Density Classification Quality of the Traffic-majority Rules

The density classification task is a famous problem in the theory of cellular automata. It is unsolvable for deterministic automata, but recently solutions for stochastic cellular automata have been found. One of them is a set of stochastic transition rules depending on a parameter $\eta$, the traffic-majority rules. Here I derive a simplified model for these cellular automata. It is valid for a subset of the initial configurations and uses random walks and generating functions. I compare its prediction with computer simulations and show that it expresses recognition quality and time correctly for a large range of $\eta$ values.Comment: 40 pages, 9 figures. Accepted by the Journal of Cellular Automata. (Some typos corrected; the numbers for theorems, lemmas and definitions have changed with respect to version 1.

Gliders and Ether in Rule 54

This is a study of the one-dimensional elementary cellular automaton rule 54 in the new formalism of "flexible time". We derive algebraic expressions for groups of several cells and their evolution in time. With them we can describe the behaviour of simple periodic patterns like the ether and gliders in an efficient way. We use that to look into their behaviour in detail and find general formulas that characterise them.Comment: 10 pages, 6 figures, 3 tables. Some errors of the printed version are correcte

One-dimensional number-conserving cellular automata

A number-conserving cellular automaton is a simplified model for a system of interacting particles. This paper contains two related constructions by which one can find all one-dimensional number-conserving cellular automata with one kind of particle. The output of both methods is a "flow function", which describes the movement of the particles. In the first method, one puts increasingly stronger restrictions on the particle flow until a single flow function is specified. There are no dead ends, every choice of restriction steps ends with a flow. The second method uses the fact that the flow functions can be ordered and then form a lattice. This method consists of a recipe for the slowest flow that enforces a given minimal particle speed in one given neighbourhood. All other flow functions are then maxima of sets of these flows. Other questions, like that about the nature of non-deterministic number-conserving rules, are treated briefly at the end.Comment: 28 pages, 6 figure

Flexible time and ether in one-dimensional cellular automata

A one-dimensional cellular automaton is an infinite row of identical machines---the cells---which depend for their behaviour only on the states of their direct neighbours.This thesis introduces a new way to think about one-dimensional cellular automata. The formalism of Flexible Time allows one to unify the states of of a finite number of cells into a single object, even if they occur at different times. This gives greater flexibility to handle the structures that occur in the development of a cellular automaton. Flexible Time makes it possible to calculate in an algebraic way the fate of a finite number of cells.In the first part of this thesis the formalism is developed in detail. Then it is applied to a specific problem of one-dimensional cellular automata, namely ether formation. The so-called ether is a periodic pattern of cells that occurs in some cellular automata: It arises from almost all randomly chosen initial configurations, and why this happens is not clear. For one of these cellular automata, the elementary cellular automaton with rule code 54, ether formation is expressed in the formalism of Flexible Time.Then a partial result about ether formation is proved: There is a certain fragment of the ether that arises with probability 1 from every random initial configuration, and it is then propagated with probability 1 to any later time. The persistence of the ether fragment is a strong argument that the ether under Rule 54 indeed arises from almost all input configurations. The result only requires that the states of the cells are chosen independently and with equal probability distributions, and that all cell states can occur. This is not yet a full proof of ether formation, but it is derived by formal means, not just by computer simulations

How Complex is Discourse Structure

• introduction: representations of discourse structure • crucial phenomena – crossed dependencies – multiple-parent structures – a combination of these: potential list structures • conclusion and outlook Markus Egg and Gisela Redeker, LREC 2010