  In this paper the Neciporuk method for proving lower bounds on the size of
Boolean formulae is reformulated in terms of one-way communication complexity.
We investigate the scenarios of probabilistic formulae, nondeterministic
formulae, and quantum formulae. In all cases we can use results about one-way
communication complexity to prove lower bounds on formula size. In the latter
two cases we newly develop the employed communication complexity bounds. The
main results regarding formula size are as follows: A polynomial size gap
between probabilistic/quantum and deterministic formulae. A near-quadratic size
gap for nondeterministic formulae with limited access to nondeterministic bits.
A near quadratic lower bound on quantum formula size, as well as a polynomial
separation between the sizes of quantum formulae with and without multiple read
random inputs. The methods for quantum and probabilistic formulae employ a
variant of the Neciporuk bound in terms of the VC-dimension. Regarding
communication complexity we give optimal separations between one-way and
two-way protocols in the cases of limited nondeterministic and quantum
communication, and we show that zero-error quantum one-way communication
complexity asymptotically equals deterministic one-way communication complexity
for total functions.
