Related papers: Cellular Automata and Bootstrap Percolation

In this paper we study monotone cellular automata in $d$ dimensions. We develop a general method for bounding the growth of the infected set when the initial configuration is chosen randomly, and then use this method to prove a lower bound…

We consider the typical asymptotic behaviour of cellular automata of higher dimension (greater than 2). That is, we take an initial configuration at random according to a Bernoulli (i.i.d) probability measure, iterate some cellular…

If A=Z/2, then A^Z is a compact abelian group. A `linear cellular automaton' is a shift-commuting endomorphism F of A^Z. If P is a probability measure on A^Z, then F `asymptotically randomizes' P if F^j P converges to the Haar measure as…

It is well-known that the spacetime diagrams of some cellular automata have a fractal structure: for instance Pascal's triangle modulo 2 generates a Sierpinski triangle. It has been shown that such patterns can occur when the alphabet is…

Let us consider the simplest model of one-dimensional probabilistic cellular automata (PCA). The cells are indexed by the integers, the alphabet is {0, 1}, and all the cells evolve synchronously. The new content of a cell is randomly…

We study discrete dynamical systems through the topological concepts of limit set, which consists of all points that can be reached arbitrarily late, and asymptotic set, which consists of all adhering values of orbits. In particular, we…

The density classification task is to determine which of the symbols appearing in an array has the majority. A cellular automaton solving this task is required to converge to a uniform configuration with the majority symbol at each site. It…

We consider two relatively natural topologizations of the set of all cellular automata on a fixed alphabet. The first turns out to be rather pathological, in that the countable space becomes neither first-countable nor sequential. Also,…

Cellular automata are a set of computational models in discrete space that have a discrete time evolution defined by neighbourhood rules. They are used to simulate many complex systems in physics and science in general. In this work,…

We investigate Boolean, totalistic cellular automata with a majority or frustrated majority vote rule, and an interaction range of variable span. These two models show a behavior which differs from the mean-field one. The majority vote…

In this paper we study the dynamics of 1- and 2- dimensional cellular automata, using a 2-adic representation of the states, we give a simple graphical technique for finding periodic solutions. We also study the continuity properties of the…

The properties of two-state nearest-neighbour cellular automata (CA) that are capable of density classification are discussed. It is shown that these CA actually conserve the total density, rather than merely classifying it. This is also…

We study two common types of time-noncontinuous updates for one dimensional stochastic cellular automata with arbitrary nearest neighbor interactions and arbitrary open boundary conditions. We first construct the stationary states using the…

This paper studies three classes of cellular automata from a computational point of view: freezing cellular automata where the state of a cell can only decrease according to some order on states, cellular automata where each cell only makes…

Stochastic processes govern the time evolution of a huge variety of realistic systems throughout the sciences. A minimal description of noisy many-particle systems within a Markovian picture and with a notion of spatial dimension is given…

How do cellular automata behave in the limit of a very large number of cells? Is there a continuum limit with simple properties? We attack this problem by mapping certain classes of automata to quantum field theories for which powerful…

We explore some aspects of phase transitions in cellular automata. We start recalling the standard formulation of statistical mechanics of discrete systems (Ising model), illustrating the Monte Carlo approach as Markov chains and stochastic…

This tutorial is about cellular automata that exhibit 'cold dynamics'. By this we mean zero entropy, stabilization of all orbits, trivial asymptotic dynamics, etc. These are purely transient irreversible dynamics, but they capture many…

Cellular automata are fully-discrete, spatially-extended dynamical systems that evolve by simultaneously applying a local update function. Despite their simplicity, the induced global dynamic produces a stunning array of richly-structured,…

This paper explores the algebraic conditions under which a cellular automaton with a non-linear local rule exhibits surjectivity and reversibility. We also analyze the role of permutivity as a key factor influencing these properties and…