Nazim Fatès

Asynchronous behavior of double-quiescent elementary cellular automata

In this paper we propose a probabilistic analysis of the relaxation time of elementary finite cellular automata (i.e., {0,1} states, radius 1 and unidimensional) for which both states are quiescent (i.e., (0,0,0) ↦ 0 and (1,1,1) ↦ 1), under α-asynchronous dynamics (i.e., each cell is updated at each time step independently with probability 0 < α ≤ 1). This work generalizes previous work in [1], in the sense that we study here a continuous range of asynchronism that goes from full asynchronism to full synchronism. We characterize formally the sensitivity to asynchronism of the relaxation times for 52 of the 64 considered automata. Our work relies on the design of probabilistic tools that enable to predict the global behaviour by counting local configuration patterns. These tools may be of independent interest since they provide a convenient framework to deal exhaustively with the tedious case analysis inherent to this kind of study. The remaining 12 automata (only 5 after symmetries) appear to exhibit interesting complex phenomena (such as polynomial/exponential/infinite phase transitions).

Stochastic Cellular Automata Solutions to the Density Classification Problem

In the density classification problem, a binary cellular automaton (CA) should decide whether an initial configuration contains more 0s or more 1s. The answer is given when all cells of the CA agree on a given state. 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 an increase of the convergence time. We experimentally demonstrate that this rule exhibits good scaling properties and that it attains qualities of classification never reached so far.

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.

Does Life Resist Asynchrony

Undoubtedly, Conway’s Game of Life — or simply Life — is one of the most amazing inventions in the field of cellular automata. Forty years after its discovery, the model still fascinates researchers as if it were an inexhaustible source of puzzles. One of the most intriguing questions is to determine what makes this rule so particular among the quasi-infinite set of rules one can search. In this chapter we analyse how the Game of Life is affected by the presence of two structural pertubations: a change in the synchrony of the updates and a modification of the links between the cells.

Translating Discrete Multi-Agents Systems into Cellular Automata: Application to Diffusion-Limited Aggregation

This paper deals with the synchronous implementation of situated Multi-Agent Systems (MAS) in order to have no execution bias and to ease their programming on massively parallel computing devices. For this purpose we investigate the translation of discrete MAS into Cellular Automata (CA). Contrarily to the sequential scheduling generally used in MAS simulations, CA are a model for massively parallel computing where the updating of the components is synchronous.

Directed percolation phenomena in asynchronous elementary cellular automata

Cellular 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, called 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 cellular automaton 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.

Robustness of cellular automata in the light of asynchronous information transmission

Cellular automata are classically synchronous: all cells are simultaneously updated. However, it has been proved that perturbations in the updating scheme may induce qualitative changes of behaviours. This paper presents a new type of asynchronism, the s-synchronism, where the transmission of information between cells is disrupted randomly. After giving a formal definition, we experimentally study the behaviour of s-synchronous models. We observe that, although many effects are similar to those induced by the perturbation of the update, novel phenomena occur. We study the qualitative variation of behaviour triggered by continuous change of the disruption probability. In particular we observe that phase transitions appear, which belong to the directed percolation universality class.

Remarks on the Cellular Automaton Global Synchronisation Problem

The global synchronisation problem consists in making a cellular automaton converge to a homogeneous blinking state from any initial condition. We here study this inverse problem for one-dimensional binary systems with periodic boundary conditions (i.e., rings). For small neighbourhoods, we present results obtained with the formulation of the problem as a SAT problem and the use of SAT solvers. Our observations suggest that it is not possible to solve this problem perfectly with deterministic systems. In contrast, the problem can easily be solved with stochastic rules.

A Robustness Approach to Study Metastable Behaviours in a Lattice-Gas Model of Swarming

Research in biology is increasingly interested in discrete dynamical systems to simulate natural phenomena with simple models. But how to take into account their robustness? We illustrate this issue by considering the behaviour of a lattice-gas model with an alignment-favouring interaction rule. This model, which has been shown to display a phase transition between an ordered and a disordered phase, follows ergodic dynamics. We present a method based on the study of stability and robustness, and show that the organised phase may result in several different behaviours. We then observe that behaviours are influenced asymptotically by the definition of the cellular lattice.

Robustness of the critical behaviour in a discrete stochastic reaction-diffusion medium

We study the steady states of a reaction-diffusion medium modelled by a stochastic 2D cellular automaton. We consider the Greenberg-Hastings model where noise and topological irregularities of the grid are taken into account. The decrease of the probability of excitation changes qualitatively the behaviour of the system from an “active” to an “extinct” steady state. Simulations show that this change occurs near a critical threshold; it is identified as a nonequilibrium phase transition which belongs to the directed percolation universality class. We test the robustness of the phenomenon by introducing persistent defects in the topology : directed percolation behaviour is conserved. Using experimental and analytical tools, we suggest that the critical threshold varies as the inverse of the average number of neighbours per cell.