We extend the # operator in a natural way and derive new types of counting complexity classes. While in the case of classes (where could be some circuit-based class like ) only proofs for acceptance of some input are being counted, one can also count proofs for rejection. The complexity classes we propose here implement this idea.
We show that in certain cases lies between and which could help understanding the relationship between and . In particular we consider and polynomial size branching programs of bounded and unbounded width. Finally we argue about negative proofs in Turing machines and how those relate to open questions.