Positive and negative proofs for circuits and branching programs

Abstract

We extend the # operator in a natural way and derive a new type of counting complexity. While #C classes (where C is some circuit-based class like NC 1) only count proofs for acceptance of some input in circuits, one can also count proofs for rejection. The here proposed Zap-C complexity classes implement this idea. We show that Zap-C lies between #C and Gap-C. In particular we consider Zap-NC1 and polynomial size branching programs of bounded and unbounded width. We find connections to planar branching programs since the duality of positive and negative proofs can be found again in the duality of graphs and their co-graphs. This links to possible applications of our contribution, like closure properties of complexity classes. © 2014 Springer International Publishing.