈ Z2 , F ◦σ k = σ k ◦F . Note that 2D CA are exactly the class of all shift-commuting functions which are (uniformly) continuous with respect to the metric d (Hedlund’s theorem from [14]). A 2D subshift S is a closed subset of the CA configuration space such that for any ~v ∈ Z2 , σ~v (S) = S. 2 For any fixed vector ~v , we denote by S~v the set of all configurations c ∈ AZ ~ v such that σ (c) = c. Remark that, for any 2D CA global map F and for any ~v , the set S~v is F -invariant, i.e., F (S~v ) ⊆ S~v . DTDS. A discrete time dynamical system (DTDS) is a pair (X, g) where X is a set equipped with a distance d and g : X 7→ X is a map which is continuous on X with respect to the metric d. When X is the configuration space of a 4
(either 1D or 2D) CA equipped with the above introduced metric, the pair (X, F ) is a DTDS. From now on, for the sake of simplicity, we identify a CA with the dynamical system induced by itself or even with its global rule F . Given a DTDS (X, g), a point c ∈ X is periodic for g if there exists an integer p > 0 such that g p (c) = c. If the set of all periodic points of g is dense in X, we say that the DTDS has dense periodic orbits (DPO). A DTDS (X, g) is (topologically) mixing if for any pair of non-empty open sets U, V ⊆ X there exists an integer n ∈ N such that for any t ≥ n we have g t (U ) ∩ V 6= ∅. A DTDS (X, g) is (topologically) strongly transitive if for any non-empty open set U ⊆ X, S it holds that n∈N g n (U ) = X. A DTDS (X, g) is open (resp., surjective) iff g is open (resp., g is surjective). Recall that two DTDS (X, g) and (X 0 , g 0 ) are isomorphic (resp., topologically conjugated ) if there exists a bijection (resp., homeomorphism) φ : X 7→ X 0 such that g 0 ◦ φ = φ ◦ g. (X 0 , g 0 ) is a factor of (X, g) if there exists a continuous and surjective map φ : X → X 0 such that g 0 ◦ φ = φ ◦ g. Remark that in that case, (X 0 , g 0 ) inherits from (X, g) some properties such as surjectivity, transitivity, mixing, and DPO. 3. Slicing constructions In this section, we introduce two powerful constructions for CA in dimension greater than 1. The idea inspiring these constructions appeared in the context of additive CA in [20] and it was formalized in [7]. In the present paper, we generalize it to arbitrary 2D CA. Moreover, we further refine it so that slices are translation invariant along some fixed direction. This confers finiteness to the set of states of the sliced CA allowing to lift even more properties. The constructions are given with respect to any direction for 2D CA, improving the ones introduced in [9]. The generalization to higher dimensions is straightforward. Fix a vector ~ν ∈ Z2 and let d~ ∈ Z2 be a normalized integer vector (i.e. a vector with co-prime coordinates) perpendicular to ~ν . Consider the line L0 generated by the vector d~ and the set L∗0 = L0 ∩ Z2 containing vectors of form ~x = td~ where t ∈ Z. Denote by ϕ : L∗0 7→ Z the isomorphism associating any ~x ∈ L∗0 with the integer ϕ(~x) = t. Consider now the family L constituted by all the lines parallel to L0 containing at least a point of integer coordinates. It is clear that L is in a one-to-one correspondence with Z. Let la be the axis given by a direction ~ea which is not contained in L0 . We enumerate the lines according to their intersection with the axis la . Formally, for any pair of lines Li , Lj , it holds that i < j iff pi < pj (pi , pj ∈ Q), where pi~ea and pj ~ea are the intersection points between the two lines and the axis la , respectively. Equivalently, Li is the line expressed in parametric form by ~x = pi~ea + td~ (~x ∈ R2 , t ∈ R) and pi = ip1 , where p1 = min {pi , pi > 0}. Remark that ∀i, j ∈ Z, if ~x ∈ Li and ~y ∈ Lj , then ~x + ~y ∈ Li+j . Let ~y1 ∈ Z2 be an arbitrary but fixed vector of L1 . For any i ∈ Z, define the vector ~yi = i~y1 which belongs to Li ∩ Z2 . Then, each line Li can be ~ Note that, for any ~x ∈ Z2 there expressed in parametric form by ~x = ~yi + td. 5
exist i, t ∈ Z, such that ~x = ~yi + td. Let us summarize the construction. We
Figure 1: Slicing of the plane according to the vector ~ν = (1, 1). have a countable collection L = {Li : i ∈ Z} of lines parallel to LS 0 inducing a partition of Z2 . Indeed, defining L∗i = Li ∩ Z2 , it holds that Z2 = i∈Z L∗i (see Figure 1). 2 Once the plane been sliced, any configuration c ∈ AZ can be viewed as S has a mapping c : i∈Z L∗i 7→ Z. For every i ∈ Z, the slice ci of the configuration c over the line Li is the mapping ci : L∗i → A. In other terms, ci is the restriction 2 of c to the set L∗i ⊂ Z2 . In this way, a configuration c ∈ AZ can be expressed as the bi-infinite one-dimensional sequence ≺ c = (. . . , c−2 , c−1 , c0 , c1 , c2 , . . .) of ∗ its slices ci ∈ ALi where the i-th component of the sequence ≺ c is ≺ c i = ci (see Figure 2). Let us stress that each slice ci is defined only over the set L∗i . Moreover, since ∀~x ∈ Z2 , ∃!i ∈ Z : ~x ∈ L∗i , for any configuration c and any vector ~x we write c(~x) = ci (~x). 2 The identification of any configuration c ∈ AZ with the corresponding biinfinite sequence of slices c ≡≺ c = (. . . , c−2 , c−1 , c0 , c1 , c2 , . . .), allows the introduction of a new one-dimensional bi-infinite CA over the alphabet AZ expressed by a global transition mapping F ∗ : (AZ )Z 7→ (AZ )Z which associates any configuration a : Z 7→ AZ with a new configuration F ∗ (a) : Z → AZ . The local rule f ∗ of this new CA we are going to define will take a certain number of configurations of AZ as input and will produce a new configuration of AZ as output. ∗ ∗ For each h ∈ Z, define the following bijective map Th : ALh 7→ AL0 which associates any slice ch over the line Lh with the slice Th (ch ) (ch : L∗h → A)
T
h −→
(Th (ch ) : L∗0 → A) ∗
defined as ∀~x ∈ L∗0 , Th (ch )(~x) = ch (~x + ~yh ). Remark that the map Th−1 : AL0 → ∗ ALh associates any slice c0 over the line L0 with the slice Th−1 (c0 ) over the line ∗ Lh such that ∀~x ∈ L∗h , Th−1 (ch )(~x) = c0 (~x − ~yh ). Denote by Φ0 : AL0 → AZ 6
Figure 2: Slicing of a 2D configuration c according to the vector ~ν = (1, 1). The components of c viewed as a 1D configuration are not from the same alphabet. the bijective mapping putting in correspondence any c0 : L∗0 → A with the configuration Φ0 (c0 ) ∈ AZ , Φ
(c0 : L∗0 → A)
Φ0 (c0 ) : Z2 → A
0 −−→
∗
Z such that ∀t ∈ Z, Φ0 (c0 )(t) := c0 (ϕ−1 (t)). The map Φ−1 AL0 associates 0 :A → −1 Z L∗ 0 any configuration a ∈ A with the configuration Φ0 (a) ∈ A in the following 2 x) = a(ϕ(~x)). Consider now the bijective map Ψ : AZ → way: ∀~x ∈ L∗0 , Φ−1 0 (a)(~ (AZ )Z defined as follows 2
∀c ∈ AZ ,
Ψ(c) = (. . . , Φ0 (T−1 (c−1 )), Φ0 (T0 (c0 )), Φ0 (T1 (c1 )), . . .) . 2
Its inverse map Ψ−1 : (AZ )Z 7→ AZ is such that ∀a ∈ (AZ )Z , −1 −1 −1 −1 −1 ≺ Ψ−1 (a) = (. . . , T−1 (Φ−1 0 (a−1 )), T0 (Φ0 (a0 )), T1 (Φ0 (a1 )), . . .) .
Starting from a configuration c, the isomorphism Ψ allows to obtain a 1D configuration a in which all components take value from the same alphabet (see Figure 3). At this point, we have all the necessary formalism to correctly define the ∗ radius r∗ local rule f ∗ : (AZ )2r +1 → AZ starting from a radius r 2D CA F . Let r1 and r2 be the indexes of the lines passing for (r, r) and (r, −r), respectively. The radius of the 1D CA is r∗ = max {r1 , r2 }. In other words, r∗ is such that L−r∗ , . . . Lr∗ are all the lines which intersect the 2D r-radius Moore neighborhood. The local rule is defined as ∀(a−r∗ , . . . , ar∗ ) ∈ (AZ )2r
∗
+1
,
f ∗ (a−r∗ , . . . , ar∗ ) = Φ0 (b)
where b : L∗0 → A is the slice obtained the simultaneous application of the local rule f of the original CA on the slices c−r∗ , . . . , cr∗ of any configuration c such 7
Figure 3: All the components of the 1D configuration a = Φ(c) are from the same alphabet. that ∀i ∈ [−r∗ , r∗ ], ci = Ti−1 (Φ−1 0 (ai )) (see Figure 4). The global map of this new CA is F ∗ : (AZ )Z → (AZ )Z and the link between F ∗ and f ∗ is given, as usual, by (F ∗ (a))i = f ∗ (ai−r∗ , . . . , ai+r∗ ) where a = (. . . , a−1 , a0 , a1 , . . .) ∈ (AZ )Z and i ∈ Z. The slicing construction can be summarized by the following 2
Theorem 1. Let (AZ , F ) be a 2D CA and let ((AZ )Z , F ∗ ) be the 1D CA obtained by the ~ν slicing construction of it, where ~ν ∈ Z2 is a fixed vector. The two CA are isomorphic by the bijective mapping Ψ. Moreover, the map Ψ−1 is 2 continuous and then (AZ , F ) is a factor of ((AZ )Z , F ∗ ). F∗
(AZ )Z −−−−→ (AZ )Z −1 yΨ Ψ−1 y 2
AZ
2
−−−−→ AZ F
Proof. It is clear that Ψ is bijective. We show that Ψ ◦ F = F ∗ ◦ Ψ, i.e., that 2 ∀i ∈ Z, ∀c ∈ AZ , Ψ(F (c))i = F ∗ (Ψ(c))i . We have Ψ(F (c))i = Φ0 (Ti (F (c)i )) where the slice F (c)i is obtained by the simultaneous application of f on the slices ci−r∗ , . . . , ci+r∗ . On the other hand F ∗ (Ψ(c))i is equal to f ∗ (Ψ(c)i−r∗ , . . . , Ψ(c)i+r∗ ) = f ∗ (Φ0 (Ti−r∗ (ci−r∗ )), . . . , Φ0 (Ti+r∗ (ci+r∗ ))) = Φ0 (b)
8
Figure 4: Local rule f ∗ of the 1D CA as sliced version of the original 2D CA. Here r = 1 and r∗ = 2. where, by definition of f ∗ , b is the slice obtained by the simultaneous application of f on the slices −1 −1 dr∗ = T−r ∗ (Ti−r ∗ (ci−r ∗ )), . . . , dr ∗ = Tr ∗ (Ti+r ∗ (ci+r ∗ ))
which gives b = Ti (F (c)i ). We now prove that Ψ−1 is a continuous map from the 1D CA configuration space (AZ )Z to the 2D CA configuration space 2 AZ , both equipped with the corresponding metric, which for the sake of simplicity is denoted by the same symbol d. Choose an arbitrary configuration a = (. . . , a−1 , a0 , a1 , . . .) ∈ (AZ )Z and a real number > 0. Let n be a positive integer such that 21n < . Consider the lines L∗i which intersect the 2D n-radius Moore neighborhood and let m be the maximum of the indexes of such lines. Setting δ = 21m , for any configuration b ∈ (AZ )Z with d(b, a) < δ, we have that bi = ai for each integer i ∈ [−m, m]. This fact implies that (Ψ−1 (b))i = (Ψ−1 (a))i for each integer i ∈ [−m, m], and then (Ψ−1 (b))i (~x) = (Ψ−1 (a))i (~x), ∗ −1 for each for (a))(~x) = (Ψ−1 (b))(~x), for i . Equivalently, we have (Ψ S any ~x ∈ L ∗ any ~x ∈ i∈[−m,m] Li , and in particular for any ~x such that |~x| ≤ n. Hence, d(Ψ−1 (b), Ψ−1 (a)) < and Ψ−1 is continuous. Remark 1. The above constructions do not depend neither on the norm nor on the sense of the vector ~ν . In other words, if ~ν is a normalized vector, all k~ν –slicing (k ∈ Z) constructions of a CA F generate the same CA F ∗ .
9
3.1. ~ν -Slicing with finite alphabet Fix a vector ~ν ∈ Z2 . For any 2D CA F , we can build an associated sliced version F ∗ with finite alphabet by considering the ~ν -slicing construction of the 2D CA restricted on the set S~v , where ~v is any vector such that ~v ⊥ ~ν . This is possible since the set S~v is F -invariant and so (S~v , F ) is a DTDS. The obtained construction leads to the following Theorem 2. Let F be a 2D CA and ~ν ∈ Z2 . For any vector ~v ∈ Z2 with ~v ⊥ ~ν , the DTDS (S~v , F ) is topologically conjugated to the 1D CA (B Z , F ∗ ) on the finite alphabet B = A|~v| obtained by the ~ν –slicing construction of F restricted on S~v . F∗ B Z −−−−→ B Z −1 yΨ Ψ−1 y S~v −−−−→ S~v F
Proof. Fix a vector ~v ⊥ ~ν . Consider the slicing construction on S~v . According to it, any configuration c ∈ S~v is identified with the corresponding bi-infinite sequence of slices. Since slices of configurations in S~v are in one-to-one correspondence with symbols of the alphabet B, the ~ν –slicing construction gives a 1D CA F ∗ : B Z → B Z such that, by Theorem 1, (S~v , F ) is isomorphic to (B Z , F ∗ ) by the bijective map Ψ : S~v → B Z . By Theorem 1, Ψ−1 is continuous. Since configurations of S~v are periodic with respect to σ~v , Ψ is continuous too. The previous result is very useful since one can use all the well-known results about 1D CA and try to lift them to F . 4. 2D Closingness The notion of closingness is of interest in 1D symbolic dynamics since it is tightly linked to several and important dynamical behaviors. In this section, we generalize it to 2D CA. As main results, we prove that closing 2D CA have DPO and that bi–closing CA are open. Notation. From now on, for any ~ν ∈ Z2 , define ~ν = −~ν . 2
Definition 1 (~ν -asymptotic configurations). Two configurations c, c0 ∈ AZ are ~ν -asymptotic if there exists q ∈ Z such that ∀~x ∈ Z2 with ~ν · ~x ≥ q it holds that c(~x) = c0 (~x).
Definition 2 (~ν -closingness). A 2D CA F is ~ν -closing if for any pair of ~ν – 2 asymptotic configurations c, c0 ∈ AZ we have that c 6= c0 implies F (c) 6= F (c0 ). A 2D CA is closing if it is ~ν -closing, for some ~ν . A 2D CA is bi-closing if it is both ~ν –closing and ~ν –closing, for some ~ν . Remark that Definitions 1 and 2 do not depend on the norm of the ~ν . Moreover, a 2D CA can be closing w.r.t. a certain direction but may not be closing w.r.t. another one as illustrated in the following example. 10
Figure 5: ~ν –slicing of a configuration c ∈ S~v on the binary alphabet A where ~ν = (1, 1) and ~v = (3, −3). The configuration Ψ(c) is on the alphabet B = A3 . Example 1. Consider the 2D CA F of radius 1 on the binary alphabet in which the local rule performs the xor operation on the two corners ~ν = (1, 1) and ~ν = (−1, −1) of the radius 1 Moore neighborhood. It is not difficult to see that F is both ~ν and ~ν –closing. On the other hand, for ~ν = (1, −1) or ~ν = (−1, 1), F is not ~ν –closing. Thanks to the ~ν -slicing construction with finite alphabet, the following results can be proved. Proposition 3. Let F be a ~ν -closing 2D CA. For any vector ~v ∈ Z2 with ~v ⊥ ~ν , let (B Z , F ∗ ) be the 1D CA of Theorem 2 which is topologically conjugated to (S~v , F ). Then F ∗ is either right or left closing. Proof. Take a vector ~v ⊥ ~ν . Assume that F ∗ is neither left nor right–closing. Then there exist two distinct either left or right asymptotic configurations a, a0 ∈ B Z such that F ∗ (a) = F ∗ (a0 ) and c = Ψ−1 (a), c0 = Ψ−1 (a0 ) ∈ S~v are two distinct ~ν –asymptotic configurations. By Theorem 2, F (c) = F (c0 ) and thus F is not ~ν –closing. Theorem 4. Any closing 2D CA has DPO. Proof. Assume that F is ~ν –closing for some ~ν ∈ Z2 . Choose a configuration 2 c ∈ AZ and let n ∈ N be an arbitrary integer. Take a vector ~v ⊥ ~ν such that |~v | ≥ 2(2n + 1). This assures that there exists a configuration c0 ∈ S~v such that d(c0 , c) < 21n . Since F is ~ν –closing, by Theorem 2 and Proposition 3, (S~v , F ) is 11
topologically conjugated to a 1D CA (B Z , F ∗ ) where F ∗ is either right or left closing and B is finite. Since closing 1D CA on finite alphabet have DPO [5, Thm. 4.4], the DTDS (S~v , F ) has DPO too. Then, there exists a periodic point p ∈ S~v such that d(c0 , p) < 21n . Hence, it also holds that d(c, p) < 21n and this concludes the proof. Corollary 5. Any closing 2D CA is surjective. Proof. Use Theorem 4 and recall that DPO implies surjectivity for DTDS on a compact space. Remark 2. The previous result is also a consequence of the Moore–Myhill Theorem [21, 22]: a 2D CA is surjective if and only if it is injective when restricted to finite configurations, i.e., configurations having the same symbol in all positions except at a finite number of them. Indeed, finite configurations are ~ν – asymptotic for every vector ~ν and ~ν –closingness is nothing but injectivity on ~ν –asymptotic configurations. Recall that a pattern P is a function from a finite domain Dom(P ) ⊆ Z2 taking values in A. The notion of cylinder can be conveniently extended to general patterns as follows: for any pattern P , consider the set n o 2 [P ] = c ∈ AZ | ∀~x ∈ Dom(P ), c(~x) = P (~x) . As in the 1D case, cylinders form a basis for the open sets. For h, t ≥ 1 and any ~ν , µ ~ ∈ Z2 normalized vectors, we say that a pattern u has a (~ν , µ ~ )–shape of size [h, t] if dom(u) = ~x ∈ Z2 | q ≤ ~ν ~x < q + h and q 0 ≤ µ ~ ~x < q 0 + t for some q, q 0 ∈ Z. Lemma 6. Consider a DTDS (X, F ), where X is a zero–dimensional ultrametric space. Let S ⊂ X be a dense set such that F (S) ⊆ S. If F is open in the relative topology R on S then F is open in the whole topology on X. Proof. Let [u] be a cylinder and c ∈ F ([u]). Since S is dense in F ([u]), for any r > 0 the ball Br (c) of center c and radius r contains a configuration c0 ∈ S. Moreover, by hypothesis, we have Br (c) = Br (c0 ). Since F is open in the relative topology R, there exists Br (c0 ) ∩ S ⊂ F [u] ∩ S. Let b ∈ Br (c0 ). Since S is dense, there is a sequence {b(n) } ∈ Br (c0 ) ∩ F ([u]) ∩ S converging to b. Since F ([u]) is closed, then b ∈ F ([u]). Thus, Br (c0 ) ⊂ F ([u]). The following result is an improvement of [9, Thm. 2] and gives a tight relation between closingness and openess. Theorem 7. Any bi–closing 2D CA is open.
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Proof. Let F be a 2D CA F which is both ~ν and ~ν –closing, for some vector ~ν ∈ Z2 which is assumed to be normalized. Let µ ~ ⊥ ~ν be a normalized vector. Fix a cylinder [u] where u is a pattern centered in the origin and having a (~ν , µ ~ )– shape of size [2h + 1, 2t + 1], for arbitrary h, t ∈ N. S Let ~k ⊥ ~ν with |~k| = 2t + 1 and denote Sn0 = Sn~k . Consider the dense set S = n∈N Sn0 endowed with the relative topology R. We prove that F ([u] ∩ S) is open in R. Choose a cylinder [v] where v is a pattern centered in the origin and having a (~ν , µ ~ )–shape of size [2l + 1, 2t + 1] with l > h. In the sequel, we show that any configuration from [v] ∩ S has a pre-image in [u]. If c ∈ [v] ∩ S there exists n such that c ∈ Sn0 ∩ [v 0 ] where [v 0 ] ⊂ [v] is the cylinder individuated by a pattern v 0 centered in the origin and having a (~ν , µ ~ )–shape of size [2l + 1, s] with Z ∗ ~ s = n|k| ≥ 2t + 1. Let (B , F ) be the 1D CA which by Theorem 2 is obtained by the ~ν –slicing construction on (Sn0 , F ). By hypothesis and by Proposition 3, (B Z , F ∗ ) is both left and right–closing. Let m > 0 be a integer from [18, Prop. 5.44]. Thus there is a cylinder [w] ⊂ [v 0 ] ⊂ [v] individuated by a pattern w centered in the origin, having a (~ν , µ ~ )–shape of size [2l+2m+1, s], and such that c ∈ [w]. Equivalently, Ψ(c) belongs to the 1D cylinder [w] ¯ = Ψ([w] ∩ Sn0 ) ⊂ B Z . Using [18, Prop. 5.44] and a completeness argument, we obtain that Ψ(c) has a pre–image in the 1D cylinder [¯ u] = Ψ([u] ∩ Sn0 ) ⊂ B Z . Hence, c has a pre-image in [u]. Therefore, for a fixed integer l > h, [ F ([u]∩S) = {[v] : F ([u]) ∩ [v] 6= ∅, and v has (~ν , µ ~ ) shape of size [2l + 1, 2t + 1]} is a union of cylinders and so F ([u] ∩ S) is open in R. Since the size of u is arbitrary, we obtain that F is open in R. Lemma 6 concludes the proof. Proposition 8. Any open 2D CA is surjective. 2
Proof. For any 2D CA of global rule F , F (AZ ) is subshift. If F is open, then 2 2 F (AZ ) has non–empty interior. It is well–known that AZ is the only subshift 2 2 with non–empty interior. Hence, we obtain F (AZ ) = AZ . 5. 2D Permutivity Notation. In this section, let ~γ ∈ {(1, 1), (−1, 1), (−1, −1), (1, −1)}. Definition 3 (Permutivity). A 2D CA of local rule f and radius r is ~γ permutive, if for each pair of matrices N, N 0 ∈ Mr with N (~x) = N 0 (~x) in all vectors ~x 6= r~γ , it holds that N (r~γ ) 6= N 0 (r~γ ) implies f (N ) 6= f (N 0 ). A 2D CA is bi-permutive iff it is both ~γ permutive and ~γ¯ -permutive. The previous definition is given assuming a r radius Moore neighborhood. It is not difficult to generalize it to suitable neighborhoods. The proofs of the results concerning permutivity with different neighborhood can also be easily adapted.
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Proposition 9. Consider a ~γ -permutive 2D CA F . For any ~ν belonging either to the same quadrant or the opposite one as ~γ , the 1D CA ((AZ )Z , F ∗ ) obtained by the ~ν –slicing construction is either rightmost or leftmost permutive. Proof. It immediately follows from the ~ν –slicing construction.
Proposition 10. Consider a ~γ -permutive 2D CA F . For any ~ν belonging to the same quadrant as ~γ , F is ~ν -closing. Proof. Choose a vector ~ν belonging to the same quadrant as ~γ . Take two ~ν – asymptotic configurations c, c0 with F (c) = F (c0 ). Let q ∈ Z be the integer such that c(~x) = c0 (~x) for any ~x with ~ν · ~x ≤ q. Consider all vectors ~y with ~ν · ~y = q + 1. Since F is ~γ –permutive, we obtain c(~y ) = c(~y ) for all these vectors. The repetition of this argument to q + 2, q + 3, . . . gives c = c0 . Proposition 11. Any ~γ -permutive 2D CA has DPO. Proof. Use Proposition 10 and Theorem 4.
Lemma 12. Let F be a 1D CA with local rule f on a possibly infinite alphabet A. If f is both rightmost and leftmost permutive, then F is strongly transitive. Proof. Choose b ∈ AZ , u ∈ A∗ , i ∈ Z, and consider the cylinder [u]i . Let t ∈ N be such that tr > i + |u| − 1 and −tr < i. The value b0 ∈ A depends only on the values of any configuration c ∈ F −t (b) in [−rt, rt]. We build a configuration a ∈ [u]i such that F t (a) = b. Fix ai = ci for each −rt ≤ i ≤ rt, assuring that a ∈ [u]i and F t (a)0 = b0 . Since the local rule f (t) of F t is both leftmost and rightmost permutive there exist symbols α−1 , α1 ∈ A, such that F t (a)−1 = b−1 and F t (a)1 = b1 , when setting a−rt−1 = α−1 , art+1 = α1 . By repeating the above procedure the thesis is obtained. Theorem 13. Any bi-permutive 2D CA is strongly transitive. Proof. Consider a 2D CA F which is both ~γ and ~γ¯ -permutive. Let ~ν a vector belonging either to the same quadrant or the opposite one as ~γ . By Theorem 1, F is a factor of the 1D CA ((AZ )Z , F ∗ ) obtained by the ~ν -slicing construction. By Proposition 9, F ∗ is both left and right permutive. Hence, by Lemma 12, F ∗ is strongly transitive. Theorem 1 conclude the proof. Lemma 14. Let F be a 1D CA with local rule f on a possibly infinite alphabet A. If f is either rightmost or leftmost permutive, then F is topologically mixing. Proof. The proof is similar to that given in [6] for CA with finite alphabet. Theorem 15. Any ~γ -permutive 2D CA is topologically mixing. Proof. Let ~ν a vector belonging either to the same quadrant or the opposite one as ~γ . By Theorem 1, F is a factor of the 1D CA ((AZ )Z , F ∗ ) obtained by the ~ν -slicing construction. By Proposition 9, F ∗ is either left or right permutive. Hence, by Lemma 14, F ∗ is topologically mixing. Theorem 1 conclude the proof. 14
Example 2. Let F be as in Example 1. This CA is both (1, 1) and (−1, −1)– permutive. Therefore, by Proposition 10 and Theorems 4, 7, 13, and 15, it is bi–closing, it has DPO, it is open, strongly transitive, and mixing. 6. Conclusions In this paper we made one step forward in the study of 2D CA dynamics. In particular, we generalized to the 2D case the notions of closingness and permutivity and we investigated their relations with other dynamical properties. This has been done by means of two constructions which associates any 2D CA with peculiar 1D CA. We strongly believe that these two constructions are useful to prove other fundamental results about 2D CA dynamics and their view as transformations of picture languages. We are currently investigating to reach these goals. 7. Acknowledgements This work has been supported by the ANR Blanc Projet “EMC”, by the “Coop´eration scientifique international R´egion Provence–Alpes–Cˆote d’Azur” Project “Automates Cellulaires, dynamique symbolique et d´ecidabilit´e”, and by the PRIN/MIUR project “Mathematical aspects and forthcoming applications of automata and formal languages”. References [1] L. Acerbi, A. Dennunzio, and E. Formenti. Conservation of some dynamical properties for operations on cellular automata. Theoretical Computer Science, 2009. In press. [2] S. Amoroso and Y. N. Patt. Decision procedures for surjectivity and injectivity of parallel maps for tesselation structures. Journal of Computer and System Sciences, 6:448–464, 1972. [3] V. Bernardi, B. Durand, E. Formenti, and J. Kari. A new dimension sensitive property for cellular automata. Theoretical Computer Science, 345:235–247, 2005. [4] F. Blanchard and A. Maass. Dynamical properties of expansive one-sided cellular automata. Israel Journal of Mathematics, 99:149–174, 1997. [5] M. Boyle and B. Kitchens. Periodic points for cellular automata. Indagationes Mathematicae, 10:483–493, 1999. [6] G. Cattaneo, A. Dennunzio, and L. Margara. Chaotic subshifts and related languages applications to one-dimensional cellular automata. Fundamenta Informaticae, 52:39–80, 2002.
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