Invariants of Fredkin-Reversible Rules

EXAMPLE Consider the iTedJcin-reversible form of the OR rule i.e. the system 

Dropping the constant I, multiplying by Si(t+ l)si-i(t), and using the fact that the square of all spin variables is equal to one (for example, Si t)si t) = -fl), this equation simplifies to 

Pomeau considered rules of the form given in equation 8.6 with j)i = OJdefined by 

Since 4 i ( j ( i+jd) - is equal to zero (whichfollows trivially from the definition 

It is instructive to observe that in terms of conventional spin variables Si = 2ai - 1 (si = 1), and written in slightly more general form, takes on the form of the familiar Ising energy 

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