Cyclic Cellular Automata Research Articles

Cyclic cellular automata and Greenberg–Hastings models on regular trees

Cyclic cellular automata and Greenberg–Hastings models on regular trees

Virtual cyclic cellular automata, finite group actions and recursive properties

The aim of these notes is three fold. First we introduce virtual cyclic cellular automata and show that the inverse of a reversible (2R+1)-cyclic cellular automaton with periodic boundary conditions is a virtual cyclic cellular automaton. These virtual automata have two special characteristics: they have active and non active cells at specific steps of times and they reflect certain periodicity. We will relate these particularities with a finite cyclic group action on the cellular automaton, and prove that the inverse transition dipolynomial is an invariant dipolynomial under this action. Secondly, we use a recursive estimation of neighbours (REN) algorithm to produce direct examples of virtual cyclic cellular automata, which moreover generalize some of the cellular automata used in applications like collective control or traffic patterns. We also propose a new REN algorithm which allows us to reinterprete a (2R+1)-cyclic cellular automata as a recursive sequence originated from the elementary cellular automaton with base rule 150, and which motivate us to introduce a new notion of a recursive Wolfram number for a (2R+1)-cyclic cellular automaton. Finally we show that this recursive Wolfram number can be computed by the new REN algorithm applied to the base rule 150 and its complementary 105 rule.

Efficient simulation execution of cellular automata on GPU

Graphics Processing Units (GPUs) can be used as convenient hardware accelerators to speed up Cellular Automata (CA) simulations, which are employed in many scientific areas. However, an important set of CA have performance constraints due to GPU memory bandwidth. Few studies have fully explored how CA implementations can take advantage of modern GPU architectures, mainly in the case of intensive memory usage. In this paper, we make a thorough study of techniques (stencil computing framework, look-up tables, and packet coding) to efficiently implement CA on GPU, taking into account its detailed architecture. Exhaustive experiments to validate these implementation techniques for a number of significant memory-bounded CA are performed. The CA analysed include the classical Game of Life, a Forest Fire model, a Cyclic cellular automaton, and the WireWorld CA. The experimental results show that implementations using the presented techniques can significantly outperform a baseline standard GPU implementation. The best performance results of all known implementations of memory bounded CA were obtained. Moreover, some of the techniques, like look-up tables or temporal blocking, are indeed relatively easy to implement or to apply when the transition rules are simple. Finally, detailed descriptions and discussions of the indicated techniques are included, which may be useful to practitioners interested in developing high performance simulations in efficient languages based on CA on GPU.

A closed formula for the inverse of a reversible cellular automaton with [formula omitted]-cyclic rule

Reversibility of cellular automata (CA) has been an extensively studied problem from both a theoretical and a practical point of view. It is known when a (2R+1)-cyclic cellular automaton with periodic boundary conditions (p.b.c.) is reversible (see Siap et al., 2013) but, as far as we know, no explicit expression is given for its inverse cellular automaton apart from the case R=1 (see Encinas and del Rey, 2007). In this paper we give a closed formula for the inverse rule of a reversible (2R+1)-cyclic cellular automaton with p.b.c. over the finite field F2 for any value of the neighbourhood radius R. It turns out that the inverse of a reversible (2R+1)-cyclic CA with p.b.c. is again a cyclic CA with p.b.c., but with a different neighbourhood radius, and this radius depends on certain numbers which need to be computed by a new algorithm we introduce. Finally, we apply our results to the case R=1 (which is the ECA with Wolfram rule number 150) to introduce an alternative and improved expression for the inverse transition dipolynomial formulated in Encinas and del Rey (2007). We also illustrate these results by giving explicit computations for the inverse transition dipolynomial of a reversible cellular automaton with penta-cyclic rule.

Asymptotic behaviour of the one-dimensional “rock–paper–scissors” cyclic cellular automaton

The one-dimensional three-state cyclic cellular automaton is a simple spatial model with three states in a cyclic “rock–paper–scissors” prey–predator relationship. Starting from a random configuration, similar states gather in increasingly large clusters; asymptotically, any finite region is filled with a uniform state that is, after some time, driven out by its predator, each state taking its turn in dominating the region (heteroclinic cycles). We consider the situation where each site in the initial configuration is chosen independently at random with a different probability for each state. We prove that the asymptotic probability that a state dominates a finite region corresponds to the initial probability of its prey. The proof methods are based on discrete probability tools, mainly particle systems and random walks.

Persistence of sums of correlated increments and clustering in cellular automata

Let T be the first return time to (−∞,0] of sums of increments given by a functional of a stationary Markov chain. We determine the asymptotic behavior of the survival probability, P(T≥t)∼Ct−1∕2 for an explicit constant C. Our analysis is based on a connection between the survival probability and the running maximum of the time-reversed process, and relies on a functional central limit theorem for Markov chains. As applications, we recover known clustering results for the 3-color cyclic cellular automaton and the Greenberg–Hastings model, and we prove a new clustering result for the 3-color firefly cellular automaton.

2D photonic crystal complete band gap search using a cyclic cellular automaton refination

We present a refination method based on a cyclic cellular automaton (CCA) that simulates a crystallization-like process, aided with a heuristic evolutionary method called differential evolution (DE) used to perform an ordered search of full photonic band gaps (FPBGs) in a 2D photonic crystal (PC). The solution is proposed as a combinatorial optimization of the elements in a binary array. These elements represent the existence or absence of a dielectric material surrounded by air, thus representing a general geometry whose search space is defined by the number of elements in such array. A block-iterative frequency-domain method was used to compute the FPBGs on a PC, when present. DE has proved to be useful in combinatorial problems and we also present an implementation feature that takes advantage of the periodic nature of PCs to enhance the convergence of this algorithm. Finally, we used this methodology to find a PC structure with a 19% bandgap-to-midgap ratio without requiring previous information of suboptimal configurations and we made a statistical study of how it is affected by disorder in the borders of the structure compared with a previous work that uses a genetic algorithm.

Cyclic cellular automata in 3D

Cyclic cellular automata in two dimensions have long been intriguing because they self organize into spirals and that behavior can be analyzed. The form for the patterns that develop is highly dependent upon the form of the neighborhood. We extend this work to three dimensional cyclic cellular automata and observe self organization dependent upon the neighborhood type. This includes neighborhood types intermediate between Von Neumann and Moore neighborhoods. We also observe that the patterns include nested shells with the appropriate forms but that the nesting is far more complex than the spirals that occur in two dimensions.

Medley of spirals from cyclic cellular automata

Cyclic cellular automata on the integer planar lattice are known to typically evolve through distinct phases ending with minimal periodic terminal states that usually appear as intertwined spirals. Here we explore the diversity of spirals that arise from nonstandard neighborhoods on the integer lattice and from looking at the automata on quasi-crystalline arrangements of cells. We see that phase transitions and development of spirals are almost ubiquitous yet the particular form of the spirals is very dependent upon the particulars of the underlying neighborhoods; in fact the spiral forms echo the neighborhoods. The quasi-crystalline illustrations provide much more subtle echoes in the spiral forms that show artifacts from the non-periodic local symmetry that occurs.

Spiral Waves in a Cyclic Cellular Automaton Model for Excitable Media

We present a cyclic cellular automaton for excitable media. The model exhibits the complex dynamics of patterns observed in excitable media. Spatial patterns dynamics provides an important example of spatio-temporal self-organization in distributed active systems. The global behavior of the cellular automaton model is described in terms of point correlation functions. The Fouries analysis pointed out the quasi-oscillatory and chaotic behavior of these functions.

Ballistic annihilation and deterministic surface growth

A model of deterministic surface growth studied by Krug and Spohn, a model of the annihilating reactionA+B→inert studied by Elskens and Frisch, a one-dimensional three-color cyclic cellular automaton studied by Fisch, and a particular automaton that has the number 184 in the classification of Wolfram can be studied via a cellular automaton with stochastic initial data called ballistic annihilation. This automaton is defined by the following rules: At timet=0, one particle is put at each integer point of ℝ. To each particle, a velocity is assigned in such a way that it may be either +1 or −1 with probabilities 1/2, independent of the velocities of the other particles. As time goes on, each particle moves along ℝ at the velocity assigned to it and annihilates when it collides with another particle. In the present paper we compute the distribution of this automaton for each timet ∈ ℕ. We then use this result to obtain the hydrodynamic limit for the surface profile from the model of deterministic surface growth mentioned above. We also show the relation of this limit process to the process which we call moving local minimum of Brownian motion. The latter is the processB min ,x ∈ ℝ, defined byB min ≔min{B y ;x−1≤y≤x+1} for everyx ∈ ℝ, whereB x ,x ∈ ℝ, is the standard Brownian motion withB 0=0.

Two applications of percolation to cellular automata

The point of this paper is to show how ideas from percolation can be used to study the asymptotic behavior of some cellular automata systems. In particular, using these ideas, we prove that the Greenberg-Hastings and cyclic cellular automata models with three colors, threshold 2, and theL ∞ neighborhood are uniformly asymptotically locally periodic ind≥2 dimensions. We also show that every lattice point is eventually “controlled by a finite clock” in the standard Greenberg-Hastings and cyclic cellular automata models in two dimensions, which is a stronger description than the already known asymptotic behavior.

Rotators, periodicity, and absence of diffusion in cyclic cellular automata

Cyclic cellular automata are two-dimensional cellular automata which generalize lattice versions of the Lorentz gas and certain biochemistry models of artificial life. We show that rotators and time reversibility play a special role in the creation of closed orbits in cyclic cellular automata. We also prove that almost every orbit is closed (periodic) and the absence of diffusion for the flipping rotator model (also known as the ant).

Asymptotic Behavior of Excitable Cellular Automata

We study two families of excitable cellular automata known as the Greenberg-Hastings model and the cyclic cellular automaton. Each family consists of focal deterministic oscillating lattice dynamics, with parallel discrete-time updating, parametrized by the range ρ of interaction, l p shape of its neighbor set, threshold θ for contact updating, and number K. of possible states per site. These models are mathematically tractable prototypes for the spatially distributed periodic wave activity of so-called excitable media observed in diverse disciplines of experimental science. Fisch, Gravner and Griffeath [Fisch et al. 1991] studied experimentally the ergodic behavior of these models on Z 2, started from random initial states. Among other phenomena, they noted the emergence of asymptotic phase diagrams (and dynamics on R 2) in the threshold-range scaling limit as ρ, θ → ∊ with θ/ρ2 constant. Mere we present several rigorous results and some experimental findings concerning various phase transitions in the asymptotic diagrams. Our efforts focus on evaluating bend(p), the limiting threshold cutoff for existence of the spirals that characterize many excitable media. Our main results are formulated in terms of spo(p), the cutoff for existence of stable periodic objects that arise asspiral cores. Somesubtle consequences of anisotropic neighbor sets (p ≠ 2) are also discussed; the case of box neighborhoods (p = ∊) is examined in detail.

Clustering in the One-Dimensional Three-Color Cyclic Cellular Automaton

cyclic cellular automaton. The author has previously shown that this process fluctuates, meaning that each lattice site changes color infinitely often, so that there is no final state for the system. The focus of the current work is on the clustering properties of this system. This paper demonstrates that the one-dimensional three-color cyclic cellular automaton clusters, and the mean cluster size, as a function of time t, is asymptotic to ct1/2, where c is an explicitly calculable constant. The method of proof also allows us to compute asymptotic estimates of the mean interparticle distance for a one-dimensional system of particles which undergo deterministic motion and which annihilate upon collision. No clustering results are known about the four-color process, but evidence is presented to suggest that the mean cluster size of such systems grows at a rate different from t1/2.

Cyclic cellular automata and related processes

A cyclic cellular automaton is a discrete-time process defined on the state space {0, 1, …, N −1} Z d . Starting from a random initial condition, at each time step, each site looks at its nearest neighbors. If the state at site x is i, then site x will change state to i + 1 (mod N) if it locates state i+1 (mod N) among its neighbors; otherwise, site x will remain in state i. Cyclic cellular automata have properties akin to cyclic particle systems, introduced by Bramson and Griffeath, which are continuous-time Markov processes on {0, 1, …, N −1} Z d with a stochastic evolution rather than a deterministic one. In one dimension, some clustering results about cyclic cellular automata are presented; the analogous problems in the context of cyclic particle systems are still unresolved. Also, we discuss the consequences of these clustering results in the context of one-dimensional systems of particles with various interaction mechanisms. Finally, we touch upon the work currently being done on two-dimensional cyclic particle systems and cellular automata.

The one-dimensional cyclic cellular automaton: A system with deterministic dynamics that emulates an interacting particle system with stochastic dynamics

Consider a cellular automaton defined on ℤ where each lattice site may take on one ofN values, referred to as colors. TheN colors are arranged in a cyclic hierarchy, meaning that colork follows colork−1 modN (k=0,...,N−1). Any two colors that are not adjacent in this hierarchy form an inert pair. In this scheme, there is symmetry in theN colors. Initialized the cellular automaton with product measure, and let time pass in discrete units. To get the configuration at timet+1 from the one at timet, each lattice site looks at the colors of its two nearest neighbors, and if it sees the color that follows its own color, then that site changes color to the color that follows; otherwise, that site does not change color. All such updates occur synchronously at timet+1. For each value ofN≥2, the fundamental question is whether each site in the cellular automaton changes color infinitely often (fluctuation) or only finitely often (fixation). We prove here that ifN≤4, then fluctuation occurs, and ifN≥5, then fixation occurs.