Abstract
We extend the # operator in a natural way and derive new types of counting complexity classes. While in the case of #C classes (where C could be some circuit-based class like NC1) only proofs for acceptance of some input are being counted, one can also count proofs for rejection. The Zap-C complexity classes we propose here implement this idea.We show that in certain cases Zap-C lies between #C and Gap-C which could help understanding the relationship between #C and Gap-C. In particular we consider Zap-NC1 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.
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