Toda's theorem

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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"[1] and was given the 1998 Gödel Prize.

Statement[edit]

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.

Definitions[edit]

#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.[2]

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 2009[3] and a complex analogue of Toda's theorem was proved by Saugata Basu in 2011.[4]

Proof[edit]

The proof is broken into two parts.

  • First, it is established that
The proof uses a variation of Valiant–Vazirani theorem. Because contains and is closed under complement, it follows by induction that .
  • Second, it is established that

Together, the two parts imply

References[edit]

  1. ^ Toda, Seinosuke (October 1991). "PP is as Hard as the Polynomial-Time Hierarchy". SIAM Journal on Computing. 20 (5): 865–877. doi:10.1137/0220053. ISSN 0097-5397.
  2. ^ 1998 Gödel Prize. Seinosuke Toda
  3. ^ Saugata Basu and Thierry Zell (2009); Polynomial Hierarchy, Betti Numbers and a Real Analogue of Toda's Theorem, in Foundations of Computational Mathematics
  4. ^ Saugata Basu (2011); A Complex Analogue of Toda's Theorem, in Foundations of Computational Mathematics