Edward J. Powley
Global Properties of Cellular Automata
PhD thesis, University of York, 2009


A cellular automaton (CA) is a discrete dynamical system, composed of a large number of simple, identical, uniformly interconnected components. CAs were introduced by John von Neumann in the 1950s, and have since been studied extensively both as models of real-world systems and in their own right as abstract mathematical and computational systems. CAs can exhibit emergent behaviour of varying types, including universal computation. As is often the case with emergent behaviour, predicting the behaviour from the specification of the system is a nontrivial task. This thesis explores some properties of CAs, and studies the correlations between these properties and the qualitative behaviour of the CA. The properties studied in this thesis are properties of the global state space of the CA as a dynamical system. These include degree of symmetry, numbers of preimages (convergence of trajectories), and distances between successive states on trajectories. While we do not obtain a complete classification of CAs according to their qualitative behaviour, we argue that these types of global properties are a better indicator than other, more local, properties.

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