Robustness of Cellular Automata in the Light of Asynchronous Information Transmission

International audienceCellular 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 beta -synchronism, where cells still update at each time step but where the transmission of information between cells is disrupted randomly. We experimentally study the behaviour of beta-synchronous models. We observe that, although many eff ects are similar to the perturbation of the update, novel phenomena occur. We particularly study phase transitions as an illustration of a qualitative variation of behaviour triggered by continuous change of the disruption probability beta

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

Asynchronous cellular automata

This text has been proposed for the Encyclopedia of Complexity and Systems Science edited by Springer Nature and should appear in 2018.International audienceThis text is intended as an introduction to the topic of asynchronous cellular automata. We start from the simple example of the Game of Life and examine what happens to this model when it is made asynchronous (Sec. 1). We then formulate our definitions and objectives to give a mathematical description of our topic (Sec. 2). Our journey starts with the examination of the shift rule with fully asynchronous updating and from this simple example, we will progressively explore more and more rules and gain insights on the behaviour of the simplest rules (Sec. 3). As we will meet some obstacles in having a full analytical description of the asynchronous behaviour of these rules, we will turn our attention to the descriptions offered by statistical physics, and more specifically to the phase transition phenomena that occur in a wide range of rules (Sec. 4). To finish this journey, we will discuss the various problems linked to the question of asynchrony (Sec. 5) and present some openings for the readers who wish to go further (Sec. 6)

Non-maximal sensitivity to synchronism in periodic elementary cellular automata: exact asymptotic measures

In [11] and [13] the authors showed that elementary cellular automata rules 0, 3, 8, 12, 15, 28, 32, 34, 44, 51, 60, 128, 136, 140, 160, 162, 170, 200 and 204 (and their conjugation, reflection, reflected-conjugation) are not maximum sensitive to synchronism, i.e. they do not have a different dynamics for each (non-equivalent) block-sequential update schedule (defined as ordered partitions of cell positions). In this work we present exact measurements of the sensitivity to synchronism for these rules, as functions of the size. These exhibit a surprising variety of values and associated proof methods, such as the special pairs of rule 128, and the connection to the bissection of Lucas numbers of rule 8

Emerging properties of financial time series in the “Game of Life”

We explore the spatial complexity of Conway’s “Game of Life,” a prototypical cellular automaton by means of a geometrical procedure generating a two-dimensional random walk from a bidimensional lattice with periodical boundaries. The one-dimensional projection of this process is analyzed and it turns out that some of its statistical properties resemble the so-called stylized facts observed in financial time series. The scope and meaning of this result are discussed from the viewpoint of complex systems. In particular, we stress how the supposed peculiarities of financial time series are, often, overrated in their importance

First steps on asynchronous lattice-gas models with an application to a swarming rule

International audienceLattice-gas cellular automata are often considered as a particular case of cellular automata in which additional constraints apply, such as conservation of particles or spatial exclusion. But what about their updating? How to deal with non-perfect synchrony? Novel definitions of asynchronism are proposed that respect the specific hypotheses of lattice-gas models. These definitions are then applied to a swarming rule in order to explore the robustness of the global emergent behaviour. In particular, we compare the synchronous and asynchronous case, and remark that anti-alignment of particles is no longer observed when a small critical amount of asynchronism is added

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)

Reversibility of Elementary Cellular Automata Under Fully Asynchronous Update

International audienceWe investigate the dynamics of Elementary Cellular Automata (ECA) under fully asynchronous update with periodic boundary conditions. We tackle the reversibility issue, that is, we want to determine whether, starting from any initial condition, it is possible to go back to this initial condition with random updates. We present analytical tools that allow us to partition the ECA space into three classes: strongly irreversible, irreversible and recurrent

Attraction Basins as Gauges of Robustness against Boundary Conditions in Biological Complex Systems

One fundamental concept in the context of biological systems on which researches have flourished in the past decade is that of the apparent robustness of these systems, i.e., their ability to resist to perturbations or constraints induced by external or boundary elements such as electromagnetic fields acting on neural networks, micro-RNAs acting on genetic networks and even hormone flows acting both on neural and genetic networks. Recent studies have shown the importance of addressing the question of the environmental robustness of biological networks such as neural and genetic networks. In some cases, external regulatory elements can be given a relevant formal representation by assimilating them to or modeling them by boundary conditions. This article presents a generic mathematical approach to understand the influence of boundary elements on the dynamics of regulation networks, considering their attraction basins as gauges of their robustness. The application of this method on a real genetic regulation network will point out a mathematical explanation of a biological phenomenon which has only been observed experimentally until now, namely the necessity of the presence of gibberellin for the flower of the plant Arabidopsis thaliana to develop normally

Independent Lazy Better-Response Dynamics on Network Games

International audienceWe study an independent best-response dynamics on network games in which the nodes (players) decide to revise their strategies independently with some probability. We provide several bounds on the convergence time to an equilibrium as a function of this probability, the degree of the network, and the potential of the underlying games. These dynamics are somewhat more suitable for distributed environments than the classical better- and best-response dynamics where players revise their strategies "sequentially'", i.e., no two players revise their strategies simultaneously