Non-uniform Cellular Automata

Gianpiero Cattaneo1 , Alberto Dennunzio1 , Enrico Formenti2, ,

and Julien Provillard3
1
Università degli Studi di Milano–Bicocca
Dipartimento di Informatica, Sistemistica e Comunicazione,
Viale Sarca 336, 20126 Milano, Italy
{cattang,dennunzio}@disco.unimib.it
2
Université de Nice-Sophia Antipolis, Laboratoire I3S,
2000 Route des Colles, 06903 Sophia Antipolis, France
enrico.formenti@unice.fr
3
Ecole Normale Supérieure de Lyon,
46, Allée d’Italie, 69364 Lyon, France
julien.provillard@ens-lyon.fr

Abstract. In this paper we begin the study the dynamical behavior

of non-uniform cellular automata and compare it to the behavior of
“classical” cellular automata. In particular we focus on surjectivity and
equicontinuity.

1 Introduction and Motivations

Cellular automata (CA) are a well-known formal model for complex systems
that is used in many scientific fields [1,2,3]. Uniformity is one of the main
characteristics of this model. Indeed, a cellular automaton is made of identi-
cal finite automata arranged on a regular lattice. The state of each automaton
is updated by a local rule on the basis of the state of the automaton itself
and of the one of a fixed set of neighbors. At each time-step, the same (here
comes uniformity) local rule is applied to all finite automata in the lattice.
For recent results on CA dynamics and an up-to-date bibliography see for in-
stance [4,5,6,7,8,9,10,11,12,13,14,15,16].
In this paper we study a more general setting relaxing the uniformity con-
straint. Assume to use CA for simulating a physical or natural phenomenon.
Relaxing the uniformity constraint can be justified in several situations:
Generality. In many phenomena, each individual locally interacts with others
but maybe these interactions depend on the individual itself or on its position
in the space.

This work has been supported by the Interlink/MIUR project “Cellular Automata:
Topological Properties, Chaos and Associated Formal Languages”, by the ANR
Blanc “Projet Sycomore” and by the PRIN/MIUR project “Formal Languages and
Automata: Mathematical and Applicative Aspects”.

Corresponding author.

A.H. Dediu, A.M. Ionescu, and C. Martı́n-Vide (Eds.): LATA 2009, LNCS 5457, pp. 302–313, 2009.

c Springer-Verlag Berlin Heidelberg 2009
 Non-uniform Cellular Automata 303

Structural stability. Assume that we are investigating the robustness of a system

w.r.t. some specific property P . If some individuals change their “standard”
behavior does the system still have property P ? What is the “largest” number
of individuals that can change their default behavior so that the system does
not change its overall evolution?

Reliability. CA are more and more used to perform (fast parallel) computations
(see for example [2]). Each cell of the CA is implemented by a simple electronic
device (FPGAs for example). Then, how reliable are computations w.r.t. failure
of some of these devices? (Here failure is interpreted as a device which behaves
differently from its “default” way).
Finally, remark that the study of these generalizations has an interest in its
own since the new model coincides with the set of continuous functions in Cantor
topology.

2 Non-uniform Cellular Automata ν-CA

In the present paper we focus on the one-dimensional case (i.e. the lattice is Z).
Remark that most of the following definitions can be easily extended to higher
dimensions. Before introducing the formal definition of ν-CA, one should recall
the definition of cellular automaton.
Let A be a finite set describing the possible states of any cell. A configuration
is a snapshot of the states of all cells in the CA, i.e., a function from Z to A.
Given a configuration x, we denote by xi ∈ A the state of the cell in position
i ∈ Z. For a fixed state a ∈ A, a configuration is a-finite if only a finite number
of automata in the CA are in a state different from a.
A (one-dimensional) CA is a structure A, r, f , where A is the above intro-
duced finite set of states, also called the alphabet, and f : A2r+1 → A is the local
rule whose radius is r. The global rule induced by a CA A, r, f  is the map
F : AZ → AZ defined by

∀x ∈ AZ , ∀i ∈ Z, F (x)i = f (xi−r , . . . , xi+r ) . (1)

This rule describes the global evolution of the CA from the generic configuration
x ∈ AZ at time-step t ∈ N to the configuration F (x) at the next time-step t + 1.
The configuration set AZ is equipped with the distance d(x, y) = 2−n where
n = min{i ≥ 0 : xi = yi or x−i = y−i }. With respect to the topology induced by
d, the configuration set is a Cantor space and F is continuous. Hence, (AZ , F )
is a discrete (time) dynamical system.

Notation. Given a configuration x ∈ AZ , for any pair i, j ∈ Z, with i ≤ j, we

denote by x[i,j] the word xi · · · xj ∈ Aj−i+1 , i.e., the portion of the configuration
x inside the interval [i, j] = {k ∈ Z : i ≤ k ≤ j}. A cylinder of block u ∈ Ak and
position i ∈ Z is the set [u]i = {x ∈ AZ : x[i,i+k−1] = u}. Cylinders are clopen
sets w.r.t. the metric d.
304 G. Cattaneo et al.

The meaning of (1) is that the same local rule f is applied to each site of
the CA. Relaxing this last constraint gives us the definition of a ν-CA. More
formally one can give the following.
Definition 1 (Non-Uniform Cellular Automaton (ν-CA)). A non-uniform
Cellular Automaton (ν-CA) is a structure A, {hi , ri }i∈Z  defined by a family of
local rules hi : A2ri +1 → A of radius ri all based on the same set of states A.
Similarly to CA, one can define the global rule of a ν-CA as the map H : AZ → AZ
given by the law
∀x ∈ AZ , ∀i ∈ Z, H(x)i = hi (xi−ri , . . . , xi+ri ) .
From now on, we identify a ν-CA (resp., CA) with the discrete dynamical system
induced by itself or even with its global rule H (resp., F ).
It is well known that the Hedlund’s Theorem [17] characterizes CA as the
class of continuous functions commuting with the shift map σ : AZ → AZ , where
∀x ∈ AZ , ∀i ∈ Z, σ(x)i = xi+1 . It is possible to give a characterization also of
the class of ν-CA since it is straightforward to prove the following.
Proposition 1. A function H : AZ → AZ is the global map of a ν-CA iff it is
continuous.
The previous proposition gives the important information that the pair (AZ , H)
is a discrete dynamical system, but in many practical cases this setting is by far
too general to be useful. Therefore, we are going to focus our attention only over
two special subclasses of ν-CA.
Definition 2 (dν-CA). A ν-CA H is a dν-CA if there exist two natural k, r
and a rule h : A2r+1 → A such that for all integers i with |i| > k it holds that
hi = h. In this case, we say that H has default rule h.
Definition 3 (rν-CA). A ν-CA H is a rν-CA if there exists an integer r such
that any local rule hi has radius r. In this case, we say that H has radius r.
The first class restricts the number of positions at which non-default rules can
appear, while the second class restricts the number of different rules but not the
number of occurrences nor it imposes the presence of a default rule. Some easy
examples follow.
Example 1. Consider the ν-CA H (1) : AZ → AZ defined as ∀x ∈ AZ , H (1) (x)i = 1
if i = 0; 0 otherwise. Remark that H (1) is a dν-CA which cannot be a CA since
it does not commute with σ. So the class of ν-CA is larger than the original class
of all CA.
Example 2. Consider the ν-CA H (2) : AZ → AZ defined as ∀x ∈ AZ , H (2) (x)i = 1
if i is even; 0 otherwise. Remark that H (2) is a rν-CA but not a dν-CA.
Example 3. Consider the ν-CA H (3) : AZ → AZ defined as ∀x ∈ AZ , H (3) (x)i =
x0 . Remark that H (3) is a ν-CA but not a rν-CA.
Focusing the study on dν-CA and rν-CA is not a great loss in generality since
(at some extent) each ν-CA can be viewed as the limit of a sequence of dν-CA.
 Non-uniform Cellular Automata 305

Proposition 2. CA  dν-CA  rν-CA  ν-CA, where CA is the set of all CA.

Proof. The inclusions ⊆ easily follow from the definitions. For the strict inclu-
sions refer to Examples 1 to 3. 


Similarly to what happens for CA one can prove the following.

Proposition 3. For every rν-CA H on the alphabet A there exists a radius 1

rν-CA H  and a bijective continuous mapping φ such that H ◦ φ = φ ◦ H  . That
is, H is topologically conjugated to H  .

Proof. Let H be a rν-CA. If H has radius r = 1 then this result is trivially

true using the identity as a conjugacy map. If r > 1, let B = Ar and define
φ : AZ → B Z as ∀i ∈ Z, φ(x)i = x[ir,(i+1)r) . Then, it is not difficult to see
that the rν-CA (B Z , H  ) of radius 1 defined as ∀x ∈ AZ , ∀i ∈ Z, H  (x)i =
hi (xi−1 , xi , xi+1 ) is topologically conjugated to H via φ, where ∀u, v, w ∈ B, ∀i ∈
Z, ∀j ∈ {0, . . . , r − 1}, (hi (u, v, w))j = hir+j (u[j,r) vw[0,j] ). 


Finally, the following result shows that every rν-CA is a subsystem of a suitable
CA. Therefore, the study of rν-CA dynamics might reveal new properties for CA
and vice-versa.
Theorem 1. Any rν-CA H : AZ → AZ of radius r is a subsystem of a CA,
i.e., there exist a CA F : B Z → B Z on a suitable alphabet B and a continuous
injection φ : AZ → B Z such that φ ◦ H = F ◦ φ.

Proof. Consider a rν-CA H : AZ → AZ of radius r. Remark that there are only

2r+1
n = |A||A| distinct functions hi : A2r+1 → A. Take a numbering (fj )1≤j≤n
of these functions and let B = A × {1, . . . , n}. Define the mapping φ : AZ → B Z
such that ∀x ∈ AZ , ∀i ∈ Z, φ(x)i = (xi , k), where k is the integer for which
H(x)i = fk (xi−r , . . . , xi+r ). Clearly, φ is injective and continuous. Now, define
a CA F : B Z → B Z using the local rule f : B 2r+1 → B such that

f ((x−r , k−r ), . . . , (x0 , k0 ), . . . , (xr , kr )) = (fk0 (x−r , . . . , xr ), k0 ) .

It is not difficult to see that φ ◦ H = F ◦ φ. 



3 CA vs. ν-CA

In this section, we investigate some differences in dynamical behavior between

CA and ν-CA. As we are going to see, many characteristics which are really
specific to the whole class of CA are lost in the larger class of ν-CA. This will be
explored by showing via counter-examples that these properties are not satisfied
by the whole class of ν-CA.
First of all, let us recall that given a ν-CA H, a configuration x ∈ AZ is an
ultimately periodic point of H if there exist p, q ∈ N such that H p+q (x) = H q (x).
If q = 0, then x is periodic, i.e., H p (x) = x. The minimum p for which H p (x) = x
306 G. Cattaneo et al.

holds is called period of x. A ν-CA is surjective (resp. injective) if its global rule
is surjective (resp. injective).
It is well known that in the case of CA the collection of all ultimately periodic
configurations is dense in the configurations space AZ . This property is not true
in the general case of ν-CA. We will show this result making reference to the
following interesting example of ν-CA.
Example 4. Let A = {0, 1} and define the following dν-CA H (4) : AZ → AZ as

xi if i = 0
∀x ∈ AZ , ∀i ∈ Z, H (4) (x)i =
xi−1 otherwise .

The first no–go result is relative to the above example.

Proposition 4. The set of ultimately periodic points of H (4) is not dense.
Proof. Let H = H (4) . Denote by P and U the sets of periodic and ultimately
periodic points, respectively. Let E = {x ∈ AZ : ∀i ∈ N, xi = x0 }. Take x ∈ P
with H p (x) = x. Remark that the set Bx = {i ∈ N : xi = x0 } is empty. Indeed,
by contradiction, assume that B = ∅ and let m = min B. It is easy to check that
∀y ∈ AZ , ∀i ∈ N, H i (y)[0,i] = y0 i+1 , hence xm = H pm (x)m = x0 , contradiction.
Thus x ∈ E and P ⊆ E.
Let y ∈ H −1 (E). We show that By = ∅. By contradiction, let n = min By .
Since H(y)n+1 = yn = y0 = H(y)0 , then H(y) ∈ / E. Contradiction,
 then y ∈ E
−1 −n −n
and
 H (E) ⊆ E. So ∀n ∈ N, H (E) ⊆ E. Moreover, U = n∈N H (P ) ⊆
−n
n∈N H (E) ⊆ E and E is not dense. 


The following proposition proves that H (4) is not surjective, despite it is based
on two local rules each of which generates a surjective CA (namely, the identity
CA and the shift CA). Moreover, unlike the CA case (see [17]), H (4) has no
configuration with an infinite number of pre-images although it is not surjective.
Proposition 5. The dν-CA H (4) is not surjective and any configuration has
either 0 or 2 pre-images.
Proof. Since ∀x ∈ AZ , H (4) (x)0 = H (4) (x)1 , configurations in the set B = {x ∈
AZ : x0 = x1 } have no pre-image. Then any x ∈ AZ \ B has 2 pre-images y and
z such that ∀i ∈/ {−1, 0}, yi = zi = xi+1 , y0 = z0 = x0 , y−1 = 0; z−1 = 1. 

In order to explore some other no-go results, we introduce an other example.
Example 5. Let A = {0, 1} and define a ν-CA H (5) : AZ → AZ by

Z 0 if i = 0
∀x ∈ A , ∀i ∈ Z, H (x)i =
(5)
xi−1 ⊕ xi+1 otherwise ,

where ⊕ is the xor operator.

The following results show that in the case of ν-CA, the Moore-Myhill theorem
on CA [18,19] is no more true.
 Non-uniform Cellular Automata 307

Proposition 6. The ν-CA H (5) is injective on the finite 0-configurations but it

is not surjective.
Proof. It is evident that H (5) is not surjective. Let x, y be two finite configura-
tions such that H (5) (x) = H (5) (y). By contradiction, assume that xi = yi , for
some i ∈ Z. Without loss of generality, assume that i ∈ N. Since xi ⊕ xi+2 =
H (5) (x)i+1 = H (5) (y)i+1 = yi ⊕ yi+2 , it holds that xi+2 = yi+2 and, by induc-
tion, ∀j ∈ N, xi+2j = yi+2j . We conclude that ∀j ∈ N, xi+2j = 1 or yi+2j = 1
contradicting the assumption that x and y are finite. 

A ν-CA H is positively expansive if there exists  > 0 such for any pair of distinct
x, y ∈ AZ , d(H n (x), H n (y)) >  for some n ∈ N. A ν-CA H is transitive if for
any distinct pair of non-empty open sets U, V ⊂ AZ , there exists n ∈ N such that
H n (U )∩V = ∅. Both of these properties are considered as standard indicators of
chaotic behavior. We will show now that, unlike the CA case, positively expansive
ν-CA are not necessarily transitive nor surjective.
Proposition 7. The ν-CA H (5) is positively expansive but it is neither transi-
tive nor surjective.
Proof. Let H = H (5) . By Proposition 6, H is not surjective and hence it is not
transitive. Let x and y be two distinct configurations. Without loss of generality,
one can assume that there exists k = min{i ∈ N, xi = yi }. If k ≤ 1, we have
d(H 0 (x), H 0 (y)) ≥ 12 . Otherwise, H(x)k−1 = xk−2 ⊕ xk = yk−2 ⊕ yk = H(y)k−1
and H(x)[0,k−2] = H(y)[0,k−2] . By induction on k ∈ N, it is easy to see that
H k−1 (x)1 = H k−1 (y)1 . Hence d(H k−1 (x), H k−1 (y)) ≥ 12 . Thus H is positively
expansive with expansivity constant 12 . 

Example 6. Let A = {0, 1} and define the ν-CA H (6) : AZ → AZ as follows
⎧
⎪
⎨xi+1 if i < 0
Z
∀x ∈ A , ∀i ∈ Z, H (x)i = x0
(6)
if i = 0
⎪
⎩
xi−1 otherwise.

Recall that for CA, the compactness of AZ and the uniformity of the local rule
allow one to prove that injective CA are surjective. The following result shows
that this does not hold in the case of ν-CA.
Proposition 8. The ν-CA H (6) is injective but not surjective.
Proof. Let H = H (6) . Concerning non-surjectivity, just remark that only con-
figurations x such that x−1 = x0 = x1 have a pre-image. Let x, y ∈ AZ with
H(x) = H(y). Then, we have ∀i > 0, xi−1 = yi−1 and ∀i < 0, xi+1 = yi+1 . So
x = y and H is injective. 


4 Surjectivity
In the context of (1D) CA, the notion of De Bruijn graph is very handy to find
fast decision algorithms for surjectivity, injectivity and openness. Here, we extend
308 G. Cattaneo et al.

this notion to work with dν-CA and find decision algorithm for surjectivity. We
stress that surjectivity is undecidable for two (or higher) dimensional dν-CA,
since surjectivity is undecidable for 2D CA [20].

Definition 4. Consider a dν-CA H of radius r and let f be its default rule. Let
k ∈ N be the largest integer such that hk = f or h−k = f . The De Bruijn graph of
H is the pair (V, E) where V = A2r ×{−k −1, . . . , k +1} and E is the set of pairs
((a, i), (b, j)) with label in A × {0, 1} and such that ∀i ∈ {0, . . . , 2r − 1}, ai = bi+1
and one of the following condition is verified
a) i = j = −k − 1; in this case the label is (f (a0 b), 0)
b) i + 1 = j; in this case the label is (hk (a0 b), 0)
c) i = j = k + 1; in this case the label is (f (a0 b), 1)

By this graph, a configuration can be seen as a bi-infinite path on vertexes

which passes once from a vertex whose second component is in [−k + 1, k − 1]
and infinite times from other vertices. The second component of vertices allows
to single out the positions of local rules different from the default one. The image
of a configuration is the sequence of first components of edge labels.

Theorem 2. Surjectivity is decidable for dν-CA.

Proof. We show that a dν-CA H is surjective iff its De Bruijn graph G recognizes

the language (A × {0})∗ (A × {1})∗ when it is considered as the graph of a finite
state automaton. Denote by (a1 , a2 ) any word of (A × {0, 1})∗. Let k be as in
Definition 4.
Assume that H is surjective and take u ∈ (A × {0})∗ (A × {1})∗ . Let n be the
number of 0’s appearing in u (in the second component). We have three cases:
1. If n = 0 then there exists v ∈ A∗ such that f (v) = u and we can construct
u by the sequence of vertices (v[0,2r) , k + 1), . . . , (v[|v|−2r,|v|) , k + 1).
2. If 0 < n < |u| then there exists v ∈ A∗ such that hk+1−n (v) = u. We can con-
struct u by the sequence of vertices (v[0,2r) , u0 ), . . . , (v[|v|−2r,|v|) , u|v|−2r−1)
where ⎧
⎨ −k − 1 if k + 1 − n + j < k
uj = k+1 if k + 1 − n + j > k
⎩
k + 1 − n + j otherwise
3. If n = |u| then there exists v ∈ A∗ such that f (v) = u and we can construct
u by the sequence of vertices (v[0,2r) , −k − 1), . . . , (v[|v|−2r,|v|) , −k − 1).
For the opposite implication, assume that G recognizes (A × {0})∗ (A × {1})∗ .
Take y ∈ AZ and let n > k. Since G recognizes (y[−n,n] n+k+1 n−k
 ,0 Z 1 ), there exists
∗
v ∈ A such that H(v)[−n,n] = y[−n,n] . Set Xn = x ∈ A , x[n,n] = y[−n,n] . For
any n
∈ N, Xn is non-empty and compact. Moreover, Xn+1 ⊆ Xn . Therefore,
X = n∈N Xn = ∅ and H(X) = {y}. Hence, H is surjective. 

 Non-uniform Cellular Automata 309

5 More on Dynamical Properties

5.1 Equicontinuity
Let H be a ν-CA. A configuration x ∈ AZ is an equicontinuity point for H if
∀ε > 0 there exists δ > 0 such that for all y ∈ AZ , d(y, x) < δ implies that
∀n ∈ N, d(H n (y), H n (x)) < ε. A ν-CA is said to be equicontinuous if the
set E of all its equicontinuous points is the whole AZ , while it is said to be
almost equicontinuous if E is residual (i.e., E can be obtained by a countable
intersection of dense open subsets). A word u ∈ Ak is s-blocking (s ≤ k) for a
CA F if there exists an offset j ∈ [0, k − s] such that for any x, y ∈ [u]0 and
any n ∈ N, F n (x)[j,j+s−1] = F n (y)[j,j+s−1] . In [21], Kůrka proved that a CA is
almost equicontinuous iff it is non-sensitive iff it admits a blocking word.
We now introduce a class of ν-CA which will be useful in the sequel. It is an
intermediate class between dν-CA and rν-CA.
Definition 5 (n-compatible rν-CA). A rν-CA H is n-compatible with a local
rule f if for any k ∈ N, there exist two integers k1 > k and k2 < −k such that
∀i ∈ [k1 , k1 + n) ∪ [k2 , k2 + n), hi = f .
In other words, a ν-CA is n-compatible with f if, arbitrarily far from the center
of the lattice, there are intervals of length n in which the local rule f is applied.
The notion of blocking word and the related results cannot be directly restated
in the context of ν-CA because some words are blocking just thanks to the
uniformity of CA. To overcome this problem we introduce the following notion.
Definition 6 (Strongly blocking word). A word u ∈ A∗ is strongly s-
blocking for a CA F of local rule f if there exists an offset p ∈ [0, |u| − s]
such that for any ν-CA H with ∀i ∈ {0, . . . , |u| − 1}, hi = f it holds that

∀x, y ∈ [u]0 , ∀n ≥ 0, H n (x)[p,p+s) = H n (y)[p,p+s) .

Roughly speaking, a word is strongly blocking if it is blocking whatever be the

perturbations involving the rules in its neighborhood. The following extends
Proposition 5.12 in [22] to strongly r-blocking words.

Proposition 9. Any r radius CA F is equicontinuous iff there exists k > 0

such that any word u ∈ Ak is strongly r-blocking for F .

Proof. If any word is strongly blocking then F is obviously equicontinuous. For

the opposite implication, by [22, Prop. 5.12], there exist p > 0 and q ∈ N such
that F q+p = F q . As a consequence, we have that ∀u ∈ A∗ , |u| > 2(q + p)r ⇒
f p+q (u) = f q (u)[pr,|u|−(2q+p)r) . Let H be a ν-CA such that hi = f for each
i ∈ {0, . . . , (2p + 2q + 1)r − 1}. For any x ∈ AZ and i ≥ 0, consider the following
words: s(i) = H i (x)[0,qr) , t(i) = H i (x)[qr,(q+p)r) , u(i) = H i (x)[(q+p)r,(q+p+1)r) ,
v (i) = H i (x)[(q+p+1)r,(q+2p+1)r) , w(i) = H i (x)[(q+2p+1)r,(2q+2p+1)r) . For all i ∈
0, . . . , q + p, u(i) is fully determined by s(0) t(0) u(0) v (0) w(0) = x[0,(2q+2p+1)r) .
Moreover, for any natural i, we have u(i+q+p) = f q+p (s(i) t(i) u(i) v (i) w(i) ) =
310 G. Cattaneo et al.

f q (s(i) t(i) u(i) v (i) w(i) )[pr,(p+1)r) = (t(i+q) u(i+q) v (i+q) )[pr,(p+1)r) = u(i+q) . Sum-
marizing, for all i ∈ N, u(i) is determined by the word x[0,(2q+2p+1)r) which
is then strongly r-blocking. Since x had been chosen arbitrarily, we have the
thesis. 

Theorem 3. Let F be a CA with local rule f admitting a strongly r-blocking
word u. Let H be a rν-CA of radius r. If H is |u|-compatible with f then H is
almost equicontinuous.
Proof. Let p and n be the offset and the length of u, respectively. For any k ∈ N,
consider the set Tu,k of configurations x ∈ AZ having the following property
P: there exist l > k and m < −k such that x[l,l+n) = x[m,m+n) = u and
∀i ∈ [l, l + n) ∪ [m, m + n) hi = f . Remark that Tu,k is open,
being a union of
cylinders. Clearly, each Tu,k is dense, thus the set Tu = k≥n Tu,k is residual.
We claim that any configuration in Tu is an equicontinuity point. Indeed, choose
arbitrarily x ∈ Tu . Set  = 2−k , where k ∈ N is such that x ∈ Tu,k . Then, there
exist k1 > k and k2 < −k − n satisfying P. Fix δ = min{2−(k1 +n) , 2−k2 } and
let y ∈ AZ be such that d(x, y) < δ. Then y[k2 ,k1 +|u|) = x[k2 ,k1 +|u|) . Since u is r-
blocking, ∀t ∈ N, H t (x) and H t (y) are equal inside the intervals [k1 +p, k1 +p+r]
and [k2 + p, k2 + p + r], then d(H t (x), H t (y)) < . 

In a similar manner one can prove the following.
Theorem 4. Let F be an equicontinuous CA of local rule f . Let k ∈ N be as in
Proposition 9. Any rν-CA k-compatible with f is equicontinuous.

5.2 Sensitivity to the Initial Conditions

Recall that a CA F is sensitive to the initial conditions (or simply sensitive) if
there exists a constant ε > 0 such that for any configuration x ∈ AZ and any δ >
0 there is a configuration y ∈ AZ such that d(y, x) < δ and d(F n (y), F n (x)) ≥ ε
for some n ∈ N.
Example 7. Let A = {0, 1, 2} and consider the CA whose local rule f : A3 → A
is defined as follows: ∀x, y ∈ A, f (x, 0, y) = 1 if x = 1 or y = 1, 0 otherwise;
f (x, 1, y) = 2 if x = 2 or y = 2, 0 otherwise; f (x, 2, y) = 0 if x = 1 or y = 1, 2
otherwise.
Proposition 10. The CA defined in Example 7 is almost equicontinuous.
Proof. Just remark that the number of 0s inside the word 20i 2 is non-decreasing.
Thus 202 is a 1-blocking word. 

The following example defines a ν-CA which is sensitive to the initial conditions
although its default rule give rise to an almost equicontinuous CA.
Example 8. Consider the dν-CA H (8) : AZ → AZ defined as follows

Z 1 if i = 0
∀x ∈ A , ∀i ∈ Z, H (x)i =
(8)
f (xi−1 , xi , xi+1 ) otherwise ,
where f and A are as in Example 7.
 Non-uniform Cellular Automata 311

Remark that positive and negative cells do not interact each other under the
action of H (8) . Therefore, in order to study the behavior of H (8) , it is sufficient
to consider the action of H (8) on AN .
Lemma 1. For any u ∈ A∗ , ∃n0 ∈ N such that ∀n > n0 , (H (8) )n (u0∞ )1 = 1.
Lemma 2. ∀u ∈ A∗ , ∀n0 ≥ 0, ∃n > n0 , (H (8) )n (u2∞ )1 = 2.
Proposition 11. The dν-CA H (8) is sensitive.
Proof. Let H = H (8) and F be the set of all a-finite configurations for a ∈ {0, 2}.
By a theorem of Knudesen [23], we can prove the statement w.r.t. F . Then, for
any u ∈ A∗ . Build x = u0∞ and y = u2∞ . By Lemma 1 and 2, there exists n
such that 1 = H n (x)1 = H n (y)1 = 2. And hence H is sensitive with sensitivity
constant  = 1/2. 

The following example shows that default rules individually defining almost
equicontinuous CA can also constitute ν-CA that have a completely different
behavior from the one in Example 8.
Example 9. Let A = {0, 1, 2} and define the local rule f : A3 → A as: ∀x, y, z ∈
A, f (x, y, z) = 2 if x = 2 or y = 2 or z = 2, z otherwise. The CA F of local
rule f is almost equicontinuous since 2 is a blocking word. The restriction of F
to {0, 1}Z gives the shift map which is sensitive. Thus F is not equicontinuous.
Define now the following dν-CA H (9) :

2 if i = 0
∀x ∈ AZ , ∀i ∈ Z, H (9) (x)i =
f (xi−1 , xi , xi+1 ) otherwise .

Proposition 12. The dν-CA H (9) is equicontinuous.

Proof. Let n ∈ N, x, y ∈ AZ be such that x[−2n,2n] = y[−2n,2n] . Since H is of
radius 1, ∀k ≤ n, H k (x)[−n,n] = H k (y)[−n,n] and ∀k > n, H k (x)[−n,n] = 22n+1 =
H k (y)[−n,n] . So, H is equicontinuous. 


5.3 Expansivity and Permutivity

Recall that a rule f : A2r+1 is leftmost (resp., rightmost) permutive if ∀u ∈
A2r , ∀b ∈ A, ∃!a ∈ A, f (au) = b (resp., f (ua) = b). This definition can be easily
extended to ν-CA. Indeed, we say that a rν-CA is leftmost (resp. rightmost)
permutive if all hi are leftmost (resp. rightmost) permutive. A ν-CA is permutive
if it is leftmost or rightmost permutive.
In a very similar way to CA, given a ν-CA H and two integers a, b ∈ N with
a < b, the column subshift (Σ[a,b] , σ) of H is defined as follows Σ[a,b] = {y ∈
N
(Ab−a+1 ) : ∃x ∈ AZ , ∀i ∈ N, yi = H i (x)[a,b] }. Consider the map I[a,b] : AZ →
Σ[a,b] defined as ∀x ∈ AZ , ∀i ∈ N, I[a,b] (x)i = H i (x)[a,b] . It is not difficult to see
that I is continuous and surjective. Moreover H ◦ I[a,b] = I[a,b] ◦ σ. Thus (ΣI , σ)
is a factor of the ν-CA (AZ , H) and we can lift some properties from (Σ[a,b] , σ)
to (AZ , H). The following result tells that something stronger happens in the
special case of leftmost and rightmost permutive rν-CA.
312 G. Cattaneo et al.

Theorem 5. Any leftmost and rightmost permutive rν-CA of radius r is conju-

gated to the full shift ((A2r )N , σ).

Proof. Just remark that the map I[1,2r] : AZ → (A2r )N is bijective. 



The requirements of the previous theorem are very strong. Indeed, there exist
ν-CA which are topologically conjugated to a full shift but that are not permu-
tive. As an example, consider the ν-CA H defined as follows

xi−1 if i ≤ 0
∀x ∈ AZ , ∀i ∈ Z, H(x)i =
xi+1 otherwise .

Then, Σ[0,1] = (A2 )N et I[0,1] is injective.

6 Conclusions

In this paper we started exploring the dynamical behavior of ν-CA. Many specific
properties for CA are no longer true for ν-CA. However, under certain conditions,
some stability forms turned out to be quite robust when altering a CA to obtain a
ν-CA. Despite of the many no-go results proved in this paper, we strongly believe
that ν-CA can be useful for many practical applications and hence deserve further
studies.

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