On monotone planar circuits

In this paper we show several results about monotone planar circuits. We show that monotone planar circuits of bounded width, with access to negated input variables, compute exactly the functions in non-uniform ACO. This provides a striking contrast to the non-planar case, where exactly NC1 is computed. We show that the circuit value problem for monotone planar circuits, with inputs on the outer face only, can be solved in LOGDCFL subset of or equal to SC, improving a LOGCFL upper bound due to Dymond and Cook. We show that for monotone planar circuits, with inputs on the outer face only, excessive depth compared to width is useless; any function computed by a monotone planar circuit of width w with inputs on the outer face can be computed by a monotone planar circuit of width O(w) and depth w(O(1)). Finally we show that monotone planar read-once circuits, with inputs on the outer face only, can be efficiently learned using membership queries.