Reduction to a One-Parameter Model

This model can be simplified by reducing it to a single parameter. Vichniac, et.al. [vich86b] for example, consider the behavior along the line p = 1 — Pi- They point out that this reduction affords pi and p2 a simple interpretation in terms of the familiar Boolean functions AND and XOR (exclusive OR). In particular, for the line p2 = 1 — pi, we can equate pi = p(XOR) and P2 = 1 — Pi = p(AND), so that 

Note that since both AND and XOR yield zero whenever ai-i and Ti+i are zero, po is equal to zero, which is consistent with the definition of the full model (equation 7.69). 

We might mention here in passing that while class-1, class-3 and class-4 (but not class-2) can all be obtained from one another with these two simple rules, class-2 behavior can only be obtained if the system is first quenched [vich86b]. That is, if the lattice is initially randomly populated with AND and XOR rules according to the prescribed value of p and is then frozen for all later times. Such quenched random rules harbor some interesting properties of their own in dimensions d 1, and are the basis of much of Kauffman s findings on random Boolean networks (see section 8.6). 

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