Surjective Multidimensional Cellular Automata Are Non-Wandering: A ...
Surjective Multidimensional Cellular Automata Are Non-Wandering: A Combinatorial ProofI Luigi Acerbib , Alberto Dennunzioa,∗∗, Enrico Formentic,∗∗ a Universit` a
degli studi di Milano-Bicocca, Dipartimento di Informatica Sistemistica e Comunicazione, viale Sarca 336, 20126 Milano (Italy) b University of Edinburgh, DTC in Neuroinformatics and Computational Neuroscience, School of Informatics, Edinburgh EH8 9AB (UK) c Universit´ e Nice-Sophia Antipolis, Laboratoire I3S, 2000 Route des Colles, 06903 Sophia Antipolis (France)
Abstract A combinatorial proof that surjective D-dimensional CA are non-wandering is given. This answers an old open question stated in [3]. Moreover, an explicit upper–bound for the return time is given. Keywords: multidimensional cellular automata, symbolic dynamics, discrete dynamical systems.
1. Introduction Cellular automata (CA) are simple formal models for complex systems. They essentially consist in an infinite number of identical finite automata arranged on a regular lattice (here ZD ). Each automaton updates its state according to a local rule which takes into account the current state of the automaton and the state of a fixed set of neighbors. This simple formal definition contrasts with the great variety of distinct dynamical behaviors. The latter allows a successful use of CA in practical applications in numerous scientific fields ranging from biology, to chemistry, or from mathematics to computer science [13]. However, almost all dynamical behaviors are undecidable [12, 15, 18, 14, 11]. Indeed, the classification of dynamical behaviors is one of the central open questions in CA domain, see for instance [5, 6, 16, 7]. In this note we consider an important feature of the dynamical behavior, namely the non-wandering property. Roughly speaking, a point c is nonI This work has been partially supported by the French National Research Agency project EMC (ANR-09-BLAN-0164) and by the PRIN 2010-11/MIUR project “Automata and Formal Languages: Mathematical and Applicative Aspects”. ∗ Corresponding author. ∗∗ Corresponding author. Email addresses: [email protected] (Luigi Acerbi), [email protected] (Alberto Dennunzio), [email protected] (Enrico Formenti)
Preprint submitted to Elsevier
January 2, 2013
wandering if there is a point arbitrarily near to c with orbit which returns near to c. Call W the set of all non-wandering points. W contains some very important characteristics of the system. Indeed, the topological entropy of a dynamical system (on a compact metric space) is concentrated on its non-wandering set [17]. In (one-dimensional) CA settings, the understanding of the properties of the non-wandering set allowed to prove an important result: surjective CA admitting an equicontinuity point have a dense set of periodic orbits (DPO) [3]. Closing CA also possess this property [4]. Indeed, it is an old-standing open problem whether DPO is shared by all surjective CA (see [1] for some reformulations of the problem, and [8, 10, 9] for the results about these properties in dimensions D > 1). Surjectivity and non-wandering are equivalent notions for D-dimensional CA. An easy compactness argument shows the implication non-wandering ⇒ surjectivity. The converse was proved for 1-dimensional CA in [3, Prop. 3.1 pag. 574] to show that surjective CA admitting an equicontinuous point have DPO. The authors used an ergodic theory argument, namely the Poincar´e recurrence Theorem (requiring an invariant measure). Remark that the proof of the implication surjectivity ⇒ non-wandering does not require the existence of an equicontinuity point and easily extends to higher dimensions. One just needs to provide an invariant measure. Indeed, such a measure can be built as in dimension 1 using the balance condition (which holds in any dimension) which characterizes surjective CA. However, the combinatorial nature of surjectivity property for CA led Blanchard and Tisseur to conjecture the existence of a purely combinatorial proof of the fact that surjective CA are non-wandering [3]. In this paper, we exhibit such a proof for any D-dimensional CA. Moreover, an upper bound on the return time is explicitly given. 2. Notations and Background 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 ∈ ZD , denote by |~x| the infinite norm (in RD ) of ~x. Let A be a finite alphabet with a number α of symbols. A D-dimensional (square) pattern P over A is a function from some finite domain dom(P ) = [−k, k]D with k ∈ N and taking values in A. D A D-dimensional configuration is a function from ZD to A. Denote AZ the D-dimensional CA configuration set equipped with the following metric d: D
∀c, c0 ∈ AZ ,
d(c, c0 ) = 2−k
k = min |~x| : ~x ∈ ZD , c(~x) 6= c0 (~x) .
where
With the induced topology, the D-dimensional configurations set is a Cantor space. For any configuration c (resp., pattern P ), c|K (resp., P|K ) denotes the restriction of c (resp., P ) to the finite subset K ⊂ ZD . In the sequel, given
2
two patterns U and P , P ≺k U means that [−k, k]D = dom(P ) ⊆ dom(U ) and U|[−k,k]D = P . D A pair (AZ , F ) is a D-dimensional CA if there exists an integer r ∈ N and D D D a map f : A[−r,r] → A such that F is a function from AZ to AZ defined as D
∀c ∈ AZ , ∀~x ∈ ZD ,
F (c)(~x) = f c|~x+[−r,r]D
,
The integer r and the map f are called the radius and the local rule of the given CA. The local rule f can be naturally extended to all (square) patterns in the following way. With a little abuse of notation, for any integer k ≥ r and any pattern P of domain dom(P ) = [−k, k]D we define f (P ) = F (c)|[−(k−r),k−r]D (with dom(f (P )) = [−(k−r), k−r]D ), where c is any configuration with c|[−k,k]D = P . For any n ∈ N and any pattern P of domain dom(P ) = [−k, k]D with k ≥ nr, we will call f n (P ) the n-image (pattern) of P . While, for any n ∈ N and any pair of patterns P, Q, we say that Q is a n-pre-image of P if f n (Q) = P , or, equivalently Q ∈ f −n (P ). Definition 1. A D-dimensional CA F is non-wandering iff for any nonempty open set U there exists an integer t > 0 such that F −t (U ) ∩ U 6= ∅. The following result expresses the balance condition for surjective D-dimensional CA. D
Theorem 1 ( [19]). Let (AZ , F ) be a surjective D-dimensional CA with local rule f and radius r. For any k ∈ N and any D-dimensional pattern P of domain [−k, k]D , it holds that D α(2(k+r)+1) (1) |f −1 (P )| = D α(2k+1) 3. The Results The property stated by the following Lemma is the core of the main result. D
Lemma 2. Let (AZ , F ) be a surjective D-dimensional CA with local rule f and radius r. For any k ∈ N and any D-dimensional pattern P of domain D [−k, k]D , there exist a non null integer t ≤ α(2k+1) and a pattern Q ∈ f −t (P ) such that Q[−k,k]D = P . D
Proof. Let t∗ = α(2k+1) . For the sake of argument, assume that there exists a pattern P such that 1. dom(P ) = [−k, k]D 2. ∀m ∈ [1, t∗ ], ∀Q ∈ f −m (P ), P 6≺k Q .
3
For any t ∈ N, define the following sets of patterns n o D S(t) = S ∈ A[−k−tr,k+tr] , ∀n ∈ [0, t], P 6≺k f n (S) , D
S(t)c = A[−k−tr,k+tr] \ S(t) n o D C(t) = S ∈ A[−k−tr,k+tr] , P ≺k S In other words, S(t) is the set of patterns S which are t-pre-images of some original pattern O 6= P of domain [−k, k]D in such a way that neither S nor any intermediate n-pre-image (0 < n < t) of O (nor trivially O) contain P in its own center. While, C(t) is the set of all patterns of domain [−k − tr, k + tr]D containing P in their own center. Let s(t) be the number of elements in S(t). We are going to compute s(t) D for all t ≤ t∗ − 1. Clearly, s(0) = α(2k+1) − 1. Furthermore, the following recurrence equation holds for s(t): s(t + 1) = b(t) · s(t) − c(t + 1)
(2)
where b(t) = α(2(k+(t+1)r)+1) c(t) = α(2(k+tr)+1)
D
D
−(2(k+tr)+1)D
−(2k+1)D
= |C(t)|
Indeed, by (1), every pattern belonging to S(t) gives rise to a number b(t) of 1-pre-image patterns, some of which contain P . Thus, to compute s(t + 1) for t + 1 ≤ t∗ − 1, we have to subtract, from the number b(t) · s(t) of such pre-image patterns, the number c(t + 1) = |C(t + 1)|, i.e., the number of all patterns of domain [−k − (t + 1)r, k + (t + 1)r]D containing P in the center. Indeed, it necessarily holds that C(t + 1) ⊆ f −1 (S(t)), since, on the contrary, there would be a pattern S ∈ f −1 (S(t)c ) ∩ C(t + 1), i.e., a pattern S containing P in its own center and with some n-image (n > 0) containing in its turns P in the center, that contradicts the initial assumption (see Figure 1). The closed form solution of the recurrence (2) is D s(t) = c(t) α(2k+1) − 1 − t (3) Indeed, by the fact that c(t + 1) = b(t) · c(t), the substitution of (3) into (2) gives D b(t) · s(t) − c(t + 1) = b(t) · c(t) α(2k+1) − 1 − t − c(t + 1) D = c(t + 1) α(2k+1) − 2 − t = s(t + 1) Taking t = t∗ −1 in (3), it follows that s(t∗ −1) = 0, and, equivalently, S(t∗ −1) = ∗ D ∅. Therefore, C(t∗ ) ⊆ f −1 (S(t∗ − 1)c ) = A(2(k+t r)+1) , i.e., all patterns of 4
domain [−k − t∗ r, k + t∗ r]D containing P have some n-image (n > 0) which in its turn contains P in the center. In other words, C(t∗ ) is contained in one of the region delimited by dotted blue curves in Figure 1.
P k+tr ... ... . . .
. . .
... ...
... P
... . . .. . .
...
P . . .. . .
. . .
No more patterns P here
No more patterns P here
No more patterns P here
Figure 1: Pre-images of patterns with domain of side [−(k + tr), k + tr] (upper row). Time t grows one unity per row. Patterns in the convex regions delimited by blue dotted curves cannot contain P in their center by the initial assumptions in the proof of Lemma 2.
Definition 2. Given a CA F and ε > 0, the return time function is defined as D ∀c ∈ AZ , RεF (c) = min t ∈ N \ {0} , Bε (c) ∩ F t (Bε (c)) 6= ∅ Remark. A D-dimensional CA is non-wandering if and only if for any > 0 and any configuration c the return time function RεF (c) < ∞. Theorem 3. For any surjective D-dimensional CA F , the return time function D is bounded, and hence F is non-wandering. In particular, ∀c ∈ AZ , ∀ε > 0 D
RεF (c) ≤ α(2k+1) where k ∈ N is such that
1 2k
< . D
Proof. Choose arbitrarily c ∈ AZ and > 0. Let k ∈ N be such that 21k < and D let P = c|[−k,k]D . By Lemma 2, there exist a non null integer t ≤ α(2k+1) and a configuration c0 ∈ F −t (B2−k (c)) such that c0|[−k,k]D = P , i.e., c0 ∈ B2−k (c). Hence, RεF (c) ≤ α(2k+1)
D
and F is non-wandering.
5
4. Conclusions In this paper, we have provided a combinatorial proof of the fact that surjective multidimensional CA are non-wandering along with an explicit upper bound for the return time. If the return time is now well understood for the case of surjective CA, the issue is almost completely open for the non-surjective case. Acknowledgements The authors warmly thanks the anonymous referees for careful reading. Referee 1 has also pointed out that some ideas about the proof of Lemma 2 were also contained in the Vincent Bernardi’s Ph.D. thesis [2] although they were never published. [1] L. Acerbi, A. Dennunzio, and E. Formenti. Conservation of some dynamcal properties for operations on cellular automata. Theoretical Computer Science, 410:3685–3693, 2009. [2] Vincent Bernardi. Lois de conservation sur automates cellulaires. PhD thesis, Universit´e de Provence, 2007. [3] F. Blanchard and P. Tisseur. Some properties of cellular automata with equicontinuity points. Ann. Inst. Henri Poincar´ e, Probabilit´ e et Statistiques, 36:569–582, 2000. [4] M. Boyle and B. Kitchens. Periodic points for cellular automata. Indagationes Mathematicae, 10:483–493, 1999. [5] G. Braga, G. Cattaneo, P. Flocchini, and C. Quaranta Vogliotti. Pattern growth in elementary cellular automata. Theoretical Computer Science, 145:1–26, 1995. [6] G. Cattaneo, A. Dennunzio, and L. Margara. Solution of some conjectures about topological properties of linear cellular automata. Theoretical Computer Science, 325:249–271, 2004. [7] A. Dennunzio, P. Di Lena, E. Formenti, and L. Margara. On the directional dynamics of additive cellular automata. Theoretical Computer Science, 410:4823–4833, 2009. [8] A. Dennunzio and E. Formenti. Decidable properties of 2d cellular automata. In Masami Ito and Masafumi Toyama, editors, Developments in Language Theory, volume 5257 of Lecture Notes in Computer Science, pages 264–275. Springer, 2008. [9] A. Dennunzio, E. Formenti, and M. Weiss. 2d cellular automata: new constructions, dynamics, and (un)decidability. Preprint, CoRR, abs/0906.0857, 2009. 6
[10] A. Dennunzio, E. Formenti, and M. Weiss. 2d cellular automata: dynamics and undecidability. In Proceedings of 6th Conference on Computability in Europe (CiE 2010), 2010. [11] P. Di Lena and L. Margara. On the undecidability of the limit behavior of cellular automata. Theoretical Computer Science, 411:1075–1084, 2010. [12] B. Durand, E. Formenti, and G. Varouchas. On undecidability of equicontinuity classification for cellular automata. Discrete Mathematics and Theoretical Computer Science, AB:117–128, 2003. [13] F. Farina and A. Dennunzio. A predator-prey cellular automaton with parasitic interactions and environmental effects. Fundamenta Informaticae, 83:337–353, 2008. [14] P. Guillon and G. Richard. Revisiting the rice theorem of cellular automata. In 27th International Symposium on Theoretical Aspects of Computer Science, STACS 2010, volume 5 of LIPIcs, pages 441–452. Schloss Dagstuhl Leibniz-Zentrum fuer Informatik, 2010. [15] J. Kari and N. Ollinger. Periodicity and immortality in reversible computing. In Edward Ochmanski and Jerzy Tyszkiewicz, editors, MFCS, volume 5162 of Lecture Notes in Computer Science, pages 419–430. Springer, 2008. [16] P. K˚ urka. Languages, equicontinuity and attractors in cellular automata. Ergodic Theory & Dynamical Systems, 17:417–433, 1997. [17] P. K˚ urka. Topological and Symbolic Dynamics. Volume 11 of Cours Sp´ ecialis´ es. Soci´ et´ e Math´ ematique de France, 2004. [18] V. Lukkarila. Sensitivity and topological mixing are undecidable for reversible one-dimensional cellular automata. Journal of Cellular Automata, 5:241–272, 2010. [19] A. Maruoka and M. Kimura. Conditions for injectivity of global maps for tessellation automata. Inf. & Control, 32:158–162, 1976.
7
degli studi di Milano-Bicocca, Dipartimento di Informatica Sistemistica e Comunicazione, viale Sarca 336, 20126 Milano (Italy) b University of Edinburgh, DTC in Neuroinformatics and Computational Neuroscience, School of Informatics, Edinburgh EH8 9AB (UK) c Universit´ e Nice-Sophia Antipolis, Laboratoire I3S, 2000 Route des Colles, 06903 Sophia Antipolis (France)
Abstract A combinatorial proof that surjective D-dimensional CA are non-wandering is given. This answers an old open question stated in [3]. Moreover, an explicit upper–bound for the return time is given. Keywords: multidimensional cellular automata, symbolic dynamics, discrete dynamical systems.
1. Introduction Cellular automata (CA) are simple formal models for complex systems. They essentially consist in an infinite number of identical finite automata arranged on a regular lattice (here ZD ). Each automaton updates its state according to a local rule which takes into account the current state of the automaton and the state of a fixed set of neighbors. This simple formal definition contrasts with the great variety of distinct dynamical behaviors. The latter allows a successful use of CA in practical applications in numerous scientific fields ranging from biology, to chemistry, or from mathematics to computer science [13]. However, almost all dynamical behaviors are undecidable [12, 15, 18, 14, 11]. Indeed, the classification of dynamical behaviors is one of the central open questions in CA domain, see for instance [5, 6, 16, 7]. In this note we consider an important feature of the dynamical behavior, namely the non-wandering property. Roughly speaking, a point c is nonI This work has been partially supported by the French National Research Agency project EMC (ANR-09-BLAN-0164) and by the PRIN 2010-11/MIUR project “Automata and Formal Languages: Mathematical and Applicative Aspects”. ∗ Corresponding author. ∗∗ Corresponding author. Email addresses: [email protected] (Luigi Acerbi), [email protected] (Alberto Dennunzio), [email protected] (Enrico Formenti)
Preprint submitted to Elsevier
January 2, 2013
wandering if there is a point arbitrarily near to c with orbit which returns near to c. Call W the set of all non-wandering points. W contains some very important characteristics of the system. Indeed, the topological entropy of a dynamical system (on a compact metric space) is concentrated on its non-wandering set [17]. In (one-dimensional) CA settings, the understanding of the properties of the non-wandering set allowed to prove an important result: surjective CA admitting an equicontinuity point have a dense set of periodic orbits (DPO) [3]. Closing CA also possess this property [4]. Indeed, it is an old-standing open problem whether DPO is shared by all surjective CA (see [1] for some reformulations of the problem, and [8, 10, 9] for the results about these properties in dimensions D > 1). Surjectivity and non-wandering are equivalent notions for D-dimensional CA. An easy compactness argument shows the implication non-wandering ⇒ surjectivity. The converse was proved for 1-dimensional CA in [3, Prop. 3.1 pag. 574] to show that surjective CA admitting an equicontinuous point have DPO. The authors used an ergodic theory argument, namely the Poincar´e recurrence Theorem (requiring an invariant measure). Remark that the proof of the implication surjectivity ⇒ non-wandering does not require the existence of an equicontinuity point and easily extends to higher dimensions. One just needs to provide an invariant measure. Indeed, such a measure can be built as in dimension 1 using the balance condition (which holds in any dimension) which characterizes surjective CA. However, the combinatorial nature of surjectivity property for CA led Blanchard and Tisseur to conjecture the existence of a purely combinatorial proof of the fact that surjective CA are non-wandering [3]. In this paper, we exhibit such a proof for any D-dimensional CA. Moreover, an upper bound on the return time is explicitly given. 2. Notations and Background 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 ∈ ZD , denote by |~x| the infinite norm (in RD ) of ~x. Let A be a finite alphabet with a number α of symbols. A D-dimensional (square) pattern P over A is a function from some finite domain dom(P ) = [−k, k]D with k ∈ N and taking values in A. D A D-dimensional configuration is a function from ZD to A. Denote AZ the D-dimensional CA configuration set equipped with the following metric d: D
∀c, c0 ∈ AZ ,
d(c, c0 ) = 2−k
k = min |~x| : ~x ∈ ZD , c(~x) 6= c0 (~x) .
where
With the induced topology, the D-dimensional configurations set is a Cantor space. For any configuration c (resp., pattern P ), c|K (resp., P|K ) denotes the restriction of c (resp., P ) to the finite subset K ⊂ ZD . In the sequel, given
2
two patterns U and P , P ≺k U means that [−k, k]D = dom(P ) ⊆ dom(U ) and U|[−k,k]D = P . D A pair (AZ , F ) is a D-dimensional CA if there exists an integer r ∈ N and D D D a map f : A[−r,r] → A such that F is a function from AZ to AZ defined as D
∀c ∈ AZ , ∀~x ∈ ZD ,
F (c)(~x) = f c|~x+[−r,r]D
,
The integer r and the map f are called the radius and the local rule of the given CA. The local rule f can be naturally extended to all (square) patterns in the following way. With a little abuse of notation, for any integer k ≥ r and any pattern P of domain dom(P ) = [−k, k]D we define f (P ) = F (c)|[−(k−r),k−r]D (with dom(f (P )) = [−(k−r), k−r]D ), where c is any configuration with c|[−k,k]D = P . For any n ∈ N and any pattern P of domain dom(P ) = [−k, k]D with k ≥ nr, we will call f n (P ) the n-image (pattern) of P . While, for any n ∈ N and any pair of patterns P, Q, we say that Q is a n-pre-image of P if f n (Q) = P , or, equivalently Q ∈ f −n (P ). Definition 1. A D-dimensional CA F is non-wandering iff for any nonempty open set U there exists an integer t > 0 such that F −t (U ) ∩ U 6= ∅. The following result expresses the balance condition for surjective D-dimensional CA. D
Theorem 1 ( [19]). Let (AZ , F ) be a surjective D-dimensional CA with local rule f and radius r. For any k ∈ N and any D-dimensional pattern P of domain [−k, k]D , it holds that D α(2(k+r)+1) (1) |f −1 (P )| = D α(2k+1) 3. The Results The property stated by the following Lemma is the core of the main result. D
Lemma 2. Let (AZ , F ) be a surjective D-dimensional CA with local rule f and radius r. For any k ∈ N and any D-dimensional pattern P of domain D [−k, k]D , there exist a non null integer t ≤ α(2k+1) and a pattern Q ∈ f −t (P ) such that Q[−k,k]D = P . D
Proof. Let t∗ = α(2k+1) . For the sake of argument, assume that there exists a pattern P such that 1. dom(P ) = [−k, k]D 2. ∀m ∈ [1, t∗ ], ∀Q ∈ f −m (P ), P 6≺k Q .
3
For any t ∈ N, define the following sets of patterns n o D S(t) = S ∈ A[−k−tr,k+tr] , ∀n ∈ [0, t], P 6≺k f n (S) , D
S(t)c = A[−k−tr,k+tr] \ S(t) n o D C(t) = S ∈ A[−k−tr,k+tr] , P ≺k S In other words, S(t) is the set of patterns S which are t-pre-images of some original pattern O 6= P of domain [−k, k]D in such a way that neither S nor any intermediate n-pre-image (0 < n < t) of O (nor trivially O) contain P in its own center. While, C(t) is the set of all patterns of domain [−k − tr, k + tr]D containing P in their own center. Let s(t) be the number of elements in S(t). We are going to compute s(t) D for all t ≤ t∗ − 1. Clearly, s(0) = α(2k+1) − 1. Furthermore, the following recurrence equation holds for s(t): s(t + 1) = b(t) · s(t) − c(t + 1)
(2)
where b(t) = α(2(k+(t+1)r)+1) c(t) = α(2(k+tr)+1)
D
D
−(2(k+tr)+1)D
−(2k+1)D
= |C(t)|
Indeed, by (1), every pattern belonging to S(t) gives rise to a number b(t) of 1-pre-image patterns, some of which contain P . Thus, to compute s(t + 1) for t + 1 ≤ t∗ − 1, we have to subtract, from the number b(t) · s(t) of such pre-image patterns, the number c(t + 1) = |C(t + 1)|, i.e., the number of all patterns of domain [−k − (t + 1)r, k + (t + 1)r]D containing P in the center. Indeed, it necessarily holds that C(t + 1) ⊆ f −1 (S(t)), since, on the contrary, there would be a pattern S ∈ f −1 (S(t)c ) ∩ C(t + 1), i.e., a pattern S containing P in its own center and with some n-image (n > 0) containing in its turns P in the center, that contradicts the initial assumption (see Figure 1). The closed form solution of the recurrence (2) is D s(t) = c(t) α(2k+1) − 1 − t (3) Indeed, by the fact that c(t + 1) = b(t) · c(t), the substitution of (3) into (2) gives D b(t) · s(t) − c(t + 1) = b(t) · c(t) α(2k+1) − 1 − t − c(t + 1) D = c(t + 1) α(2k+1) − 2 − t = s(t + 1) Taking t = t∗ −1 in (3), it follows that s(t∗ −1) = 0, and, equivalently, S(t∗ −1) = ∗ D ∅. Therefore, C(t∗ ) ⊆ f −1 (S(t∗ − 1)c ) = A(2(k+t r)+1) , i.e., all patterns of 4
domain [−k − t∗ r, k + t∗ r]D containing P have some n-image (n > 0) which in its turn contains P in the center. In other words, C(t∗ ) is contained in one of the region delimited by dotted blue curves in Figure 1.
P k+tr ... ... . . .
. . .
... ...
... P
... . . .. . .
...
P . . .. . .
. . .
No more patterns P here
No more patterns P here
No more patterns P here
Figure 1: Pre-images of patterns with domain of side [−(k + tr), k + tr] (upper row). Time t grows one unity per row. Patterns in the convex regions delimited by blue dotted curves cannot contain P in their center by the initial assumptions in the proof of Lemma 2.
Definition 2. Given a CA F and ε > 0, the return time function is defined as D ∀c ∈ AZ , RεF (c) = min t ∈ N \ {0} , Bε (c) ∩ F t (Bε (c)) 6= ∅ Remark. A D-dimensional CA is non-wandering if and only if for any > 0 and any configuration c the return time function RεF (c) < ∞. Theorem 3. For any surjective D-dimensional CA F , the return time function D is bounded, and hence F is non-wandering. In particular, ∀c ∈ AZ , ∀ε > 0 D
RεF (c) ≤ α(2k+1) where k ∈ N is such that
1 2k
< . D
Proof. Choose arbitrarily c ∈ AZ and > 0. Let k ∈ N be such that 21k < and D let P = c|[−k,k]D . By Lemma 2, there exist a non null integer t ≤ α(2k+1) and a configuration c0 ∈ F −t (B2−k (c)) such that c0|[−k,k]D = P , i.e., c0 ∈ B2−k (c). Hence, RεF (c) ≤ α(2k+1)
D
and F is non-wandering.
5
4. Conclusions In this paper, we have provided a combinatorial proof of the fact that surjective multidimensional CA are non-wandering along with an explicit upper bound for the return time. If the return time is now well understood for the case of surjective CA, the issue is almost completely open for the non-surjective case. Acknowledgements The authors warmly thanks the anonymous referees for careful reading. Referee 1 has also pointed out that some ideas about the proof of Lemma 2 were also contained in the Vincent Bernardi’s Ph.D. thesis [2] although they were never published. [1] L. Acerbi, A. Dennunzio, and E. Formenti. Conservation of some dynamcal properties for operations on cellular automata. Theoretical Computer Science, 410:3685–3693, 2009. [2] Vincent Bernardi. Lois de conservation sur automates cellulaires. PhD thesis, Universit´e de Provence, 2007. [3] F. Blanchard and P. Tisseur. Some properties of cellular automata with equicontinuity points. Ann. Inst. Henri Poincar´ e, Probabilit´ e et Statistiques, 36:569–582, 2000. [4] M. Boyle and B. Kitchens. Periodic points for cellular automata. Indagationes Mathematicae, 10:483–493, 1999. [5] G. Braga, G. Cattaneo, P. Flocchini, and C. Quaranta Vogliotti. Pattern growth in elementary cellular automata. Theoretical Computer Science, 145:1–26, 1995. [6] G. Cattaneo, A. Dennunzio, and L. Margara. Solution of some conjectures about topological properties of linear cellular automata. Theoretical Computer Science, 325:249–271, 2004. [7] A. Dennunzio, P. Di Lena, E. Formenti, and L. Margara. On the directional dynamics of additive cellular automata. Theoretical Computer Science, 410:4823–4833, 2009. [8] A. Dennunzio and E. Formenti. Decidable properties of 2d cellular automata. In Masami Ito and Masafumi Toyama, editors, Developments in Language Theory, volume 5257 of Lecture Notes in Computer Science, pages 264–275. Springer, 2008. [9] A. Dennunzio, E. Formenti, and M. Weiss. 2d cellular automata: new constructions, dynamics, and (un)decidability. Preprint, CoRR, abs/0906.0857, 2009. 6
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