Abstract
Well-known complexity classes such as NP, co-NP, ⊕P (PARITY-P), and PP are produced by considering a nondeterministic polynomial time Turing machine N and defining acceptance in terms of the number of accepting paths in N. That is, they are subclasses of P #P[1]. Other interesting classes such as MOD k P and C + P are also subclasses of P #P[1]. Many relations among these classes are unresolved. Of course, these classes coincide if P = PSPACE. However, we develop a simple combinatorial technique for constructing oracles that separate counting classes. Our results suggest that it will be difficult to resolve the unknown relationships among different counting classes. In addition to presenting new oracle separations, we simplify several previous constructions.
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