Abstract:
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 niterations
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.
