Asynchronism Induces Second Order Phase Transitions in Elementary Cellular Automata

Cellular automata are widely used to model natural or artificial systems. Classically they are run with perfect synchrony, i.e., the local rule is applied to each cell at each time step. A possible modification of the updating scheme consists in applying the rule with a fixed probability, called the synchrony rate. For some particular rules, varying the synchrony rate continuously produces a qualitative change in the behaviour of the cellular automaton. We investigate the nature of this change of behaviour using Monte-Carlo simulations. We show that this phenomenon is a second-order phase transition, which we characterise more specifically as belonging to the directed percolation or to the parity conservation universality classes studied in statistical physics

Experimental study of Elementary Cellular Automata dynamics using the density parameter

International audienceClassifying cellular automata in order to capture the notion of chaos algorithmically is a challenging problem than can be tackled in many ways.We here give a classification based on the computation of a macroscopic parameter, the $d$-spectrum, and show how our classifying scheme can be used to separate the chaotic ECA from the non-chaotic ones

Aesthetics and randomness in cellular automata

International audienceWe propose two images obtained with an asynchronous and a stochastic cellular automaton. Deterministic cellular automata are now well-studied models and even if there is still so much to understand, their main properties are now largely explored. By contrast, the universe of asynchronous and stochastic is mainly a terra incognita. Only a few islands of this vast continent have been discovered so far. The two examples below present space-time diagrams of one-dimensional cellular automata with nearest-neighbour interaction. The cells are arranged in a ring, that is, the right neighbour of the rightmost cell is the leftmost cell, and vice versa; in formal words, indices are taken in Z/nZ, where n is the number of cells. The space-time diagrams are obtained with the FiatLux software. Time goes from bottom to top: the successive states of the system are stacked one on the other

Remarks on the cellular automaton global synchronisation problem – deterministic vs. stochastic models

International audienceIn the global synchronisation problem, one is asked to find a cellular automaton which has the property that every initial condition evolves into a homogeneous blinking state. We study this simple inverse problem for the case of one-dimensional systems with periodic boundary conditions. Two paradoxical observations are made: (a) despite the apparent simplicity of finding rules with good statistical results, there exist no perfect deterministic solutions to this problem, (b) if we allow the use of randomness in the local rule, constructing ``perfect" stochastic solutions is easy. For the stochastic case, we give some rules for which the mean time of synchronisation varies quadratically with the number of cells and ask if this result can be improved.To explore more deeply the deterministic rules, we code our problem as a SAT problem and USE SAT solvers to find rules that synchronise a large set of initial conditions (in appendix)

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

Is there something like ''modellability'' ? - Reflections on the robustness of discrete models of complex systems

International audienceExtended abstract of the talk given in Universidad de Concepcion, Chile, Octobre 21st., 2013. Invitation by Pr. Julio Aracen

Does Life resist asynchrony ?

We study Conway's Game of Life with an asynchronous updating

An asynchronous cellular system that solves the parity problem

We present stochastic solutions to the cellular automata parity problem. The model is an interacting particles system where cells are updated by pairs, randomly chosen at each time step. We analyse the convergence properties of two rules and show that they possess the required properties to classify the parity of the initial configurations. We present a formal analysis of the classification time, as well as numerical simulations, to establish that the classification time scales quadratically with the number of cells

Critical phenomena in cellular automata: perturbing the update, the transitions, the topology

International audienceWe survey the effect of perturbing the regular structure of a cellular automaton. We are interested in critical phenomena, i.e., when a continuous variation in the local rules of a cellular automaton triggers a qualitative change of its global behaviour. We focus on three types of perturbations: (a) when the updating is made asynchronous, (b) when the transition rule is made stochastic, (c) when the topological defects are introduced. It is shown that although these perturbations have various effects on CA models, they generally produce the same effects, which are identified as first-order or second-order phase transitions. We present open questions related to this topic and discuss implications on the activity of modelling