Response curves of deterministic and probabilistic cellular automata in one and two dimensions

One of the most important problems in the theory of cellular automata (CA) is determining the proportion of cells in a specific state after a given number of time iterations. We approach this problem using patterns in preimage sets- that is, the set of blocks which iterate to the desired output. This allows us to construct a response curve- a relationship between the proportion of cells in state 1 after n-iterations as a function of the initial proportion. We derive response curve formulae for many two-dimensional deterministic CA rules with L-neighbourhood. For all remaining rules, we find experimental response curves. We also use preimage sets to classify surjective rules. In the last part of the thesis, we consider a special class of one-dimensional probabilistic CA rules. We find response surface formula for these rules and experimental response surfaces for all remaining rules. Acknowledgements I would like to sincerely thank my supervisor, Dr. Henryk FuM, for his support and advice throughout the development of this thesis. This work would not have been possible without his enthusiasm and guidance. It has been a pleasure and privilege to work with him during this process. I knew very little of cellular automata theory prior to beginning my studies, but his love of this topic is contagious and I have thoroughly enjoyed studying in this area of mathematics. lowe him an immense debt of gratitude. I also owe thanks to Dr. Babak Farzad, Dr. Ron Kerman, Dr. Joffre Mercie