Topological properties of cellular automata on trees

We prove that there do not exist positively expansive cellular automata defined on the full k-ary tree shift (for k>=2). Moreover, we investigate some topological properties of these automata and their relationships, namely permutivity, surjectivity, preinjectivity, right-closingness and openness.Comment: In Proceedings AUTOMATA&JAC 2012, arXiv:1208.249

A family of sand automata

We study some dynamical properties of a family of two-dimensional cellular automata: those that arise from an underlying one-dimensional sand automaton whose local rule is obtained using a Latin square. We identify a simple sand automaton Γ whose local rule is algebraic, and classify this automaton as having equicontinuity points, but not being equicontinuous. We also show that it is not surjective. We generalise some of these results to a wider class of sand automata

A compact topology for sand automata

In this paper, we exhibit a strong relation between the sand automata configuration space and the cellular automata configuration space. This relation induces a compact topology for sand automata, and a new context in which sand automata are homeomorphic to cellular automata acting on a specific subshift. We show that the existing topological results for sand automata, including the Hedlund-like representation theorem, still hold. In this context, we give a characterization of the cellular automata which are sand automata, and study some dynamical behaviors such as equicontinuity. Furthermore, we deal with the nilpotency. We show that the classical definition is not meaningful for sand automata. Then, we introduce a suitable new notion of nilpotency for sand automata. Finally, we prove that this simple dynamical behavior is undecidable

A guided tour of asynchronous cellular automata

Research on asynchronous cellular automata has received a great amount of attention these last years and has turned to a thriving field. We survey the recent research that has been carried out on this topic and present a wide state of the art where computing and modelling issues are both represented.Comment: To appear in the Journal of Cellular Automat

Additive Cellular Automata Over Finite Abelian Groups: Topological and Measure Theoretic Properties

We study the dynamical behavior of D-dimensional (D >= 1) additive cellular automata where the alphabet is any finite abelian group. This class of discrete time dynamical systems is a generalization of the systems extensively studied by many authors among which one may list [Masanobu Ito et al., 1983; Giovanni Manzini and Luciano Margara, 1999; Giovanni Manzini and Luciano Margara, 1999; Jarkko Kari, 2000; Gianpiero Cattaneo et al., 2000; Gianpiero Cattaneo et al., 2004]. Our main contribution is the proof that topologically transitive additive cellular automata are ergodic. This result represents a solid bridge between the world of measure theory and that of topology theory and greatly extends previous results obtained in [Gianpiero Cattaneo et al., 2000; Giovanni Manzini and Luciano Margara, 1999] for linear CA over Z_m i.e. additive CA in which the alphabet is the cyclic group Z_m and the local rules are linear combinations with coefficients in Z_m. In our scenario, the alphabet is any finite abelian group and the global rule is any additive map. This class of CA strictly contains the class of linear CA over Z_m^n, i.e.with the local rule defined by n x n matrices with elements in Z_m which, in turn, strictly contains the class of linear CA over Z_m. In order to further emphasize that finite abelian groups are more expressive than Z_m we prove that, contrary to what happens in Z_m, there exist additive CA over suitable finite abelian groups which are roots (with arbitrarily large indices) of the shift map. As a consequence of our results, we have that, for additive CA, ergodic mixing, weak ergodic mixing, ergodicity, topological mixing, weak topological mixing, topological total transitivity and topological transitivity are all equivalent properties. As a corollary, we have that invertible transitive additive CA are isomorphic to Bernoulli shifts. Finally, we provide a first characterization of strong transitivity for additive CA which we suspect it might be true also for the general case

Basic properties for sand automata

Presented at MFCS 2005 (Gdansk, POLAND). Long version with complete proofs published in Theoretical Computer Science, 2006, under the title "From Sandpiles to Sand Automata".International audienceWe prove several results about the relations between injectivity and surjectivity for sand automata. Moreover, we begin the exploration of the dynamical behavior of sand automata proving that the property of nilpotency is undecidable. We believe that the proof technique used for this last result might reveal useful for many other results in this context

Avalanches in the Weakly Driven Frenkel-Kontorova Model

A damped chain of particles with harmonic nearest-neighbor interactions in a spatially periodic, piecewise harmonic potential (Frenkel-Kontorova model) is studied numerically. One end of the chain is pulled slowly which acts as a weak driving mechanism. The numerical study was performed in the limit of infinitely weak driving. The model exhibits avalanches starting at the pulled end of the chain. The dynamics of the avalanches and their size and strength distributions are studied in detail. The behavior depends on the value of the damping constant. For moderate values a erratic sequence of avalanches of all sizes occurs. The avalanche distributions are power-laws which is a key feature of self-organized criticality (SOC). It will be shown that the system selects a state where perturbations are just able to propagate through the whole system. For strong damping a regular behavior occurs where a sequence of states reappears periodically but shifted by an integer multiple of the period of the external potential. There is a broad transition regime between regular and irregular behavior, which is characterized by multistability between regular and irregular behavior. The avalanches are build up by sound waves and shock waves. Shock waves can turn their direction of propagation, or they can split into two pulses propagating in opposite directions leading to transient spatio-temporal chaos. PACS numbers: 05.70.Ln,05.50.+q,46.10.+zComment: 33 pages (RevTex), 15 Figures (available on request), appears in Phys. Rev.

The algebraic entropy of one-dimensional finitary linear cellular automata

The aim of this paper is to present one-dimensional finitary linear cellular automata $S$ on $\mathbb Z_m$ from an algebraic point of view. Among various other results, we: (i) show that the Pontryagin dual $\widehat S$ of $S$ is a classical one-dimensional linear cellular automaton $T$ on $\mathbb Z_m$; (ii) give several equivalent conditions for $S$ to be invertible with inverse a finitary linear cellular automaton; (iii) compute the algebraic entropy of $S$, which coincides with the topological entropy of $T=\widehat S$ by the so-called Bridge Theorem. In order to better understand and describe the entropy we introduce the degree $\mathrm{deg}(S)$ and $\mathrm{deg}(T)$ of $S$ and $T$.Comment: 21 page

MFCS\u2798 Satellite Workshop on Cellular Automata

For the 1998 conference on Mathematical Foundations of Computer Science (MFCS\u2798) four papers on Cellular Automata were accepted as regular MFCS\u2798 contributions. Furthermore an MFCS\u2798 satellite workshop on Cellular Automata was organized with ten additional talks. The embedding of the workshop into the conference with its participants coming from a broad spectrum of fields of work lead to interesting discussions and a fruitful exchange of ideas. The contributions which had been accepted for MFCS\u2798 itself may be found in the conference proceedings, edited by L. Brim, J. Gruska and J. Zlatuska, Springer LNCS 1450. All other (invited and regular) papers of the workshop are contained in this technical report. (One paper, for which no postscript file of the full paper is available, is only included in the printed version of the report). Contents: F. Blanchard, E. Formenti, P. Kurka: Cellular automata in the Cantor, Besicovitch and Weyl Spaces K. Kobayashi: On Time Optimal Solutions of the Two-Dimensional Firing Squad Synchronization Problem L. Margara: Topological Mixing and Denseness of Periodic Orbits for Linear Cellular Automata over Z_m B. Martin: A Geometrical Hierarchy of Graph via Cellular Automata K. Morita, K. Imai: Number-Conserving Reversible Cellular Automata and Their Computation-Universality C. Nichitiu, E. Remila: Simulations of graph automata K. Svozil: Is the world a machine? H. Umeo: Cellular Algorithms with 1-bit Inter-Cell Communications F. Reischle, Th. Worsch: Simulations between alternating CA, alternating TM and circuit families K. Sutner: Computation Theory of Cellular Automat