Cellular automaton supercolliders

Gliders in one-dimensional cellular automata are compact groups of non-quiescent and non-ether patterns (ether represents a periodic background) translating along automaton lattice. They are cellular-automaton analogous of localizations or quasi-local collective excitations travelling in a spatially extended non-linear medium. They can be considered as binary strings or symbols travelling along a one-dimensional ring, interacting with each other and changing their states, or symbolic values, as a result of interactions. We analyse what types of interaction occur between gliders travelling on a cellular automaton `cyclotron' and build a catalog of the most common reactions. We demonstrate that collisions between gliders emulate the basic types of interaction that occur between localizations in non-linear media: fusion, elastic collision, and soliton-like collision. Computational outcomes of a swarm of gliders circling on a one-dimensional torus are analysed via implementation of cyclic tag systems

From quantum cellular automata to quantum lattice gases

A natural architecture for nanoscale quantum computation is that of a quantum cellular automaton. Motivated by this observation, in this paper we begin an investigation of exactly unitary cellular automata. After proving that there can be no nontrivial, homogeneous, local, unitary, scalar cellular automaton in one dimension, we weaken the homogeneity condition and show that there are nontrivial, exactly unitary, partitioning cellular automata. We find a one parameter family of evolution rules which are best interpreted as those for a one particle quantum automaton. This model is naturally reformulated as a two component cellular automaton which we demonstrate to limit to the Dirac equation. We describe two generalizations of this automaton, the second of which, to multiple interacting particles, is the correct definition of a quantum lattice gas.Comment: 22 pages, plain TeX, 9 PostScript figures included with epsf.tex (ignore the under/overfull \vbox error messages); minor typographical corrections and journal reference adde

Growth of surfaces generated by a probabilistic cellular automaton

A one-dimensional cellular automaton with a probabilistic evolution rule can generate stochastic surface growth in $(1 + 1)$ dimensions. Two such discrete models of surface growth are constructed from a probabilistic cellular automaton which is known to show a transition from a active phase to a absorbing phase at a critical probability associated with two particular components of the evolution rule. In one of these models, called model $A$ in this paper, the surface growth is defined in terms of the evolving front of the cellular automaton on the space-time plane. In the other model, called model $B$, surface growth takes place by a solid-on-solid deposition process controlled by the cellular automaton configurations that appear in successive time-steps. Both the models show a depinning transition at the critical point of the generating cellular automaton. In addition, model $B$ shows a kinetic roughening transition at this point. The characteristics of the surface width in these models are derived by scaling arguments from the critical properties of the generating cellular automaton and by Monte Carlo simulations.Comment: 17 pages text (LaTeX) + 14 figures (PostScript). To appear in Int. J. Mod. Phys. C (1999

One Dimensional nnary Density Classification Using Two Cellular Automaton Rules

Suppose each site on a one-dimensional chain with periodic boundary condition may take on any one of the states $0,1,..., n-1$, can you find out the most frequently occurring state using cellular automaton? Here, we prove that while the above density classification task cannot be resolved by a single cellular automaton, this task can be performed efficiently by applying two cellular automaton rules in succession.Comment: Revtex, 4 pages, uses amsfont

Cellular automaton supercomputing

Many of the models now used in science and engineering are over a century old. And most of them can be implemented on modern digital computers only with considerable difficulty. Some new basic models are discussed which are much more directly suitable for digital computer simulation. The fundamental principle is that the models considered herein are as suitable as possible for implementation on digital computers. It is then a matter of scientific analysis to determine whether such models can reproduce the behavior seen in physical and other systems. Such analysis was carried out in several cases, and the results are very encouraging

On the decomposition of stochastic cellular automata

In this paper we present two interesting properties of stochastic cellular automata that can be helpful in analyzing the dynamical behavior of such automata. The first property allows for calculating cell-wise probability distributions over the state set of a stochastic cellular automaton, i.e. images that show the average state of each cell during the evolution of the stochastic cellular automaton. The second property shows that stochastic cellular automata are equivalent to so-called stochastic mixtures of deterministic cellular automata. Based on this property, any stochastic cellular automaton can be decomposed into a set of deterministic cellular automata, each of which contributes to the behavior of the stochastic cellular automaton.Comment: Submitted to Journal of Computation Science, Special Issue on Cellular Automata Application

Damaging 2D Quantum Gravity

We investigate numerically the behaviour of damage spreading in a Kauffman cellular automaton with quenched rules on a dynamical $\phi^3$ graph, which is equivalent to coupling the model to discretized 2D gravity. The model is interesting from the cellular automaton point of view as it lies midway between a fully quenched automaton with fixed rules and fixed connectivity and a (soluble) fully annealed automaton with varying rules and varying connectivity. In addition, we simulate the automaton on a fixed $\phi^3$ graph coming from a 2D gravity simulation as a means of exploring the graph geometry.Comment: 6 pages, COLO-HEP-332;LPTHE-Orsay-93-5

On the reversibility and the closed image property of linear cellular automata

When $G$ is an arbitrary group and $V$ is a finite-dimensional vector space, it is known that every bijective linear cellular automaton $\tau \colon V^G \to V^G$ is reversible and that the image of every linear cellular automaton $\tau \colon V^G \to V^G$ is closed in $V^G$ for the prodiscrete topology. In this paper, we present a new proof of these two results which is based on the Mittag-Leffler lemma for projective sequences of sets. We also show that if $G$ is a non-periodic group and $V$ is an infinite-dimensional vector space, then there exist a linear cellular automaton $\tau_1 \colon V^G \to V^G$ which is bijective but not reversible and a linear cellular automaton $\tau_2 \colon V^G \to V^G$ whose image is not closed in $V^G$ for the prodiscrete topology