  Random instances of feedforward Boolean circuits are studied both
analytically and numerically. Evaluating these circuits is known to be a
P-complete problem and thus, in the worst case, believed to be impossible to
perform, even given a massively parallel computer, in time much less than the
depth of the circuit. Nonetheless, it is found that for some ensembles of
random circuits, saturation to a fixed truth value occurs rapidly so that
evaluation of the circuit can be accomplished in much less parallel time than
the depth of the circuit. For other ensembles saturation does not occur and
circuit evaluation is apparently hard. In particular, for some random circuits
composed of connectives with five or more inputs, the number of true outputs at
each level is a chaotic sequence. Finally, while the average case complexity
depends on the choice of ensemble, it is shown that for all ensembles it is
possible to simultaneously construct a typical circuit together with its
solution in polylogarithmic parallel time.
