Toda's theorem is a result in computational complexity theory that was proven by Seinosuke Toda in his paper "PP is as Hard as the Polynomial-Time Hierarchy"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 that is correct more than half the time. The class P#P consists of all the problems that 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.
An analogous result in the complexity theory over the reals (in the sense of Blum–Shub–Smale real Turing machines) was proved by Saugata Basu and Thierry Zell in 2009and a complex analogue of Toda's theorem was proved by Saugata Basu in 2011.
The proof is broken into two parts.
Together, the two parts imply
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Seinosuke Toda is a computer scientist working at the Nihon University in Tokyo. Toda earned his Ph.D. from the Tokyo Institute of Technology in 1992, under the supervision of Kojiro Kobayashi. He was a recipient of the 1998 Gödel Prize for proving Toda's theorem in computational complexity theory, which states that every problem in the polynomial hierarchy has a polynomial-time Turing reduction to a counting problem.
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