Finding Efficient Circuits for Ensemble Computation

Given a Boolean function as input, a fundamental problem is to find a Boolean circuit with the least number of elementary gates (AND, OR, NOT) that computes the function. The problem generalises naturally to the setting of multiple Boolean functions: find the smallest Boolean circuit that computes all the functions simultaneously. We study an NP-complete variant of this problem titled Ensemble Computation and, especially, its relationship to the Boolean satisfiability (SAT) problem from both the theoretical and practical perspectives, under the two monotone circuit classes: OR-circuits and SUM-circuits. Our main result relates the existence of nontrivial algorithms for CNF-SAT with the problem of rewriting in subquadratic time a given OR-circuit to a SUM-circuit. Furthermore, by developing a SAT encoding for the ensemble computation problem and by employing state-of-the-art SAT solvers, we search for concrete instances that would witness a substantial separation between the size of optimal OR-circuits and optimal SUM-circuits. Our encoding allows for exhaustively checking all small witness candidates. Searching over larger witness candidates presents an interesting challenge for current SAT solver technology.