Parallel complexity of random Boolean circuits

Abstract

Random instances of feedforward Boolean circuits are studied bothanalytically and numerically. Evaluating these circuits is known to be aP-complete problem and thus, in the worst case, believed to be impossible to perform, evengiven a massively parallel computer, in a time much less than the depth of the circuit.Nonetheless, it is found that, for some ensembles of random circuits, saturation to a fixedtruth value occurs rapidly so that evaluation of the circuit can be accomplished in muchless parallel time than the depth of the circuit. For other ensembles saturation does notoccur and circuit evaluation is apparently hard. In particular, for some random circuitscomposed of connectives with five or more inputs, the number of true outputs at each levelis a chaotic sequence. Finally, while the average case complexity depends on the choiceof ensemble, it is shown that for all ensembles it is possible to simultaneouslyconstruct a typical circuit together with its solution in polylogarithmic paralleltime.