Three Variable Multiple Output Tree Circuits

This article treats the tree circuit synthesis problem for families F = {f
₁
, f
₂
, . . ., f
_m
} of Boolean functions f
_j
of the same three variables. In addition to the development of criteria for determining the most economical of the three possible tree circuit decompositions: f
_j
(X
₃
, X
₂
, X
₁
) = F
_3j
,(G
_3j
(X
₂
, X
₁
), H
_3j
(X
₂
, X
₁
), X
₃
) f
_j
(x
₃
, X
₂
, X
₁
) = F
_2j
(G
_2j
(x
₃
, x
₁
), H
_2j
(x
₃
, X
₁
), X
₂
) f
_j
(X
₃
, X
₂
, X
₁
) = F
_1j
(G
_1j
(X
₃
, X
₂
), H
_1j
(X
₃
, X
₂
), X
₁
) (j = 1, 2, ..., m) of a given family, certain upper bounds B
_T
(3, m) are obtained on the ``tree circuit cost'' T(F) of such a family; these have the property that regardless of the members of F, the inequality T(F)≤B
_T
(3, m) holds and furthermore, a family F exists which actually attains this upper bound. These upper bounds or estimates are known to have a wide application in switching theory generally, and in particular in the theory of tree circuits and in the decomposition theory of Boolean functions.