An Example Elementary Rule

Let us first of all consider the simple additive rule R90, defined by 

In our algebraic formalism, the time evolution of this system is represented by multiplying the generating function by the dipolynomial 

An initial configuration consisting of single nonzero site therefore evolves after t iterations to a configuration given by 

As we shall immediately, this algebraic formalism is deceptively deep a great deal can be deduced from first principles alone. For example, we can easily prove the following characterization of Garden-ot -Eden configurations  

Lemma 1 Configurations containing an odd number of sites with value 1 can never occur in the evolution of the CA evolving according to additive rule R90 (equation 5.34) they can occur only as initial states. 

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