For any Boolean function f, let L(f)
be its formula size complexity in the basis {∧, [oplus ] 1}.
For every n and every k[les ]n/2, we describe
a probabilistic distribution on formulas in the basis {∧, [oplus ] 1}
in some given set of n variables and of size at most
[lscr ](k)=4k.
Let pn,k(f) be
the probability that the formula chosen from the distribution
computes the function f. For every function f with
L(f)[les ][lscr ](k)α,
where α=log4(3/2), we have
pn,k(f)>0. Moreover,
for every function
f, if pn,k(f)>0,
then
formula here
where c>1 is an absolute constant. Although the
upper and lower bounds are exponentially
small in [lscr ](k), they are quasi-polynomially related whenever
[lscr ](k)[ges ]lnΩ(1)n. The construction
is a step towards developing a model appropriate for investigation of the
properties of a
typical (random) Boolean function of some given complexity.