Parallel Complexity of Random Boolean Circuits

TL;DR

This paper investigates the parallel complexity of random Boolean circuits, revealing that some ensembles saturate quickly allowing fast evaluation, while others remain hard, with implications for understanding computational difficulty in parallel processing.

Contribution

It demonstrates that for certain random Boolean circuit ensembles, evaluation can be done in polylogarithmic parallel time, contrasting with the P-complete worst-case complexity.

Findings

Some ensembles saturate rapidly, enabling quick evaluation.

Circuits with five or more inputs exhibit chaotic output sequences.

Typical circuits and solutions can be constructed in polylogarithmic parallel time.

Abstract

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.