Two remarks on the power of counting
The relationship between the polynomial hierarchy and Valiant's class #P is at present unknown. We show that some low portions of the polynomial hierarchy, namely deterministic polynomial algorithms using an NP oracle at most a logarithmic number of times, can be simulated by one #P computation. We also show that the class of problems solvable by polynomial-time nondeterministic Turing machines which accept whenever there is an odd number of accepting computations is idempotent, that is, closed under usage of oracles from the same class.
KeywordsCounting problems oracle computation polynomial hierarchy parity problems machine simulation
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