Fast Evaluation of Boolean Circuits Based on Two-Players Game and Optical Connectivity Circuits

In this work we consider the problem of fast parallel evaluation of boolean circuits - namely to evaluate a boolean circuit C, with input leaf values, faster than its depth, which would practically require log depth iterations to complete. Finding a general parallel algorithm that can evaluate any circuit using log depth iterations is known as the Circuit Value Problem (CVP). The CVP and its approximations are known to be P-complete and therefore, a heuristically solution that practically works for all “real-computations” is sought. In this work we propose a new algorithm based on a two players game that can reduce the evaluation time of a boolean circuit C by upto min(h, min(log d, log co - d)) iterations where h is the maximal number of and-or alternations along any path in C and d (and co - d) is the algebraic degree (and co-degree) of C. This improves the theoretical bound of the MRK algorithm (Miller, Ramachandran and Kaltofen 86) for the case of parallel evaluation of boolean circuits. More importantly we show, via experiments, that for circuits emanating from real programs, the proposed algorithm can practically evaluate circuits in log - depth iterations. Each iteration can be evaluated in parallel using a connectivity step, and although it can be implemented using log-depth boolean circuits, we consider an optical switching realization that is based on Optical Ring Resonators (ORR). Due to quantum effects, propagating a light beam through a sequence of ORRs can be done with zero latency, thus making ORRs ideal for implementing the connectivity step required by the proposed algorithm. In order to obtain the needed experiments, we have extended the LLVM compiler to transform C-code into boolean circuits and then simulated the optical evaluation of these circuits using the proposed two player game. Our experiments indeed show that circuits emanating from real applications can be evaluated in log-depth iterations of the proposed algorithm and that the optical implementation is feasible.