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2d cellular automata

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2D cellular automata: dynamics and undecidability Alberto Dennunzio 1 , Enrico Formenti 2 , and Michael Weiss 2 1 Universit` a degli Studi di Milan o–Bic occa Dipartimento di Informatica, Sistemistica e Comunicazione, Viale Sarca 336, 20126 Milano (Italy) [email protected] mib.it michael.weiss@cui .unige.ch 2 Universit´ e de Nice-So phia Antip olis, L abora toire I3S , 2000 Route des Colles, 06903 Sophia Antipolis (France). [email protected] Abstract. In this paper we introduce the notion of quasi-expansivity for 2D CA and we show that it shares many properties with expansivity (that holds only for 1D CA). Similarly, we introduce the notions of quasi- sensitivity and prove that the classical dichotomy theorem holds in this new setting. Moreover, we show a tight relation between closingness and openness for 2D CA. Finally, the undecidability of closingness property for 2D CA is proved. Keywords: cellular automata, symbolic dynamics, (un-)decidability, tilings. 1 In troduction Cellular automata (CA) are a widely used formal model for complex systems with applications in many different fields ranging from physics to biology, com- puter science, mathematics, etc.. Although applications mainly concern two or higher dimensional CA, the study of the dynamical behavior has been mostly carried on in dimension 1. Only few results are known for dimension 2, and prac- tically speaking, a syste matic study of 2D CA dynamics has just start ed (see for example [21, 7]). This paper contr ibutes the following main results: – properties characterizing quasi-expansive 2D CA; – topolog ical entrop y of quasi -expa nsiv e 2D CA is infinit e; – a dichotomy for quasi-sensitivity. – a tight relation between closingness and openness; – undec idabili ty of closin gness for 2D CA; It is well-known that there is no positively expansive 2D CA [20]. However, the absence of positively expansive 2D CA seems, at a certain extent, more an artifact of Cantor metric than an intrinsic property of CA. In this paper we Corresponding author.   a   r    X    i   v   :    0    9    0    6  .    0    8    5    7   v    2    [   c   s  .    F    L    ]    2    9    S   e   p    2    0    0    9

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