Toda's theorem

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Toda's theorem was proven by Seinosuke Toda in his paper "PP is as Hard as the Polynomial-Time Hierarchy" (1991) and was given the 1998 Gödel Prize. The theorem states that the entire polynomial hierarchy PH is contained in PPP; this implies a closely related statement, that PH is contained in P#P. #P is the problem of exactly counting the number of solutions to a polynomially-verifiable question (that is, to a question in NP), while loosely speaking, PP is the problem of giving an answer which is correct at least half the time. The class P#P consists of all the problems which can be solved in polynomial time if you have access to instantaneous answers to any counting problem in #P (polynomial time relative to a #P oracle). Thus Toda's theorem implies that for any problem in the polynomial hierarchy there is a deterministic polynomial-time Turing reduction to a counting problem.1

The proof is broken into two parts.

  • First, it is established that
\Sigma^P \cdot \mathsf{BP} \cdot \oplus \mathsf{P} \subseteq \mathsf{BP} \cdot \oplus \mathsf{P}
The proof uses a variation of Valiant-Vazirani theorem. Because \mathsf{BP} \cdot \oplus \mathsf{P} contains \mathsf{P} and is closed under complement, it follows by induction that \mathsf{PH} \subseteq \mathsf{BP} \cdot \oplus \mathsf{P}.
  • Second, it is established that
\mathsf{P} \cdot \oplus \mathsf{P} \subseteq \mathsf{P}^{\sharp P}

Together, the two parts imply

\mathsf{PH} \subseteq \mathsf{BP} \cdot \oplus \mathsf{P} \subseteq \mathsf{P} \cdot \oplus \mathsf{P} \subseteq \mathsf{P}^{\sharp P}

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