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

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

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

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

Stochastic Cellular Automata Solutions to the Density Classification Problem - When randomness helps computing

Extended version of the Stacs 2011 proceedings paper.International audienceIn the density classification problem, a binary cellular automaton should decide whether an initial configuration contains more 0s or 1s. The answer is given when all cells of the CA agree on a given state (0 or 1). This problem is known for having no exact solution in the case of binary deterministic one-dimensional CA. We investigate how randomness in CA may help us solve the problem. We analyse the behaviour of stochastic CA rules that perform the density classification task. We show that describing stochastic rules as a ''blend'' of deterministic rules allows us to derive quantitative results on the classification time and the classification time of previously studied rules. We introduce a new rule whose effect is to spread defects and to wash them out. This stochastic rule solves the problem with an arbitrary precision, that is, its quality of classification can be made arbitrarily high, though at the price of a longer time to converge. We experimentally demonstrate that this rule exhibits good scaling properties and that it attains qualities of classification never reached so far

Directed Percolation Phenomena in Asynchronous Elementary Cellular Automata

8 pagesInternational audienceCellular automata are discrete dynamical systems that are widely used to model natural 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, call the synchrony rate. It has been shown in a previous work that varying the synchrony rate continuously could produce a discontinuity in the behaviour of the CA. This works aims at investigating the nature of this change of behaviour using intensive numerical simulations. We apply a two-step protocol to show that the phenomenon is a phase transition whose critical exponents are in good agreement with the predicted values of directed percolation