One-dimensional majority rules

Consider the one-dimensional majority rule defined on a radius-r neighborhood  

We recognize that 0id majority merely a special case of the generalized threshold rule defined in equation 5.121. In particulm-, majority may be defined using that more general definition by setting 

Since theorem 5 applies to this system, we immediately conclude that majority can yield only either fixed points or cycles of period two. It turns out that one can actually prove a stronger result for this particular rule. Let a finite state be any state consisting of a finite number of nonzero sites. Then we have the following theorem. 

Theorem 7 ([tch82], [goles84], [golesQO]) All finite initiai states evolve, under the map majority) to a fixed point in a finite number of steps. 

Proof The proof consists of two parts. The first part, which we leave to the reader to show, is to observe that if a state f is a fixed point of majority then it must be of the following form  

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A bound on the transient length of the one-dimensional majority rule (equation 5.142) follows in straightforward fashion from our more general results of the previous section. 

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