Remarks on the Cellular Automaton Global Synchronisation Problem

Part 2: Regular PapersInternational audienceThe 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

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)

On the Synchronisation Problem over Cellular Automata

Cellular automata are a discrete, synchronous, and uniform dynamical system that give rise to a wide range of dynamical behaviours. In this paper, we investigate whether this system can achieve synchronisation. We study the cases of classical bi-infinite configurations, periodic configurations, and periodic configurations of prime period. In the two former cases, we prove that only a "degenerated" form of synchronisation - there exists a fix-point - is possible. In the latter case, we give an explicit construction of a cellular automaton for which any periodic configuration of prime period eventually converges to cycle of two uniform configurations. Our construction is based upon sophisticated tools: aperiodic NW-deterministic tilings and partitioned intervals

Revisiting the Rice Theorem of Cellular Automata

A cellular automaton is a parallel synchronous computing model, which consists in a juxtaposition of finite automata whose state evolves according to that of their neighbors. It induces a dynamical system on the set of configurations, i.e. the infinite sequences of cell states. The limit set of the cellular automaton is the set of configurations which can be reached arbitrarily late in the evolution. In this paper, we prove that all properties of limit sets of cellular automata with binary-state cells are undecidable, except surjectivity. This is a refinement of the classical "Rice Theorem" that Kari proved on cellular automata with arbitrary state sets.Comment: 12 pages conference STACS'1

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

Models for the modern power grid

This article reviews different kinds of models for the electric power grid that can be used to understand the modern power system, the smart grid. From the physical network to abstract energy markets, we identify in the literature different aspects that co-determine the spatio-temporal multilayer dynamics of power system. We start our review by showing how the generation, transmission and distribution characteristics of the traditional power grids are already subject to complex behaviour appearing as a result of the the interplay between dynamics of the nodes and topology, namely synchronisation and cascade effects. When dealing with smart grids, the system complexity increases even more: on top of the physical network of power lines and controllable sources of electricity, the modernisation brings information networks, renewable intermittent generation, market liberalisation, prosumers, among other aspects. In this case, we forecast a dynamical co-evolution of the smart grid and other kind of networked systems that cannot be understood isolated. This review compiles recent results that model electric power grids as complex systems, going beyond pure technological aspects. From this perspective, we then indicate possible ways to incorporate the diverse co-evolving systems into the smart grid model using, for example, network theory and multi-agent simulation.Comment: Submitted to EPJ-ST Power Grids, May 201

Cellular Automata on Group Sets

We introduce and study cellular automata whose cell spaces are left-homogeneous spaces. Examples of left-homogeneous spaces are spheres, Euclidean spaces, as well as hyperbolic spaces acted on by isometries; uniform tilings acted on by symmetries; vertex-transitive graphs, in particular, Cayley graphs, acted on by automorphisms; groups acting on themselves by multiplication; and integer lattices acted on by translations. For such automata and spaces, we prove, in particular, generalisations of topological and uniform variants of the Curtis-Hedlund-Lyndon theorem, of the Tarski-F{\o}lner theorem, and of the Garden-of-Eden theorem on the full shift and certain subshifts. Moreover, we introduce signal machines that can handle accumulations of events and using such machines we present a time-optimal quasi-solution of the firing mob synchronisation problem on finite and connected graphs.Comment: This is my doctoral dissertation. It consists of extended versions of the articles arXiv:1603.07271 [math.GR], arXiv:1603.06460 [math.GR], arXiv:1603.07272 [math.GR], arXiv:1701.02108 [math.GR], arXiv:1706.05827 [math.GR], and arXiv:1706.05893 [cs.FL

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

Numerical Study of Random Corrosion Characteristics of Metal Based On the Cellular Automata Method

In the production process of coal chemical companies, the corrosion of metal equipment and the resulting shortening of its service life can cause safety hazards. Simulation modeling of pit emergence and development during corrosion evolution provides a new approach to corrosion research. By analyzing the effect of different parameters on causing corrosion to occur, it is possible to reflect the influence of complex physico-chemical systems. In this paper, the simulation of a meta-cellular automaton model of pit growth under diffusion and the introduction of a passivation probability to correct the chemical reaction rate are developed; The effect of reaction passivation probability, chemical reaction rate and diffusion coefficient on the degree of corrosion was also analyzed by means of quantitative analysis. The results show that for metal corrosion loss processes, the degree of corrosion damage decreases with increasing probability of reactive passivation and increases with increasing chemical reaction rate, increasing electrolyte concentration and increasing time step. The CA model was applied to simulate the growth and change of pitting corrosion of metal materials with their corrosion protection layer under damaged conditions. The corrosion model can simulate the corrosion morphology change characteristics similar to the real metal to the corrosion pit evolution simulation related research has certain scientific, validity, reference

Numerical Study of Random Corrosion Characteristics of Metal Based On the Cellular Automata Method

In the production process of coal chemical companies, the corrosion of metal equipment and the resulting shortening of its service life can cause safety hazards. Simulation modeling of pit emergence and development during corrosion evolution provides a new approach to corrosion research. By analyzing the effect of different parameters on causing corrosion to occur, it is possible to reflect the influence of complex physico-chemical systems. In this paper, the simulation of a meta-cellular automaton model of pit growth under diffusion and the introduction of a passivation probability to correct the chemical reaction rate are developed; The effect of reaction passivation probability, chemical reaction rate and diffusion coefficient on the degree of corrosion was also analyzed by means of quantitative analysis. The results show that for metal corrosion loss processes, the degree of corrosion damage decreases with increasing probability of reactive passivation and increases with increasing chemical reaction rate, increasing electrolyte concentration and increasing time step. The CA model was applied to simulate the growth and change of pitting corrosion of metal materials with their corrosion protection layer under damaged conditions. The corrosion model can simulate the corrosion morphology change characteristics similar to the real metal to the corrosion pit evolution simulation related research has certain scientific, validity, reference