Evaluating Monotone Circuits on Cylinders, Planes and Tori

We revisit monotone planar circuits MPCVP, with special attention to circuits with cylindrical embeddings. MPCVP is known to be in NC3 in general, and in LogDCFL for the special case of upward stratified circuits. We characterize cylindricality, which is stronger than planarity but strictly generalizes upward planarity, and make the characterization partially constructive. We use this construction, and four key reduction lemmas, to obtain several improvements. We show that monotone circuits with embeddings that are stratified cylindrical, cylindrical, planar one-input-face and focused can be evaluated in LogDCFL, AC1(LogDCFL), LogCFL and AC1(LogDCFL) respectively. We note that the NC3 algorithm for general MPCVP is in AC1(LogCFL) = SAC2. Finally, we show that monotone circuits with toroidal embeddings can, given such an embedding, be evaluated in NC.

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