Abstract
One-dimensional nearest-neighbor cellular automata defined over Z 2 are characterized in terms of a set of eight non-additive basis operators which act on the automaton state space. Every evolution rule for such automata can be expressed as an operator which is a direct sum of the basis operators. This approach allows decomposition of automata rules into additive and non-additive parts. As a result it is simple to determine fixed points (those states for which the rule reduces to the identity), and shift cycles (sets of states on which the rule reduces to a shift). Sets of states on which any given nearest-neighbor automaton reduces to an identity or a shift are characterized, and maximum tree heights are computed for certain cases. The formalism is used to explore and generalize a relationship recently studied by Ito. We conclude by indicating a connection to the invariance matrix approach to cellular automata which has been developed by Jen.
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