Density classification on infinite lattices and trees

Consider an infinite graph with nodes initially labeled by independent Bernoulli random variables of parameter p. We address the density classification problem, that is, we want to design a (probabilistic or deterministic) cellular automaton or a finite-range interacting particle system that evolves on this graph and decides whether p is smaller or larger than 1/2. Precisely, the trajectories should converge to the uniform configuration with only 0's if p1/2. We present solutions to that problem on the d-dimensional lattice, for any d>1, and on the regular infinite trees. For Z, we propose some candidates that we back up with numerical simulations

Restricted density classification in one dimension

The density classification task is to determine which of the symbols appearing in an array has the majority. A cellular automaton solving this task is required to converge to a uniform configuration with the majority symbol at each site. It is not known whether a one-dimensional cellular automaton with binary alphabet can classify all Bernoulli random configurations almost surely according to their densities. We show that any cellular automaton that washes out finite islands in linear time classifies all Bernoulli random configurations with parameters close to 0 or 1 almost surely correctly. The proof is a direct application of a "percolation" argument which goes back to Gacs (1986).Comment: 13 pages, 5 figure

Ergodicity versus non-ergodicity for Probabilistic Cellular Automata on rooted trees

In this article we study a class of shift-invariant and positive rate probabilistic cellular automata (PCA) on rooted d-regular trees $\mathbb{T}^d$. In a first result we extend the results of [10] on trees, namely we prove that to every stationary measure $\nu$ of the PCA we can associate a space-time Gibbs measure $\mu_{\nu}$ on $\mathbb{Z} \times \mathbb{T}^d$. Under certain assumptions on the dynamics the converse is also true. A second result concerns proving sufficient conditions for ergodicity and non-ergodicity of our PCA on d-ary trees for $d\in \{ 1,2,3\}$ and characterizing the invariant product Bernoulli measures.Comment: 17 page

A note on the Density Classification Problem in Two Dimensions

International audienceThe density classification problem is explored experimentally in the case of two-dimensional grids. We compare the performance of deterministic and stochastic CA, as well as interacting particle systems. The question of how to design a rule that would attain an arbitrary precision is examined and we show that it seems more difficult to solve than in the one-dimensional case

Diagnostic décentralisé à l'aide d'automates cellulaires

ISBN 9782364937192National audienceWe address the problem of detecting failures in a distributed network. If some components can break down over time, how can we detect that the failure rate has exceeded a given threshold wi- thout any central authority ? Our aim is to have an estimate of the global state of the network, only through local interactions of components with their neighbours. In particular, we wish to reach a consensus on an alert state when the failure rate exceeds a given threshold. We use the model of cellular automata in order to pro- pose solutions in the case of a network with a grid structure. We compare three methods of self- organisation that are partly inspired by physical and biological phenomena. As an application, we envision sensor networks or any type of de- centralised system.Nous nous intéressons au problème du diagnostic de défaillances dans un réseau distribué. Lorsque les composants du réseau sont suscep-tibles de tomber en panne, comment détecter le moment où le taux de composants défaillants dépasse un certain seuil sans faire appel à une autorité centrale ? Notre objectif est d'avoir une estimation de l'état général du réseau par le seul biais d'interactions locales des composants avec leurs voisins. En particulier, nous souhai-tons qu'un consensus émerge sous forme d'état d'alerte lorsque le taux de défaillance dépasse un certain seuil. Nous utilisons le modèle des automates cellulaires pour proposer des solutions dans le cas d'un réseau ayant une structure de grille. Nous comparons trois méthodes d'auto-organisation du réseau, en partie inspirées de phénomènes physiques ou biologiques. Comme domaine d'application, nous avons en vue les ré-seaux de capteurs ou tout système fonctionnant de manière décentralisée

Dynamical Phase Transitions in Graph Cellular Automata

Discrete dynamical systems can exhibit complex behaviour from the iterative application of straightforward local rules. A famous example are cellular automata whose global dynamics are notoriously challenging to analyze. To address this, we relax the regular connectivity grid of cellular automata to a random graph, which gives the class of graph cellular automata. Using the dynamical cavity method (DCM) and its backtracking version (BDCM), we show that this relaxation allows us to derive asymptotically exact analytical results on the global dynamics of these systems on sparse random graphs. Concretely, we showcase the results on a specific subclass of graph cellular automata with ``conforming non-conformist'' update rules, which exhibit dynamics akin to opinion formation. Such rules update a node's state according to the majority within their own neighbourhood. In cases where the majority leads only by a small margin over the minority, nodes may exhibit non-conformist behaviour. Instead of following the majority, they either maintain their own state, switch it, or follow the minority. For configurations with different initial biases towards one state we identify sharp dynamical phase transitions in terms of the convergence speed and attractor types. From the perspective of opinion dynamics this answers when consensus will emerge and when two opinions coexist almost indefinitely.Comment: 15 page

Cellular automata for the self-stabilisation of colourings and tilings

International audienceWe examine the problem of self-stabilisation, as introduced by Dijkstra in the 1970's, in the context of cellular automata stabilising on k-colourings, that is, on infinite grids which are coloured with k distinct colours in such a way that adjacent cells have different colours. Suppose that for whatever reason (e.g., noise, previous usage, tampering by an adversary), the colours of a finite number of cells in a valid k-colouring are modified, thus introducing errors. Is it possible to reset the system into a valid k-colouring with only the help of a local rule? In other words, is there a cellular automaton which, starting from any finite perturbation of a valid k-colouring, would always reach a valid k-colouring in finitely many steps? We discuss the different cases depending on the number of colours, and propose some deterministic and probabilistic rules which solve the problem for k = 3. We also explain why the case k = 3 is more delicate. Finally, we propose some insights on the more general setting of this problem, passing from k-colourings to other tilings (subshifts of finite type)

Self-stabilisation of cellular automata on tilings

Given a finite set of local constraints, we seek a cellular automaton (i.e., a local and uniform algorithm) that self-stabilises on the configurations that satisfy these constraints. More precisely, starting from a finite perturbation of a valid configuration, the cellular automaton must eventually fall back to the space of valid configurations where it remains still. We allow the cellular automaton to use extra symbols, but in that case, the extra symbols can also appear in the initial finite perturbation. For several classes of local constraints (e.g., $k$-colourings with $k\neq 3$, and North-East deterministic constraints), we provide efficient self-stabilising cellular automata with or without additional symbols that wash out finite perturbations in linear or quadratic time, but also show that there are examples of local constraints for which the self-stabilisation problem is inherently hard. We note that the optimal self-stabilisation speed is the same for all local constraints that are isomorphic to one another. We also consider probabilistic cellular automata rules and show that in some cases, the use of randomness simplifies the problem. In the deterministic case, we show that if finite perturbations are corrected in linear time, then the cellular automaton self-stabilises even starting from a random perturbation of a valid configuration, that is, when errors in the initial configuration occur independently with a sufficiently low density.Comment: 43 pages, 28 figure

Coalescence in fully asynchronous elementary cellular automata

International audienceCellular automata (CA) are discrete mathematical systems formed by a set of cells arranged in a regular fashion. Each of these cells is in a particular state and evolves according to a local rule depending on the state of the cells in its neighbourhood. In spite of their apparent simplicity, these dynamical systems are able to display a complex emerging behaviour, and the macroscopic structures they produce are not always predictable despite complete local knowledge. While studying the robustness of CA to the introduction of asynchronism in their updating scheme, a phenomenon called coalescence was observed for the first time: for some asynchronous CA, the application of the same local rule on any two di↵erent initial conditions following the same sequence of updates quickly led to the same non-trivial configuration. Afterwards, it was experimentally found that some CA would always coalesce whilst others would never coalesce, and that some of them exhibit a phase transition between a coalescing and non-coalescing behaviour. However, a formal explanation of non-trivial rapid coalescence has yet to be found, and this is the purpose of this project, where we try to characterise and explain this phenomenon both qualitatively and analytically. In particular, we analytically study trivial coalescence, find lower bounds for the coalescence time of ECA 154 and ECA 62, and give some first steps towards finding their upper bounds in order to prove that they have, respectively, quadratic and linear coalescence time