Cellular automata and dynamical systems

Abstract

In this thesis we investigate the theoretical nature of the mathematical structures termed cellular automata. Chapter 1: Reviews the origin and history of cellular automata in order to place the current work into context. Chapter 2: Develops a cellular automata framework which contains the main aspects of cellular automata structure which have appeared in the literature. We present a scheme for specifying the cellular automata rules for this general model and present six examples of cellular automata within the model. Chapter 3: Here we develop a statistical mechanical model of cellular automata behaviour. We consider the relationship between variations within the model and their relationship to dynamical systems. We obtain results on the variance of the state changes, scaling of the cellular automata lattice, the equivalence of noise, spatial mixing of the lattice states and entropy, synchronous and asynchronous cellular automata and the equivalence of the rule probability and the time step of a discrete approximation to a dynamical system. Chapter 4: This contains an empirical comparison of cellular automata within our general framework and the statistical mechanical model. We obtain results on the transition from limit cycle to limit point behaviour as the rule probabilities are decreased. We also discuss failures of the statistical mechanical model due to failure of the assumptions behind it. Chapter 5: Here a practical application of the preceding work to population genetics is presented. We study this in the context of some established population models and show it may be most useful in the field of epidemiology. Further generalisations of the statistical mechanical and cellular automata models allow the modelling of more complex population models and mobile populations of organisms. Chapter 6: Reviews the results obtained in the context of the open questions introduced in Chapter 1. We also consider further questions this work raises and make some general comments on how these may apply to related fields