Toffoli's own scheme [21] increases the dimensionality of the automaton rather than the number of generations spanned by its evolutionary rule. However, the simplest solution to the problem of exhibiting explicitly reversible rules of evolution may well lie in increasing the number of states of the automaton, by forming their cartesian products, whose rule of evolution needs to span but a single generation:
Thus a single cell of a 
automaton may hold two successive
generations of a 
automaton by forming pairs whose components
are to be interpreted as
substitution of these values into Eq. 3 reproduces Eq. 1, in its turn reversible by the rule
Similar formulas would allow the formation of reversible 
rules
from arbitrary 
rules, the more so by taking advantage of
additional alternatives for 
. Nor is it difficult to invent schemes
whereby the equivalent of three generations of evolution could be
incorporated into 
automata, and so on and on for even longer
histories.
To understand possible complications which can arise, consider
Fredkin's idea applied to a 
automaton via a rule in which
sequential letters replace subscript indices for improved legibility:
Evolution loses the value of c, preserves d intact, yet permits the
recovery of a if b were known explicitly. Supposing a third cell
provides two cells in the second generation,
from which c can now be recovered via
with the aid of the explicitly
known d and f. Since d is already known, the cell 
has
been reconstructed, as could be done for all the rest of the cells of a
configuration.
In each case evolution ``remembers'' the second component of the right cell, but the left ancestor is the one recovered by the reversed rule. Typical of neighborhoods with half integral radius, this asymmetry seems to have no further significance. It is often seen in rules with shifting.
