Three Variable Multiple Output Tree Circuits

This article treats the tree circuit synthesis problem for families F = {f1, f2, . . ., fm} of Boolean functions fj of the same three variables. In addition to the development of criteria for determining the most economical of the three possible tree circuit decompositions: fj(X3, X2, X1) = F3j,(G3j(X2, X1), H3j(X2, X1), X3) fj(x3, X2, X1) = F2j(G2j(x3, x1), H2j(x3, X1), X2) fj(X3, X2, X1) = F1j(G1j(X3, X2), H1j(X3, X2), X1) (j = 1, 2, ..., m) of a given family, certain upper bounds BT(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)¿BT(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.